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arXiv:2505.21302 · The Journal of Chemical Physics

Reduced density matrices and phase-space distributions in thermofield dynamics

Bartosz Błasiak, Dominik Brey, Rocco Martinazzo, Irène Burghardt

Published Jan 27, 2026• 3 citations

Abstract

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.

DOI: 10.1063/5.0308440

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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.

We present a streamlined implementation of a computational framework for constructing and analyzing reduced dynamical maps for complex system--bath models at finite temperature. The methodology is based on three established ingredients of quantum dynamics: the Choi--Jamiołkowski isomorphism for the representation of quantum channels, thermofield (TFD) purification of thermal environments, and tensor-train (TT) propagation of the resulting enlarged pure state. The reduced map is obtained from a single unitary propagation in a thermofield-doubled Hilbert space and represented in matrix form through the Choi--Jamiołkowski isomorphism. The TFD evolution is implemented in the TT representation, enabling efficient propagation of high-dimensional purified thermal states. We illustrate the methodology for exciton transfer in the Fenna--Matthews--Olson complex with site-dependent structured spectral densities represented by discretized bosonic environments. The resulting maps are used to analyze decoherence, relaxation, and finite-memory effects, and to assess the crossover to an effectively time-local description. The proposed approach provides a route to compute reduced propagators and to post-process them into memory kernels, transfer tensors, and effective kinetic rate descriptions for complex molecular systems.

The Wigner phase-space representation of thermal excitations used in thermo-field-dynamics (TFD) is investigated. The phase-space description is obtained by introducing a reduced density operator in which the fictitious degrees of freedom of TFD are traced over. A new closed-form expression for the Wigner function of the harmonic oscillator thermal excitation is derived, thereby showing that the thermal excitation may be described as a temperature-dependent radial spreading of the coordinates and momenta in phase space. The population distribution in these stationary excitations is determined and shown to reduce to a Boltzmann distribution for the thermal vacuum only.

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.

The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the

\\alpha

--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual

2\\times2

Bogoliubov matrix is extended to a

4\\times4

matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.