Exact Real Time Dynamics of Quantum Spin Systems Using the Positive-P Representation

Abstract

We discuss a scheme for simulating the real time quantum quench dynamics of interacting quantum spin systems within the positive-P formalism. As model systems we study the transverse field Ising model as well as the Heisenberg model undergoing a quench away from the classical ferromagnetic ordered state and antiferromagnetic Neel state, depending on the sign of the Heisenberg exchange interaction. The connection to the positive-P formalism as it is used in quantum optics is established by mapping the spin operators on to Schwinger bosons. In doing so, the dynamics of the interacting quantum spin system is mapped onto a set of Ito stochastic differential equations (SDEs) the number of which scales linearly with the number of spins, N, compared to an exact solution through diagonalization that in the case of the Heisenberg model would require matrices exponentially large in N . This mapping is exact and can be extended to higher dimensional interacting systems as well as to systems with an explicit coupling to the environment.

27 pages, 10 figures

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• Stochastic Differential Equations for Quantum Dynamics of Spin-Boson Networks
• Simulation of the Dynamics of Many-Body Quantum Spin Systems Using Phase-Space Techniques
• Non-Equilibrium Quantum Spin Dynamics from Classical Stochastic Processes
• Quantum dynamics of long-range interacting systems using the positive-P and gauge-P representations
• Multi-time correlations in the positive-P, Q, and doubled phase-space representations
• Grassmann phase-space methods for fermions: uncovering classical probability structure
• Time-Evolving Weiss Fields in the Stochastic Approach to Quantum Spins
• Correspondence between open bosonic systems and stochastic differential equations
• Stochastic correction to the Maxwell-Bloch equations via the positive $P$ representation
• Phase-Space Methods for Many-Body Quantum Optics
• Benchmarking quantum phase-space methods for near-resonant light propagation
• Trajectory-Resolved Weiss Fields for Quantum Spin Dynamics