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arXiv:1510.05905 · Physical review. B./Physical review. B

Distortion of a reduced equilibrium density matrix: Influence on quantum emulation

Iris Schwenk, Michael Marthaler

Published Jan 15, 2016• 3 citations
arXivHTMLPDF

Abstract

We study a system coupled to external degrees of freedom, called bath, where we assume that the total system consisting of system and bath is in equilibrium. An expansion in the coupling between system and bath leads to a general form of the reduced density matrix of the system as a function of the bath self energy. The coupling to the bath results in a renormalization of the energies of the system and in a change of the eigenbasis. This theory is applicable to quantum emulators in thermal equilibrium. Undesired external degrees of freedom can affect their reliability. We study the influence of bosonic degrees of freedom on the state of a six qubit system.

DOI
10.1103/physrevb.93.014305

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We study a system coupled to external degrees of freedom, called bath, where we assume that the total system consisting of system and bath is in equilibrium. An expansion in the coupling between system and bath leads to a general form of the reduced density matrix of the system as a function of the bath self energy. The coupling to the bath results in a renormalization of the energies of the system and in a change of the eigenbasis. This theory is applicable to quantum emulators in thermal equilibrium. Undesired external degrees of freedom can affect their reliability. We study the influence of bosonic degrees of freedom on the state of a six qubit system.
Calculation of the equilibrium state of an open quantum system interacting with a bath remains a challenge to this day, mostly due to a huge number of bath degrees of freedom. Here, we present an analytical expression for the reduced density operator in terms of an effective Hamiltonian for a high temperature case. Comparing with numerically exact results, we show that our theory is accurate for slow baths and up to intermediate system-bath coupling strengths. Our results demonstrate that the equilibrium state does not depend on the shape of spectral density in the slow bath regime. The key quantity in our theory is the effective coupling between the states, which depends exponentially on the ratio of the reorganization energy to temperature and, thus, has opposite temperature dependence than could be expected from the small polaron transformation.
We study the problem of the basis of an open quantum system, under a quantum chaotic environment, which is preferred in view of its stationary reduced density matrix (RDM), that is, the basis in which the stationary RDM is diagonal. It is shown that, under an initial condition composed of sufficiently many energy eigenstates of the total system, such a basis is given by the eigenbasis of a renormalized self-Hamiltonian of the system, in the limit of large Hilbert space of the environment. Here, the renormalized self-Hamiltonian is given by the unperturbed self-Hamiltonian plus a certain average of the interaction Hamiltonian over the environmental degrees of freedom. Numerical simulations performed in two models, both with the kicked rotor as the environment, give results consistent with the above analytical predictions.
The exact form of a quantum propagator of a quantum oscillator, interacting with a bosonic bath consisting of N -distinguished quantum oscillators with different frequencies, is obtained in the Heisenberg picture. A reduced density matrix for the oscillator is obtained. The kernel or Greenʼs function connecting the initial density matrix of the oscillator to the density matrix in an arbitrary time is obtained, and its connection to the Feynman–Vernon influence functional is discussed. A weak coupling regime and squared mean values for position, momentum, and energy of the oscillator are obtained in equilibrium.
Thermalization in open systems coupled to macroscopic environments is usually analyzed from the perspective of relaxation of the reduced state of the system to the equilibrium state. Less emphasis is given to the change of the state of the bath. However, as previously shown for some specific models, during the thermalization the environment may undergo a nontrivial dynamics, indicated by the the change of its von Neumann entropy, at timescales even longer than the relaxation time of the system; here such a behavior is called post-thermalization. We provide a more detailed analysis of this phenomenon by simulating the full dynamics of a variety of systems together with their environment. In particular, the post-thermalization is qualitatively explained as a result of reconversion of the initially built up correlation between the system and the bath into the correlation between the degrees of freedom in the environment. We also present exemplary systems in which such a reconversion is suppressed due to non-Markovian dynamics or the presence of interactions.
We study the structure of the time evolution of the density matrix in contact with a thermal bath in a standard projection operator sheme. The reduced density matrix of the system in the steady state is obtained by tracing out the degree of freedom of the thermal bath from the equilibrium density matrix of the total system. This reduced density matrix is modified by the interaction, and is different from that of the equilibrium of the system alone. We explicitly calculate the contribution of each term in quantum master equation to the realization of the steady state density matrix, and make clear roles of each term. By making use of the role of each term, the properties of the commonly used quantum master equation are examined.
We consider a model of environment-induced dissipationless decoherence of a quantum system where the system is coupled to the bath degrees of freedom via the system Hamiltonian itself. We solve exactly for the reduced density operator of the system for an arbitrary spectral density of the thermal bath and also write down an exact master equation in the Lindblad form. We compare and contrast the above results with those obtained by considering the system frequencies to be randomly modulated as in stochastic models. We observe that a coupling to the bath as above necessarily induces a Kerr-like coherent contribution in the reduced dynamics of the system. This Kerr-like term is a reflection of the quantum nature of the bath and cannot be obtained from stochastic models. For the special case of a harmonic oscillator we consider the influence of decoherence and its relation with phase diffusion. Our numerical results exhibit oscillations in the evolution of system variables which overall is a signature of the quantum nature of the environment.