Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

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“…The reason for not getting the Lagrangian and generalized linear momentum from the first integral (A.1) reported in Refs. [23,24] is as follows:…”

Section: Discussionmentioning

confidence: 99%

“…Before proceeding further, we wish to point out that an explicit time independent integral for the damped harmonic oscillator (2) was given by López and López [23,24]. However, no explicit Hamiltonian could be associated with it.…”

Section: Case: 1 α 2 < 4λmentioning

confidence: 99%

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

Chandrasekar

Senthilvelan

Lakshmanan

2007

Journal of Mathematical Physics

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“…There are two approaches to attack this problem. The first one consists on to look for the phenomenological classical dissipative system and to get its associated Hamiltonian, then to proceed to do the usual quantization of the system [1,2]. The other one, which it is more fundamental, uses the matrix density approach for the whole system and makes the trace over the environment variables [3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model

López¹,

López²

2012

JMP

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We study the phenomenon of decoherence during the operation of one qubit transformation, controlled-not (CNOT) and controlled-controlled-not (C 2 NOT) quantum gates in a quantum computer model formed by a linear chain of three nuclear spins system. We make this study with different type of environments, and we determine the associated decoherence time as a function of the dissipative parameter. We found that the dissipation parameter to get a well defined quantum gates (without significant decoherence) must be within the range of . We also study the behavior of the purity parameter for these gates and different environments and found linear or quadratic decays of this parameter depending on the type of environments.

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“…Different types of approaches have been developed to understand the phenomenon of decoherence that arises in the open quantum systems which it is related to the lost of the interference terms of the product of the quantum wave function [1][2][3][4][5][6][7][8], that is, the non diagonal elements of the reduced density matrix go to zero value. Since the complete insulate quantum system is almost impossible to have, decoherence becomes an intrinsic phenomenon related to the quantum principles and maybe related to the "emergent reality" of the classical world [9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…For a thermalized mixed environmental system one can sees that where we have made the change of variables 2 …”

Section: Introductionmentioning

confidence: 99%

Quasi non-Markovian Approach to the Study of Decoherence of a Controlled-Not Quantum Gate in a Chain of Few Nuclear Spins Quantum Computer

Vázquez¹,

Vázquez²

2012

JMP

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We develop in the weak coupling approximation a quasi-non-Markovian master equation and study the phenomenon of decoherence during the operation of a controlled-not (CNOT) quantum gate in a quantum computer model formed by a linear chain of three nuclear spins system with second neighbor Ising interaction between them. We compare with the behavior of the Markovian counterpart for temperature different from zero (thermalization) and at zero temperature for low and high dissipation rates. At high dissipation there is a very small difference between Markovian and quasi no-Markovian at any temperature which is unlikely to be measured, and at low dissipation there is a difference which is likely to be measured at any temperature.

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Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Cited by 11 publications

References 15 publications

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model

Quasi non-Markovian Approach to the Study of Decoherence of a Controlled-Not Quantum Gate in a Chain of Few Nuclear Spins Quantum Computer

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