Jukka Kiukas, Juha-Pekka Pellonpää, Jussi Schultz
We consider state reconstruction from the measurement statistics of phase space observables generated by photon number states. The results are obtained by inverting certain infinite matrices. In particular, we obtain reconstruction formulas, each of which involves only a single phase space observable.
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We investigate up to the fourth order normalized factorial moments of free-propagating and pulsed single photons displaced in phase space in a phase-averaged manner. Due to their loss independence, these moments offer expedient methods for quantum optical state characterization. We examine quantum features of the prepared displaced states, retrieve information on their photon-number content and study the reliability of the state reconstruction method used.
Displaced single-photon entanglement is a simple form of optical entanglement, obtained by sending a photon on a beamsplitter and subsequently applying a displacement operation. We show that it can generate, through a momentum transfer in the pulsed regime, an optomechanical entangled state involving macroscopically distinct mechanical components, even if the optomechanical system operates in the single-photon weak coupling regime. We discuss the experimental feasibility of this approach and show that it might open up a way for testing unconventional decoherence models.
Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state |1> with coherent states on a high-reflection beamsplitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals highly non-classical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. This is the first time complete tomographic reconstruction has been performed on a highly non-classical optical state.
Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon number distribution of a linear optical interferometric network, which is fed with single-photon Fock state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fo…
We propose a scheme for quantum key distribution (QKD) protocol with dual-rail displaced photon states. Displaced single photon states carry bit value of code which may be extracted while coherent states carry nothing and they only provide inconclusive outcome. Developed QKD protocol works with experimental attendant noise to observe presence of malicious Eve. Pulses with large amplitudes unlike conventional QKD relying on faint laser pulses are used that may approximate it to standard telecommunication communication and may show resistance to eavesdropping even in settings with high attenuati…
Nonclassical properties of photon added and subtracted displaced Fock states have been studied using various witnesses of lower- and higher-order nonclassicality. Compact analytic expressions are obtained for the nonclassicality witnesses. Using those expressions, it is established that these states and the states that can be obtained as their limiting cases (except coherent states) are highly nonclassical as they show the existence of lower- and higher-order antibunching and sub-Poissonian photon statistics, in addition to the nonclassical features revealed through the Mandel $Q_M$ parameter,…
Analyzing performance of $f$-deformed displaced Fock state in continuous-variable quantum teleportation
本文研究了非高斯资源在连续变量量子隐形传态中的成功概率，推导了基于$f$-变形平移Fock态等纠缠信道的解析表达式，发现非线性替换能超越阈值提升保真度，且光子扣除平移Fock态展现最大传输效率。
Based on correlations of coherently displaced photon-numbers, we derive entanglement criteria for the purpose to verify non-Gaussian entanglement. Our construction method enables us to verify bipartite and multipartite entanglement of complex states of light. An important advantage of our technique is that the certified entanglement even persists in the presence of arbitrarily high, constant losses. We exploit experimental correlation schemes for the two-mode and multimode scenarios, which allow us to directly measure the desired observables. To detect entanglement of a given state, a genetic …
In this paper, we study some quantum properties of a superposition of displaced squeezed two-mode vacuum and single-photon states, such as the second-order correlation function, the Cauchy-Schwartz inequality, quadrature squeezing, quasiprobability distribution functions and purity. This type of states includes two mechanisms, namely, interference in phase space and entanglement. We show that these states can exhibit sub-Piossonian statistics, squeezing and deviate from the classical Cauchy-Schwartz inequality. Moreover, the amount of entanglement in the system can be increased by increasing t…
Non-Gaussian and nonclassical states and processes are already found to be important resources for performing various tasks related to quantum gravity and quantum information processing. The effect of non-Gaussianity inducing operators on the nonclassicality of quantum states has also been studied rigorously. Considering these facts, a quantitative analysis of the nonclassical and non-Gaussian features is performed here for photon added displaced Fock state, as a test case, using a set of measures like entanglement potential, Wigner Yanese skew information, Wigner logarithmic negativity and re…
