Reduced density matrices and phase-space distributions in thermofield dynamics

Thermofield dynamics (TFD) is a powerful framework for accounting for thermal effects in a wave function setting and has been extensively used in physics and quantum optics. TFD relies on a duplicated state space and creates a correlated two-mode thermal state via a Bogoliubov transformation acting on the vacuum state. However, a very useful variant of TFD uses the vacuum state as the initial condition and transfers the Bogoliubov transformation into the propagator. This variant, referred to here as the inverse Bogoliubov transformation (iBT) variant, has recently been applied to vibronic coupling problems and coupled-oscillator Hamiltonians in a chemistry context, where the method is combined with efficient tensor network methods for high-dimensional quantum propagation. In the iBT-TFD representation, the mode expectation values are clearly defined and easy to calculate, but the thermalized reduced particle distributions, such as the reduced 1-particle densities or Wigner distributions, are highly non-trivial due to the Bogoliubov back-transformation of the original thermal TFD wave function. Here, we derive formal expressions for the reduced 1-particle density matrix (1-RDM) that use the correlations between the real and tilde modes encoded in the associated reduced 2-particle density matrix. We apply this formalism to define the 1-RDM and the Wigner distributions in the special case of a thermal harmonic oscillator. Moreover, we discuss several approximate schemes that can be extended to higher-dimensional distributions. These methods are demonstrated for the thermal reduced 1-particle density of an anharmonic oscillator.