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Israel Journal of Chemistry

Generic Construction of Kraus Operators: d‐level Systems in a Thermal Bosonic Bath

Asoka Biswas, Paul W. Brumer

Published Mar 22, 2012• 4 citations
Publisher

Abstract

Abstract The dynamical evolution of open systems can be conveniently expressed in the operator sum representation in terms of Kraus operators. Here we introduce an expression for the canonical Kraus representation in terms of the eigenstates |γ〉 and eigenvalues Eγ of the total Hamiltonian, thus making a stronger connection to the tools in chemistry. The method is exact, and hence valid for non‐Markovian evolution of the system to all orders of the coupling to the bath. Significantly, it provides a starting point for developing approximate Kraus operators based upon perturbative approximations to |γ〉 and Eγ. As an example, the explicit form of these operators is derived for a d‐level system interacting with a single‐mode thermal bath.

DOI
10.1002/ijch.201100114

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Abstract The dynamical evolution of open systems can be conveniently expressed in the operator sum representation in terms of Kraus operators. Here we introduce an expression for the canonical Kraus representation in terms of the eigenstates |γ〉 and eigenvalues Eγ of the total Hamiltonian, thus making a stronger connection to the tools in chemistry. The method is exact, and hence valid for non‐Markovian evolution of the system to all orders of the coupling to the bath. Significantly, it provides a starting point for developing approximate Kraus operators based upon perturbative approximations to |γ〉 and Eγ. As an example, the explicit form of these operators is derived for a d‐level system interacting with a single‐mode thermal bath.
In order to describe the evolution of a quantum system that is coupled to a reservoir, a non-phenomenological Kraus map is constructed. At time zero, system and reservoir are not entangled. In the perturbative series for the density operator of the system all reservoir correlation functions are factorised into products of pair-correlation functions. This allows for a resummation of the perturbative series up to infinite order. The density operator can be expressed in terms of an auxiliary system operator that satisfies an analytically tractable integral equation. Hence, the difficulties caused by integral kernels of Nakajima-Zwanzig type are circumvented. Assuming an interaction between system and reservoir of the Jaynes-Cummings form, one shows that the Kraus map is capable of generating Rabi oscillations of a two-level atom. If the reservoir is a continuum, the Kraus map reproduces the Wigner-Weisskopf theory of spontaneous emission.
Starting from unitary dynamics we study the evolution in time of a non-relativistic quantum system that exchanges energy with a thermal reservoir of harmonic oscillators. System and reservoir are assumed to be initially decorrelated. Reservoir correlation functions are factorized by means of a Kraus compliant version of Wick's theorem. As a result, the non-Markovian Kraus map for the system density operator can be completely expressed in terms of system potentials and reservoir pair correlation functions. An infinite hierarchy for the evolution operators of the Kraus map is derived. The system density operator is obtained as a time-ordered exponential containing a non-Markovian counterpart of the standard Markovian generator for dissipative dynamics. We specify a condition on this non-Markovian generator for return to thermal equilibrium. We set up a non-Markovian perturbation theory that preserves both trace and positivity of the system density operator.
We develop an operator formulation of statistical quasiparticle theory for open quantum systems with nonlinear system-bath coupling, extending beyond the linear coupling setting of the standard Gaussian influence functional formulation. For Gaussian baths, the statistical quasiparticles, termed dissipatons, are explicitly constructed as collective operators in the thermofield bath space. Our construction provides a microscopic operator basis for the dissipaton algebra and generates hierarchical equations of motion directly from the Liouville equation. Nonlinear interactions are incorporated through a recursive generalized Wick's theorem, enabling automatic generation of hierarchical equations for polynomial couplings. The resulting framework connects explicit bath operator representations to nonperturbative, non-Markovian dynamics and unifies the treatment of linear and nonlinear system-bath interactions.
We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an NNN-dimensional system. Moreover, applications of our result are illustrated through several examples.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N 2×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×N 2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a “best possible” master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.
We set up the fermionic thermal entangled state approach to address Fermi operator master equations for the amplitude- and phase-decay processes and find the explicit Kraus operator-sum representations (KOSR) describing time-evolution of Fermi systems. As an application of KOSR, the evolution law of the two-level atomic system interacting with a photon field in these two decay processes is analytically derived, which shows that the interaction system has a quantum jump as a result of dissipation, but it always remains unchanged in the phase-decay process.