Quantum technologies in space

Rainer Kaltenbaek, Antonio Acin, Laszlo Bacsardi, Paolo Bianco, Philippe Bouyer, Eleni Diamanti, Christoph Marquardt, Yasser Omar, Valerio Pruneri, et al.

Recently, the European Commission supported by many European countries has announced large investments towards the commercialization of quantum technology (QT) to address and mitigate some of the biggest challenges facing today’s digital era – e.g. secure communication and computing power. For more than two decades the QT community has been working on the development of QTs, which promise landmark breakthroughs leading to commercialization in various areas. The ambitious goals of the QT community and expectations of EU authorities cannot be met solely by individual initiatives of single countries, and therefore, require a combined European effort of large and unprecedented dimensions comparable only to the Galileo or Copernicus programs. Strong international competition calls for a coordinated European effort towards the development of QT in and for space, including research and development of technology in the areas of communication and sensing. Here, we aim at summarizing the state of the art in the development of quantum technologies which have an impact in the field of space applications. Our goal is to outline a complete framework for the design, development, implementation, and exploitation of quantum technology in space.

Microsphere kinematics from the polarization of tightly focused nonseparable light

Stefan Berg-Johansen, Martin Neugebauer, Andrea Aiello, Gerd Leuchs, Peter Banzer, Christoph Marquardt

Recently, it was shown that vector beams can be utilized for fast kinematic sensing via measurements of their global polarization state [Optica2, 864 (2015)10.1364/OPTICA.2.000864]. The method relies on correlations between the spatial and polarization degrees of freedom of the illuminating field which result from its nonseparable mode structure. Here, we extend the method to the nonparaxial regime. We study experimentally and theoretically the far-field polarization state generated by the scattering of a dielectric microsphere in a tightly focused vector beam as a function of the particle position. Using polarization measurements only, we demonstrate position sensing of a Mie particle in three dimensions. Our work extends the concept of back focal plane interferometry and highlights the potential of polarization analysis in optical tweezers employing structured light.

Agile and versatile quantum communication: Signatures and secrets

Stefan Richter, Matthew Thornton, Imran Khan, Hamish Scott, Kevin Jaksch, Ulrich Vogl, Birgit Stiller, Gerd Leuchs, Christoph Marquardt, et al.

Agile cryptography allows for a resource-efficient swap of a cryptographic core in case the security of an underlying classical cryptographic algorithm becomes compromised. Conversely, versatile cryptography allows the user to switch the cryptographic task without requiring any knowledge of its inner workings. In this paper, we suggest how these related principles can be applied to the field of quantum cryptography by explicitly demonstrating two quantum cryptographic protocols, quantum digital signatures (QDS) and quantum secret sharing (QSS), on the same hardware sender and receiver platform. Crucially, the protocols differ only in their classical postprocessing. The system is also suitable for quantum key distribution (QKD) and is highly compatible with deployed telecommunication infrastructures, since it uses standard quadrature phase-shift keying encoding and heterodyne detection. For the first time, QDS protocols are modified to allow for postselection at the receiver, enhancing protocol performance. The cryptographic primitives QDS and QSS are inherently multipartite, and we prove that they are secure not only when a player internal to the task is dishonest, but also when (external) eavesdropping on the quantum channel is allowed. In our first proof-of-principle demonstration of an agile and versatile quantum communication system, the quantum states are distributed at GHz rates. A 1-bit message may be securely signed using our QDS protocols in less than 0.05 ms over a 2-km fiber link and in less than 0.2 s over a 20-km fiber link. To our knowledge, this also marks the first demonstration of a continuous-variable direct QSS protocol.

Distillation of squeezing using an engineered pulsed parametric down-conversion source

Thomas Dirmeier, Johannes Tiedau, Imran Khan, Vahid Ansari, Christian R. Müller, Christine Silberhorn, Christoph Marquardt, Gerd Leuchs

Hybrid quantum information processing combines the advantages of discrete and continues variable protocols by realizing protocols consisting of photon counting and homodyne measurements. However, the mode structure of pulsed sources and the properties of the detection schemes often require the use of optical filters in order to combine both detection methods in a common experiment. This limits the efficiency and the overall achievable squeezing of the experiment. In our work, we use photon subtraction to implement the distillation of pulsed squeezed states originating from a genuinely spatially and temporally single-mode parametric down-conversion source in non-linear waveguides. Due to the distillation, we witness an improvement of 0.17 dB from an initial squeezing value of −1.648 ± 0.002 dB, while achieving a purity of 0.58, and confirm the non-Gaussianity of the distilled state via the higher-order cumulants. With this, we demonstrate the source’s suitability for scalable hybrid quantum network applications with pulsed quantum light.

Binary homodyne detection for observing quadrature squeezing in satellite links

Christian R. Müller, Kaushik P. Seshadreesan, C. Peuntinger, C. Marquardt

Optical satellite links open up new prospects for realizing quantum physical experiments over unprecedented length scales. We analyze and affirm the feasibility of detecting quantum squeezing in an optical mode with homodyne detection of only one bit resolution, as is found in satellites already in orbit. We show experimentally that, in combination with a coherent displacement, a binary homodyne detector can still detect quantum squeezing efficiently even under high loss. The sample overhead in comparison to nondiscretized homodyne detection is merely a factor of 3.3.

Efficient generation of temporally shaped photons using nonlocal spectral filtering

V. Averchenko, D. Sych, C. Marquardt, G. Leuchs

Optical satellite links open up new prospects for realizing quantum physical experiments over unprecedented length scales. We analyze and affirm the feasibility of detecting quantum squeezing in an optical mode with homodyne detection of only one bit resolution, as is found in satellites already in orbit. We show experimentally that, in combination with a coherent displacement, a binary homodyne detector can still detect quantum squeezing efficiently even under high loss. The sample overhead in comparison to nondiscretized homodyne detection is merely a factor of 3.3.

Fading channel estimation for free-space continuous-variable secure quantum communication

László Ruppert, Christian Peuntinger, Bettina Heim, Kevin Günthner, Vladyslav C. Usenko, Dominique Elser, Gerd Leuchs, Radim Filip, Christoph Marquardt

We investigate estimation of fluctuating channels and its effect on security of continuous-variable quantum key distribution. We propose a novel estimation scheme which is based on the clusterization of the estimated transmittance data. We show that uncertainty about whether the transmittance is fixed or not results in a lower key rate. However, if the total number of measurements is large, one can obtain using our method a key rate similar to the non-fluctuating channel even for highly fluctuating channels. We also verify our theoretical assumptions using experimental data from an atmospheric quantum channel. Our method is therefore promising for secure quantum communication over strongly fluctuating turbulent atmospheric channels.

Nonlinear power dependence of the spectral properties of an optical parametric oscillator below threshold in the quantum regime

Golnoush Shafiee, Dmitry V. Strekalov, Alexander Otterpohl, Florian Sedlmeir, Gerhard Schunk, Ulrich Vogl, Harald Schwefel, Gerd Leuchs, Christoph Marquardt

Photon pairs and heralded single photons, obtained from cavity- assisted parametric down conversion (PDC), play an important role in quantum communications and technology. This motivated a thorough study of the spectral and temporal properties of parametric light, both above the Optical Parametric Oscillator (OPO) threshold, where the semiclassical approach is justified, and deeply below it, where the linear cavity approximation is applicable. The pursuit of a higher two- photon emission rate leads into an interesting intermediate regime where the OPO still operates considerably below the threshold but the nonlinear cavity phenomena cannot be neglected anymore. Here, we investigate this intermediate regime and show that the spectral and temporal properties of the photon pairs, as well as their emission rate, may significantly differ from the widely accepted linear model. The observed phenomena include frequency pulling and broadening in the temporal correlation for the down converted optical fields. These factors need to be taken into account when devising practical applications of the high-rate cavity-assisted SPDC sources.

R&D advances for quantum communication systems

Gerd Leuchs, Christoph Marquardt, Luis Sanchez-Soto, Dmitry V. Strekalov, Alan E. Willner

Understanding the nature of light leads to the question of how the principles of quantum physics can be harnessed in practical optical communication. A deeper understanding of fundamental physics has always advanced technology. However, the quantum principles certainly have a distinctly limiting character when looked upon from the engineering point of view. A particle cannot have well-defined momentum and position at the same time. An informative measurement will unpredictably alter the state of a quantum object. One cannot reliably clone an arbitrary quantum state. These and a number of other similar principles give rise to what is commonly known as the quantum “no-go theorems”—a disconcerting term when it comes to building something practical. And yet a search for novel principles of communication enabled by quantum physics began already in its early days and has only intensified since. On this path physicists are faced with a remarkable challenge: to turn a series of negative statements into new technological recipes.

Squeezed vacuum states from a whispering gallery mode resonator

Alexander Otterpohl, Florian Sedlmeir, Ulrich Vogl, Thomas Dirmeier, Golnoush Shafiee, Gerhard Schunk, Dmitry Strekalov, Harald G. L. Schwefel, Tobias Gehring, et al.

Squeezed vacuum states enable optical measurements below the quantum limit and hence are a valuable resource for applications in quantum metrology and also quantum communication. However, most available sources require high pump powers in the milliwatt range and large setups that hinder real-world applications. Furthermore, degenerate operation of such systems presents a challenge. Here, we use a compact crystalline whispering gallery mode resonator made of lithium niobate as a degenerate parametric oscillator. We demonstrate about 1.4 dB noise reduction below the shot-noise level for only 300 μW of pump power in degenerate single-mode operation. Furthermore, we report a record pump threshold as low as 1.35 μW. Our results show that the whispering gallery-based approach presents a promising platform for a compact and efficient source for nonclassical light.

Efficient generation of temporally shaped photons using nonlocal spectral filtering

Valentin Averchenko, Denis Sych, Christoph Marquardt, Gerd Leuchs

We study the generation of single-photon pulses with the tailored temporal shape via nonlocal spectral fil- tering. A shaped photon is heralded from a time-energy entangled photon pair upon spectral filtering and time-resolved detection of its entangled counterpart. We show that the temporal shape of the heralded photon is defined by the time-inverted impulse response of the spectral filter and does not depend on the heralding instant. Thus one can avoid post-selection of particular heralding instants and achieve substantially higher heralding rate of shaped photons as compared to the generation of photons via nonlocal temporal modulation. Furthermore, the method can be used to generate shaped photons with a coherence time in the ns-μs range and is particularly suitable to produce photons with the exponentially rising temporal shape required for efficient interfacing to a single quantum emitter in free space.

The standard quantum limit of coherent beam combining

Christian Müller, Florian Sedlmeir, Vitaliy O. Martynov, Christoph Marquardt, Alexey V. Andrianov , Gerd Leuchs

Coherent beam combining refers to the process of generating a bright output beam by merging independent input beams of individually diffusing relative phases by locking them to each other. We report the first quantum mechanical noise limit calculations for coherent beam combining and compare our results to quantum-limited amplification. Our coherent beam combining scheme is based on an optical Fourier transformation which renders the scheme compatible with integrated optics combined with feed-back stabilization of the relative phases. The scheme can be layed out for an arbitrary number of input beams and approaches the shot noise limit for a large number of inputs.

Quantum-limited measurements of intensity noise levels in Yb-doped fiber amplifiers

Alexandra Popp, Victor Distler, Kevin Jaksch, Florian Sedlmeir, Christian Müller, Nicoletta Haarlammert, Thomas Schreiber, Christoph Marquardt, Andreas Tünnermann, et al.

We investigate the frequency-resolved intensity noise spectrum of an Yb-doped fiber amplifier down to the fundamental limit of quantum noise. We focus on the kHz and low MHz frequency regime with special interest in the region between 1 and 10 kHz. Intensity noise levels up to ≥60 dB above the shot noise limit are found, revealing great optimization potential. Additionally, two seed lasers with different noise characteristics were amplified, showing that the seed source has a significant impact and should be considered in the design of high power fiber amplifiers.

Distillation of Squeezing using an engineered PDC source

Thomas Dirmeier, Johannes Tiedau, Imran Khan, Vahid Ansari, Christian Müller, Christine Silberhorn, Christoph Marquardt, Gerd Leuchs

We demonstrate homodyne detection of quantum states originating from a genuinely spatially and temporally singlemode parametric downconversion source in non-linear waveguides. By using single photon subtraction, we implement the distillation of squeezed states witnessing an improvement of 0.1 dB from an intial squeezing value of 1.62 ± 0.01 dB, while achieving a purity of 0.58, and confirm the non-Gaussianity of the distilled state via the higher order cumulants. With this we demonstrate the sources suitability for scalable hybrid quantum network applications.

Space QUEST mission proposal: experimentally testing decoherence due to gravity

Siddarth Koduru Joshi, Jacques Pienaar, Timothy C. Ralph, Luigi Cacciapuoti, Will McCutcheon, John Rarity, Dirk Giggenbach, Jin Gyu Lim, Vadim Makarov, et al.

Models of quantum systems on curved space-times lack sufficient experimental verification. Some speculative theories suggest that quantum correlations, such as entanglement, may exhibit different behavior to purely classical correlations in curved space. By measuring this effect or lack thereof, we can test the hypotheses behind several such models. For instance, as predicted by Ralph et al [5] and Ralph and Pienaar [1], a bipartite entangled system could decohere if each particle traversed through a different gravitational field gradient. We propose to study this effect in a ground to space uplink scenario. We extend the above theoretical predictions of Ralph and coworkers and discuss the scientific consequences of detecting/failing to detect the predicted gravitational decoherence. We present a detailed mission design of the European Space Agency's Space QUEST (Space-Quantum Entanglement Space Test) mission, and study the feasibility of the mission scheme.

Towards terahertz detection and calibration through spontaneous parametric down-conversion in the terahertz idler-frequency range generated by a 795 nm diode laser system

Vladimir V. Kornienko, Galiya Kh. Kitaeva, Florian Sedlmeir, Gerd Leuchs, Harald G. L. Schwefel

We study a calibration scheme for terahertz wave nonlinear-optical detectors based on spontaneous parametric down-conversion. Contrary to the usual low wavelength pump in the green, we report here on the observation of spontaneous parametric down-conversion originating from an in-growth poled lithium niobate crystal pumped with a continuous wave 50 mW, 795 nm diode laser system, phase-matched to a terahertz frequency idler wave. Such a system is more compact and allows for longer poling periods as well as lower losses in the crystal. Filtering the pump radiation by a rubidium-87 vapor cell allowed the frequency-angular spectra to be obtained down to similar to 0.5 THz or similar to 1 nm shift from the pump radiation line. The presence of an amplified spontaneous emission "pedestal" in the diode laser radiation spectrum significantly hampers the observation of spontaneous parametric down-conversion spectra, in contrast to conventional narrowband gas lasers. Benefits of switching to longer pump wavelengths are pointed out, such as collinear optical-terahertz phase-matching in bulk crystals. (c) 2018 Author(s).

Satellite-Based QKD

Imran Khan, Bettina Heim, Andreas Neuzner, Christoph Marquardt

A global network of spacecraft and ground stations, distributing secret encryption keys by meansof quantum technology, could meet emerging and long-term threats to data security.

