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Investigation of dissipation in a quantum LC circuit

Davoud Nasr Esfahani, Fardin Kheirandish

Published Mar 1, 2010• 0 citations
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Abstract

In this article we consider the resistance of a quantum LC circuit as a heat bath. The heat bath can be modeled by a collection of quantum harmonic oscillators with a continuum of frequencies. By using the minimal coupling method between the circuit and the field describing the environment, the process of energy dissipation and probability transitions between the energy levels of the quantum circuit, are obtained.

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In this article we consider the resistance of a quantum LC circuit as a heat bath. The heat bath can be modeled by a collection of quantum harmonic oscillators with a continuum of frequencies. By using the minimal coupling method between the circuit and the field describing the environment, the process of energy dissipation and probability transitions between the energy levels of the quantum circuit, are obtained.
We consider a resistance element (R) as an environment of an LC circuit, which is treated as the main system modeled by a Hamiltonian containing an oscillator with a continuous range of frequencies. Using a minimal coupling method which is appropriate for dissipative quantum systems, we investigate the quantum dynamics of this system. The Heisenberg equation for such a dissipative system contains a dissipative term proportional to Ohmic damping. Some transition probabilities indicating the way energy flows between the subsystems are obtained.
We discuss the concept and realization of a heat bath in solid state quantum systems. We demonstrate that, unlike a true resistor, a finite one-dimensional Josephson junction array or analogously a transmission line with non-vanishing frequency spacing, commonly considered as a reservoir of a quantum circuit, does not strictly qualify as a Caldeira-Leggett type dissipative environment. We then consider a set of quantum two-level systems as a bath, which can be realized as a collection of qubits. We show that only a dense and wide distribution of energies of the two-level systems can secure long Poincare recurrence times characteristic of a proper heat bath. An alternative for this bath is a collection of harmonic oscillators, for instance, in the form of superconducting resonators.
Quantum mechanics of a general one dimensional dissipative system investigated by it's coupling to a Klein-Gordon field as the environment using a minimal coupling method. Heisenberg equation for such a dissipative system containing a dissipative term proportional to velocity obtained. As an example, quantum dynamics of a damped harmonic oscillator as the prototype of some important one dimensional dissipative models investigated consistently. Some transition probabilities indicating the way energy flows between the subsystems obtained.
The dynamics of a two-state system coupled to the dissipative degrees of freedom of the environment appears in many different contexts. In this paper we have given a general formulation of the problem as well as a detailed comparison of two a priori very different heat baths: (a) a collection of harmonic oscillators coupled to the system, and (b) a collection of fermions coupled to the system. Consideration of such heat baths is motivated by the assertion that the weakly excited states of the environment can be modeled by a collection of elementary excitations following Landau. Our conclusion is that the case of Ohmic dissipation of harmonic-oscillator heat bath is very similar to the fermionic bath, although a crucially important dimensionless parameter \ensuremath{\alpha} assumes a different range of values in the two different cases.