Quantum phase properties of photon added and subtracted displaced Fock states (and a set of quantum states which can be obtained as the limiting cases of these states) are investigated from a number of perspectives, and it is shown that the quantum phase properties are dependent on the quantum state engineering operations performed. Specifically, the analytic expressions for quantum phase distributions and angular $Q$ distribution as well as measures of quantum phase fluctuation and phase dispersion are obtained. The uniform phase distribution of the initial Fock states is observed to be trans…
Detecting Nonclassicality and quantum non-Gaussianity of photon subtracted displaced Fock state
本文利用多种测度定量研究了光子扣除平移Fock态的非经典性与量子非高斯性，发现Fock参数的作用强于光子扣除数，并证明Wigner负性仅能由高效率探测器揭示。
We consider the phase sensing via weak optical coherent state at quantum limit precision. A new detection scheme for the phase estimation is proposed which is inspired by the suboptimal quantum measurement in coherent optical communication. We theoretically analyze a performance of our detection scheme, which we call the displaced-photon counting, for phase sensing in terms of the Fisher information and show that the displaced-photon counting outperforms the static homodyne and heterodyne detections in wide range of the target phase. The proof-of-principle experiment is performed with linear o…
We propose an approach that enables to use it for construction of rotations of coherent states, in particular, it gives a possibility to construct Hadamard gate for the coherent states. Our approach is based on representation of arbitrary one-mode pure state in free-travelling fields, in particular superposition of coherent states (SCSs), in terms of displaced number states with arbitrary amplitude of displacement. Studied optical scheme is based on alternation of photon additions and displacement operators (in general case, photon additions and displacements are required) with seed coherent s…
We develop a new method of representation of quantum states in terms of the displaced number states. We call it representation, where is an amplitude of the base displaced states. In particular, representation was obtained for set of the displaced number states with different amplitude of the displacement, two-mode squeezed vacuum (TMSV) and superposition of vacuum and single photon. The treatment is employed for building of elementary linear operators, in particular, two-qubit control-sign gate, Hadamard matrix between two two-dimensional Hilbert spaces. Main idea underlying the method is ext…
SU(1,1)-displaced coherent states, photon counting and squeezing
本文研究 Perelomov SU(1,1) 位移相干态的光子计数、压缩及光场量子化性质，指出探测器效率与表示参数 k 的关系，并给出可用于态制备的位移算子。
We study su(2) and su(1,1) displaced number states. Those states are eigenstates of density-dependent interaction systems of quantized radiation field with classical current. Those states are intermediate states interpolating between number and displaced number states. Their photon number distribution, statistical and squeezing properties are studied in detail. It is show that these states exhibit strong nonclassical properties.
Realistic continuous-variable quantum teleportation using a displaced Fock state channel
本文研究基于位移福克态资源的连续变量量子隐形传态，发现其对单模输入场的平均保真度仅达经典阈值，并分析了贝尔测量误差和光纤损耗等退相干因素对保真度的影响及参数优化条件。
In this paper we discuss the quantum properties for superposition of squeezed displaced number states against multiphoton Jaynes-Cummings model (JCM). In particular, we investigate atomic inversion, photon-number distribution, purity, quadrature squeezing, Mandel $Q$ parameter and Wigner function. We show that the quadrature squeezing for three-photon absorption case can exhibit revivals and collapses typical to those occurring in the atomic inversion for one-photon absorption case. Also we prove that for odd number absorption parameter there is a connection between the evolution of the atomic…
We show that entanglement of multiple atoms can arise via resonant interaction with a displaced thermal field with a macroscopic photon-number. The cavity field acts as the catalyst, which is disentangled with the atomic system after the operation. Remarkably, the entanglement speed does not decrease as the average photon-number of the mixed thermal state increases. The atoms may evolve to a highly entangled state even when the photon-number of the cavity mode approaches infinity.
Displaced photon helicity states: a quantum realization of Arago-Fresnel interference
该研究基于阿拉戈-菲涅耳定律，通过计算位移光子自旋态组合光束的斯托克斯参数，实现了量子阿拉戈-菲涅耳干涉，可纯光学测定未知入射光子的自旋。
Nonlinear displaced Kerr state and its nonclassical properties
本文构造了非线性位移克尔态，研究其光子统计、压缩等非经典性质，并通过低克尔参数近似将其非经典性转化为双模纠缠。
We show that a nonlinear asymmetric directional coupler composed of a linear waveguide and a nonlinear waveguide operating by nondegenerate parametric amplification is an effective source of single-mode squeezed light. This is has been demonstrated, under certain conditions and for specific modes, for incident coherent beams in terms of the quasiprobability functions, photon-number distribution and phase distribution.