Renyi relative entropies of quantum Gaussian states

Kaushik P Seshadreesan, Ludovico Lami, Mark M Wilde

The quantum Renyi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Renyi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) Renyi relative entropy as well as the more recent sandwiched Renyi relative entropy, finding formulas that are expressed solely in terms of the mean vectors and covariance matrices of the underlying quantum Gaussian states. Our development handles the hitherto elusive case for the Petz-Renyi relative entropy when the Renyi parameter is larger than one. Finally, we also derive a formula for the max-relative entropy of two quantum Gaussian states, and we discuss some applications of the formulas derived here.

Binary Homodyne Detection for Observing Quadrature Squeezing in Satellite Links

Christian R. Mueller, Kaushik P Seshadreesan, Christian Peuntinger, Christoph Marquardt

Optical satellite links open up new prospects for realizing quantum physical experiments over unprecedented length scales. We analyze and affirm the feasibility of detecting quantum squeezing in an optical mode with homodyne detection of only one bit resolution, as is found in satellites already in orbit. We show experimentally that, in combination with a coherent displacement, a binary homodyne detector can still detect quantum squeezing efficiently even under high loss. The sample overhead in comparison to non-discretized homodyne detection is merely a factor of 3.3.

Stabilization of transmittance fluctuations caused by beam wandering in continuous-variable quantum communication over free-space atmospheric channels

Vladyslav C. Usenko, Christian Peuntinger, Bettina Heim, Kevin Günthner, Ivan Derkach, Dominique Elser, Christoph Marquardt, Radim Filip, Gerd Leuchs

Transmittance fluctuations in turbulent atmospheric channels result in quadrature excess noise which limits applicability of continuous-variable quantum communication. Such fluctuations are commonly caused by beam wandering around the receiving aperture. We study the possibility to stabilize the fluctuations by expanding the beam, and test this channel stabilization in regard of continuous-variable entanglement sharing and quantum key distribution. We perform transmittance measurements of a real free-space atmospheric channel for different beam widths and show that the beam expansion reduces the fluctuations of the channel transmittance by the cost of an increased overall loss. We also theoretically study the possibility to share an entangled state or to establish secure quantum key distribution over the turbulent atmospheric channels with varying beam widths. We show the positive effect of channel stabilization by beam expansion on continuous-variable quantum communication as well as the necessity to optimize the method in order to maximize the secret key rate or the amount of shared entanglement. Being autonomous and not requiring adaptive control of the source and detectors based on characterization of beam wandering, the method of beam expansion can be also combined with other methods aiming at stabilizing the fluctuating free-space atmospheric channels.

Quantum Limits of Coherent Beam Combining

Gerd Leuchs, Christian Müller, Florian Sedlmeir, Sourav Chatterjee, Nicoletta Haarlammert, Cesar Jauregui, Thomas Schreiber, Jens Limpert, Andreas Tünnermann, et al.

Coherent beam combining refers to the process of generating a bright output beam by merging independent input beams with locked relative phases. We report the first quantum mechanical noise limit calculations for coherent beam combining and compare our results to quantum-limited amplification. We highlight a favourable noise scaling for both pure and noisy input states.

Printing of Large-Scale, Flexible, Long-Term Stable Dielectric Mirrors with Suppressed Side Interferences

Carina Bronnbauer, Arne Riecke, Marius Adler, Julian Hornich, Gerhard Schunk, Christoph J. Brabec, Karen Forberich

Dielectric mirrors are wavelength-selective mirrors which are based on thin film interference effects. Their optical band can precisely be adjusted in width, position, and reflectance by the refractive index of the applied materials, the layers' thicknesses, and the amount of deposited layers. Nowadays, they are a well-known light management tool for efficiency enhancement in, for example, semitransparent organic solar cells (OSCs) and light guiding in organic light-emitting diodes (OLEDs). However, most of the dielectric mirrors are still fabricated by lab-scale techniques such as spin-coating or physical vapor deposition under vacuum. Large-scale, fully printed (maximum 20 x 20 cm(2)) dielectric mirrors with adjustable reflectance characteristics are fabricated, using temperatures of maximum 50 degrees C and alcohol-based inks. According to the moderate processing conditions they can be easily deposited not only on rigid glass substrates but also on flexible foils. They show high stability against humidity, light irradiation, and temperature, positioning themselves as good candidates for applications in OLEDs and OSCs. Eventually, by simulations and experiments it is verified that a moderate degree of variations in layer thickness and surface roughness can suppress side interference fringes, while not impacting the main transmittance minimum or the main reflection maximum, respectively.

Quantum cryptography with twisted photons through an outdoor underwater channel

Frédéric Bouchard, Alicia Sit, Felix Hufnagel, Aazad Abbas, Yingwen Zhang, Khabat Heshami, Robert Fickler, Christoph Marquardt, Gerd Leuchs, et al.

Quantum communication has been successfully implemented in optical fibres and through free-space. Fibre systems, though capable of fast key and low error rates, are impractical in communicating with destinations without an established fibre link. Free-space quantum channels can overcome such limitations and reach long distances with the advent of satellite-to-ground links. However, turbulence, resulting from local fluctuations in refractive index, becomes a major challenge by adding errors and losses. Recently, an interest in investigating the possibility of underwater quantum channels has arisen. Here, we investigate the effect of turbulence on an underwater quantum channel using twisted photons in outdoor conditions. We study the effect of turbulence on transmitted error rates, and compare different quantum cryptographic protocols in an underwater quantum channel, showing the feasibility of high-dimensional encoding schemes. Our work may open the way for secure high-dimensional quantum communication between submersibles, and provides important input for potential submersibles-to-satellite quantum communication.

Sensitivity limits of millimeter-wave photonic radiometers based on efficient electro-optic upconverters

Gabriel Santamaria Botello, Florian Sedlmeir, Alfredo Rueda, Kerlos Atia Abdalmalak, Elliott R. Brown, Gerd Leuchs, Sascha Preu, Daniel Segovia-Vargas, Dmitry V. Strekalov, et al.

Conventional ultra-high sensitivity detectors in the millimeter-wave range are usually cooled as their own thermal noise at room temperature would mask the weak received radiation. The need for cryogenic systems increases the cost and complexity of the instruments, hindering the development of, among others, airborne and space applications. In this work, the nonlinear parametric upconversion of millimeter-wave radiation to the optical domain inside high-quality (Q) lithium niobate whispering-gallery mode (WGM) resonators is proposed for ultra-low noise detection. We experimentally demonstrate coherent upconversion of millimeter-wave signals to a 1550 nm telecom carrier, with a photon conversion efficiency surpassing the state-of-the-art by 2 orders of magnitude. Moreover, a theoretical model shows that the thermal equilibrium of counterpropagating WGMs is broken by overcoupling the millimeter-wave WGM, effectively cooling the upconverted mode and allowing ultra-low noise detection. By theoretically estimating the sensitivity of a correlation radiometer based on the presented scheme, it is found that room-temperature radiometers with better sensitivity than state-of-the-art high-electron-mobility transistor (HEMT)-based radiometers can be designed. This detection paradigm can be used to develop room-temperature instrumentation for radio astronomy, earth observation, planetary missions, and imaging systems. (C) 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Temporal and spectral properties of quantum light

B. Stiller, U. Seyfarth, G. Leuchs, C. Fabre, V. Sandoghdar, N. Treps, L.F. Cugliandolo

Superiority of heterodyning over homodyning: An assessment with quadrature moments

Y. S. Teo, C. R. Mueller, H. Jeong, Z. Hradil, J. Rehacek, L. L. Sanchez-Soto

We examine the moment-reconstruction performance of both the homodyne and heterodyne (doublehomodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant nonclassical states. Finally, we present an extension of our theories to two-mode sources, which can be straightforwardly generalized to all other multimode sources.

Joint measurement of complementary observables in moment tomography

Yong Siah Teo, Christian R. Mueller, Hyunseok Jeong, Zdenek Hradil, Jaroslav Rehacek, Luis L. Sanchez-Soto

Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning (HOM) and heterodyning (HET) that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that HET can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.

High-dimensional intracity quantum cryptography with structured photons

Alicia Sit, Frederic Bouchard, Robert Fickler, Jeremie Gagnon-Bischoff, Hugo Larocque, Khabat Heshami, Dominique Elser, Christian Peuntinger, Kevin Guenthner, et al.

Quantum key distribution (QKD) promises information-theoretically secure communication and is already on the verge of commercialization. The next step will be to implement high-dimensional protocols in order to improve noise resistance and increase the data rate. Hitherto, no experimental verification of high-dimensional QKD in the singlephoton regime has been conducted outside of the laboratory. Here, we report the realization of such a single-photon QKD system in a turbulent free-space link of 0.3 km over the city of Ottawa, taking advantage of both the spin and orbital angular momentum photonic degrees of freedom. This combination of optical angular momenta allows us to create a 4-dimensional quantum state; wherein, using a high-dimensional BB84 protocol, a quantum bit error rate of 11% was attained with a corresponding secret key rate of 0.65 bits per sifted photon. In comparison, an error rate of 5% with a secret key rate of 0.43 bits per sifted photon is achieved for the case of 2-dimensional structured photons. We thus demonstrate that, even through moderate turbulence without active wavefront correction, high-dimensional photon states are advantageous for securely transmitting more information. This opens the way for intracity high-dimensional quantum communications under realistic conditions. (C) 2017 Optical Society of America

Unconstrained Capacities of Quantum Key Distribution and Entanglement Distillation for Pure-Loss Bosonic Broadcast Channels

Masahiro Takeoka, Kaushik P. Seshadreesan, Mark M. Wilde

We consider quantum key distribution (QKD) and entanglement distribution using a single-sender multiple-receiver pure-loss bosonic broadcast channel. We determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they are allowed arbitrary public classical communication. A practical implication of our result is that the capacity region demonstrated drastically improves upon rates achievable using a naive time-sharing strategy, which has been employed in previously demonstrated network QKD systems. We show a simple example of a broadcast QKD protocol overcoming the limit of the point-to-point strategy. Our result is thus an important step toward opening a new framework of network channel-based quantum communication technology.

Polarization-Selective Out-Coupling of Whispering-Gallery Modes

Florian Sedlmeir, Matthew R. Foreman, Ulrich Vogl, Richard Zeltner, Gerhard Schunk, Dmitry V. Strekalov, Christoph Marquardt, Gerd Leuchs, Harald G. L. Schwefel

Whispering-gallery mode (WGM) resonators are an important platform for linear, nonlinear, and quantum optical experiments. In such experiments, independent control of in-coupling and out-coupling rates to different modes can lead to higher conversion efficiencies and greater flexibility in the generation of nonclassical states based on parametric down-conversion. In this work, we introduce a scheme that enables selective out-coupling of WGMs belonging to a specific polarization family, while the orthogonally polarized modes remain largely unperturbed. Our technique utilizes material birefringence in both the resonator and the coupler such that a negative (positive) birefringence allows for polarization-selective coupling to TE (TM) WGMs. We formulate a refined coupling condition suitable for describing the case where the refractive indices of the resonator and the coupler are almost the same, from which we derive a criterion for polarization-selective coupling. Finally, we experimentally demonstrate our proposed method using a lithium niobate disk resonator coupled to a lithium niobate prism, where we show a 22-dB suppression of coupling to TM modes relative to TE modes.

Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum

Masahiro Takeoka, Kaushik P. Seshadreesan, Chenglong You, Shuro Izumi, Jonathan P. Dowling

In the lore of quantum metrology, one often hears (or reads) the following no-go theorem: If you put a vacuum into one input port of a balanced Mach-Zehnder interferometer, then no matter what you put into the other input port, and nomatter what your detection scheme, the sensitivity can never be better than the shot-noise limit (SNL). Often the proof of this theorem is cited to be inC. Caves, Phys. Rev. D23, 1693 (1981), but upon further inspection, no such claim is made there. Quantum-Fisher-information-based arguments suggestive of this no-go theorem appear elsewhere in the literature, but are not stated in their full generality. Here we thoroughly explore this no-go theorem and give a rigorous statement: the no-go theorem holds whenever the unknown phase shift is split between both of the arms of the interferometer, but remarkably does not hold when only one arm has the unknown phase shift. In the latter scenario, we provide an explicit measurement strategy that beats the SNL. We also point out that these two scenarios are physically different and correspond to different types of sensing applications.

Free-space quantum links under diverse weather conditions

D. Vasylyev, A. A. Semenov, W. Vogel, K. Guenthner, A. Thurn, O. Bayraktar, Ch. Marquardt

Free-space optical communication links are promising channels for establishing secure quantum communication. Here we study the transmission of nonclassical light through a turbulent atmospheric link under diverse weather conditions, including rain or haze. To include these effects, the theory of light transmission through atmospheric links in the elliptic-beam approximation presented by Vasylyev et al. [D. Vasylyev et al., Phys. Rev. Lett. 117, 090501 (2016)] is further generalized. It is demonstrated, with good agreement between theory and experiment, that low-intensity rain merely contributes additional deterministic losses, whereas haze also introduces additional beam deformations of the transmitted light. Based on these results, we study theoretically the transmission of quadrature squeezing and Gaussian entanglement under these weather conditions.

Quantum communication with coherent states of light

Imran Khan, Dominique Elser, Thomas Dirmeier, Christoph Marquardt, Gerd Leuchs

Quantum communication offers long-term security especially, but not only, relevant to government and industrial users. It is worth noting that, for the first time in the history of cryptographic encoding, we are currently in the situation that secure communication can be based on the fundamental laws of physics ( information theoretical security) rather than on algorithmic security relying on the complexity of algorithms, which is periodically endangered as standard computer technology advances. On a fundamental level, the security of quantum key distribution (QKD) relies on the non-orthogonality of the quantum states used. So even coherent states are well suited for this task, the quantum states that largely describe the light generated by laser systems. Depending on whether one uses detectors resolving single or multiple photon states or detectors measuring the field quadratures, one speaks of, respectively, a discrete- or a continuous-variable description. Continuous-variable QKD with coherent states uses a technology that is very similar to the one employed in classical coherent communication systems, the backbone of today's Internet connections. Here, we review recent developments in this field in two connected regimes: (i) improving QKD equipment by implementing front-end telecom devices and (ii) research into satellite QKD for bridging long distances by building upon existing optical satellite links. This article is part of the themed issue 'Quantum technology for the 21st century'.

Temporal shaping of single photons enabled by entanglement

Valentin Averchenko, Denis Sych, Gerhard Schunk, Ulrich Vogl, Christoph Marquardt, Gerd Leuchs

We present a method to produce pure single photons with an arbitrary designed temporal shape in a heralded way. As an indispensable resource, the method uses pairs of time-energy entangled photons. One photon of a pair undergoes temporal amplitude-phase modulation according to the desired shape. Subsequent frequency-resolved detection of the modulated photon heralds its entangled counterpart in a pure quantum state. The temporal shape of the heralded photon is indirectly affected by the modulation in the heralding arm. We derive conditions for which the shape of the heralded photon is given by the modulation function. The method can be implemented with various sources of time-energy entangled photons. In particular, using entangled photons from parametric down-conversion the method provides a simple means to generate pure shaped photons with an unprecedented broad range of temporal durations, from tenths of femtoseconds to microseconds. This shaping of single photons will push forward the implementation of scalable multidimensional quantum information protocols, efficient photon-matter coupling, and quantum control at the level of single quanta.

