We theoretically study the propagation of an interacting Bose-Einstein condensate in a two-dimensional disorder potential, following the principle of an atom laser. The constructive interference between time-reversed scattering paths gives rise to coherent backscattering, which may be observed under the form of a sharp cone in the disorder-averaged angular backscattered current. As is found by the numerical integration of the Gross-Pitaevskii equation, this coherent backscattering cone is inversed when a non-vanishing interaction strength is present, indicating a crossover from constructive to destructive interferences. Numerical simulations based on the Truncated Wigner method allow one to go beyond the mean-field approach and show that dephasing renders this signature of antilocalisation hidden behind a structureless and dominant incoherent contribution as the interaction strength is increased and the injected density decreased, in a regime of parameters far away from the mean-field limit. However, despite a partial dephasing, we observe that this weak antilocalisation scenario prevails for finite interaction strengths, opening the way for an experimental observation with $^{87}$Rb atoms.
We study the transport properties of an ultracold gas of Bose-Einstein condensate that is coupled from a magnetic trap into a one-dimensional waveguide. Our theoretical approach to tackle this problem is based on the truncated Wigner method for which we assume the system to consist of two semi-infinite non-interacting leads and a finite interacting scattering region with two constrictions modelling an atomic quantum dot. The transmission is computed in the steady-state regime and we find a good agreement between truncated Wigner and Matrix-Product State calculations. We also identify clear sig…
We consider quasi-stationary scattering of interacting bosonic matter waves in one-dimensional waveguides, as they arise in guided atom lasers. We show how the truncated Wigner (tW) method, which corresponds to the semiclassical description of the bosonic many-body system on the level of the diagonal approximation, can be utilized in order to describe such many-body bosonic scattering processes. Special emphasis is put on the discretization of space at the exact quantum level, in order to properly implement the semiclassical approximation and the tW method, as well as on the discussion of the …
We develop and utilize the SU(3) truncated Wigner approximation (TWA) in order to analyze far-from-equilibrium quantum dynamics of strongly interacting Bose gases in an optical lattice. Specifically, we explicitly represent the corresponding Bose--Hubbard model at an arbitrary filling factor with restricted local Hilbert spaces in terms of SU(3) matrices. Moreover, we introduce a discrete Wigner sampling technique for the SU(3) TWA and examine its performance as well as that of the SU(3) TWA with the Gaussian approximation for the continuous Wigner function. We directly compare outputs of thes…
The discrete truncated Wigner approximation (DTWA) is a powerful tool for analyzing dynamics of quantum spin systems. Since the DTWA includes the leading-order quantum corrections to a mean-field approximation, it is naturally expected that the DTWA becomes more accurate when the range of interactions of the system increases. However, quantitative corroboration of this expectation is still lacking mainly because it is generally difficult in a large system to evaluate a timescale on which the DTWA is quantitatively valid. In order to investigate how the validity timescale depends on the interac…
We discuss stochastic phase-space methods within the truncated Wigner approximation and show explicitly that they can be used to solve non-equilibrium dynamics of bosonic atoms in one-dimensional traps. We consider systems both with and without an optical lattice, and address different approximations in the stochastic synthesization of quantum statistical correlations of the initial atomic field. We also present a numerically efficient projection method for analyzing correlation functions of the simulation results, and demonstrate physical examples of non-equilibrium quantum dynamics of solito…
The truncated Wigner approximation is an established approach that describes the dynamics of weakly interacting Bose gases beyond the mean-field level. Although it allows a quantum field to be expressed by a stochastic c-number field, the simulation of the time evolution is still very demanding for most applications. Here, we develop a numerically inexpensive approximation by decomposing the c-number field into a variational ansatz function and a residual field. The dynamics of the ansatz function is described by a tractable set of coupled ordinary stochastic differential equations for the res…
We present a systematic approach for the semiclassical treatment of many-body dynamics of interacting, open spin systems. Our approach overcomes some of the shortcomings of the recently developed discrete truncated Wigner approximation (DTWA) based on Monte-Carlo sampling in a discrete phase space that improves the classical treatment by accounting for lowest-order quantum fluctuations. We provide a rigorous derivation of the DTWA by embedding it in a continuous phase space, thereby introducing a hybrid discrete-continuous truncated Wigner approximation (DCTWA). We derive a set of operator-dif…
The quantum phase transition from the Mott insulator state to the superfluid in the Bose-Hubbard model is investigated. We research one, two and three dimensional lattices in the truncated Wigner approximation. We compute both kinetic and potential energy and they turn out to have a power law behaviour as a function of the transition rate, with the power equal to 1/3. The same applies to the total energy in a system with a harmonic trap, which is usually present in the experimental set-up. These observations are in agreement with the experiment of [8], where such scalings were also observed an…
