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Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum
Abstract
We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.
... IV B. On the detection side, ideally, we also want to measure the photon-number-cat states to obtain unbiased estimation of the phase-error rates, e X m,m . While this is not directly measurable in practice, we employ the homodyne tomography technique and estimate the photon-numbercat state measurement via quadrature measurement results [35][36][37][38][39][40], which shall be discussed in Sec. IV A. ...
... Nevertheless, we can construct unbiased estimators with the available states and detection settings to evaluate these values. On the detection side, we apply the homodyne tomography technique to evaluate the photon-number observables [35][36][37][38][39][40]. The homodyne tomography allows unbiased estimation of the expected value of a variety of observables, including the photon-number observables, of measuring an unknown quantum state. ...
... Due to the lack of photon-numberresolving detectors, these operators are not directly measurable. Nevertheless, we can apply homodyne tomography and obtain unbiased estimation [35][36][37][38][39][40]. For a systematic review, we recommend the tutorial textbook of Ref. [33]. ...
Continuous-variable quantum key distribution (CV QKD) using optical coherent detectors is practically favorable due to its low implementation cost, flexibility of wavelength division multiplexing, and compatibility with standard coherent communication technologies. However, the security analysis and parameter estimation of CV QKD are complicated due to the infinite-dimensional latent Hilbert space. Also, the transmission of strong reference pulses undermines the security and complicates the experiments. In this work, we tackle these two problems by presenting a time-bin-encoding CV protocol with a simple phase-error-based security analysis valid under general coherent attacks. With the key encoded into the relative intensity between two optical modes, the need for global references is removed. Furthermore, phase randomization can be introduced to decouple the security analysis of different photon-number components. We can hence tag the photon number for each round, effectively estimate the associated privacy using a carefully designed coherent-detection method, and independently extract encryption keys from each component. Simulations manifest that the protocol using multi-photon components increases the key rate by two orders of magnitude compared to the one using only the single-photon component. Meanwhile, the protocol with four-intensity decoy analysis is sufficient to yield tight parameter estimation with a short-distance key-rate performance comparable to the best Bennett-Brassard-1984 implementation.
... This is not enough to characterize the full wave function unambiguously. Instead, a conventional method, known as quantum state tomography [4][5][6][7][8][9], has been established to estimate the wave function through a large set of projective measurements. This strategy, however, still presents drawbacks in terms of simplicity, versatility, and directness. ...
The measurement of a wave function plays a pivotal role in quantum physics and presents a distinctive challenge in experiment. Recent works have shown that both the real and imaginary components of the wave function can be extracted by employing weak or strong measurements, thereby enabling the determination of its amplitude and phase. Here, we propose a simple approach for reconstructing the wave function utilizing the spin-orbit interaction of light at the air-glass interface. By directly measuring the amplitude and employing spatial differentiation to capture the phase gradient, it becomes possible to successfully reconstruct an unknown wave function. To demonstrate its feasibility, we experimentally measure the pure wave function of photons with a Gaussian state. Furthermore, we conduct measurements on a custom state featuring a targeted phase jump to examine the accuracy of our methodology. The measured results show a distribution with high contrast and considerable accuracy, with a fidelity that can exceed 85%. We believe that this work contributes valuable insights into the practical applications of spin-orbit interaction, including optical image processing, wave-front sensing, and quantitative phase imaging.
... The dark areas in the density matrix correspond to elements that are not covered by the finite number of measured subdiagonals.Measuring the full quantum state of photoelectrons is a nontrivial problem due to the continuous nature of the photoelectron energy distribution. This problem is similar to that encountered in quantum-optics experiments aiming, for example, at measuring the density matrix of squeezed quantum states of light(34). Different methods such as maximum-likelihood or maximum-entropy reconstructions have been developed to estimate the quantum state based on a finite number of measurements(35), and machine-learning techniques have recently been applied to QST(36). Here we employ Bayesian estimation using a Hamiltonian Monte Carlo method to extract the density matrix from our measurements (more details are presented in the SM). ...
... Time-domain balanced homodyne detectors (TBHDs) are crucial components in quantum information fields, such as quantum tomography and continuous-variable quantum key distribution (CVQKD) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. These detectors can be used to measure the quadratures of pulsed quantum signals. ...