Progress towards practical device-independent quantum key distribution with spontaneous parametric down-conversion sources, on-off photodetectors, and entanglement swapping

Kaushik P. Seshadreesan, Masahiro Takeoka, Masahide Sasaki

Device-independent quantum key distribution (DIQKD) guarantees unconditional security of a secret key without making assumptions about the internal workings of the devices used for distribution. It does so using the loophole-free violation of a Bell's inequality. The primary challenge in realizing DIQKD in practice is the detection loophole problem that is inherent to photonic tests of Bell' s inequalities over lossy channels. We revisit the proposal of Curty and Moroder [Phys.Rev.A 84, 010304(R) (2011)] to use a linear optics-based entanglement-swapping relay (ESR) to counter this problem. We consider realistic models for the entanglement sources and photodetectors: more precisely, (a) polarization-entangled states based on pulsed spontaneous parametric down-conversion sources with infinitely higher-order multiphoton components and multimode spectral structure, and (b) on-off photodetectors with nonunit efficiencies and nonzero dark-count probabilities. We show that the ESR-based scheme is robust against the above imperfections and enables positive key rates at distances much larger than what is possible otherwise.

30 years of squeezed light generation

Ulrik L. Andersen, Tobias Gehring, Christoph Marquardt, Gerd Leuchs

Squeezed light generation has come of age. Significant advances on squeezed light generation have been made over the last 30 years-from the initial, conceptual experiment in 1985 till today's top-tuned, application-oriented setups. Here we review the main experimental platforms for generating quadrature squeezed light that have been investigated in the last 30 years.

Quantum-polarization state tomography

Oemer Bayraktar, Marcin Swillo, Carlota Canalias, Gunnar Bjork

We propose and demonstrate a method for quantum-state tomography of qudits encoded in the quantum polarization of N-photon states. This is achieved by distributing N photons nondeterministically into three paths and their subsequent projection, which for N = 1 is equivalent to measuring the Stokes (or Pauli) operators. The statistics of the recorded N-fold coincidences determines the unknown N-photon polarization state uniquely. The proposed, fixed setup manifestly rules out any systematic measurement errors due to moving components and allows for simple switching between tomography of different states, which makes it ideal for adaptive tomography schemes.

Free-Space Quantum Signatures Using Heterodyne Measurements

Callum Croal, Christian Peuntinger, Bettina Heim, Imran Khan, Christoph Marquardt, Gerd Leuchs, Petros Wallden, Erika Andersson, Natalia Korolkova

Unconstrained distillation capacities of a pure-loss bosonic broadcast channel

Masahiro Takeoka, Kaushik P. Seshadreesan, Mark M. Wilde

Frequency tuning of single photons from a whispering-gallery mode resonator to MHz-wide transitions

G. Schunk, U. Vogl, F. Sedlmeir, D. V. Strekalov, A. Otterpohl, V. Averchenko, H. G. L. Schwefel, G. Leuchs, Ch. Marquardt

Heralded source of bright multi-mode mesoscopic sub-Poissonian light

T. Sh. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, G. Leuchs

In a direct detection scheme, we observed 7.8 dB of twin-beam squeezing for multi-mode two-color squeezed vacuum generated via parametric downconversion. Applying postselection, we conditionally prepared a sub-Poissonian state of light containing 6.3 . 10(5) photons per pulse on the average with the Fano factor 0.63 +/- 0.01. The scheme can be considered as the heralded preparation of pulses with the mean energy varying between tens and hundreds of fJ and the uncertainty considerably below the shot-noise level. Such pulses can be used in metrology (for instance, for radiometer calibration), as well as for probing multi-mode non-linear optical effects. (C) 2016 Optical Society of America

Evading Vacuum Noise: Wigner Projections or Husimi Samples?

C. R. Mueller, C. Peuntinger, T. Dirmeier, I. Khan, U. Vogl, Ch. Marquardt, G. Leuchs, L. L. Sanchez-Soto, Y. S. Teo, et al.

Nonlinear and quantum optics with whispering gallery resonators

Dmitry V. Strekalov, Christoph Marquardt, Andrey B. Matsko, Harald G. L. Schwefel, Gerd Leuchs

Attacks on practical quantum key distribution systems (and how to prevent them)

Nitin Jain, Birgit Stiller, Imran Khan, Dominique Elser, Christoph Marquardt, Gerd Leuchs

With the emergence of an information society, the idea of protecting sensitive data is steadily gaining importance. Conventional encryption methods may not be sufficient to guarantee data protection in the future. Quantum key distribution (QKD) is an emerging technology that exploits fundamental physical properties to guarantee perfect security in theory. However, it is not easy to ensure in practice that the implementations of QKD systems are exactly in line with the theoretical specifications. Such theory-practice deviations can open loopholes and compromise security. Several such loopholes have been discovered and investigated in the last decade. These activities have motivated the proposal and implementation of appropriate countermeasures, thereby preventing future attacks and enhancing the practical security of QKD. This article introduces the so-called field of quantum hacking by summarising a variety of attacks and their prevention mechanisms.

Quantum uncertainty in the beam width of spatial optical modes

Vanessa Chille, Peter Banzer, Andrea Aiello, Gerd Leuchs, Christoph Marquardt, Nicolas Treps, Claude Fabre

We theoretically investigate the quantum uncertainty in the beam width of transverse optical modes and, for this purpose, define a corresponding quantum operator. Single mode states are studied as well as multimode states with small quantum noise. General relations are derived, and specific examples of different modes and quantum states are examined. For the multimode case, we show that the quantum uncertainty in the beam width can be completely attributed to the amplitude quadrature uncertainty of one specific mode, which is uniquely determined by the field under investigation. This discovery provides us with a strategy for the reduction of the beam width noise by an appropriate choice of the quantum state. (C) 2015 Optical Society of America

Highly efficient generation of single-mode photon pairs from a crystalline whispering-gallery-mode resonator source

Michael Foertsch, Gerhard Schunk, Josef U. Fuerst, Dmitry Strekalov, Thomas Gerrits, Martin J. Stevens, Florian Sedlmeir, Harald G. L. Schwefel, Sae Woo Nam, et al.

We report a highly efficient source of narrow-band photon pairs based on parametric down-conversion in a crystalline-whispering-gallery-mode resonator. Remarkably, each photon of a pair is detected in a single spatial and temporal mode, as witnessed by Glauber's autocorrelation function. We explore the phase-matching conditions in spherical geometries, and determine the requirements for single-mode operation. Understanding these conditions has allowed us to experimentally demonstrate a single-mode pair-detection efficiency of 1.13 x 10(6) pairs/s per mW pump power per 26.8 MHz bandwidth.

Measuring the Propagation of Information and Entanglement in Dispersive Media

Jeremy B. Clark, Ryan T. Glasser, Quentin Glorieux, Ulrich Vogl, Tian Li, Kevin M. Jones, Paul D. Lett

Although it is widely accepted that information cannot travel faster than the speed of light in vacuum, the behavior of quantum correlations and entanglement propagating through actively pumped dispersive media has not been thoroughly studied. Here we investigate the behavior of quantum correlations and information in the presence of a nonlinear dispersive gaseous medium. We show that the quantum correlations can be advanced by a small fraction of the correlation time while the entanglement is preserved even in the presence of noise added by phase insensitive gain. Additionally, although we observe an advance of the peak of the quantum mutual information between the modes, we find that the degradation of the mutual information due to the added noise appears to prevent an advancement of the mutual information's leading tail. In contrast, we show that both the leading and trailing tails of the mutual information in a slow light system can be significantly delayed in the presence of four-wave mixing (4WM) and electromagnetically induced transparency.

Quantum hacking of continuous-variable quantum key distribution systems: realtime Trojan-horse attacks

B. Stiller, I. Khan, N. Jain, P. Jouguet, S. Kunz-Jacques, E. Diamanti, Ch. Marquardt, G. Leuchs

We experimentally demonstrate successful Trojan-horse attacks on laboratory and commercial continuous-variable quantum key distribution systems with binary and Gaussian modulation. Furthermore, we analyze appropriate countermeasures regarding their spectral performance. (C) 2014 Optical Society of America

Near-infrared single-photon spectroscopy of a whispering gallery mode resonator using energy-resolving transition edge sensors

Michael Foertsch, Thomas Gerrits, Martin J. Stevens, Dmitry Strekalov, Gerhard Schunk, Josef U. Fuerst, Ulrich Vogl, Florian Sedlmeir, Harald G. L. Schwefel, et al.

We demonstrate a method to perform spectroscopy of near-infrared single photons without the need of dispersive elements. This method is based on a photon energy resolving transition edge sensor and is applied for the characterization of widely wavelength tunable narrow-band single photons emitted from a crystalline whispering gallery mode resonator. We measure the emission wavelength of the generated signal and idler photons with an uncertainty of up to 2 nm.

A robust quantum receiver for phase shift keyed signals

C. R. Mueller, Ch Marquardt

The impossibility of perfectly discriminating non-orthogonal quantum states imposes far-reaching consequences both on quantum and classical communication schemes. We propose and numerically analyze an optimized quantum receiver for the discrimination of phase encoded signals. Our scheme outperforms the standard quantum limit and approaches the Helstrom bound for any signal power. The discrimination is performed via an optimized, feedback-mediated displacement prior to a photon counting detector. We provide a detailed analysis of the influence of excess noise and technical imperfections on the average error probability. The results demonstrate the receiver's robustness and show that it can outperform any classical receiver over a wide range of realistic parameters.

Quantum nature of Gaussian discord: Experimental evidence and role of system-environment correlations

Vanessa Chille, Niall Quinn, Christian Peuntinger, Callum Croal, Ladislav Mista Jr., Christoph Marquardt, Gerd Leuchs, Natalia Korolkova

We provide experimental evidence of quantum features in bipartite states classified as entirely classical according to a conventional criterion based on the Glauber P function but possessing nonzero Gaussian quantum discord. Their quantum nature is experimentally revealed by acting locally on one part of the discordant state. We experimentally verify and investigate the effect of discord increase under the action of local loss and link it to the entanglement with the environment. Adding an environmental system purifying the state, we unveil the flow of quantum correlations within a global pure system using the Koashi-Winter inequality. For a discordant state generated by splitting a state in which the initial squeezing is destroyed by random displacements, we demonstrate the recovery of entanglement highlighting the role of system-environment correlations.

Intracavity Squeezing Can Enhance Quantum-Limited Optomechanical Position Detection through Deamplification

V. Peano, H. G. L. Schwefel, Ch. Marquardt, F. Marquardt

It has been predicted and experimentally demonstrated that by injecting squeezed light into an optomechanical device, it is possible to enhance the precision of a position measurement. Here, we present a fundamentally different approach where the squeezing is created directly inside the cavity by a nonlinear medium. Counterintuitively, the enhancement of the signal-to-noise ratio works by deamplifying precisely the quadrature that is sensitive to the mechanical motion without losing quantum information. This enhancement works for systems with a weak optomechanical coupling and/or strong mechanical damping. This can allow for larger mechanical bandwidth of quantum-limited detectors based on optomechanical devices. Our approach can be straightforwardly extended to quantum nondemolition qubit detection.

Depolarized guided acoustic wave Brillouin scattering in hollow-core photonic crystal fibers

Wenjia Elser Nee Zhong, Birgit Stiller, Dominique Elser, Bettina Heim, Christoph Marquardt, Gerd Leuchs

By performing quantum-noise-limited optical heterodyne detection, we observe polarization noise in light after propagation through a hollow-core photonic crystal fiber (PCF). We compare the noise spectrum to the one of a standard fiber and find an increase of noise even though the light is mainly transmitted in air in a hollow-core PCF. Combined with our simulation of the acoustic vibrational modes in the hollow-core PCF, we are offering an explanation for the polarization noise with a variation of guided acoustic wave Brillouin scattering (GAWBS). Here, instead of modulating the strain in the fiber core as in a solid core fiber, the acoustic vibrations in hollow-core PCF influence the effective refractive index by modulating the geometry of the photonic crystal structure. This induces polarization noise in the light guided by the photonic crystal structure. (C) 2015 Optical Society of America

Quantum-like nonseparable structures in optical beams

Andrea Aiello, Falk Toeppel, Christoph Marquardt, Elisabeth Giacobino, Gerd Leuchs

When two or more degrees of freedom become coupled in a physical system, a number of observables of the latter cannot be represented by mathematical expressions separable with respect to the different degrees of freedom. In recent years it appeared clear that these expressions may display the same mathematical structures exhibited by multiparty entangled states in quantum mechanics. In this work, we investigate the occurrence of such structures in optical beams, a phenomenon that is often referred to as 'classical entanglement'. We present a unified theory for different kinds of light beams exhibiting classical entanglement and we indicate several possible extensions of the concept. Our results clarify and shed new light upon the physics underlying this intriguing aspect of classical optics.

Advanced quantum noise correlations

Ulrich Vogl, Ryan T. Glasser, Jeremy B. Clark, Quentin Glorieux, Tian Li, Neil V. Corzo, Paul D. Lett

We use the quantum correlations of twin beams of light to investigate the fundamental addition of noise when one of the beams propagates through a fastlight medium based on phase-insensitive gain. The experiment is based on two successive four-wave mixing processes in rubidium vapor, which allow for the generation of bright two-mode-squeezed twin beams followed by a controlled advancement while maintaining the shared quantum correlations between the beams. The demonstrated effect allows the study of irreversible decoherence in a medium exhibiting anomalous dispersion, and for the first time shows the advancement of a bright nonclassical state of light. The advancement and corresponding degradation of the quantum correlations are found to be operating near the fundamental quantum limit imposed by using a phase-insensitive amplifier.

Classical entanglement in polarization metrology

Falk Toeppel, Andrea Aiello, Christoph Marquardt, Elisabeth Giacobino, Gerd Leuchs

Quantum approaches relying on entangled photons have been recently proposed to increase the efficiency of optical measurements. We demonstrate here that, surprisingly, the use of classical light with entangled degrees of freedom can also bring outstanding advantages over conventional measurements in polarization metrology. Specifically, we show that radially polarized beams of light allow to perform real-time single-shot Mueller matrix polarimetry. Our results also indicate that quantum optical procedures requiring entanglement without nonlocality can be actually achieved in the classical optics regime.

Entanglement Creation by Locally Splitting a Discordant State

Callum Croal, Ladislav Mista Jr., Vanessa Chille, Christian Peuntinger, Christoph Marquardt, Gerd Leuchs, Natalia Korolkova

We introduce and experimentally implement counter-intuitive entanglement creation by locally splitting a classical mode that is part of a larger discordant state. Possible applications are quantum advantage in information encoding and assisted dense coding. (C) 2014 Optical Society of America

Atomic mercury vapor inside a hollow-core photonic crystal fiber

Ulrich Vogl, Christian Peuntinger, Nicolas Y. Joly, Philip St. J. Russell, Christoph Marquardt, Gerd Leuchs

We demonstrate high atomic mercury vapor pressure in a kagome-style hollow-core photonic crystal fiber at room temperature. After a few days of exposure to mercury vapor the fiber is homogeneously filled and the optical depth achieved remains constant. With incoherent optical pumping from the ground state we achieve an optical depth of 114 at the 6(3)P(2) - 6(3)D(3) transition, corresponding to an atomic mercury number density of 6 x 10(10) cm(-3). The use of mercury vapor in quasi one-dimensional confinement may be advantageous compared to chemically more active alkali vapor, while offering strong optical nonlinearities in the ultraviolet region of the optical spectrum. (C) 2014 Optical Society of America

M-ary Phase Shift Keying Receiver Beating the Standard Quantum Limit for any Signal Power

Christian R. Mueller, Andreas Leven, Henning Buelow, Laurent Schmalen, Gerd Leuchs, Christoph Marquardt

We analyze a novel quantum receiver for M-ary phase-shift keyed signals that outperforms the standard quantum limit for any signal power. We provide a detailed analysis of its robustness against realistic imperfections and discuss the achievable mutual information.