A method is introduced which allows to measure normal-ordered moments of the displaced photon-number operator up to high orders. It is based on unbalanced homodyne correlation measurements, the local oscillator being replaced by a displaced dephased laser. The measured moments yield a simple approximation of the full quantum state, which is obtained without the need of photon-number resolution. Quantum properties of light are efficiently certified through the direct detection of normal-ordered observables, which is illustrated for a weakly squeezed vacuum and a single-photon-added thermal stat…
Various nonclassical and quantum phase properties of photon added then subtracted displaced Fock state have been examined systematically and rigorously. Higher-order moments of the relevant bosonic operators are computed to test the nonclassicality of the state of interest, which reduces to various quantum states (having applications in quantum optics, metrology and information processing) in different limits ranging from the coherent (classical) state to the Fock (most nonclassical) states. The nonclassical features are discussed using Klyshko's, Vogel's, and Agarwal-Tara's criteria as well a…
We address conditional de-Gaussification of continuous variable states by inconclusive photon subtraction (IPS) and review in details its application to bipartite twin-beam state of radiation. The IPS map in the Fock basis has been derived, as well as its counterpart in the phase-space. Teleportation assisted by IPS states is analyzed and the corresponding fidelity evaluated as a function of the involved parameters. Nonlocality of IPS states is investigated by means of different tests including displaced parity, homodyne detection, pseudospin, and displaced on/off photodetection. Dissipation a…
We present the reconstruction of the Wigner function of a classical phase-sensitive state, a pulsed coherent state, by measurements of the distributions of detected-photons of the state displaced by a coherent probe field. By using a hybrid photodetector operated above its photon-resolving regime, we obtained both the statistics at different values of the probe field and the values of the probe phase required to reconstruct the Wigner function.
We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their disp…
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between 2 measurement settings, each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the 4 total measurements are carried out by homodyne detection. The largest vio…
Near-optimal discrimination of displaced squeezed binary signals using displacement, inverse-squeezing, and photon-number-resolving detection
本文提出反压缩Kennedy接收器，用于区分二进制相位调制位移压缩真空态，通过反压缩增强光子数对比度，实现超越标准量子限和Helstrom界，且在实际参数下保持鲁棒性。
Exploiting higher-order correlation functions for photon-statistics-based characterization and reconstruction of arbitrary Gaussian states
本文提出利用高阶相关函数表征和重构任意高斯态的方法，通过分析六阶矩分解公式揭示高斯态分类的条件，并展示如何结合额外损耗信息实现状态重构。
This paper presents some results on some aspects of the two-level atom interacting with a single-mode with the privileged field mode being in the squeezed displaced Fock state (SDFS). The exact results are employed to perform a careful investigation of the temporal evolution of the atomic inversion, entropy and phase distribution. It is shown that the interference between component states leads to non-classical oscillations in the photon number distribution. At mid revival time the field is almost in the pure state. We have briefly discussed the evolution of the Q function of the cavity field.…
Squeezed Displaced Schrödinger-cat state as a signature of the PT-symmetry phase transition
该研究提出利用压缩平移薛定谔猫态作为标志来捕获电子系统中的 PT 对称性相变，发现该态在 PT 对称性破缺时以高保真度出现，并揭示了量子 Fisher 信息和非经典性在例外点处的急剧下降现象。
Full-Period Optical Phase Estimation with Heisenberg Scaling Using Displaced Squeezed States and Gaussian Measurements
提出固定能量约束下单模高斯态和高斯测量的全周期光学相位估计两阶段优化策略，经推导广义克拉美罗界，位移压缩态在第一阶段可降低优化界。
We report a direct measurement of the Wigner function characterizing the quantum state of a light mode. The experimental scheme is based on the representation of the Wigner function as an expectation value of a displaced photon number parity operator. This allowed us to scan the phase space point-by-point, and obtain the complete Wigner function without using any numerical reconstruction algorithms.
We present an experimental realisation of the direct scheme for measuring the Wigner function of a single quantized light mode. In this method, the Wigner function is determined as the expectation value of the photon number parity operator for the phase space displaced quantum state.
