Phase space representation of quantum dynamics (0905.3384v3)

Abstract: We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose Hubbard model, Dicke model and others.

• The paper studies quantum dynamics using phase space methods, introducing the truncated Wigner approximation (TWA) and its extension with quantum corrections.
• The truncated Wigner approximation (TWA) models quantum dynamics using classical trajectories, encoding quantum effects in initial conditions.
• This phase space approach offers a computationally efficient method for simulating non-equilibrium quantum dynamics in complex systems where direct methods are intractable.

Overview of the Paper: Phase Space Representation of Quantum Dynamics

This paper by Anatoli Polkovnikov presents a paper of quantum dynamics using the phase space methodology, a framework traditionally developed for classical mechanics, applicable to systems with many degrees of freedom. The paper explicitly examines expansions around the classical limits and discusses perturbative methods where quantum dynamics are expanded in terms of quantum fluctuations. These classical limits are explored in three configurations: (i) the corpuscular limit for coordinate-momentum systems, (ii) the wave limit for interacting bosons, and (iii) the Bloch limit for spin systems.

Semiclassical Approximation and Quantum Corrections

The paper notably introduces the truncated Wigner approximation (TWA), a semiclassical approach where quantum effects are primarily encoded in the initial conditions. This technique allows for the practical modeling of quantum dynamics using classical trajectories. The extension of TWA incorporates quantum corrections, which manifest as stochastic quantum jumps or nonlinear responses, thus generalizing the representation for all the mentioned classical limits.

Key Concepts and Techniques

Polkovnikov identifies several theoretical components crucial to the phase space formalism:

• Wigner Function and Weyl Symbols: The Wigner function acts as a quasi-probability distribution function in phase space, supporting the transition from quantum operators to classical analogue functions.
• Moyal Product and Brackets: These guide the interactions between operator symbols in phase space, providing a framework for representing the quantum Liouville equation in terms of Moyal brackets.
• Bopp Operators: These operators extend the phase space formulation to evolve specific quantum dynamical systems, accommodating explicit quantum fluctuation contributions.

Notable Results and Implications

The paper maintains a rigorous analysis of non-equilibrium quantum dynamics, applying the discussed methods across various conceptual models, including the Bose-Hubbard model and sine-Gordon models. The implications are vast, suggesting that these methods offer a computationally efficient approach for systems where direct quantum mechanical computations become untenable due to increased complexity.

The results indicate that TWA and its subsequent quantum corrections possess a formidable range of accuracy across parameter-rich models, even when considering ultracold atomic systems or interacting spin models. This flexibility underscores the potentially broad applicability of the methods outside pure quantum systems to include hybrid quantum-classical systems.

Future Directions and Theoretical Speculations

While the paper is extensive, Polkovnikov opens the floor for future exploration. Potential advancements could further refine the systematic approach of quantum corrections, extend phase space techniques to fermionic systems through Grassmann variables, or explore novel interactions not wholly captured by existing frameworks.

Essentially, the work lays foundational elements for quantum phase space methodologies that could redefine non-equilibrium quantum simulations. This is crucial given the current drives to solve larger quantum systems that bottleneck classical computational methods. Theoretical advancements in the phase space representation in systems with interacting particles and hybrid quantum classes with complex symmetries could enrich the landscape of quantum simulations.

In sum, Polkovnikov's paper fundamentally emphasizes the utility of phase space techniques for simulating complex quantum systems. By aligning quantum systems closer to classical analogs within this framework, a newfound understanding and tractable approach for analyzing non-equilibrium quantum dynamics emerge, bolstering both experimentation and computation in probing quantum mechanical phenomena.