Cascaded phase-preserving multilevel amplitude regeneration

Tobias Roethlingshoefer, Georgy Onishchukov, Bernhard Schmauss, Gerd Leuchs

The performance of cascaded in-line phase-preserving amplitude regeneration using nonlinear amplifying loop mirrors has been studied in numerical simulations. As an example of a spectrally efficient modulation format with two amplitude states and multiple phase states, the regeneration performance of a star-16QAM format, basically an 8PSK format with two amplitude levels, was evaluated. An increased robustness against amplified spontaneous emission and nonlinear phase noise was observed resulting in a significantly increased transmission distance. (C) 2014 Optical Society of America

All-Optical Signal Regeneration of Advanced Modulation Formats

Tobias Roethlingshoefer, Thomas Richter, Birgit Stiller, Georgy Onishchukov, Colja Schubert, Bernhard Schmauss, Gerd Leuchs

With growing demand on transmission capacity, spectral-efficient multilevel modulation formats such as quadrature amplitude modulation (QAM) become of great interest. One of their weaknesses is high sensitivity to noise accumulation, especially in long-haul transmission systems. Our investigations have shown the possibility of all-optical regeneration of multiple amplitude and phase states. Processing of amplitude noise in several amplitude states is based on periodicity of interference conditions in modified nonlinear fiber Sagnac interferometers. Their staircase-like power transfer characteristic can be used for phase-preserving amplitude regeneration of multiple amplitude states. Processing of QAM with up to three non-zero amplitude states, e. g. 16QAM, has been demonstrated in numerical simulations. Furthermore, simultaneous amplitude regeneration of a star-8QAM format with two amplitude states was performed experimentally. Recently, it has also been shown that phase-sensitive amplification for multiple phase states can be realized in fiber optical parametric amplifiers using four-wave mixing (FWM) with a high-order idler. Our numerical simulations and experimental results for star-8QAM revealed that with some modifications, this approach can be used not only for reduction of phase noise in multilevel phase-shift keying but also for signals with multiple amplitude states. The transmission improvement using a cascade of these two regenerator types has been demonstrated in numerical simulations and experiments. Numerical investigations confirm also the possibility to combine both approaches in a single device by using the highly nonlinear fiber in the Sagnac interferometer loop simultaneously for phase-sensitive amplification in one propagation direction.

Quantum mutual information of an entangled state propagating through a fast-light medium

Jeremy B. Clark, Ryan T. Glasser, Quentin Glorieux, Ulrich Vogl, Tian Li, Kevin M. Jones, Paul D. Lett

It is widely accepted that information cannot travel faster than c, the speed of light in vacuum(1-3). Here, we investigate the behaviour of quantum correlations and information in the presence of dispersion. To do so we send one half of an entangled state of light through a gain-assisted slow-or fast-light medium and detect the transmitted quantum correlations and quantum mutual information(4-6). We show that quantum correlations can be advanced by a small fraction of the correlation time, even in the presence of noise added by phase-insensitive gain. Additionally, although the peak of the quantum mutual information between the modes can be advanced, we find that the degradation of the mutual information due to added noise appears to prevent an advancement of the leading edge. In contrast, we demonstrate a significant delay of both the leading and trailing edges of the mutual information in a slow-light system.

All-optical phase-preserving multilevel amplitude regeneration

Tobias Roethlingshoefer, Thomas Richter, Colja Schubert, Georgy Onishchukov, Bernhard Schmauss, Gerd Leuchs

The possibility of all-optical phase-preserving amplitude regeneration for star-8QAM is demonstrated using a modified nonlinear optical loop mirror. Experiments show a reduction in amplitude noise on both amplitude levels simultaneously, considering two different types of signal distortions: deterministic low-frequency amplitude modulation and broadband amplitude noise. Furthermore, using this amplitude regeneration, the robustness against nonlinear phase noise from fiber nonlinearity in a transmission line is increased. The scheme suppresses the conversion of amplitude noise to nonlinear phase noise. This is shown for simultaneous amplitude regeneration of the two amplitude states as well as for amplitude regeneration of the high-power states only. If the transmission is limited by nonlinear phase noise, single-level operation at the more critical higher-power state will benefit because of the wider plateau region. Numerical simulations confirm the experimental results. (C) 2014 Optical Society of America.

Phase regeneration of a star-8QAM signal in a phase-sensitive amplifier with conjugated pumps

B. Stiller, G. Onishchukov, B. Schmauss, G. Leuchs

We demonstrate numerically phase regeneration of a star8QAM signal with two amplitude and four phase states in a phase-sensitive amplifier. In a dual-stage setup, two phase-conjugated idlers are generated in a first stage consisting of two fiber-optic parametric phase-insensitive amplifiers operated in highly nonlinear gain regime. These are used as pumps in the second, phase-sensitive amplification stage which enables efficient phase regeneration via a degenerate four-wave-mixing process. The latter can be operated in two different operation modes: without format conversion or with phase-shifted amplitude levels. In both regimes, we observe high phase-regeneration efficiency for all amplitude levels: the initial phase noise with 0.2 rad standard deviation is reduced by a factor of 5. (C) 2014 Optical Society of America

Near-Infrared Spectroscopy on a Single Photon Level

Michael Foertsch, Thomas Gerrits, Martin J. Stevens, Josef U. Fuerst, Gerhard Schunk, Florian Sedlmeir, Harald G. L. Schwefel, Dmitry Strekalov, Sae Woo Nam, et al.

We present a spectroscopy technique on the single photon level by combining a widely wavelength tunable single photon source based on a crystalline whispering gallery mode resonator with an energy resolving transition edge sensor.

Decorrelated PDC Source at Telecom Wavelengths with Identical Signal and Idler

Georg Harder, Vahid Ansari, Benjamin Brecht, Thomas Dirmeier, Christoph Marquardt, Christine Silberhorn

Multipolar hierarchy of efficient quantum polarization measures

P. de la Hoz, A. B. Klimov, G. Bjork, Y. -H. Kim, C. Mueller, Ch. Marquardt, G. Leuchs, L. L. Sanchez-Soto

We advocate a simple multipole expansion of the polarization density matrix. The resulting multipoles appear as successive moments of the Stokes variables and can be obtained from feasible measurements. In terms of these multipoles we construct a whole hierarchy of measures that accurately assess higher-order polarization fluctuations.

A versatile source of single photons for quantum information processing

Michael Foertsch, Josef U. Fuerst, Christoffer Wittmann, Dmitry Strekalov, Andrea Aiello, Maria V. Chekhova, Christine Silberhorn, Gerd Leuchs, Christoph Marquardt

The generation of high-quality single-photon states with controllable narrow spectral bandwidths and central frequencies is key to facilitate efficient coupling of any atomic system to non-classical light fields. Such an interaction is essential in numerous experiments for fundamental science and applications in quantum communication and information processing, as well as in quantum metrology. Here we implement a fully tunable, narrow-band and efficient single-photon source based on a whispering gallery mode resonator. Our disk-shaped, monolithic and intrinsically stable resonator is made of lithium niobate and supports a cavity-assisted spontaneous parametric down-conversion process. The generated photon pairs are emitted into two highly tunable resonator modes. We verify wavelength tuning over 100 nm of both modes with controllable bandwidth between 7.2 and 13 MHz. Heralding of single photons yields anti-bunching with g((2))(0) < 0.2.

Complex polarization in non z-cut whispering gallery mode resonators

F. Sedlmeir, M. Hauer, J. Fuerst, D. V. Strekalov, H. G. L. Schwefel

Optimal working points for continuous-variable quantum channels

Imran Khan, Christoffer Wittmann, Nitin Jain, Nathan Killoran, Norbert Luetkenhaus, Christoph Marquardt, Gerd Leuchs

The most important ability of a quantum channel is to preserve the quantum properties of transmitted quantum states. We experimentally demonstrate a continuous-variable system for efficient benchmarking of quantum channels. We probe the tested quantum channels for a wide range of experimental parameters such as amplitude, phase noise, and channel lengths up to 40 km. The data is analyzed using the framework of effective entanglement. We subsequently are able to deduce an optimal point of operation for each quantum channel with respect to the rate of distributed entanglement. This procedure is a promising candidate for benchmarking quantum nodes and individual links in large quantum networks of different physical implementations.

Measurement-induced amplification of optical cat-like states

A. Laghaout, J. S. Neergaard-Nielsen, J. Rigas, C. Kragh, A. Tipsmark, U. L. Andersen

Experimental characterization of an uniaxial angle cut whispering gallery mode resonator

Florian Sedlmeir, Martin Hauer, Josef U. Fuerst, Gerd Leuchs, Harald G. L. Schwefel

The usual configuration of uniaxial whispering gallery mode resonators is a disk shaped geometry where the optic axis points along the symmetry axis, a so called z-cut resonator. Recently x-cut resonators, where the optic axis lies in the equatorial plane, became of interest as they enable extremely broadband second harmonic generation. In this paper we report on the properties of a more generalized system, the so called angle-cut resonator, where the optic axis exhibits an arbitrary angle against the symmetry axis. We show experimentally that the modal structure and quality factors are similar to common resonators but that the polarization properties differ quite significantly: due to the asymmetry the polarization depends on the equatorial position and is, in general, elliptical. (C) 2013 Optical Society of America

The polarization properties of a tilted polarizer

Jan Korger, Tobias Kolb, Peter Banzer, Andrea Aiello, Christoffer Wittmann, Christoph Marquardt, Gerd Leuchs

Polarizers are key components in optical science and technology. Thus, understanding the action of a polarizer beyond oversimplifying approximations is crucial. In this work, we study the interaction of a polarizing interface with an obliquely incident wave experimentally. To this end, a set of Mueller matrices is acquired employing a novel procedure robust against experimental imperfections. We connect our observation to a geometric model, useful to predict the effect of polarizers on complex light fields. (C) 2013 Optical Society of America

Distributing Entanglement with Separable States

Christian Peuntinger, Vanessa Chille, Ladislav Mista Jr., Natalia Korolkova, Michael Foertsch, Jan Korger, Christoph Marquardt, Gerd Leuchs

We experimentally demonstrate a protocol for entanglement distribution by a separable quantum system. In our experiment, two spatially separated modes of an electromagnetic field get entangled by local operations, classical communication, and transmission of a correlated but separable mode between them. This highlights the utility of quantum correlations beyond entanglement for the establishment of a fundamental quantum information resource and verifies that its distribution by a dual classical and separable quantum communication is possible.

Quantum polarization tomography of bright squeezed light

C. R. Mueller, B. Stoklasa, C. Peuntinger, C. Gabriel, J. Rehacek, Z. Hradil, A. B. Klimov, G. Leuchs, Ch Marquardt, et al.

We reconstruct the polarization sector of a bright polarization squeezed beam starting from a complete set of Stokes measurements. Given the symmetry that underlies the polarization structure of quantum fields, we use the unique SU(2) Wigner distribution to represent states. In the limit of localized bright states, the Wigner function can be approximated by an inverse three-dimensional Radon transform. We compare this direct reconstruction with the results of a maximum likelihood estimation, thus finding excellent agreement.

Tools for detecting entanglement between different degrees of freedom in quadrature squeezed cylindrically polarized modes

C. Gabriel, A. Aiello, S. Berg-Johansen, Ch Marquardt, G. Leuchs

Quadrature squeezed cylindrically polarized modes contain entanglement not only in the polarization and spatial electric field variables but also between these two degrees of freedom [C. Gabriel et al., Phys. Rev. Lett. 106, 060502 (2011)]. In this paper we present tools to generate and detect this entanglement. Experimentally we demonstrate the generation of quadrature squeezing in cylindrically polarized modes by mode transforming a squeezed Gaussian mode. Specifically, -1.2dB +/- 0.1 dB of amplitude squeezing are achieved in the radially and azimuthally polarized mode. Furthermore, theoretically it is shown how the entanglement contained within these modes can be measured and how strong the quantum correlations are, depending on the measurement scheme.

Entanglement of Gaussian states and the applicability to quantum key distribution over fading channels

Vladyslav C. Usenko, Bettina Heim, Christian Peuntinger, Christoffer Wittmann, Christoph Marquardt, Gerd Leuchs, Radim Filip

Entanglement properties of Gaussian states of light as well as the security of continuous variable quantum key distribution with Gaussian states in free-space fading channels are studied. These qualities are shown to be sensitive to the statistical properties of the transmittance distribution in the cases when entanglement is strong or when channel excess noise is present. Fading, i.e. transmission fluctuations, caused by beam wandering due to atmospheric turbulence, is a frequent challenge in free-space communication. We introduce a method of fading discrimination and subsequent post-selection of the corresponding sub-states and show that it can improve the entanglement resource and restore the security of the key distribution over a realistic fading link. Furthermore, the optimal post-selection strategy in combination with an optimized entangled resource is shown to drastically increase the protocol's robustness to excess noise, which is confirmed for experimentally measured fading channel characteristics. The stability of the result against finite data ensemble size and imperfect channel estimation is also addressed.

Experimental demonstration of the geometric spin Hall effect of light in highly focused vector beams

Martin Neugebauer, Peter Banzer, Thomas Bauer, Norbert Lindlein, Andrea Aiello, Christoph Marquardt, Jan Korger, Gerd Leuchs

We demonstrate the geometric spin Hall effect of light by focusing a specially polarization tailored beam of light, resulting in the generation of purely transverse angular momentum in the focal plane. (C) 2011 Optical Society of America

On the use of a Continuous Wave Nd:YAG Laser in the Generation of Random Numbers

M. Sabuncu

The Nd:YAG laser is a solid state laser that has many applications in the industry, science and medicine. We in this paper use a continuous wave (CW) Nd:YAG laser emitting light at 1064 nm to generate random numbers. We first give a recipe of an experimental setup that is capable of measuring and recording the vacuum fluctuations with the help of our cw Nd:YAG laser. The basic scheme simply involves the Nd:YAG laser, detectors and some radio frequency electronic components. The laser source used in the demonstration provides us with a quantum noise limited CW at a radio frequency sideband of 14 MHz. The final data obtained after appropriate electronic signal processing are digitized voltage fluctuations that correspond to the vacuum fluctuations in the optical domain. The data randomly lies above or below zero due to its quantum mechanical probabilistic nature. We assign a '1' to the data that is positive and a '0' to the data that is negative, hence we are able to generate a random bit sequence by using a cw Nd:YAG laser. This result adds yet another interesting application to the many number of broad applications of the Nd:YAG laser system.