We designed and experimentally demonstrated a silicon photonics-integrated time-domain balanced homodyne detector (TBHD), containing an optical part of dimensions of 1.5 mm × 0.4 mm. To automatically and accurately balance the detector, new variable optical attenuators were used, and a common mode rejection ratio of 86.9 dB could be achieved. In the quantum tomography experiment, the density matrix and Wigner function of a coherent state were reconstructed with 99.97% fidelity. The feasibility of this TBHD in a continuous-variable quantum key distribution (CVQKD) system was also demonstrated. Our TBHD technologies are expected to be used in silicon photonics-integrated CVQKD system and silicon photonics-integrated BB84 heterodyne system.
... The spectral distribution of the field is observed to be centred around 1308.5 nm in the experiment (see Supplementary Material). Further investigations can be carried out through the reconstruction of the Wigner function [39]. Figure. ...
The generation of broadband squeezed states of light lies at the heart of high-speed continuous-variable quantum information. Traditionally, optical nonlinear interactions have been employed to produce quadrature-squeezed states. However, the harnessing of electrically pumped semiconductor lasers offers distinctive paradigms to achieve enhanced squeezing performance. We present evidence that quantum dot lasers enable the realization of broadband amplitude-squeezed states at room temperature across a wide frequency range, spanning from 3 GHz to 12 GHz. Our findings are corroborated by a comprehensive stochastic simulation in agreement with the experimental data. The evolution of photonics-based quantum information technologies is currently on the brink of initiating a revolutionary transformation in data processing and communication protocols [1, 2]. A cornerstone within this realm will be the quantum emitter. In recent years, there has been a substantial upsurge in both theoretical and experimental investigations centred around semiconductor quantum dot (QD) nanostructures [3]. A particular emphasis has been placed on self-assembled QDs embedded into microcavities, which facilitate the generation of single photons with high purity and indistinguishability [4-6]. As a result, such sources assume a pivotal role in quantum computing [7, 8] as well as the discrete variables (DV) quantum key distribution (QKD) [9]. In stark contrast to the DV QKD, which requires single-photon sources and detectors, continuous variable (CV) QKD leverages lasers and balanced detection to continuously retrieve the light's quadrature components during key distillation. This approach benefits from readily available equipment and seamless integration into existing optical telecommunications networks [10]. One of the CV QKD protocols, GG02 [11], is widely acclaimed for its security due to the no-cloning theorem of coherent states [12]. Nevertheless, a recent study has delved into the use of squeezed states to achieve even higher levels of security and robustness [13]. This innovative approach strives to completely eliminate information leakage to potential eavesdroppers in a pure-loss channel and to minimize it in a symmetric noisy channel. Within this cutting-edge protocol, information can be exclusively encoded through a Gaussian modulation of amplitude-squeezed states, which are commonly referred to as photon-number squeezed states. These states demonstrate reduced fluctuations in photon number ∆n 2 < n with respect to coherent states, albeit encountering enhanced phase fluctuations due to the minimum-uncertainty principle. Over the past years, squeezed states of light have been frequently generated using χ (2) or χ (3) nonlinear interactions via parametric down-conversion and four-wave mixing [14]. A variety of nonlinear materials have been applied in these processes, including LiNbO 3 (PPLN) [15], KTiOPO 4 (PPKTP) [16], silicon [17], atomic vapour [18], disk resonator [19], and Si 3 N 4 [20]. Recent advancements have also facilitated the transition from traditional benchtop instruments to a more compact single-chip design [21-25]. As opposed to that, Y. Yamamoto et al. [26] initially proposed an alternative to produce amplitude-squeezed states directly with off-the-shelf semiconductor lasers using a "quiet" pump, i.e. a constant-current source. A striking peculiarity of semiconductor lasers is their ability to be pumped by injection current supplied via an electrical circuit. Unlike optical pumping, electrical pumping is not inherently a Poisson point process due to the Coulomb interaction and allows for reducing pump noise below the shot-noise level [26]. Notably, this method can take full advantage of the mature fabrication processes in the semiconductor industry, thereby significantly boosting its feasibility. In other words, its efficacy hinges on the improved performance of recently developed semiconductor lasers, characterized by their compact footprint, ultralow intensity noise and narrow frequency linewidth. While subsequent experiments involving various types of laser diodes have gained widespread interest in this domain, including commercial quantum well (QW) lasers [27, 28], vertical-cavity surface-emitting lasers [29], and semiconductor microcavity lasers [30], the observed bandwidth has, until now, remained relatively limited. The broadest achieved bandwidth has reached 1.1 GHz with a QW transverse junction stripe laser operating at a cryogenic temperature of 77 K [31]. This limitation indeed poses strict constraints on the practical implementation of room-temperature conditions and hinders the realization of high-speed quantum communications. While a recent study did anticipate the theoretical potential of producing broadband amplitude-squeezed states in interband cascade lasers [32], it is worth noting that, prior to this Letter, no experimental demonstration of such phenomenon had been presented.