It is a long-standing goal to generate robust deterministic states of light with unique quantum properties, such as squeezing, sub-Poissonian statistics and entanglement. It is of interest to consider whether such quantum states of light could be generated by exploiting interactions with free electrons, going beyond their already ubiquitous use in generating classical light. This question is motivated by developments in electron microscopy, which present a new platform for manipulating photons through their interaction with quantum free electrons. Here, we explore the shaping of photon statist…
Using the normally ordered Gaussian form of displaced-squeezed thermal field characteristic of average photon number n, we introduce the photon-added squeezed thermo state (PASTS) and investigate its statistical properties, such as Mandel's Q-parameter, number distribution (as a Legendre polynomial), the Wigner function. We then study its decoherence in a photon-loss channel in term of the negativity of WF by deriving the analytical expression of WF for PASTS. It is found that the WF with single photon-added is always partial negative for the arbitrary values of n and the squeezing parameter r…
Inspired by the recent experiment of Hamsen et al. [Phys. Rev. Lett. 118, 133604 (2017)], which demonstrated two-photon blockade in a driven nonlinear system (composed of a harmonic cavity with a driven atom), we show that two-photon blockade and other nonstandard types of photon blockade and photon-induced tunneling can be generated in a driven harmonic cavity without an atom or any other kind of nonlinearity, but instead coupled to a nonlinear (i.e., squeezed) reservoir. We also simulate these single- and two-photon effects with squeezed coherent states and displaced squeezed thermal states.
We present the reconstruction of the Wigner function of some classical pulsed optical states obtained by direct measurement of the detected-photon probability distributions of the state displaced by a coherent field. We use a photodetector endowed with internal gain, which is operated in the non- photon-resolving regime. The measurements are performed up to mesoscopic intensities (up to more than 30 photons per pulse). The method can be applied to characterize nonclassical continuous-variable states.
Optimizing photon-number distributions of Gaussian states in the presence of loss: Towards minimizing the impact of loss in Gaussian boson sampling
本文研究了损失对高斯态光子数统计的影响，提出并通过评估几种方法来减少损失态的光子数分布与无损态的偏差，通过重新定义高斯态参数来恢复目标分布，探讨了优化保真度、相空间函数、低阶矩和真空重叠等关键属性的方法。
We discuss a phase space description of the photon number distribution of non classical states which is based on Husimi's $Q(α)$ function and does not rely in the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution . This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.
The observation of genuine quantum features of nano-mechanical motion is a key goal for both fundamental and applied quantum science. To this end, a promising approach is the stabilization of nonclassical features in the presence of dissipation, by means of the tunable coupling with a photonic environment. Here we present a scheme that combines dissipative squeezing with a mechanical nonlinearity to stabilize arbitrary approximations of (displaced) mechanical Fock state of any number. We consider an optomechanical system driven by three control lasers--at the cavity resonance and at the two me…
Experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams are reported. Our scheme is based on on/off avalanche photodetection assisted by maximum-likelihood estimation and does not involve photon counting. Reconstructions of the distribution for both semiclassical and quantum states of light are reported for single-mode as well as for multimode beams.
We calculate exactly the quantum mechanical, temporal characteristic function $χ(η)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of $χ(η)$ allows only the determination of the time development of arbitrary functions of equal-time products of creation and annihilation photon operators. We calculate, in particular, the fluctuations in photon number, quadrature operators, and quadrature variance. We contrast the very important difference between the nonclassicality criteria based on the one-time charact…
We show that conditional output measurement on a beam splitter may be used to produce photon-added states for a large class of signal-mode quantum states, such as thermal states, coherent states, squeezed states, displaced photon-number states, and coherent phase states. Combining a mode prepared in such a state and a mode prepared in a photon-number state, the state of the mode in one of the output channels of the beam splitter ``collapses'' to a photon-added state, provided that no photons are detected in the other output channel. We present analytical and numerical results, with special emp…
A method for reconstructing joint photon-number distributions of twin beams from the experimental photocount histograms is suggested and experimentally implemented. Contrary to the standard reconstruction methods, it incorporates spatial noise reduction based on spatial pairing of photons. Superior performance of the method above the usual one for the maximum-likelihood approach is demonstrated.
We address the reconstruction of the full photon distribution of multimode fields generated by seeded parametric down-conversion (PDC). Our scheme is based on on/off avalanche photodetection assisted by maximum-likelihood (MaxLik) estimation and does not involve photon counting. We present a novel constrained MaxLik method that incorporates the request of finite energy to improve the rate of convergence and, in turn, the overall accuracy of the reconstruction.