Studying free-space transmission statistics and improving free-space quantum key distribution in the turbulent atmosphere

C. Erven, B. Heim, E. Meyer-Scott, J. P. Bourgoin, R. Laflamme, G. Weihs, T. Jennewein

The statistical fluctuations in free-space links in the turbulent atmosphere are important for the distribution of quantum signals. To that end, we first study statistics generated by the turbulent atmosphere in an entanglement-based free-space quantum key distribution (QKD) system. Using the insights gained from this analysis, we study the effect of link fluctuations on the security and key generation rate of decoy state QKD concluding that it has minimal effect in the typical operating regimes. We then investigate the novel idea of using these turbulent fluctuations to our advantage in QKD experiments. We implement a signal-to-noise ratio filter (SNRF) in our QKD system which rejects measurements during periods of low transmission efficiency, where the measured quantum bit error rate is temporarily elevated. Using this, we increase the total secret key generated by the system from 78 009 bits to 97 678 bits, representing an increase of 25.2% in the final secure key rate, generated from the same raw signals. Lastly, we present simulations of a QKD exchange with an orbiting low earth orbit satellite and show that an SNRF will be extremely useful in such a situation, allowing many more passes to extract a secret key than would otherwise be possible.

Probabilistic cloning of coherent states without a phase reference

Christian R. Mueller, Christoffer Wittmann, Petr Marek, Radim Filip, Christoph Marquardt, Gerd Leuchs, Ulrik L. Andersen

We present a probabilistic cloning scheme operating independently of any phase reference. The scheme is based solely on a phase-randomized displacement and photon counting, omitting the need for nonclassical resources and nonlinear materials. In an experimental implementation, we employ the scheme to clone coherent states from a phase covariant alphabet and demonstrate that the cloner is capable of outperforming the hitherto best-performing deterministic scheme. An analysis of the covariances between the output states shows that uncorrelated clones can be approached asymptotically. This simultaneously demonstrates how the effect of loss on coherent states can be compensated via noiseless preamplification.

A High-Speed Secure Quantum Random Number Generator Based on Vacuum States

Christian Gabriel, Christoffer Wittmann, Bastian Hacker, Wolfgang Mauerer, Elanor Huntington, Metin Sabuncu, Christoph Marquardt, Gerd Leuchs

A high-speed continuous-variable quantum random bit generator with an expected effective bit generation rate of up to 10GBit/s is presented. The obtained bit sequences are truly random and unique, i.e. they cannot be known by an adversary. (c) 2011 Optical Society of America

Four-state discrimination scheme beyond the heterodyne limit

Christian R. Mueller, Mario A. Usuga, Christoffer Wittmann, Masahiro Takeoka, Christoph Marquardt, Ulrik L. Andersen, Gerd Leuchs

We propose and experimentally demonstrate a hybrid discrimination scheme for the quadrature phase shift keying protocol, which outperforms heterodyne detection for any signal power. The discrimination is composed of a quadrature measurement, feed forward and photon detection. (C) 2011 Optical Society of America

Quadrature phase shift keying coherent state discrimination via a hybrid receiver

C. R. Mueller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch Marquardt, U. L. Andersen, G. Leuchs

We propose and experimentally demonstrate a near-optimal discrimination scheme for the quadrature phase shift keying (QPSK) protocol. We show in theory that the performance of our hybrid scheme is superior to the standard scheme-heterodyne detection-for all signal amplitudes and underpin the predictions with our experimental results. Furthermore, our scheme provides hitherto the best performance in the domain of highly attenuated signals. The discrimination is composed of a quadrature measurement, a conditional displacement and a threshold detector.

Concentrating the phase of a coherent state by means of probabilistic amplification

Mario A. Usuga, Christian R. Mueller, Christoffer Wittmann, Petr Marek, Radim Filip, Christoph Marquardt, Gerd Leuchs, Ulrik L. Andersen

We discuss the recent implementation of phase concentration of an optical coherent state by use of a probabilistic noiseless amplifier. The operation of the amplifier is described pictorially with phase space diagrams, and the experimental results are outlined.

Classical and quantum properties of cylindrically polarized states of light

Annemarie Holleczek, Andrea Aiello, Christian Gabriel, Christoph Marquardt, Gerd Leuchs

We investigate theoretical properties of beams of light with non-uniform polarization patterns. Specifically, we determine all possible configurations of cylindrically polarized modes (CPMs) of the electromagnetic field, calculate their total angular momentum and highlight the subtleties of their structure. Furthermore, a hybrid spatio-polarization description for such modes is introduced and developed. In particular, two independent Poincare spheres have been introduced to represent simultaneously the polarization and spatial degree of freedom of CPMs. Possible mode-to-mode transformations accomplishable with the help of Bconventional polarization and spatial phase retarders are shown within this representation. Moreover, the importance of these CPMs in the quantum optics domain due to their classical features is highlighted. (C) 2011 Optical Society of America

After-gate attack on a quantum cryptosystem

C. Wiechers, L. Lydersen, C. Wittmann, D. Elser, J. Skaar, Ch Marquardt, V. Makarov, G. Leuchs

We present a method to control the detection events in quantum key distribution systems that use gated single-photon detectors. We employ bright pulses as faked states, timed to arrive at the avalanche photodiodes outside the activation time. The attack can remain unnoticed, since the faked states do not increase the error rate per se. This allows for an intercept-resend attack, where an eavesdropper transfers her detection events to the legitimate receiver without causing any errors. As a side effect, afterpulses, originating from accumulated charge carriers in the detectors, increase the error rate. We have experimentally tested detectors of the system id3110 (Clavis2) from ID Quantique. We identify the parameter regime in which the attack is feasible despite the side effect. Furthermore, we outline how simple modifications in the implementation can make the device immune to this attack.

Quantum Light from a Whispering-Gallery-Mode Disk Resonator

J. U. Fuerst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, Ch Marquardt, G. Leuchs

Optical parametric down-conversion has proven to be a valuable source of nonclassical light. The process is inherently able to produce twin-beam correlations along with individual intensity squeezing of either parametric beam, when pumped far above threshold. Here, we present for the first time the direct observation of intensity squeezing of -1.2 dB of each of the individual parametric beams in parametric down-conversion by use of a high quality whispering-gallery-mode disk resonator. In addition, we observed twin-beam quantum correlations of -2.7 dB with this cavity. Such resonators feature strong optical confinement and offer tunable coupling to an external optical field. This work exemplifies the potential of crystalline whispering-gallery-mode resonators for the generation of quantum light. The simplicity of this device makes the application of quantum light in various fields highly feasible.

Entangling Different Degrees of Freedom by Quadrature Squeezing Cylindrically Polarized Modes

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Foertsch, D. Elser, U. L. Andersen, et al.

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.

Device Calibration Impacts Security of Quantum Key Distribution

Nitin Jain, Christoffer Wittmann, Lars Lydersen, Carlos Wiechers, Dominique Elser, Christoph Marquardt, Vadim Makarov, Gerd Leuchs

Characterizing the physical channel and calibrating the cryptosystem hardware are prerequisites for establishing a quantum channel for quantum key distribution (QKD). Moreover, an inappropriately implemented calibration routine can open a fatal security loophole. We propose and experimentally demonstrate a method to induce a large temporal detector efficiency mismatch in a commercial QKD system by deceiving a channel length calibration routine. We then devise an optimal and realistic strategy using faked states to break the security of the cryptosystem. A fix for this loophole is also suggested.

Quantum Effects in Optical Fibers

Gerd Leuchs, Christoph Marquardt

The quantization of the light field is important for understanding some aspects of optical fiber systems: the quantum limit of amplifiers, the dynamics of solitons below shot noise and the appearance of non-classical light.

Superlinear threshold detectors in quantum cryptography

Lars Lydersen, Nitin Jain, Christoffer Wittmann, Oystein Maroy, Johannes Skaar, Christoph Marquardt, Vadim Makarov, Gerd Leuchs

We introduce the concept of a superlinear threshold detector, a detector that has a higher probability to detect multiple photons if it receives them simultaneously rather than at separate times. Highly superlinear threshold detectors in quantum key distribution systems allow eavesdropping the full secret key without being revealed. Here, we generalize the detector control attack, and analyze how it performs against quantum key distribution systems with moderately superlinear detectors. We quantify the superlinearity in superconducting single-photon detectors based on earlier published data, and gated avalanche photodiode detectors based on our own measurements. The analysis shows that quantum key distribution systems using detector(s) of either type can be vulnerable to eavesdropping. The avalanche photodiode detector becomes superlinear toward the end of the gate. For systems expecting substantial loss, or for systems not monitoring loss, this would allow eavesdropping using trigger pulses containing less than 120 photons per pulse. Such an attack would be virtually impossible to catch with an optical power meter at the receiver entrance.

Geometric Spin Hall Effect of Light at polarizing interfaces

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, G. Leuchs

The geometric Spin Hall Effect of Light (geometric SHEL) amounts to a polarization-dependent positional shift when a light beam is observed from a reference frame tilted with respect to its direction of propagation. Motivated by this intriguing phenomenon, the energy density of the light beam is decomposed into its Cartesian components in the tilted reference frame. This illustrates the occurrence of the characteristic shift and the significance of the effective response function of the detector. We introduce the concept of a tilted polarizing interface and provide a scheme for its experimental implementation. A light beam passing through such an interface undergoes a shift resembling the original geometric SHEL in a tilted reference frame. This displacement is generated at the polarizer and its occurrence does not depend on the properties of the detection system. We give explicit results for this novel type of geometric SHEL and show that at grazing incidence this effect amounts to a displacement of multiple wavelengths, a shift larger than the one introduced by Goos-Hanchen and Imbert-Fedorov effects.

Birefringence and dispersion of cylindrically polarized modes in nanobore photonic crystal fiber

T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Foertsch, P. Banzer, A. Brenn, et al.

We demonstrate experimentally and theoretically that a nanoscale hollow channel placed centrally in the solid-glass core of a photonic crystal fiber strongly enhances the cylindrical birefringence (the modal index difference between radially and azimuthally polarized modes). Furthermore, it causes a large split in group velocity and group velocity dispersion. We show analytically that all three parameters can be varied over a wide range by tuning the diameters of the nanobore and the core. (C) 2010 Optical Society of America

Noise-powered probabilistic concentration of phase information

Mario A. Usuga, Christian R. Mueller, Christoffer Wittmann, Petr Marek, Radim Filip, Christoph Marquardt, Gerd Leuchs, Ulrik L. Andersen

Phase-insensitive optical amplification of an unknown quantum state is known to be a fundamentally noisy operation that inevitably adds noise to the amplified state(1-5). However, this fundamental noise penalty in amplification can be circumvented by resorting to a probabilistic scheme as recently proposed and demonstrated in refs 6-8. These amplifiers are based on highly non-classical resources in a complex interferometer. Here we demonstrate a probabilistic quantum amplifier beating the fundamental quantum limit using a thermal-noise source and a photon-number-subtraction scheme(9). The experiment shows, surprisingly, that the addition of incoherent noise leads to a noiselessly amplified output state with a phase uncertainty below the uncertainty of the state before amplification. This amplifier might become a valuable quantum tool in future quantum metrological schemes and quantum communication protocols.

A bridge between the single-photon and squeezed-vacuum states

Nitin Jain, S. R. Huisman, Erwan Bimbard, A. I. Lvovsky

The two modes of the Einstein-Podolsky-Rosen quadrature entangled state generated by parametric down-conversion interfere on a beam splitter of variable splitting ratio. Detection of a photon in one of the beam splitter output channels heralds preparation of a signal state in the other, which is characterized using homodyne tomography. By controlling the beam splitting ratio, the signal state can be chosen anywhere between the single-photon and squeezed state. (C) 2010 Optical Society of America

A novel method for polarization squeezing with Photonic Crystal Fibers

Josip Milanovic, Mikael Lassen, Ulrik L. Andersen, Gerd Leuchs

Photonic Crystal Fibers can be tailored to increase the effective Kerr nonlinearity, while producing smaller amounts of excess noise compared to standard silicon fibers. Using these features of Photonic Crystal Fibers we create polarization squeezed states with increased purity compared to standard fiber squeezing experiments. Explicit we produce squeezed states in counter propagating pulses along the same fiber axis to achieve near identical dispersion properties. This enables the production of polarization squeezing through interference in a polarization type Sagnac interferometer. We observe Stokes parameter squeezing of -3.9 +/- 0.3dB and anti-squeezing of 16.2 +/- 0.3dB. (C) 2010 Optical Society of America

Concentration of Phase-Information

Mario Usuga, Christian Mueller, Christoffer Wittmann, Petr Marek, Radim Filip, Christoph Marquardt, Ulrik L. Andersen, Gerd Leuchs

We demonstrate a probabilistic scheme capable of increasing the phase information of weak coherent states by addition of thermal noise and subsequent photon-counting based postselection. We present experimental and theoretical results. (C) 2010 Optical Society of America

Discrimination of optical coherent states using a photon number resolving detector

Christoffer Wittmann, Ulrik L. Andersen, Gerd Leuchs

The discrimination of non-orthogonal quantum states with reduced or without errors is a fundamental task in quantum measurement theory. In this work, we investigate a quantum measurement strategy capable of discriminating two coherent states probabilistically with significantly smaller error probabilities than can be obtained using non-probabilistic state discrimination. We find that appropriate postselection of the measurement data of a photon number resolving detector can be used to discriminate two coherent states with small error probability. We compare our new receiver to an optimal intermediate measurement between minimum error discrimination and unambiguous state discrimination.

Witnessing effective entanglement over a 2km fiber channel

Christoffer Wittmann, Josef Fuerst, Carlos Wiechers, Dominique Elser, Hauke Haeseler, Norbert Luetkenhaus, Gerd Leuchs

We present a fiber-based continuous-variable quantum key distribution system. In the scheme, a quantum signal of two non-orthogonal weak optical coherent states is sent through a fiber-based quantum channel. The receiver simultaneously measures conjugate quadratures of the light using two homodyne detectors. From the measured Q-function of the transmitted signal, we estimate the attenuation and the excess noise caused by the channel. The estimated excess noise originating from the channel and the channel attenuation including the quantum efficiency of the detection setup is investigated with respect to the detection of effective entanglement. The local oscillator is considered in the verification. We witness effective entanglement with a channel length of up to 2km. (C) 2010 Optical Society of America

Experimental demonstration of squeezed-state quantum averaging

Mikael Lassen, Lars Skovgaard Madsen, Metin Sabuncu, Radim Filip, Ulrik L. Andersen

We propose and experimentally demonstrate a universal quantum averaging process implementing the harmonic mean of quadrature variances. The averaged variances are prepared probabilistically by means of linear optical interference and measurement-induced conditioning. We verify that the implemented harmonic mean yields a lower value than the corresponding value obtained for the standard arithmetic-mean strategy. The effect of quantum averaging is experimentally tested for squeezed and thermal states as well as for uncorrelated and partially correlated noise sources. The harmonic-mean protocol can be used to efficiently stabilize a set of squeezed-light sources with statistically fluctuating noise levels.

