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arXiv:1411.7303 · Physica Scripta

Operator approach to quantum optomechanics

C. Ventura-Velázquez, B. M. Rodríguez-Lara, H. M. Moya-Cessa

Published May 13, 2015• 15 citations
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Abstract

We study the mirror-field interaction in several frameworks: when it is driven, when it is affected by an environment, and when a two-level atom is introduced in the cavity. By using operator techniques, we show either how these problems may be solved or how the Hamiltonians involved, via sets of unitary transformations, may be taken to known Hamiltonians for which there exist approximate solutions.

DOI
10.1088/0031-8949/90/6/068010

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We study the mirror-field interaction in several frameworks: when it is driven, when it is affected by an environment, and when a two-level atom is introduced in the cavity. By using operator techniques, we show either how these problems may be solved or how the Hamiltonians involved, via sets of unitary transformations, may be taken to known Hamiltonians for which there exist approximate solutions.
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The field of quantum optomechanics is newly grooming research field, availed good attention in the last couple of years. Here, we theoretically study the system of optomechanics containing a two-level atom, which is coupled to the cavity field, and driven coherently by external fields. Analytical results for the system’s operator dynamics, steady state solutions and transmissivity of optomechanical system are calculated. Transmission (optical response) from the optomechanical system shows some useful information about the current optomechanical system. Particularly, [Formula: see text] = [Formula: see text]-g0([Formula: see text] + [Formula: see text]) is a crucial quantity in optomechanics, focused as main parameters in this paper. Optical transmission is studied in two regions. The first region (case) (i) when [Formula: see text] = [Formula: see text] - [Formula: see text], and in second region (case), (ii) [Formula: see text] = [Formula: see text] + [Formula: see text]. The transmission is examined and discussed with respect to the mechanical frequency of the oscillating mirror.
We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.
We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.
We show that the interaction between a movable mirror with a quantized field tha t interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operator s (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a unitary transformation, into a Hamiltonian that is equivalent to the ion-laser Hamiltonian. We would like to stress that the transformation in terms of SGO already simplifies e nough the Hamiltonian in the sense that, in an exact way, it eliminates one of the three-subsystems, namely the quantized fie ld.