The semi-classical discrete truncated Wigner approximation (dTWA) has recently been proposed as a simulation method for spin-$1/2$ systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmark on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the s…
We study the effect of quantum fluctuations on the dynamics of a quasi-one-dimensional Bose gas in an optical lattice at zero-temperature using the truncated Wigner approximation with a variety of basis sets for the initial fluctuation modes. The initial spatial distributions of the quantum fluctuations are very different when using a limited number of plane-wave (PW), simple-harmonic-oscillator (SHO) and self-consistently determined Bogoliubov (SCB) modes. The short-time transport properties of the Bose gas, characterized by the phase coherence in the PW basis are distinct from those gained u…
By means of the discrete truncated Wigner approximation we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition these transitions separate a phase with nonvanishing magnetization along the ordering direction from a symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\propto 1/r^α$ decaying algebraically as a function of distance $r$ and a two-dimensional system with…
We present a general method by which linear quantum Hamiltonian dynamics with exponentially many degrees of freedom is replaced by approximate classical nonlinear dynamics with the number of degrees of freedom (phase space dimensionality) scaling polynomially in the system size. This method is based on generalization of the truncated Wigner approximation (TWA) to a higher dimensional phase space, where phase space variables are associated with a complete set of quantum operators spanning finite size clusters. The method becomes asymptotically exact with the increasing cluster size. The crucial…
Although highly successful, the Truncated Wigner Approximation (TWA) does not account for genuine many-body quantum interference between different solutions of the mean-field equations of a bosonic many-body (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 $\hbar$. 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 …
In this work we present the formal background used to develop the methods used in earlier works to extend the truncated Wigner representation of quantum and atom optics in order to address multi-time problems. The truncated Wigner representation has proven to be of great practical use, especially in the numerical study of the quantum dynamics of Bose condensed gases. In these cases, it allows for the simulation of effects which are missed entirely by other approximations, such as the Gross-Pitaevskii equation, but does not suffer from the severe instabilities of more exact methods. The numeric…
Nonadiabatic molecular dynamics occur in a wide range of chemical reactions and femtochemistry experiments involving electronically excited states. These dynamics are hard to treat numerically as the system's complexity increases and it is thus desirable to have accurate yet affordable methods for their simulation. Here, we introduce a linearized semiclassical method, the generalized discrete truncated Wigner approximation (GDTWA), which is well-established in the context of quantum spin lattice systems, into the arena of chemical nonadiabatic systems. In contrast to traditional continuous map…
We study the transport of an interacting Bose--Einstein condensate through a 1D correlated disorder potential. We use for this purpose the truncated Wigner method, which is, as we show, corresponding to the diagonal approximation of a semiclassical van Vleck-Gutzwiller representation of this many-body transport process. We also argue that semiclassical corrections beyond this diagonal approximation are vanishing under disorder average, thus confirming the validity of the truncated Wigner method in this context. Numerical calculations show that, while for weak atom-atom interaction strength And…
We compare the continuous and discrete truncated Wigner approximations of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization technique [R. Mukherjee {\em et al.}, Phys. Rev. A {\bf 97}, 043606 (2018)]. We find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction.
We theoretically study the propagation of a guided atom laser across an Aharonov-Bohm ring which is exposed to a synthetic gauge field. The presence of disorder within the ring gives rise to Al'tshuler-Aronov-Spivak oscillations, seen in the disorder average of the transmission as a function of the effective gauge flux that is contained within the ring. Those oscillations are induced by coherent backscattering and represent a manifestation of weak localization. Through analytical and numerical calculations that are based on the mean-field Gross-Pitaevskii approximation for the propagating Bose…
We investigate the real-time evolution of lattice bosons in two spatial dimensions whose dynamics is governed by a Markovian quantum master equation. We employ the Wigner-Weyl phase space quantization and derive the functional integral for open quantum many-body systems that governs the time evolution of the Wigner function. Using the truncated Wigner approximation, in which quantum fluctuations are only taken into account in the initial state whereas the dynamics is governed by classical evolution equations, we study the buildup of long-range correlations due to the action of non-Hermitean qu…
We show that metastable phases of an antiferromagnetic spin-1 condensate in a simple model with pure contact interactions can exhibit a rotonlike minimum in the excitation spectrum. The introduction of magnetic field gives rise to the instability of roton modes, which can lead to spontaneous emergence of regular periodic, polygonal, polyhedral or crystalline patterns, as shown in numerical simulations within the truncated Wigner approximation. An explanation of the occurrence of rotonlike instability is given based on the energy and spin conservation laws.