Low-Threshold Optical Parametric Oscillations in a Whispering Gallery Mode Resonator

J. U. Fuerst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, Ch. Marquardt, G. Leuchs

In whispering gallery mode (WGM) resonator light is guided by continuous total internal reflection along a curved surface. Fabricating such resonators from an optically nonlinear material one takes advantage of their exceptionally high quality factors and small mode volumes to achieve extremely efficient optical frequency conversion. Our analysis of the phase-matching conditions for optical parametric down-conversion (PDC) in a spherical WGM resonator shows their direct relation to the sum rules for photons' angular momenta and predicts a very low parametric oscillation threshold. We realized such an optical parametric oscillator (OPO) based on naturally phase-matched PDC in lithium niobate. We demonstrated a single-mode, strongly nondegenerate OPO with a threshold of 6: 7 mu W and linewidth under 10 MHz. This work demonstrates the remarkable capabilities of WGM-based OPOs.

Avoiding the blinding attack in QKD reply

Lars Lydersen, Carlos Wiechers, Christoffer Wittmann, Dominique Elser, Johannes Skaar, Vadim Makarov

Continuous-variable entanglement distillation of non-Gaussian mixed states

Ruifang Dong, Mikael Lassen, Joel Heersink, Christoph Marquardt, Radim Filip, Gerd Leuchs, Ulrik L. Andersen

Many different quantum-information communication protocols such as teleportation, dense coding, and entanglement-based quantum key distribution are based on the faithful transmission of entanglement between distant location in an optical network. The distribution of entanglement in such a network is, however, hampered by loss and noise that is inherent in all practical quantum channels. Thus, to enable faithful transmission one must resort to the protocol of entanglement distillation. In this paper we present a detailed theoretical analysis and an experimental realization of continuous variable entanglement distillation in a channel that is inflicted by different kinds of non-Gaussian noise. The continuous variable entangled states are generated by exploiting the third order nonlinearity in optical fibers, and the states are sent through a free-space laboratory channel in which the losses are altered to simulate a free-space atmospheric channel with varying losses. We use linear optical components, homodyne measurements, and classical communication to distill the entanglement, and we find that by using this method the entanglement can be probabilistically increased for some specific non-Gaussian noise channels.

Microwave whispering-gallery resonator for efficient optical up-conversion

D. V. Strekalov, H. G. L. Schwefel, A. A. Savchenkov, A. B. Matsko, L. J. Wang, N. Yu

Conversion of microwave radiation into the optical range has been predicted to reach unity quantum efficiency in whispering-gallery resonators made from an optically nonlinear crystal and supporting microwave and optical modes simultaneously. In this work, we theoretically explore and experimentally demonstrate a resonator geometry that can provide the required phase matching for such a conversion at any desired frequency in the sub-THz range. We show that such a ring-shaped resonator not only allows for the phase matching but also maximizes the overlap of the interacting fields. As a result, unity-efficient conversion is expected in a resonator with feasible parameters.

Quadrature measurements of a bright squeezed state via sideband swapping

Jessica Schneider, Oliver Gloeckl, Gerd Leuchs, Ulrik L. Andersen

The measurement of an arbitrary quadrature of a bright quantum state of light is a commonly requested action in many quantum information protocols, but it is experimentally challenging with previously proposed schemes. We suggest that the quadrature be measured at a specific sideband frequency of a bright quantum state by transferring the sideband modes under interrogation to a vacuum state and subsequently measuring the quadrature via homodyne detection. The scheme is implemented experimentally, and it is successfully tested with a bright squeezed state of light. (C) 2009 Optical Society of America

Feasibility of free space quantum key distribution with coherent polarization states

D. Elser, T. Bartley, B. Heim, Ch Wittmann, D. Sych, G. Leuchs

We demonstrate for the first time the feasibility of free space quantum key distribution with continuous variables under real atmospheric conditions. More specifically, we transmit coherent polarization states over a 100m free space channel on the roof of our institute's building. In our scheme, signal and local oscillator (LO) are combined in a single spatial mode, which auto-compensates atmospheric fluctuations and results in an excellent interference. Furthermore, the LO acts as a spatial and spectral filter, thus allowing unrestrained daylight operation.

Nonparaxial polarizers

Andrea Aiello, Christoph Marquardt, Gerd Leuchs

We develop a theoretical description for polarizers that goes beyond the paraxial approximation. By combining existing theories for fields with nonplanar wavefronts, we are able to derive a simple power series expansion expressing the electric field of a light beam after a polarizer as a linear function of the field and its spatial derivatives evaluated before the polarizer. The first few terms of such expansion are explicitly given, and their physical meaning is discussed. (C) 2009 Optical Society of America

Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, G. Leuchs

This Colloquium examines the field of the Einstein, Podolsky, and Rosen (EPR) gedanken experiment, from the original paper of Einstein, Podolsky, and Rosen, through to modern theoretical proposals of how to realize both the continuous-variable and discrete versions of the EPR paradox. The relationship with entanglement and Bell's theorem are analyzed, and the progress to date towards experimental confirmation of the EPR paradox is summarized, with a detailed treatment of the continuous-variable paradox in laser-based experiments. Practical techniques covered include continuous-wave parametric amplifier and optical fiber quantum soliton experiments. Current proposals for extending EPR experiments to massive-particle systems are discussed, including spin squeezing, atomic position entanglement, and quadrature entanglement in ultracold atoms. Finally, applications of this technology to quantum key distribution, quantum teleportation, and entanglement swapping are examined.

Transverse Angular Momentum and Geometric Spin Hall Effect of Light

Andrea Aiello, Norbert Lindlein, Christoph Marquardt, Gerd Leuchs

We present a novel fundamental phenomenon occurring when a polarized beam of light is observed from a reference frame tilted with respect to the direction of propagation of the beam. This effect has a purely geometric nature and amounts to a polarization-dependent shift or split of the beam intensity distribution evaluated as the time-averaged flux of the Poynting vector across the plane of observation. We demonstrate that such a shift is unavoidable whenever the beam possesses a nonzero transverse angular momentum. This latter result has general validity and applies to arbitrary systems such as, e.g., electronic and atomic beams.

Continuous Variable Entanglement Distillation of Non-Gaussian States

Mikael Lassen, Ruifang Dong, Joel Heersink, Christoph Marquardt, Radim Filip, Gerd Leuchs, Ulrik L. Andersen

We experimentally demonstrate distillation of continuous variable entangled light that has undergone non-Gaussian attenuation loss. The continuous variable entanglement is generated with optical fibers and sent through a lossy channel, where the transmission is varying in time. By employing simple linear optical components, a measurement induced operation and classical communication, we demonstrate that the entanglement is probabilistically increased.

Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States

M. Lassen, G. Leuchs, U. L. Andersen

We report the first experimental characterization of the first-order continuous variable orbital angular momentum states. Using a spatially nondegenerate optical parametric oscillator (OPO) we produce quadrature entanglement between the two first-order Laguerre-Gauss modes. The family of orbital angular momentum modes is mapped on an orbital Poincare sphere, where the mode's position on the sphere is spanned by the three orbital parameters. Using a nondegenerate OPO we produce squeezing of these parameters, and as an illustration, we reconstruct the "cigar-shaped" uncertainty volume on the orbital Poincare sphere.

Correlation measurement of squeezed light

Leonid A. Krivitsky, Ulrik L. Andersen, Ruifang Dong, Alexander Huck, Christoffer Wittmann, Gerd Leuchs

We study the implementation of a correlation measurement technique for the characterization of squeezed light which is nearly free of electronic noise. With two different sources of squeezed light, we show that the sign of the covariance coefficient, revealed from the time-resolved correlation data, is witnessing the presence of squeezing in the system. Furthermore, we estimate the degree of squeezing using the correlation method and compare it to the standard homodyne measurement scheme. We show that the role of electronic detector noise is minimized using the correlation approach as opposed to homodyning where it often becomes a crucial issue.

Electronic noise-free measurements of squeezed light

Leonid A. Krivitsky, Ulrik L. Andersen, Ruifang Dong, Alexander Huck, Christoffer Wittmann, Gerd Leuchs

We study the implementation of a correlation measurement technique for the characterization of squeezed light. We show that the sign of the covariance coefficient revealed from the time-resolved correlation data allow us to distinguish among squeezed, coherent, and thermal states. In contrast to the traditional method of characterizing squeezed light, involving measurement of the variation of the difference photocurrent, the correlation measurement method allows one to eliminate the contribution of the electronic noise, which becomes a crucial issue in experiments with dim sources of squeezed light. (C) 2008 Optical Society of America

Experimental Demonstration of Continuous Variable Entanglement Distillation

R. F. Dong, M. Lassen, Ch. Marquardt, R. Filip, U. L. Andersen, G. Leuchs

We experimentally demonstrate distillation of continuous variable entanglement through an attenuating quantum channel. Through a simple protocol based entirely on linear optics and homodyne detection we probabilistically filter out a highly entangled state from a less non-entangled mixed state. (C) 2008 Optical Society of America

Demonstration of a Quantum Nondemolition Sum Gate

Jun-ichi Yoshikawa, Yoshichika Miwa, Alexander Huck, Ulrik L. Andersen, Peter van Loock, Akira Furusawa

The sum gate is the canonical two-mode gate for universal quantum computation based on continuous quantum variables. It represents the natural analogue to a qubit C-NOT gate. In addition, the continuous-variable gate describes a quantum nondemolition (QND) interaction between the quadrature components of two light modes. We experimentally demonstrate a QND sum gate, employing the scheme by R. Filip, P. Marek, and U. L. Andersen [Phys. Rev. A 71, 042308 (2005)], solely based on off-line squeezed states, homodyne measurements, and feedforward. The results are verified by simultaneously satisfying the criteria for QND measurements in both conjugate quadratures.

Experimental Demonstration of Macroscopic Quantum Coherence in Gaussian States

Ch. Marquardt, U. L. Andersen, G. Leuchs, Y. Takeno, M. Yukawa, H. Yonezawa, A. Furusawa

We experimentally proved the existence of macroscopic coherences in Gaussian quantum states using a recently proposed criterion [I]. We show interferences with remarkable distances in phase space for squeezed, entangled and even for coherent states. (C) 2008 Optical Society of America

Experimental entanglement distillation of mesoscopic quantum states

Ruifang Dong, Mikael Lassen, Joel Heersink, Christoph Marquardt, Radim Filip, Gerd Leuchs, Ulrik L. Andersen

The distribution of entangled states between distant parties in an optical network is crucial for the successful implementation of various quantum communication protocols such as quantum cryptography, teleportation and dense coding(1-3). However, owing to the unavoidable loss in any real optical channel, the distribution of loss-intolerant entangled states is inevitably afflicted by decoherence, which causes a degradation of the transmitted entanglement. To combat the decoherence, entanglement distillation, a process of extracting a small set of highly entangled states from a large set of less entangled states, can be used(4-14). Here we report on the distillation of deterministically prepared light pulses entangled in continuous variables that have undergone non-Gaussian noise. The entangled light pulses(15-17) are sent through a lossy channel, where the transmission is varying in time similarly to light propagation in the atmosphere. By using linear optical components and global classical communication, the entanglement is probabilistically increased.

Experimental continuous-variable cloning of partial quantum information

Metin Sabuncu, Gerd Leuchs, Ulrik L. Andersen

The fidelity of a quantum transformation is strongly linked with the prior partial information of the state to be transformed. We illustrate this interesting point by proposing and demonstrating the superior cloning of coherent states with prior partial information. More specifically, we propose two simple transformations that under the Gaussian assumption optimally clone symmetric Gaussian distributions of coherent states as well as coherent states with known phases. Furthermore, we implement for the first time near-optimal state-dependent cloning schemes relying on simple linear optics and feedforward.

Experimentally feasible quantum erasure-correcting code for continuous variables

J. Niset, U. L. Andersen, N. J. Cerf

We devise a scheme that protects quantum coherent states of light from probabilistic losses, thus achieving the first continuous-variable quantum erasure-correcting code. If the occurrence of erasures can be probed, then the decoder enables, in principle, a perfect recovery of the original light states. Otherwise, if supplemented with postselection based on homodyne detection, this code can be turned into an efficient erasure-filtration scheme. The experimental feasibility of the proposed protocol is carefully addressed.

Experimental evidence for Raman-induced limits to efficient squeezing in optical fibers

Ruifang Dong, Joel Heersink, Joel F. Corney, Peter D. Drummond, Ulrik L. Andersen, Gerd Leuchs

We report new experiments on polarization squeezing using ultrashort photonic pulses in a single pass of a birefringent fiber. We measure what is to our knowledge a record squeezing of -6.8 +/- 0.3 dB in optical fibers, which when corrected for linear losses is -10.4 +/- 0.8 dB. The measured polarization squeezing as a function of optical pulse energy, which spans a wide range from 3.5-178.8 pJ, shows a very good agreement with the quantum simulations, and for the first time we see the proof experimentally that Raman effects limit and reduce squeezing at high pulse energy. (c) 2008 Optical Society of America.

Quantum filtering of optical coherent states

C. Wittmann, D. Elser, U. L. Andersen, R. Filip, P. Marek, G. Leuchs

We propose and experimentally demonstrate nondestructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors, i.e., vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on-off detection, phase stabilized, and phase randomized homodyne detection. We find that on-off detection, optimal in the ideal theoretical setting, is superior to the homodyne strategy also in a practical setting.

Optical amplification at the quantum limit

Ulrik L. Andersen, Gerd Leuchs

In this paper we evaluate experimentally measured signal-to-noise ratio performance of different types of optical amplifiers [V. Josse, M. Sabuncu, N. Cerf, et al., Phys. Rev. Lett. 96 163602 ( 2006); Z. Y. Ou, S. F. Pereira and H. J. Kimble, Phys. Rev. Lett. 70 3239 ( 1993)]. We pay particular attention to the low gain regime where the quantum limit of the amplifier varies with gain. Independent of the amplifier type the noise performance is remarkably close to the quantum limit. Low gain optical amplifiers are of interest for quantum communication.

Guided acoustic wave Brillouin scattering in photonic crystal fibers

D. Elser, Ch Wittmann, U. L. Andersen, O. Gloeckl, S. Lorenz, Ch Marquardt, G. Leuchs

In silica glass fibers, thermally excited acoustic phonons scatter light into the beam propagating in the forward direction. At acoustic frequencies up to several hundreds of megahertz, the wave vectors of the phonons interacting with the light propagate essentially transversally to the fiber axis. This effect is known as Guided Acoustic Wave Brillouin Scattering (GAWBS) and leads to phase and polarization noise in the guided light. For fiber-based quantum optics experiments, this excess noise is a major limitation. In Photonic Crystal Fibers (PCFs), light is guided by a microstructure simultaneously acting as a 2D transversal phononic crystal which modifies the acoustic noise spectrum. We demonstrate a GAWBS-noise reduction in commercially available PCFs. This gives rise to the prospect of fiber-based quantum optic devices exhibiting less excess noise, thus resulting in higher quantum state purity. Further improvement can be achieved by tailoring the photonic microstructure such that a reduction of phonon noise by design is achieved.

Demonstration of deterministic and high fidelity squeezing of quantum information

Jun-ichi Yoshikawa, Toshiki Hayashi, Takayuki Akiyama, Nobuyuki Takei, Alexander Huck, Ulrik L. Andersen, Akira Furusawa

By employing a recent proposal [R. Filip, P. Marek, and U.L. Andersen, Phys. Rev. A 71, 042308 (2005)] we experimentally demonstrate a universal, deterministic, and high-fidelity squeezing transformation of an optical field. It relies only on linear optics, homodyne detection, feedforward, and an ancillary squeezed vacuum state, thus direct interaction between a strong pump and the quantum state is circumvented. We demonstrate three different squeezing levels for a coherent state input. This scheme is highly suitable for the fault-tolerant squeezing transformation in a continuous variable quantum computer.

