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On the Interaction Between Two Coupled Coherent Oscillators.

GERALD J. IAFRATE, Mark Croft

Published Jun 1, 1974• 0 citations
OpenAlex

Abstract

The quantum dynamics for two coupled harmonic oscillators is presented. Using the coupled Boson representation, the assumed problem is shown to be isomorphic with a perturbed angular momentum oscillator. The current operator is obtained and its associated expectation values with respect to the number states, coherent states (Glauber states) and atomic coherent states are given. In the limit of weak coupling the results obtained using the coherent states are shown to manifest a Josephson phase relationship. (Author)

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Interaction between two coupled oscillators

GERALD J. IAFRATE, Mark Croft

The quantum dynamics for two coupled harmonic oscillators is presented. Using the coupled-boson representation, the assumed problem is shown to be isomorphic with a perturbed angular-momentum oscillator. The current operator is obtained and its associated expectation values with respect to the number states, coherent states (Glauber states), and atomic coherent states are given. The eigenvalue spectrum of the current operator is seen to be finite and discrete. An interesting correspondence between this analysis and past work on Josephson tunneling and quantum interference is discussed.

Physical Review A•Oct 1, 1975•9 citations

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Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.

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Exact dynamics and squeezing in two harmonic modes coupled through angular momentum

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Abstract We investigate the exact dynamics of a system of two independent harmonic oscillators coupled through their angular momentum. The exact analytic solution of the equations of motion for the field operators is derived, and the conditions for dynamical stability are obtained. As for the application, we examine the emergence of squeezing and mode entanglement for an arbitrary separable coherent initial state. It is shown that close to instability, the system develops considerable entanglement, which is accompanied with simultaneous squeezing in the coordinate of one oscillator and the momentum of the other oscillator. In contrast, for weak coupling away from instability, the generated entanglement is small, with weak alternating squeezing in the coordinate and momentum of each oscillator. Approximate expressions describing these regimes are also provided.

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A new coherent paired state with possible applications to fluctuation-dissipation phenomena

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The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.

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Spin Number Coherent States and the Problem of Two Coupled Oscillators*

D. Ojeda-Guillén, R. D. Mota, V. D. Granados

Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.

arXiv:1206.1555•Communications in Theoretical Physics•Jul 1, 2015•9 citations

Quantum M echanics o f Two Coupled Oscillators.

M. Sebawe Abdalla

Summary. -- From the poir.t of view of quar.tum ~echanics, the problem of two coupled oscillators with two different coupling parameters is considered. By using the accurate definition of Dirac operators the wave function in both coherent state and number state (SchrSdinger picture) are obtained. The Green's function and the expectation value of the energy are calculated; the transition amplitude between the coherent states when the coupling parameters are different and equal, as well as the eigenstates, are given. The constants of the motion for such a system have been also considered.

•Jan 1, 1994•0 citations
The quantum dynamics for two coupled harmonic oscillators is presented. Using the coupled Boson representation, the assumed problem is shown to be isomorphic with a perturbed angular momentum oscillator. The current operator is obtained and its associated expectation values with respect to the number states, coherent states (Glauber states) and atomic coherent states are given. In the limit of weak coupling the results obtained using the coherent states are shown to manifest a Josephson phase relationship. (Author)
The quantum dynamics for two coupled harmonic oscillators is presented. Using the coupled-boson representation, the assumed problem is shown to be isomorphic with a perturbed angular-momentum oscillator. The current operator is obtained and its associated expectation values with respect to the number states, coherent states (Glauber states), and atomic coherent states are given. The eigenvalue spectrum of the current operator is seen to be finite and discrete. An interesting correspondence between this analysis and past work on Josephson tunneling and quantum interference is discussed.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
Abstract We investigate the exact dynamics of a system of two independent harmonic oscillators coupled through their angular momentum. The exact analytic solution of the equations of motion for the field operators is derived, and the conditions for dynamical stability are obtained. As for the application, we examine the emergence of squeezing and mode entanglement for an arbitrary separable coherent initial state. It is shown that close to instability, the system develops considerable entanglement, which is accompanied with simultaneous squeezing in the coordinate of one oscillator and the momentum of the other oscillator. In contrast, for weak coupling away from instability, the generated entanglement is small, with weak alternating squeezing in the coordinate and momentum of each oscillator. Approximate expressions describing these regimes are also provided.
The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.
Summary. -- From the poir.t of view of quar.tum ~echanics, the problem of two coupled oscillators with two different coupling parameters is considered. By using the accurate definition of Dirac operators the wave function in both coherent state and number state (SchrSdinger picture) are obtained. The Green's function and the expectation value of the energy are calculated; the transition amplitude between the coherent states when the coupling parameters are different and equal, as well as the eigenstates, are given. The constants of the motion for such a system have been also considered.