Abstract Although highly successful, the truncated Wigner approximation (TWA) does not account for genuine many-body (MB) quantum interference between different solutions of the mean-field equations of a bosonic MB system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck’s constant ℏ . The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being compared with ‘exact’ quantum calculations that do not involve any a priori approximation. A rather prominent cause for such quantum effects that are not accounted for by the TWA is the constructive interference between the contributions of symmetry-related trajectories, which would occur in the presence of discrete symmetries provided the phase-space distribution of the initial state and the observable to be evaluated feature a strong localization about the corresponding symmetry subspaces. Here we show how one can conceive an augmented version of the TWA which can account for this particular effect. This augmented TWA effectively amounts to complementing conventional TWA calculations by separate truncated Wigner simulations that are restricted to symmetric subspaces and involve weight factors that account for the dynamical stability of sampling trajectories with respect to perpendicular deviations from those subspaces. We illustrate the validity of this method at pre- as well as post-Ehrenfest time scales in prototypical Bose–Hubbard systems displaying chaotic classical dynamics, where it also reveals the existence of additional MB interference effects.
Abstract Although highly successful, the truncated Wigner approximation (TWA) does not account for genuine many-body (MB) quantum interference between different solutions of the mean-field equations of a bosonic MB system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck’s constant ℏ . The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being compared with ‘exact’ quantum calculations that do not involve any a priori approximation. A rather prominent cause for such quantum effects that are not accounted for by the TWA is the constructive interference between the contributions of symmetry-related trajectories, which would occur in the presence of discrete symmetries provided the phase-space distribution of the initial state and the observable to be evaluated feature a strong localization about the corresponding symmetry subspaces. Here we show how one can conceive an augmented version of the TWA which can account for this particular effect. This augmented TWA effectively amounts to complementing conventional TWA calculations by separate truncated Wigner simulations that are restricted to symmetric subspaces and involve weight factors that account for the dynamical stability of sampling trajectories with respect to perpendicular deviations from those subspaces. We illustrate the validity of this method at pre- as well as post-Ehrenfest time scales in prototypical Bose–Hubbard systems displaying chaotic classical dynamics, where it also reveals the existence of additional MB interference effects.
Although highly successful, the truncated Wigner approximation (TWA) leaves out many-body quantum interference between mean-field Gross-Pitaevskii solutions as well as other quantum effects, and is therefore essentially classical. Turned around, if a system's quantum properties deviate from TWA, they must be exhibiting some quantum phenomenon, such as localization, diffraction, or tunneling. Here, we examine a particular interference effect arising from discrete symmetries, which can significantly enhance quantum observables with respect to the TWA prediction, and derive an augmented TWA in order to incorporate them. Using the Bose-Hubbard model for illustration, we further show strong evidence for the presence of dynamical localization due to remaining differences between the TWA predictions and quantum results.
The truncated Wigner approximation (TWA) enables us to investigate bosonic quantum many-body dynamics, including open quantum systems described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the TWA, the Weyl-Wigner transformation, a way of mapping from quantum-mechanical operators to c -numbers, of the GKSL equation leads to the Fokker–Planck equation, which we calculate by reducing it to the corresponding stochastic differential equations. However, the Fokker–Planck equation is not always reduced to the stochastic differential equations depending on details of jump operators. In this work, we clarify the condition for obtaining the stochastic differential equations from the Fokker–Planck equation and derive analytical expressions of these equations for a system with an arbitrary Hamiltonian with jump operators that do not couple different states. This result leads to remarkable simplification of the complicated calculations required in the conventional procedures of the TWA. In the course of the derivation, we formulate the GKSL equation by using the path-integral representation based on the Weyl-Wigner transformation, which gives us a clear interpretation of the relation between the TWA and quantum fluctuations and allows us to calculate the non-equal time correlation functions in the TWA. In the benchmark calculations, we numerically confirm that the relaxation dynamics of physical quantities including the non-equal time correlation functions obtained in our formulation agrees well with the exact ones in the numerically solvable models. Since our formulation is widely applicable to bosonic systems such as the Bose-Hubbard model, it facilitates studies of the open quantum many-body phenomenon.