Quantum reconstruction of an intense polarization squeezed optical state

Ch. Marquardt, J. Heersink, R. Dong, M. V. Chekhova, A. B. Klimov, L. L. Sanchez-Soto, U. L. Andersen, G. Leuchs

We perform a reconstruction of the polarization sector of the density matrix of an intense polarization squeezed beam starting from a complete set of Stokes measurements. By using an appropriate quasidistribution, we map this onto the Poincare space, providing a full quantum mechanical characterization of the measured polarization state.

An efficient source of continuous variable polarization entanglement

Ruifang Dong, Joel Heersink, Jun-Ichi Yoshikawa, Oliver Gloeckl, Ulrik L. Andersen, Gerd Leuchs

We have experimentally demonstrated the efficient creation of highly entangled bipartite continuous variable polarization states. Exploiting an optimized scheme for the production of squeezing using the Kerr non-linearity of a glass fibre we generated polarization squeezed pulses with a mean classical excitation in (S)over-cap(3). Polarization entanglement was generated by interfering two independent polarization squeezed fields on a symmetric beam splitter. The resultant beams exhibit strong quantum noise correlations in the dark (S)over-cap(1)-(S)over-cap(2) polarization plane. To verify entanglement generation, we characterized the quantum correlations of the system for two different sets of conjugate Stokes parameters. The quantum correlations along the squeezed and the anti-squeezed Stokes parameters were observed to be - 4.1 +/- 0.3 and - 2.6 +/- 0.3 dB below the shot noise level, respectively. The degree of correlations was found to depend critically on the beam-splitting ratio of the entangling beam splitter. Carrying out measurements on a different set of conjugate Stokes parameters, correlations of -3.6 +/- 0.3 and -3.4 +/- 0.3 dB have been observed. This result is more robust against asymmetries in the entangling beam splitter, even in the presence of excess noise.

Generation of polarization squeezing with periodically poled KTP at 1064 nm

Mikael Lassen, Metin Sabuncu, Preben Buchhave, Ulrik L. Andersen

We report the experimental demonstration of directly produced polarization squeezing at 1064 nm from a type I optical parametric amplifier (OPA) based on a periodically poled KTP crystal (PPKTP). The orthogonal polarization modes of the polarization squeezed state are both defined by the OPA cavity mode, and the birefringence induced by the PPKTP crystal is compensated by a second, but inactive, PPKTP crystal. Stokes parameter squeezing of 3.6 dB and anti squeezing of 9.4 dB is observed. (c) 2007 Optical Society of America.

Experimental quantum cloning with continuous variables

Ulrik L. Andersen, Vincent Josse, Norbert Luetkenhaus, Gerd Leuchs

In this chapter we present a scheme for optimal Gaussian cloning of optical coherent states. Its optical realization is based entirely on simple linear optical elements and homodyne detection. This is in contrast to previous proposals where parametric processes were suggested to be used for optimal Gaussian cloning. The optimality of the presented scheme is only limited by detection inefficiencies. Experimentally we achieved a cloning fidelity of up to 65%, which almost touches the optimal value of 2/3.

Experimental demonstration of macroscopic quantum coherence in Gaussian states

Christoph Marquardt, Ulrik L. Andersen, Gerd Leuchs, Yuishi Takeno, Mitsuyoshi Yukawa, Hidehiro Yonezawa, Akira Furusawa

We witness experimentally the presence of macroscopic coherence in Gaussian quantum states using a recently proposed criterion [E. G. Cavalcanti and M. D. Reid, Phys. Rev. Lett. 97 170405 (2006)]. The macroscopic coherence stems from interference between macroscopically distinct states in phase space, and we prove experimentally that a coherent state contains these features with a distance in phase space of 0.51 +/- 0.02 shot noise units. This is surprising because coherent states are generally considered being at the border between classical and quantum states, not yet displaying any nonclassical effect. For squeezed and entangled states the effect may be larger but depends critically on the state purity.

Distillation of squeezing from non-Gaussian quantum states

J. Heersink, Ch. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, U. L. Andersen

We show that single copy distillation of squeezing from continuous variable non-Gaussian states is possible using linear optics and conditional homodyne detection. A specific non-Gaussian noise source, corresponding to a random linear displacement, is investigated experimentally. Conditioning the signal on a tap measurement, we observe probabilistic recovery of squeezing.

Verifying continuous-variable entanglement of intense light pulses

O Glockl, UL Andersen, G Leuchs

Three different methods have been discussed to verify continuous variable entanglement of intense light beams. We demonstrate all three methods using the same setup to facilitate the comparison. The nonlinearity used to generate entanglement is the Kerr effect in optical fibers. Due to the brightness of the entangled pulses, standard homodyne detection is not an appropriate tool for the verification. However, we show that by using large asymmetric interferometers on each beam individually, two noncommuting variables can be accessed and the presence of entanglement verified via joint measurements on the two beams. Alternatively, we witness entanglement by combining the two beams on a beam splitter that yields certain linear combinations of quadrature amplitudes which suffice to prove the presence of entanglement.

Nonunity gain quantum nondemolition measurements based on measurement and repreparation

Jessica Schneider, Oliver Gloeckl, Gerd Leuchs, Ulrik L. Andersen

We demonstrate experimentally a nonunity gain quantum nondemolition measurement based on a simple homodyne measurement and recreation strategy. Although the output state is an amplified version of the input state, the device meets standard criteria for QND measurements: the transfer coefficient was measured to 1,78, and the conditional variance was measured to 0.66. (c) 2006 Optical Society of America.

All-fibre source of continuous variable entangled light

A Nandan, M Sabuncu, J Heersink, O Glockl, G Leuchs, UL Andersen

We experimentally demonstrate the production of continuous variable entanglement of femto second laser pulses from an all-fibre asymmetric Sagnac interferometer exploiting the Kerr effect. Contrary to the experiment of Silberhorn et at. [Phys. Rev. Lett. 86,4267 (2001)], the asymmetric coupler in the Sagnac loop is fully integrated in the fibre, making the source extremely compact, reliable and robust. Employing a simple detection scheme, comprising a beam splitter and two intensity detectors, we clearly observe quantum correlations of conjugate quadratures, hereby witnessing entanglement.

Reduction of guided acoustic wave Brillouin scattering in photonic crystal fibers

D. Elser, U. L. Andersen, A. Korn, O. Gloeckl, S. Lorenz, Ch. Marquardt, G. Leuchs

Guided acoustic wave Brillouin scattering (GAWBS) generates phase and polarization noise of light propagating in glass fibers. This excess noise affects the performance of various experiments operating at the quantum noise limit. We experimentally demonstrate the reduction of GAWBS noise in a photonic crystal fiber in a broad frequency range by tailoring the acoustic modes using the photonic also as a phononic crystal. We compare the noise spectrum to the one of a standard fiber and observe a tenfold noise reduction in the frequency range up to 200 MHz. Based on our measurement results as well as on numerical simulations, we establish a model for the reduction of GAWBS noise in photonic crystal fibers.

THE QUANTUM PROPERTIES OF MULTIMODE OPTICAL AMPLIFIERS REVISITED

G. Leuchs, U. L. Andersen, C. Fabre

There are a number of physically different realizations of an optical amplifier and yet they all share the same fundamental quantum limit as far as their noise characteristics are concerned. We review the underlying mathematical formalism without the restriction of a minimum number of modes being involved, its physical implications and relate it to the phenomenological models for the various amplifiers.

Quantum information processing and communication : Strategic report on current status, visions and goals for research in Europe

Peter Zoller, Th. Beth, D. Binosi, R. Blatt, H. Briegel, D. Bruss, T. Calarco, J. Ignacio Cirac, D. Deutsch, et al.

Quantum information processing and communication - Strategic report on current status, visions and goals for research in Europe

P Zoller, T Beth, D Binosi, R Blatt, H Briegel, D Bruss, T Calarco, JI Cirac, D Deutsch, et al.

We present an excerpt of the document "Quantum Information Processing and Communication: Strategic report on current status, visions and goals for research in Europe", which has been recently published in electronic form at the website of FET (the Future and Emerging Technologies Unit of the Directorate General Information Society of the European Commission, http://www.cordis.lu/ist/fet/qipc-sr.htm). This document has been elaborated, following a former suggestion by FET, by a committee of QIPC scientists to provide input towards the European Commission for the preparation of the Seventh Framework Program. Besides being a document addressed to policy makers and funding agencies (both at the European and national level), the document contains a detailed scientific assessment of the state-of-the-art, main research goals, challenges, strengths, weaknesses, visions and perspectives of all the most relevant QIPC sub-fields, that we report here.

The effect of dissipation on nonclassical states of the radiation field

G Leuchs, UL Andersen

We point out similarities in the evolution of different types of nonclassical light fields. Generally, Fock and cat states are considered to decay much faster under dissipation than do squeezed states. We connect measurements of the intensity correlation function of nonclassical light to the second moment of the photon number distribution function. It is shown that all these nonclassical fields behave in a similar manner when one is looking at an appropriate property. Finally it is shown how the electronic-detection noise floor, typically present in measurements of the variance of a photocurrent, can be eliminated based upon a procedure that is well-established for intensity correlation measurements.

Experimental purification of coherent states

Ulrik Andersen, R Filip, J Fiurasek, V Josse, Gerd Leuchs

We propose a scheme for optimal Gaussian purification of coherent states from several imperfect copies. The proposal is experimentally demonstrated for the case of two copies of a coherent state sent through independent noisy channels. Our purification protocol relies on only linear optics and an ancilla vacuum state, rendering this approach an interesting alternative to the more complex protocols of entanglement distillation and quantum error correction.

Unconditional quantum cloning of coherent states with linear optics

Ulrik L. Andersen, V Josse, Gerd Leuchs

A scheme for optimal Gaussian cloning of optical coherent states is proposed and experimentally demonstrated. Its optical realization is based entirely on simple linear optical elements and homodyne detection. The optimality of the presented scheme is limited only by detection inefficiencies. Experimentally, we achieved a cloning fidelity of about 65%, which almost touches the optimal value of 2/3.

Demonstration of the spatial separation of the entangled quantum sidebands of an optical field

EH Huntington, GN Milford, C Robilliard, TC Ralph, O Glockl, Ulrik L. Andersen, S Lorenz, Gerd Leuchs

Quantum optics experiments on "bright" beams are based on the spectral analysis of field fluctuations and typically probe correlations between radio-frequency sideband modes. However, the extra degree of freedom represented by this dual-mode picture is generally ignored. We demonstrate the experimental operation of a device which can be used to separate the quantum sidebands of an optical field. We use this device to explicitly demonstrate the quantum entanglement between the sidebands of a squeezed beam.

Efficient polarization squeezing in optical fibers

J Heersink, V Josse, G Leuchs, UL Andersen

We report on a novel and efficient source of polarization squeezing that uses a single pass through an optical fiber. Using the fiber's two orthogonal polarization axes produces two identical squeezed beams. Combining these in a Stokes measurement generates polarization squeezing of up to 5.1 &PLUSMN; 0.3 dB. Furthermore, this scheme enables us to directly measure, for both polarizations, the noise of any given quadrature. &COPY; 2005 Optical Society of America.

Continuous-variable quantum key distribution using polarization encoding and post selection

S Lorenz, N Korolkova, G Leuchs

We present an experimental demonstration of a quantum key distribution protocol using coherent polarization states. Post selection is used to ensure a low error rate and security against beam-splitting attacks even in the presence of high losses. Signal encoding and readout in polarization bases avoids the difficult task of sending a local oscillator with the quantum channel. This makes our setup robust and easy to implement. A shared key was established for losses up to 64%.

Sub-shot-noise phase quadrature measurement of intense light beams

O Glockl, Ulrik L. Andersen, S Lorenz, Christine Silberhorn, N Korolkova, Gerd Leuchs

We present a setup for performing sub-shot-noise measurements of the phase quadrature of intense pulsed light without the use of a separate local oscillator. A Mach-Zehnder interferometer with an unbalanced arm length is used to detect the fluctuations of the phase quadrature at a single sideband frequency. With this setup, the nonseparability of a pair of quadrature-entangled beams is demonstrated experimentally. (C) 2004 Optical Society of America.

Experimental demonstration of continuous variable quantum erasing

UL Andersen, O Glockl, S Lorenz, G Leuchs, R Filip

We experimentally demonstrate the concept of continuous variable quantum erasing. The amplitude quadrature of the signal state is labeled to another state via a quantum nondemolition interaction, leading to a large uncertainty in the determination of the phase quadrature due to the inextricable complementarity of the two observables. We show that by erasing the amplitude quadrature information we are able to recover the phase quadrature information of the signal state.

Squeezing based on nondegenerate frequency doubling internal to a realistic laser

UL Andersen, P Tidemand-Lichtenberg, P Buchhave

We investigate theoretically the quantum fluctuations of the fundamental field in the output of a nondegenerate second-harmonic generation process occurring inside a laser cavity. Due to the nondegenerate character of the nonlinear medium, a field orthogonal to the laser field is for some operating conditions independent of the fluctuations produced by the laser medium. We show that this fact may lead to perfect squeezing for a certain polarization mode of the fundamental field. The experimental feasibility of the system is also discussed.

All-fibre source of amplitude squeezed light pulses

M Meissner, C Marquardt, J Heersink, T Gaber, A Wietfeld, G Leuchs, UL Andersen

An all-fibre source of amplitude squeezed solitons utilizing the self-phase modulation in an asymmetric Sagnac interferometer is experimentally demonstrated. The asymmetry of the interferometer is passively controlled by an integrated fibre coupler, allowing for the optimization of the noise reduction. We have carefully studied the dependence of the amplitude noise on the asymmetry and the power launched into the Sagnac interferometer. Qualitatively, we find good agreement between the experimental results, a semi-classical theory and earlier numerical calculations (Schmitt et al 1998 Phys. Rev. Lett. 81 2446). The stability and flexibility of this all-fibre source makes it particularly well suited to applications in quantum information science.

Polarization squeezing of intense pulses with a fiber-optic Sagnac interferometer

J Heersink, T Gaber, S Lorenz, O Glockl, N Korolkova, Gerd Leuchs

We report on the generation of polarization squeezing of intense, short light pulses using an asymmetric fiber-optic Sagnac interferometer. The Kerr nonlinearity of the fiber is exploited to produce independent amplitude squeezed pulses. The polarization squeezing properties of spatially overlapped amplitude squeezed and coherent states are discussed. The experimental results for a single-amplitude squeezed beam are compared to the case of two phase-matched, spatially overlapped amplitude squeezed pulses. For the latter, noise variances of -3.4 dB below shot noise in the S-0 and the S-1 and of -2.8 dB in the S-2 Stokes parameters were observed, which is comparable to the input squeezing magnitude. Polarization squeezing, that is, squeezing relative to a corresponding polarization minimum uncertainty state, was generated in S-1.

