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arXiv:0905.3127 · Physical Review A

Commuting Heisenberg operators as the quantum response problem: Time-normal averages in the truncated Wigner representation

B. Berg, Lev I. Plimak, Anatoli Polkovnikov, M. K. Olsen, Michael Fleischhauer, Wolfgang P. Schleich

Published Sep 29, 2009• 43 citations
arXivHTMLPDF

Abstract

The applicability of the so-called truncated Wigner approximation $(\ensuremath{-}W)$ is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the $\ensuremath{-}W$ approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The $\ensuremath{-}W$ equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo's renowned formula relating the linear-response function to two-time commutators suffices. The $\ensuremath{-}W$ is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.

DOI
10.1103/physreva.80.033624

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The applicability of the so-called truncated Wigner approximation (\ensuremath−W)(\ensuremath{-}W)(\ensuremath−W) is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the \ensuremath−W\ensuremath{-}W\ensuremath−W approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The \ensuremath−W\ensuremath{-}W\ensuremath−W equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo's renowned formula relating the linear-response function to two-time commutators suffices. The \ensuremath−W\ensuremath{-}W\ensuremath−W is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.
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.
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.
Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-1/21/21/2 systems using the continuous SU(2)\mathrm{SU}(2)SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.
A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.
The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time, termed iTWA. The evolution of the canonical density matrix in imaginary time is mapped to a partial differential equation of its Wigner function. Truncation at the Fokker-Planck level leads to a set of stochastic differential equations, which can be efficiently simulated even for large systems. We show that for general Ising Hamiltonians the approximation becomes exact for large imaginary times subject only to sampling errors. Thus the iTWA is ideal to determine the ground state of spin glasses or to find solutions to quadratic unconstrained binary optimization problems (QUBO) on a controlled approximation level. We illustrate this for MaxCut on random, unweighted 3-regular graphs, encoded in an anti-ferromagnetic Ising Hamiltonian, for which finding the exact ground state and even approximations to it beyond a certain accuracy is know to be NP hard. Furthermore, in order to assess the quality of the method also for general spin models, we analyze the ground-state quantum phase transition of the transverse-field Ising model in one and two spatial dimensions, finding reasonably good agreement with the exact behavior.