We present a detailed discussion of a general theory of phase-space distributions, introduced recently by the authors [J. Phys. A {\bf 31}, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generalized coherent states and the method of harmonic analysis are used to construct explicitly a family of phase-space functions which are postulated to satisfy the Stratonovich-Weyl correspondence with a generalized traciality condition. The symbol calculus for the phase-space functions is given by …
We analyze the photon statistics of a weakly driven optomechanical system and discuss the effect of photon blockade under single photon strong coupling conditions. We present an intuitive interpretation of this effect in terms of displaced oscillator states and derive analytic expressions for the cavity excitation spectrum and the two photon correlation function $g^{(2)}(0)$. Our results predict the appearance of non-classical photon correlations in the combined strong coupling and sideband resolved regime, and provide a first detailed understanding of photon-photon interactions in strong coup…
Here we present a reconstruction of the Positive Operator-Value Measurement of a photon-number-resolving detector comprised of three 50:50 beamsplitters in a tree configuration, terminated with four single-photon avalanche detectors. The four detectors' outputs are processed by an electronic board that discriminates detected photon number states from 0 to 4 and implements a "smart counting" routine to compensate for dead time issues at high count rates.
An iterative algorithm for reconstructing the photon distribution from the random phase homodyne statistics is discussed. This method, derived from the maximum-likelihood approach, yields a positive definite estimate for the photon distribution with bounded statistical errors even if numerical compensation for the detector imperfection is applied.
Quantum Rotation Diversity in Displaced Squeezed Binary Phase-Shift Keying
本文提出一种利用量子旋转多样性提升光学量子通信在湍流信道中性能的方案，通过优化旋转角和能量分配实现四阶有效多样性。
The output of a photodetector consists of a current pulse whose charge has the statistical distribution of the actual photon numbers convolved with a Bernoulli distribution. Photodetectors are characterized by a nonunit quantum efficiency, i.e. not all the photons lead to a charge, and by a finite resolution, i.e. a different number of detected photons leads to a discriminable values of the charge only up to a maximum value. We present a detailed comparison, based on Monte Carlo simulated experiments and real data, among the performances of detectors with different upper limits of counting cap…
We experimentally demonstrate the reconstruction of a photon number conditioned state without using a photon number discriminating detector. By using only phase randomized homodyne measurements, we reconstruct up to the three photon subtracted squeezed vacuum state. The reconstructed Wigner functions of these states show regions of pronounced negativity, signifying the non-classical nature of the reconstructed states. The techniques presented allow for complete characterization of the role of a conditional measurement on an ensemble of states, and might prove useful in systems where photon cou…
We present a study of optical quantum states generated by subtraction of photons from the thermal state. Some aspects of their photon number and quadrature distributions are discussed and checked experimentally. We demonstrate an original method of up to ten photon subtracted state preparation with use of just one single-photon detector. All the states where measured with use of balanced homodyne technique, and the corresponding density matrices where reconstructed. The fidelity between desired and reconstructed states exceeds 99%
Estimation with ultimate quantum precision of the transverse displacement between two photons via two-photon interference sampling measurements
本文提出基于横向动量采样测量的双光子干涉方案，可在估计两光子横向位移时实现终极量子精度，且不受波包重叠影响，为超分辨纳米成像开辟新范式。
The quantum interference of entangled photons forms a key phenomenon underlying various quantum-optical technologies. It is known that the quantum interference patterns of entangled photon pairs can be reconstructed classically by the time-reversal method; however, the time-reversal method has been applied only to time-frequency-entangled two-photon systems in previous experiments. Here, for the first time, we apply the time-reversal method to the position-wavevector-entangled two-photon systems: the two-photon Young interferometer and the two-photon beam focusing system. We experimentally dem…
We demonstrate a practical possibility of loss compensation in measured photocounting statistics in the presence of dark counts and background radiation noise. It is shown that satisfactory results are obtained even in the case of low detection efficiency and large experimental errors.
Coherent photon-emitter interfaces offer a way to mediate efficient nonlinear photon-photon interactions, much needed for quantum information processing. Here we experimentally study the case of a two-level emitter, a quantum dot, coupled to a single optical mode in a nanophotonic waveguide. We carry out few-photon transport experiments and record the statistics of the light to reconstruct the scattering matrix elements of 1- and 2-photon components. This provides direct insight to the complex nonlinear photon interaction that contains rich many-body physics.
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