We show that the spreading of the center-of-mass density of ultracold attractively interacting bosons can become superballistic in the presence of decoherence, via single-, two- and/or three-body losses. In the limit of weak decoherence, we analytically solve the numerical model introduced in [Phys. Rev. A 91, 063616 (2015)]. The analytical predictions allow us to identify experimentally accessible parameter regimes for which we predict superballistic spreading of the center-of-mass density. Ultracold attractive Bose gases form weakly bound molecules; quantum matter-wave bright solitons. Our c…
We formulate the time-dependent Bogoliubov dynamics of colliding Bose-Einstein condensates in terms of a positive-P representation of the Bogoliubov field. We obtain stochastic evolution equations for the field which converge to the full Bogoliubov description as the number of realisations grows. The numerical effort grows linearly with the size of the computational lattice. We benchmark the efficiency and accuracy of our description against Wigner distribution and exact positive-P methods. We consider its regime of applicability, and show that it is the most efficient method in the common sit…
We study the phase-space representation of dynamics of bosons in the semiclassical regime where the occupation number of the modes is large. To this end, we employ the van Vleck-Gutzwiller propagator to obtain an approximation for the Green's function of the Wigner distribution. The semiclassical analysis incorporates interference of classical paths and reduces to the truncated Wigner approximation (TWA) when the interference is ignored. Furthermore, we identify the Ehrenfest time after which the TWA fails. As a case study, we consider a single-mode quantum nonlinear oscillator, which displays…
We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein condensate atom interferometer with nonlinear losses. We use stochastic equations in a truncated Wigner representation to treat quantum noise. Our approach includes the multi-mode spatial evolution of spinor components and describes the many-body dynamics of a mesoscopic quantum system.
We have studied the interaction between multiple, competing spatial modes that are excited by a quantum quench of an antiferromagnetic spinor Bose-Einstein condensate. We observed Hanbury Brown-Twiss correlations and associated super-Poissonian noise in the mode populations. The decay of these correlations was consistent with experimentally observed spin domain patterns. Data were compared with a real-space Bogoliubov theory as well as numerical solution of the coupled Gross-Pitaevskii equations that were seeded by quantum noise via the truncated Wigner approximation. The spatial modes that we…
By studying the coarsening dynamics of a one-dimensional spin-1 Bose-Hubbard model in a superfluid regime, we analytically find an unconventional universal dynamical scaling for the growth of the spin correlation length, which is characterized by the exponential integral unlike the conventional power-law or simple logarithmic behavior, and numerically confirmed with the truncated Wigner approximation.
In this work we look at the quantum dynamics of the process known as either transport without transit (TWT), or coherent transfer of atomic population (CTAP), of a Bose-Einstein condensate from one well of a lattice potential to another, non-adjacent well, without macroscopic occupation of the well between the two. This process has previously been analysed and in this work we extend those analyses by considering the effects of quantum statistics on the dynamics and entanglement properties of the condensate modes in the two relevant wells. In order to do this, we go beyond the mean-field analys…
We demonstrate that ultracold interacting bosonic atoms in an optical lattice show sub-Poissonian on-site and inter-site atom number fluctuations. The experimental observations agree with numerical predictions of the truncated Wigner approximation. The correlations persist in the presence of multi-mode atom dynamics and even over large spatially extended samples involving several sites and large populations.
We experimentally and numerically investigate thermalization processes of a trapped $^{87}$Rb Bose gas, initially prepared in a non-equilibrium state through partial Bragg diffraction of a Bose-Einstein condensate (BEC). The system evolves in a Gaussian potential, where we observe the destruction of the BEC due to collisions, and subsequent growth of a new condensed fraction in an oscillating reference frame. Furthermore, we occasionally observe the presence of defects, which we identify as gray solitons. We simulate the evolution of our system using the truncated Wigner method and compare the…
We demonstrate the application of the Metropolis-Hastings algorithm to sampling of classical thermal states of one-dimensional Bose-Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only…
We consider the dynamics of a quantum impurity after a sudden interaction quench into a one-dimensional degenerate Bose gas. We use the Keldysh path integral formalism to derive a truncated Wigner like approach that takes the back action of the impurity onto the condensate into account already on the mean-field level and further incorporates thermal and quantum effects up to one-loop accuracy. This framework enables us not only to calculate the real space trajectory of the impurity but also the absorption spectrum. We find that quantum corrections and thermal effects play a crucial role for th…
We discuss the non-equilibrium time evolution of the phase field in the sine-Gordon model using two very different approaches: the truncated Wigner approximation and the truncated conformal space approach. We demonstrate that the two approaches agree for a period covering the first few oscillations, thereby giving a solid theoretical prediction in the framework of sine-Gordon model, which is thought to describe the dynamics of two bosonic condensates in quasi-one-dimensional traps coupled via a Josephson tunneling term. We conclude, however, that the recently observed phase-locking behavior ca…
The truncated Wigner and positive-P phase-space representations are used to study the dynamics of a one-dimensional Bose gas. This allows calculations of the breathing quantum dynamics of higher-order solitons with 10^{3}-10^{5} particles, as in realistic Bose-Einstein condensation (BEC) experiments. Although classically stable, these decay quantum mechanically. Our calculations show that there are large nonlocal correlations. These also violate the Cauchy-Schwarz inequality, showing the presence of nonclassical quantum entanglement.