A major goal of the semiclassical theory is to understand and treat a quantum system in the regime of large quantum numbers using information from its classical limit. In this framework, quantum phenomena are addressed through a path integral perspective, i.e. the weighted interference of phases accumulated along classical trajectories. While this formalism has been successful in explaining interference phenomena, for instance, in mesoscopic single-particle quantum systems, the recent extension of the semiclassical theory allows now also the treatment of interacting quantum many-body systems. This thesis aims to contribute to these ongoing efforts to apply semiclassical techniques to quantum many-body systems. It is focused on bosonic quantum many-body systems, where the thermodynamic limit of a large number of particles represents an alternative version of a semiclassical limit. Within the derivation of the corresponding semiclassical theory, mean-field equations, i.e. effective nonlinear wave equations describing the matter wave, re-emerge as Hamilton's equations of motion of an abstract Hamilton function, defining the above thermodynamic limit as the classical limit. Within this thesis, two different topics are studied. The first one deals with the so-called “Out-of-Time-Order Correlator” (OTOC), the expectation value of the squared commutator of two local operators at different times. The OTOC provides a direct probe for the presence of chaos in the classical limit (if any) of a quantum many-body system. This is based on that for short times the OTOC directly relates to the stability of classical solutions upon changes in their initial conditions. One of the defining properties of chaos is that this stability displays an exponential growth, with the rate called the classical Lyapunov exponent. Immediately, this implies an exponential growth of the OTOC for short times, from which the Lyapunov exponent can be extracted. For later times, one observes a saturation of the OTOC as a consequence of unitary quantum time evolution. The Ehrenfest time, the time scale which marks the onset of quantum interference, separates these different dynamical behaviors. In this thesis a thorough understanding of the concept of chaos in the classical limit of quantum many-body systems is provided, and with that the underlying interference mechanisms involved in the early exponential growth and the later saturation of OTOCs are identified. It is found that the pre-Ehrenfest time exponential growth is given through the interference of multiple contributions, all of them essentially following only a single solution of mean-field equations. Conversely the post-Ehrenfest time behavior stems from the contributions of fundamentally different mean-field solutions, which display correlated dynamics only for a limited amount of time, of the order of Ehrenfest time. The second topic covered in this thesis deals with the coherent transport of cold bosonic atoms through an Aharonov-Bohm ring structure. This setup consists of two semi-infinite wave guides attached on the opposite sides of a wave guide ring which is pierced by a synthetic gauge field. Within the ring, the atoms are further subject to both a weak disorder potential and particle-particle interaction. In the non-interacting case, the disorder-averaged transmission probability as a function of the encircled flux displays the well-known Al'tshuler-Aronov-Spivak (AAS) oscillations, i.e. oscillations with a frequency twice as large as Aharonov-Bohm oscillations. This thesis provides further insight into the influence of a weak particle-particle interaction on AAS oscillations, for which numerical results predict an interaction-based inversion of peaks. Here, using semiclassical techniques to solve the corresponding mean-field problem, this inversion is successfully reproduced, and a first indicator of the relevant mechanism leading to the inversion is found through contributions of self-averaging scattering path constellations.
Far out-of-equilibrium many-body quantum dynamics in isolated systems necessarily generates interferences beyond an Ehrenfest timescale, where quantum and classical expectation values diverge. Of great recent interest is the role these interferences play in the spreading of quantum information across the many degrees of freedom, i.e., scrambling. Ultracold atomic gases provide a promising setting to explore these phenomena. Theoretically speaking, the heavily-relied-upon truncated Wigner approximation leaves out these interferences. We develop a semiclassical theory which bridges classical and quantum concepts in many-body bosonic systems and properly incorporates such missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it is shown that this theory captures post-Ehrenfest quantum interference phenomena very accurately, and contains relevant phase information to perform many-body spectroscopy with high precision.
A phase-space method, suitable to study the many-body spin dynamics in generic spin lattice models is introduced, discussed, and tested. This technique, which we refer to as the dTWA, is based on the discrete sampling of the Wigner function instead of the continuous sampling in traditional Truncated Wigner Approximation (TWA) methods. The power of the method is demonstrated by comparisons with analytical and numerically exact calculations, which show a significant improvement compared to the TWA, e.g. by capturing revivals of spin-observables. The dTWA is demonstrated to not only correctly reproduce dynamics of one-particle observables but also two-point correlations and spin squeezing, a property relying on entanglement. This method thus opens the possibility to study the quantum dynamics in long-range spin models, accessible to recent state-of-the-art experiments, in regimes where other numerical techniques are inapplicable.
We review the truncated Wigner approximation (TWA) for spins as a computationally inexpensive numerical approximation method to describe interacting and / or dissipative many-body spin systems. Using the Wigner-Moyal mapping from Hilbert space to a suitable phase space, the many-body density matrix is represented by a c-number distribution, the Wigner function. The gauge freedom in continuous phase space can be exploited to find positive Wigner functions for a large class of spin states, including entangled ones. Employing different sets of correspondence rules, we derive equations of motion for the Wigner function, which, applying controlled approximations, can be mapped to stochastic differential equations. This allows a computationally inexpensive simulation of expectation values. Using a phase-space analog of the quantum regression theorem also multi-time correlations and spectra can be obtained. To illustrate the potential of the method, we benchmark the TWA for spins with some exactly solvable problems of interacting, dissipative spin systems, and then discuss its application to collective processes, such as the superradiant emission of light. Extending the TWA to imaginary time furthermore provides a tool to approximately calculate thermal and ground states of spin Hamiltonians. Finally, we show that the TWA stochastic equations can equivalently be derived within a path-integral approach, provided that the operator products in the dissipator are rigorously mapped onto the curved phase space.