A pulsed source of continuous variable polarization entanglement

O Glockl, J Heersink, N Korolkova, Gerd Leuchs, S Lorenz

We have experimentally demonstrated polarization entanglement using continuous variables in an ultra-short pulsed laser system at the telecommunications wavelength of 1.5 mum. Exploiting the Kerr non-linearity of a glass fibre we generated polarization squeezed pulses with (S) over cap (2) as the only non-zero Stokes parameter, thus (S) over cap (1) and (S) over cap (3) being the conjugate pair. Polarization entanglement was generated by interference of the polarization squeezed field with a vacuum on a 50:50 beamsplitter. The two resultant beams exhibit strong quantum noise correlations in (S) over cap (1) and (S) over cap (3). The sum noise signal of (S) over cap (3) was at the respective shot noise level and the difference noise signal of (S) over cap (1) fell -2.9 dB below this value.

Quantum interferometry with intense optical pulses

Gerd Leuchs, Christine Silberhorn, O Glockl, Christoph Marquardt, N Korolkova

For intense optical pulses the optical Kerr interaction in matter such as an optical fiber is large enough to generate quantum states of light with significant non-classical properties. On this basis pairs of entangled light pulses have been generated. This entanglement can be used for novel schemes in high precision interferometry and for quantum communication protocols such as quantum dense coding.

Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state

O Glockl, S Lorenz, Christoph Marquardt, J Heersink, M Brownnutt, Christine Silberhorn, Q Pan, P van Loock, N Korolkova, et al.

We present a protocol for performing entanglement swapping with intense pulsed beams. In a first step, the generation of amplitude correlations between two systems that have never interacted directly is demonstrated. This is verified in direct detection with electronic modulation of the detected photocurrents. The measured correlations are better than expected from a classical reconstruction scheme. In an entanglement swapping process, a four-partite entangled state is generated. We prove experimentally that the amplitudes of the four optical modes are quantum correlated 3 dB below shot noise, which is consistent with the presence of genuine four-party entanglement.

Soliton backaction-evading measurement using spectral filtering

F Konig, B Buchler, T Rechtenwald, Gerd Leuchs, A Sizmann

We report an a backaction-evading (BAE) measurement of the photon number of fiber-optical solitons operating in the quantum regime. We employ a recently proposed different detection scheme based on spectral filtering of colliding optical solitons. The measurements of the BAE criteria demonstrate significant quantum state, preparation and transfer of the input signal to the signal and probe outputs exiting the apparatus, displaying the quantum-nondemolition behavior of the experiment.

Mode structure and photon number correlations in squeezed quantum pulses

T Opatrny, N Korolkova, Gerd Leuchs

The question of an efficient multimode description of optical pulses is studied. We show that a relatively very small number of nonmonochromatic modes can be sufficient for a complete quantum description of pulses with Gaussian quadrature statistics. For example, a three-mode description was enough to reproduce the experimental data of photon number correlations in optical solitons [S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, Phys. Rev. Lett. 81, 786 (1998)]. This approach is very useful for a detailed understanding of squeezing properties of soliton pulses with the main potential for quantum communication with continuous variables. We show how homodyne detection and/or measurements of photon number correlations can be used to determine the quantum state of the multimode field. We also discuss a possible way of physical separation of the nonmonochromatic modes.

Polarization squeezing and continuous-variable polarization entanglement

N Korolkova, Gerd Leuchs, R Loudon, TC Ralph, Christine Silberhorn

A concept of polarization entanglement for continuous variables is introduced. For this purpose the Stokes-parameter operators and the associated Poincare sphere, which describe the quantum-optical polarization properties of light, are defined and their basic properties are reviewed. The general features of the Stokes operators are illustrated by evaluation of their means and variances for a range of simple polarization states. Some of the examples show polarization squeezing, in which the variances of one or more Stokes parameters are smaller than the coherent-state value. The main object of the paper is the application of these concepts to bright squeezed light. It is shown that a light beam formed by interference of two orthogonally polarized quadrature-squeezed beams exhibits squeezing in some of the Stokes parameters. Passage of such a primary polarization-squeezed beam through suitable optical components generates a pair of polarization-entangled light beams with the nature of a two-mode squeezed state. Implementation of these schemes using the double-fiber Sagnac interferometer provides an efficient method for the generation of bright nonclassical polarization states. The important advantage of these nonclassical polarization states for quantum communication is the possibility of experimentally determining all of the relevant conjugate variables of both squeezed and entangled fields using only linear optical elements followed by direct detection.

Quantum key distribution with bright entangled beams

Christine Silberhorn, N Korolkova, Gerd Leuchs

We suggest a quantum cryptographic scheme using continuous EPR-like correlations of bright optical beams. For binary key encoding, the continuous information is discretized in a novel way by associating a respective measurement, amplitude, or phase, with a bit value "1" or "0." The secure key distribution is guaranteed by the quantum correlations. No predetermined information is sent through the quantum channel contributing to the security of the system.

Direct experimental test of non-separability and other quantum techniques using continuous variables of light

N Korolkova, Christine Silberhorn, O Glockl, S Lorenz, Christoph Marquardt, Gerd Leuchs

We present schemes for the generation and evaluation of continuous variable entanglement of bright optical beams and give a brief overview of a variety of optical techniques and quantum communication applications on this basis. A new entanglement-based quantum interferometry scheme with bright beams is suggested. The performance of the presented schemes is independent of the relative interference phase which is advantageous for quantum communication applications.

Continuous variable quantum cryptography: Beating the 3 dB loss limit

Christine Silberhorn, TC Ralph, Norbert Lütkenhaus, Gerd Leuchs

We demonstrate that secure quantum key distribution systems based on continuous variable implementations can operate beyond the apparent 3 dB loss limit that is implied by the beam splitting attack. The loss limit was established for standard minimum uncertainty states such as coherent states. We show that, by an appropriate postselection mechanism, we can enter a region where Eve's knowledge on Alice's key falls behind the information shared between Alice and Bob, even in the presence of substantial losses.

The role of coherent effects in mode bistability of a gas laser in inhomogeneous medium

G Wasik, Gerd Leuchs

We have studied microscopic effects leading to mode competition in an inhomogeneously broadened medium of a gas laser with a weakly anisotropic cavity. The polarization bistability observed in some lasers, which is believed to be caused by hole burning, we attribute to microscopic coherent effects induced by two light fields. Due to these effects one field can be coupled with another. Hole burning is found to be important, but not crucial. The exact solution of the equation of motion for field amplitudes is compared with the third order approximation of perturbation theory. The role of four wave mixing, important for bistability of axial modes in a homogeneously broadened medium, is discussed.

Time evolution of a quantum soliton in a Kerr medium

N Korolkova, R Loudon, G Gardavsky, MW Hamilton, Gerd Leuchs

The Q-function of the quantum soliton in a fibre is derived in a new form suitable for the assessment of the possibilities for experimental observations of specific quantum-soliton effects. The characteristic effects of soliton evolution are associated with a cubic term in the nonlinear phase angle and appear at distances well beyond the attenuation length (long range). At the accessible distances, the nonlinear dynamics of a fundamental soliton follows essentially the single-mode dynamics (short and middle range). However, for the higher-order solitons, the form of the evolution parameter suggests strong deviations of soliton dynamics from the single-mode with a square of soliton number. The use of enhanced-nonlinearity fibres and higher-order solitons might make the experimental studies of the specific quantum-soliton features more viable.

Bright EPR-entangled beams for quantum communication

Christine Silberhorn, PK Lam, N Korolkova, Gerd Leuchs

Bright Einstein-Podolsky-Rosen (EPR) entanglement is obtained in linear interference of two amplitude-squeezed beams produced in Kerr nonlinear interactions. The correlation of the amplitude (phase) quadratures are measured to be 4.0 +/- 0.2(4.0 +/- 0.4) dB below the quantum noise limit. Continuous variable EPR-paradox was demonstrated with measured product of the conditional variances of 0.64 +/- 0.08, well below the quantum limit of unity.

Squeezed light from microstructured fibres: towards free-space quantum cryptography

S Lorenz, Christine Silberhorn, N Korolkova, RS Windeler, Gerd Leuchs

Amplitude-squeezed pulsed light has been produced using a microstructured silica fibre. By spectrally filtering after the non-linear propagation in the fibre a squeezing value of -1.7 dB has been measured. A quantum key distribution scheme based on squeezed light from such microstructured fibres is proposed.

Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber

Christine Silberhorn, PK Lam, O Weiss, F Konig, N Korolkova, Gerd Leuchs

We report on the generation of a continuous variable Einstein-Podolsky-Rosen (EPR) entanglement using an optical fiber interferometer. The Kerr nonlinearity in the fiber is exploited for the generation of two independent squeezed beams. These interfere at a beam splitter and EPR entanglement is obtained between the output beams. The correlation of the amplitude (phase) quadratures is measured to be 4.0 +/- 0.2 (4.0 +/- 0.4) dB below the quantum noise limit. The sum criterion for these squeezing variances 0.80 +/- 0.03 < 2 verifies the nonseparability of the state. The product of the inferred uncertainties for one beam (0.64 +/- 0.08) is well below the EPR limit of unity.

The optical Kerr effect and quantum optics in fibers

A Sizmann, Gerd Leuchs

Scheme for the generation of entangled solitons for quantum communication

Gerd Leuchs, TC Ralph, Christine Silberhorn, N Korolkova

Quantum-communication needs a source of entanglement which is accessible to non-local measurements. It is shown that an entangled pair of light beams may be generated from two bright amplitude squeezed beams. This is applicable to light pulses and to optical solitons in fibres. A possible experimental scheme for stroboscopic quantum teleportation is discussed.

Photon-number squeezed solitons from an asymmetric fiber-optic sagnac interferometer

S Schmitt, J Ficker, M Wolff, F Konig, A Sizmann, Gerd Leuchs

Direct photon-number squeezing is demonstrated, for the first time to our knowledge, in a nonlinear fiber-optic interferometer. Launching 126-fs solitons into a highly asymmetric Sagnac loop, the maximum photocurrent noise reduction was 3.9 +/- 0.2 dB below shot noise, corresponding to 6.0 +/- 0.9 dB, when corrected for linear losses. The loop is a model system for squeezing generated by the interference of two pulses after nonlinear propagation through a fiber.

Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime

F Konig, S Spalter, IL Shumay, A Sizmann, T Fauster, Gerd Leuchs

The nonlinear optical Kerr effect, acting on optical pulses in fibres, creates spectral sidebands and noise correlations between these sidebands. The reduction of photon-number fluctuations of these pulses below the shot-noise limit by spectral filtering is well established in the anomalous dispersion regime which allows for soliton formation. Here it is demonstrated that a significant quantum-noise reduction with spectral filtering can also be reached for pulses in the normal dispersion regime. The filter function was optimized and the power dependence of the noise reduction was investigated. The best squeezing result is (1.2 +/- 0.2) dB (corresponding to (2.6 +/- 0.7) dB inferred for 100% detection efficiency).

Noise-free quantum-nondemolition measurement using optical solitons

JM Courty, S Spalter, F Konig, A Sizmann, Gerd Leuchs

The collision of solitons in optical fibers is a promising scheme for the quantum-nondemolition measurement of the photon number using the cross Kerr effect. In a realistic experimental situation, the sensitivity of the measurement is limited not only by self-phase-modulation noise of the probe but also by the phase noise added due to the vibrational excitations of the fiber. Former schemes proposed to overcome either of these problems are not compatible with each other. In this paper the sensitivity of different type of detection is carefully studied. We propose a solution based on the use of a phase-shifting cavity. We show that this provides a conceptually and practically simple solution eliminating simultaneously both sources of excess noise. [S1050-2947(98)04406-0].

Observation of multimode quantum correlations in fiber optical solitons

S Spalter, N Korolkova, F Konig, A Sizmann, Gerd Leuchs

Quantum correlations of photon numbers in different spectral components of ultrashort optical solitons have been observed experimentally. These correlations an crucial for the understanding and characterization of the internal quantum structure of soliton pulses and contribute significantly to soliton squeezing by spectral filtering. The accessible information on the nonclassical state of the correlated spectral components is discussed with the example of two modes. The method may be generalized to obtain a complete quantum description of a multimode field. [S0031-9007(98)06700-3].

Photon-number noise of spectrally filtered optical solitons

A Sizmann, S Spalter, M Burk, U Strossner, M Bohm, Gerd Leuchs

Propagation of quantum properties of sub-picosecond solitons in a fiber

S Spalter, M Burk, U Strossner, A Sizmann, Gerd Leuchs

We present new results on photon number squeezing of spectrally filtered solitons in fibers. The impact of frequency low-, high-, and bandpass filtering on noise reduction has been measured as a function of fiber length for 130-fs pulses close to the soliton energy. For short fibers our results agree qualitatively with theoretical predictions. For longer fibers, however, the measured squeezing increases to an unexpectedly large value. Spectral filtering of a strongly Raman-shifted, higher energy pulse squeezed the directly detected photocurrent fluctuations down to 3.8+/-0.2 dB (59%) below the shot noise level. The measured noise reductions are broadband from 5 to 90 MHz. (C) 1997 Optical Society of America.

Self-starting soliton-modelocked femtosecond Cr(4+):YAG laser using an antiresonant Fabry-Perot saturable absorber

S Spalter, M Bohm, M Burk, B Mikulla, R Fluck, ID Jung, G Zhang, U Keller, A Sizmann, et al.

We set up a self-starting passively modelocked femtosecond Cr(4+):YAG laser using a low-finesse antiresonant Fabry-Perot saturable absorber (A-FPSA). Stable transform-limited pulses were produced without the critical cavity alignment necessary for Kerr lens modelocking. We achieved 114-fs pulses with 94 mW output power at the telecommunication wavelength 1.5 mu m. The pulse formation is demonstrated to be due to soliton modelocking. At higher intracavity energy multiple pulsing was observed and is explained by the saturation behaviour of the A-FPSA. In the multipulsing regime a pulse width as short as 95 fs and an averaged output power of up to 395 mW have been obtained.

Photon number squeezing of spectrally filtered sub-picosecond optical solitons

S Spalter, M Burk, U Strossner, M Bohm, A Sizmann, Gerd Leuchs

Photon number squeezing of spectrally filtered optical solitons has been observed in the sub-picosecond regime, where the Raman effect changes the internal quantum structure of each pulse. 160 fs pulses were propagated down a 10 m optical fibre of anomalous group velocity dispersion. At energies below the soliton energy subsequent spectral filtering with an optical band-pass squeezed the directly detected photocurrent fluctuations to 3.2 +/- 0.1 dB (52%) below the shot-noise limit. This implies 4.5 dB (65%) of soliton photon number squeezing, if detection losses are taken into account. Using a spectral edge filter the spectral components of the pulses are shown to contribute asymmetrically to the squeezing due to stimulated Raman scattering.

Quantum non-demolition measurements with optical solitons

S Spalter, Peter van Loock, A Sizmann, Gerd Leuchs

The optical Kerr effect provides an ideal quantum non-demolition interaction. Experiments and proposals using this interaction are reviewed with special emphasis on optical soliton pulses propagating in fibres. The performance of a quantum non-demolition experiment using the optical Kerr effect may be reduced by self-phase modulation of the probe pulse. Proposals to overcome this limitation are discussed.