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Stochastically averaged subdynamics of two-level systems (TLS) coupled to a colored noise: a nonperturbative time-ordered cluster cumulant method

Sheikh Hannan Mandal, Rathindranath Ghosh, Debashis Mukherjee

Published May 31, 2003• 0 citations
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Abstract

Department of Physical Chemistry, Indian Association for the Cultivation of Science, Kolkata- 700 032, India E-Mail : pcdm@mahendra.iacs.res.in Present address: Department of Physical Science, R. K. M. Sikshanamandira, Belurmath, Howrah-711 202, India Present address : Department of Chemistry, R. K. M. Residential College, Narendrapur, Kolkata-700 103, India Also at : Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560 064, India Manuscript received 4 October 2002 We present here a non-perturbative cumulant expansion method of studying stochastically averaged dynamics of a TLS coupled to a gaussian colored noise bath. It uses a factorized cluster Ansatz for the evolution operator, U(t) suggested in an earlier formalism (Ref. 22). The factorization separates the operators involving only the system variables from those containing the intractions involving the bath variables. The stochastically averaged U(t) is obtained in a compact way by way of introducing bath-normal ordered products, which have zero bath averages for the bath-variables satisfying gaussian factorization property. We use a Wick-like operator reordering theorem (Ref. 25) to have a bath-normal ordered cluster expansion representation of U(t) and subsequently Novikov's theorem (Ref. 27) to get equations for the cluster amplitudes without free bath variables.The subdynamics of the system where the bath-variables are averaged out is governed in our method by a time-local effective hamiltonian involving system variables only. The method is efficacious when the bath is strongly colored and its coupling with the system is strong, e.g. when nonperturbative effects are prominent. The efficacy of the method is illustrated by computing power spectra from a model system which mimics nonadiabatic vibronic coupling simulated by colored noise.

DOI
10.5281/zenodo.5839236

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We have developed in this paper a nonperturbative cluster-expansion strategy for generating the stochastically averaged time-evolution operator for quantum systems driven by Ornstein-Uhlenbeck (OU) colored noise. The method induces a boson mapping of the (real or complex) stochastic variable f, and interprets the stochastic average of a pair of variables f at two different times as the expectation value of the time-ordered product of the associated bosons with respect to the boson vacuum \ensuremath{\Vert}0B{0}_{\mathit{B}}0B​〉. The stochastic evolution is thus mapped onto a deterministic evolution in an expanded Fock space. The evolution of the system from the groups of state of interest is monitored using our recently developed nonperturbative time-dependent multireference coupled-cluster (TDMRCC) method. In this, the evolution operator U is written in a factorized form Uex{\mathit{U}}_{\mathrm{ex}}Uex​UM{\mathit{U}}_{\mathit{M}}UM​, where UM{\mathit{U}}_{\mathit{M}}UM​ evolves in the space of the starting functions (model space) and Uex{\mathit{U}}_{\mathrm{ex}}Uex​ brings in the virtual functions. Uex{\mathit{U}}_{\mathrm{ex}}Uex​ and UM{\mathit{U}}_{\mathit{M}}UM​ are both written as normal ordered exponentials involving cluster operators. In the present context, the TDMRCC method translates into one for generating the evolution operator UB{\mathit{U}}_{\mathit{B}}UB​ for the Hamiltonian containing the additional boson variables, and the stochastic averaging of UB{\mathit{U}}_{\mathit{B}}UB​ is realized as the expectation value 〈0B{0}_{\mathit{B}}0B​\ensuremath{\Vert}UB{\mathit{U}}_{\mathit{B}}UB​\ensuremath{\Vert}0B{0}_{\mathit{B}}0B​〉.We call the TDMRCC method involving the expanded Fock space a cluster cumulant method. We have analyzed the relation of the cluster cumulant approach with the Fox-Kubo operator cumulant expansion and the method of marginal averages involving the Fokker-Planck operator of the OU process. It has been shown that an order by order expansion of our equations in the power of stochastic coupling generates the perturbative cumulant results of Fox and Kubo. It is also demonstrated that the traditional Fokker-Planck method of marginal averages uses a Kubo-Schr\"odinger operator which is related to our boson mapped Hamiltonian by a transformation which converts the Fokker-Planck operator \ensuremath{\Gamma} to a manifestly Hermitian form. However, the method of marginal averages involves a linear expansion of UB{\mathit{U}}_{\mathit{B}}UB​ in terms of eigenfunctions of \ensuremath{\Gamma}, while the cluster cumulant method uses a cluster expansion, which is nonlinear in the expansion parameters. The former is thus analogous to a configuration-interaction expansion, while the latter is analogous to a coupled-cluster strategy. The method is illustrated by applying it to a simple yet nontrivial problem, viz., a harmonic oscillator linearly perturbed by a (real or complex) stochastic variable. Survivalities of both the ground and the first excited states have been considered for a wide range of coupling strengths and noise correlation times. Our method provides exact results in this case; these have been used as benchmarks for assessing performance of the second- and fourth-order cumulant results.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
Reduced equations of motion for a two-level system strongly coupled to a harmonic oscillators bath are constructed by extending the hierarchy of equations introduced by Tanimura and Kubo [J. Phys. Soc. Jpn. 58 (1989) 101]. The set of equations treats the bath in a nonperturbative manner and is applicable to a low-temperature system with taking into account the correlation time of noise. By numerically calculating linear absorption spectra for different temperatures, we demonstrate that the present theory is not afflicted with the dynamical positivity problem that occurs at low temperatures without the rotating wave approximation. Remarkable changes are found in the spectra when the temperature is lower than the resonant energy of the two-level system.
A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.