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Chemical Physics Letters

Stochastic averaging in quantum dynamics driven by colored noise: a nonperturbative cluster cumulant approach

Sruti Guha, Debashis Mukherjee

Published Feb 1, 1992• 2 citations
Publisher
DOI
10.1016/0009-2614(92)86107-s

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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}${0}_{\mathit{B}}$〉. 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 ${\mathit{U}}_{\mathrm{ex}}$${\mathit{U}}_{\mathit{M}}$, where ${\mathit{U}}_{\mathit{M}}$ evolves in the space of the starting functions (model space) and ${\mathit{U}}_{\mathrm{ex}}$ brings in the virtual functions. ${\mathit{U}}_{\mathrm{ex}}$ and ${\mathit{U}}_{\mathit{M}}$ 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 ${\mathit{U}}_{\mathit{B}}$ for the Hamiltonian containing the additional boson variables, and the stochastic averaging of ${\mathit{U}}_{\mathit{B}}$ is realized as the expectation value 〈${0}_{\mathit{B}}$\ensuremath{\Vert}${\mathit{U}}_{\mathit{B}}$\ensuremath{\Vert}${0}_{\mathit{B}}$〉.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 ${\mathit{U}}_{\mathit{B}}$ 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.

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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.
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.
We propose a quantum model consisting of two remote qubits interacting with two correlated colored noises and establish an exact stochastic Schr\"odinger equation for this open quantum system. It is shown that the quantum dynamics of the qubit system is profoundly modulated by the mutual correlation between baths and the bath memory capability through dissipation and fluctuation. We report a physical effect on generating inner correlation and entanglement of two distant qubits arising from the strong bath-bath correlation.