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-
1/2 systems using the continuous
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.
We develop the path integral theory for master equations of general Lindblad form (positive semigroups), describing Markovian open quantum systems. First the Hamiltonian path integral expression for the propagator is derived, which exhibits nicely the decoherence of pairs of phase space histories. A very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional. For the important class of (effective) Hamiltonians quadratic in the momenta, we derive the Lagrangian version of the path integral propagator. We then evaluate the path integral approximately in a stationary phase approximation, leading to a Van Vleck-type propagator valid under semiclassical conditions. We also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals. Finally, connections to the active field of stochastic pure-state descriptions of open quantum systems are established, here in particular to linear quantum state diffusion.
In the semiclassical limit, the functional-integral description of a quantum-mechanical system is dominated by certain classical paths of motion. We identify the relevant paths for a double-well potential and study how they are affected by coupling to a dissipative environment. When the coupling exceeds a critical value, the system becomes localized in one or the other of the two wells. The classical-path method reproduces previous computations of the critical coupling. This description is applicable for all values of the dissipative coupling and for any low-lying state of the system.