Time evolution of coupled spin systems in a generalized Wigner representation

Although the origin of quantum optics can be traced back to the beginning of the 20th century, when the fundamental ideas about the quantum nature of the interaction between light and matter were put forward, the splendid blossoming of this part of physics began half a century later, after the invention of masers and lasers. It is remarkable that after another half a century the tree of quantum optics is not only very strong and spreading, but all its branches continue to grow, showing new beautiful blossoms and giving very useful fruits.

Quantum statistical properties of interacting optical fields in Kerr nonlinear coupler can be regarded as one of the most fundamental problems in quantum optics. As the analytical solution to these systems is not always possible, a semi-analytic or direct numerical solution has to be employed to obtain more complete description of their evolution. This paper investigates the quantum properties of a four-mode coupler composed of two Kerr nonlinear waveguides via phase-space representation. In this system, the electromagnetic field is described by its Hamiltonian and the time evolution of the system is described by Von-Neumann equation. Following the standard techniques, the master equation for the density matrix of the system is obtained and converted to the corresponding classical Fokker-Planck equation using both positive-P and Wigner representations. The corresponding set of Langevin Stochastic equations are then obtained from the Fokker-Planck equation using Ito rules. Finally, the system is integrated numerically and averaged over many trajectories to get the relevant information. The dynamics of the mean fields, the effects of self-action nonlinearity, cross-action coupling and initial energy of the coherent light on the evolution of the field quadrature variances are investigated. Further, the influences of multimode interaction on the dynamics of the squeezed states are discussed. We show that the system can produce squeezed states in both quadrature variances for all calculated parameters. In addition, the multimode interaction extends the range of squeezing at the expense of its amplitude.

We consider tripartite entangled states for continuous variable systems of EPR type, which generalize the famous bipartite CV EPR states (eigenvectors of conjugate choices X1 − X2, P1 + P2, of the systems' relative position and total momentum variables). We give the regularized forms of such tripartite EPR states in second-quantized formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r → ∞, or tanh r → 1− (approaching the EPR states in the bipartite case), our regularized tripartite EPR states show singular behaviour not only in the approach to the EPR-type region (s → 1 in our notation), but also for an additional, auxiliary regime of the regulator . While the s → 1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalized CHSH inequalities. Violations of the tripartite CHSH bound (B3 ⩽ 2) are established, with in the canonical regime (s → 1+), as well as in the auxiliary regime .

Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.

The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.

Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.

Single-quantum coherence filter for strongly coupled spin systems for localized (1)H NMR spectroscopy.

In this paper, we propose a scheme to protect quantum entanglement and coherence from a non-Markovian noisy environment. By applying two quantum weak measurements before and after sending the quantum state into the noisy channel, the quantum state can be “pushed” closer to a decoherence free state thus suffer less decoherence in the time evolution. After the time evolution the second weak measurement can partially retrieve the original information encoded in the quantum system. Our study is based on a non-Markovian dynamic equation which allows us to investigate the impact of the memory effect on the performance of the protection scheme. We analyze several factors that may affect the protection efficiency. The results suggest that two measurement strengths should be chosen in a linear relation but the ratio is not one. Besides, we also show the memory effect can drastically improve the protection efficiency.

We introduce the study of dynamical quantum noise in Bose-Einstein\ncondensates through numerical simulation of stochastic partial differential\nequations obtained using phase space representations. We derive evolution\nequations for a single trapped condensate in both the positive-$P$ and Wigner\nrepresentations, and perform simulations to compare the predictions of the two\nmethods. The positive-$P$ approach is found to be highly susceptible to the\nstability problems that have been observed in other strongly nonlinear, weakly\ndamped systems. Using the Wigner representation, we examine the evolution of\nseveral quantities of interest using from a variety of choices of initial state\nfor the condensate, and compare results to those for single-mode models.\n

We derive a compact expression for the second-order correlation function of a quantum state in terms of its Wigner function, thereby establishing a direct link between and the state’s shape in phase space. We conduct an experiment that simultaneously measures through direct photocounting and reconstructs the Wigner function via homodyne tomography. The results confirm our theoretical predictions.This article is part of the theme issue ‘The quantum theory of light’.

On the full quantum description of nonlinear dynamic systems in the phase-space representation

Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm—utilizing a decomposition of nonunitary operators approach—that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.