Efficient acquiring information from a quantum state is important for research in fundamental quantum physics and quantum information applications. Instead of using standard quantum state tomography method with reconstruction algorithm, weak values were proposed to directly measure density matrix elements of quantum state. Recently, similar to the concept of weak value, modular values were introduced to extend the direct measurement scheme to nonlocal quantum wavefunction. However, this method still involves approximations, which leads to inherent low precision. Here, we propose a new scheme which enables direct measurement for ideal value of the nonlocal density matrix element without taking approximations. Our scheme allows more accurate characterization of nonlocal quantum states, and therefore has greater advantages in practical measurement scenarios.
The high photostability of DNAs and RNAs is inextricably related to the photochemical and photophysical properties of their building blocks: nucleobases and nucleosides, which can dissipate the absorbed UV light...
Quantum state tomography is an essential component of modern quantum technology. In application to continuous-variable harmonic-oscillator systems, such as the electromagnetic field, existing tomography methods typically reconstruct the state in discrete bases, and are hence limited to states with relatively low amplitudes and energies. Here, we overcome this limitation by utilizing a feed-forward neural network to obtain the density matrix directly in the continuous position basis. An important benefit of our approach is the ability to choose specific regions in the phase space for detailed reconstruction. This results in a relatively slow scaling of the amount of resources required for the reconstruction with the state amplitude, and hence allows us to dramatically increase the range of amplitudes accessible with our method.
A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.
Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schrödinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with partial exponential speedup.
This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.
Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.
The probability of a configuration is given in classical theory by the Boltzmann formula exp[−VhT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
The concept of quantum transition is critically examined from the perspective of the modern quantum theory of measurement. Historically rooted in the famous quantum jump of the Old Quantum Theory, the transition idea survives today in experimental jargon due to (1) the notion of uncontrollable disturbance of a system by measurement operations and (2) the wave-packet reduction hypothesis in several forms. Explicit counterexamples to both (1) and (2) are presented in terms of quantum measurement theory. It is concluded that the idea of transition, or quantum jump, can no longer be rationally comprehended within the framework of contemporary physical theory.
We discuss the profound influence which the Wigner distribution function has had in many areas of physics during its fifty years of existence.
A common approach to quantum physics is enshrouded in a jargon which treats state vectors as attributes of physical systems and the concept of state preparation as a filtration scheme wherein a process involving measurement selects from a primordial assembly of systems those bearing some prescribed vector of interest. By contrast, the empirical experiences with which quantum theory is actually concerned relate measurement and preparation in quite an opposite manner. Reproducible preparation schemes are logically and temporally anterior to measurement acts. Measurement extracts numbers from systems prepared in a specified manner; these data are then regularized by the theory by means of a state concept which is in turn used to characterize succinctly the given mode of preparation. The present paper offers, in a simple spin model, a method for determining the quantum state that represents any reproducible preparation.
In quantum mechanics, the state of an individual particle (or system) is unobservable, i.e., it cannot be determined experimentally, even in principle. However, the notion of measuring a state is meaningful if it refers to anensemble of similarly prepared particles, i.e., the question may be addressed: Is it possible to determine experimentally the state operator (density matrix) into which a given preparation procedure puts particles. After reviewing the previous work on this problem, we give simple procedures, in the line of Lamb's operational interpretation of quantum mechanics, for measuring a translational state operator (whether pure or mixed), via its Wigner function. These procedures closely parallel methods that might be used in classical mechanics to determine a true phase space probability distribution; thus, the Wigner function simulates such a distribution not only formally, but operationally also.
There is no way to determine what the wave function (or state vector) of a system is—if arbitrarily given, there is no way to measure its wave function. Clearly, such a measurement would have to result in afunction of several variables, not in a relatively small set ofnumbers .... In order to verify the [quantum] theory in its generality, at least a succession of two measurements are needed. There is in general no way to determine the original state of the system, but having produced a definite state by a first measurement, the probabilities of the outcomes of a second measurement are then given by the theory.
E. P. Wigner(1)
Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.
Mode-locked lasers provide a train of high-intensity light pulses that can be used to pump nonlinear media in order to produce squeezed light. The squeezed light produced would consist of a train of short-duration pulses. It is shown here that a homodyne detector using a pulsed local oscillator can be used to observe the squeezing even when the response time of the photodetector is much longer than the local oscillator or squeezed light pulse width.