We propose a realistic experiment to demonstrate a dynamic Kosterlitz-Thouless transition in ultra-cold atomic gases in two dimensions. With a numerical implementation of the Truncated Wigner Approximation we simulate the time evolution of several correlation functions, which can be measured via matter wave interference. We demonstrate that the relaxational dynamics is well-described by a real-time renormalization group approach, and argue that these experiments can guide the development of a theoretical framework for the understanding of critical dynamics.
We report on the out-of-equilibrium dynamics of a Bose-Einstein condensate (BEC) placed in an optical lattice whose phase is suddenly modulated. The frequency and the amplitude of modulation are chosen to ensure a negative renormalized tunneling rate. Under these conditions, staggered states are nucleated by a spontaneous four wave mixing mechanism. The nucleation time is experimentally studied as a function of the renormalized tunnel rate, the atomic density and the modulation frequency. Our results are quantitatively well accounted for by a Truncated Wigner approach and reveal the nucleation…
We propose to test the concept of seeded vacuum decay in cosmology using an analogue gravity Bose-Einstein condensate system. The role of the nucleation seed is played by a vortex within the condensate. We present two complementary theoretical analyses that demonstrate seeded decay is the dominant decay mechanism of the false vacuum. First, we adapt the standard instanton methods to the Gross-Pitaevskii equation. Second, we use the truncated Wigner method to study vacuum decay.
We theoretically study the entanglement between phonons spontaneously generated in atomic Bose-Einstein condensates by analog Hawking and dynamical Casimir processes. The quantum evolution of the system is numerically modeled by a truncated Wigner method based on a full microscopic description of the condensate and state non-separability is assessed by applying a generalized Peres-Horodecki criterion. The peculiar distribution of entanglement is described in both real and momentum spaces and its robustness against increasing initial temperature is investigated. Viable strategies to experimenta…
We propose an ultracold atom analogue of early universe vacuum decay using all three states of a spin-1 Bose gas. We consider a one-dimensional system with both radio frequency and optical Raman coupling between internal states. An advantage of our proposal is the lack of a time-modulated coupling, which can lead to instabilities. Within the elaborate phase structure of the system we identify an effective Klein-Gordon field and use Gross-Pitaevskii simulations within the truncated Wigner approximation to model the decay of its false vacuum. We examine the dependence of the rate of vacuum decay…
We investigate a classical phase-space approach of matter-wave propagation based on the Truncated Wigner Equation (TWE). We show that such description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing mode frequency for long-ranged interactions decaying as $1/r^2$ in 3D and for a…
We numerically examine slow and hierarchical relaxation dynamics of interacting bosons described by a tilted two-band Bose-Hubbard model. The system is found to exhibit signatures of quantum chaos within the spectrum and the validity of the eigenstate thermalization hypothesis for relevant physical observables is demonstrated for certain parameter regimes. Using the truncated Wigner representation in the semiclassical limit of the system, dynamics of relevant observables reveal hierarchical relaxation and the appearance of prethermalized states is studied from the perspective of statistics of …
A binary mixture of interacting Bose-Einstein condensates (BEC) in the presence of fragmentation-driving external lattice potentials forms two interdependent mean-field lattices made of each component. These effective mean-field lattices, like ordinary optical lattices, can induce additional fragmentation and phase coherence loss of BECs between lattice sites. In this study, we consider the nonequilibrium dynamics of two hyperfine states of one-dimensional Bose-Einstein condensates, subjected to state-dependent optical lattices. Our numerical calculations using the truncated Wigner approximati…
Using the truncated Wigner approximation (TWA) we study quench dynamics of two-dimensional lattice systems consisting of interacting spinless fermions with potential disorder. First, we demonstrate that the semiclassical dynamics generally relaxes faster than the full quantum dynamics. We obtain this result by comparing the semiclassical dynamics with exact diagonalization and Lanczos propagation of one-dimensional chains. Next, exploiting the TWA capabilities of simulating large lattices, we investigate how the relaxation rates depend on the dimensionality of the studied system. We show that …
We investigate the non-equilibrium dynamics of the bosonic Hubbard model starting from inhomogeneous superfluid or Mott insulator initial states using the truncated Wigner approximation (TWA). We find that the relaxation of the system develops in two steps for sufficiently large interaction strengths: after a fast relaxation the system gets caught in metastable prethermalized states that precede the true equilibrium state. We find that the lifetime of these prethermalized states increases by several orders of magnitude as we increase the on-site interaction strength beyond a threshold value. W…
