Reduced Density Matrices and Phase-Space Distributions in Thermofield Dynamics

TL;DR

This work tackles the challenge of extracting reduced 1-particle distributions within the iBT-TFD framework, where the Bogoliubov back-transformation entangles real and tilde modes. It derives exact expressions for the 1-RDM and Wigner distribution from the iBT 2-RDM in the thermal harmonic oscillator and discusses their realization in MCTDH/SOP wavefunctions. Two practical approximations are proposed: neglecting real–tilde correlations and a moment-expansion method that reconstructs densities from a finite set of moments; these are demonstrated on a thermalized anharmonic oscillator. The results show that the moment-expansion approach captures the essential width and qualitative behavior of the thermally excited densities, offering a scalable route to phase-space analysis in high-dimensional quantum dynamics with potential extensions to higher-dimensional distributions and connections to recent path-integral treatments.

Abstract

Thermofield dynamics (TFD) is a powerful framework to account for thermal effects in a wavefunction 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 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 wavefunction. Here we derive formal expressions for the reduced 1-particle density matrix (1-RDM) that uses the correlations between the real and tilde modes encoded in the associated reduced 2-particle density matrix (2-RDM). 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.

Figures (4)

• Figure 1: Two-mode squeezing effect on the diagonal part of the 2-RDM under the iBT formalism. (a) Gaussian 2-RDM; (b) Analytical result of the iBT-transformed 2-RDM from (a); (c-f) actual diagonal 2-RDMs from MCTDH calculations for the weakly quartic oscillator of Eq. (\ref{['e:example-h0_1']}) at different times. $\delta$ denotes the location of the shifted initial condition in the iBT position space and $\delta'$ is the transformed initial condition in the physical position space. Black axes in panels (a-d) help to emphasize the shift. In (c-d), the state at $t = 0$ is shown, while (e-f) shows the state at 0.9 ps in the complementary representations.
• Figure 2: Time evolution of the reduced 1-particle density of the anharmonic oscillator at $T =$ 0 K (l.h.s.) and $T =$ 300K (r.h.s.). The 1-particle densities $\rho^{(1)}_{\rm iBT}(z)$ and $\rho^{(1)}_{\rm iBT}(\tilde{z})$ are also shown for comparison. At non-zero temperature, the latter do not have a physical meaning.
• Figure 3: Exact and approximate reduced 1-particle density of the anharmonic oscillator. (a) Exact time evolution, (b) uncorrelated approximation, (c) deviation of the uncorrelated approximation from the exact result, (d) moment approximation, (e) deviation of the moment approximation from the exact result.
• Figure 4: First moment (left) and variance (right) of the physical mode calculated with the two approximations discussed in the main text, i.e., the uncorrelated approximation and the moment approximation.