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Journal of Nanosciences: Current Research
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Decoherence of Driven Coupled Harmonic Oscillator

Kenfack SC1*, Nguimeya G.-P1, Talla PK2, Fotue AJ1, Fobasso MFC1, Tiotsop M1, Djomou JRD1, Ekosso CM1 and Fai LC1

1Faculty of Science, Mesoscopic and Multilayer Structures Laboratory, Department of Physics, University of Dschang, P.O. Box 479 Dschang, Cameroon

2Laboratory of Mechanics and modeling of Physical systems, Faculty of Science, University of Dshang, PO Box 67 Dschang, Cameroon

*Corresponding Author:
Kenfack Sadem Christian
Faculty of Science, Mesoscopic and Multilayer Structures Laboratory
Department of Physics, University of Dschang
PO Box 479 Dschang, Cameroon
Tel: 00237 678 00 59 00
E-mail: [email protected]

Received Date: June 06, 2016; Accepted Date: September 23, 2016; Published Date: November 04, 2016

Citation: Kenfack SC, Nguimeya GP, Talla PK, Fotue AJ et al. (2016) Decoherence of Driven Coupled Harmonic Oscillator. J Nanosci Curr Res 1: 104.

Copyright: © 2016 Kenfack SC, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We consider a pair of linearly coupled harmonic oscillators to explore the decoherence phenomenon induces by interaction of a quantum system with a classical environment using Feynman path’s integral method. We determine the DCHO propagator afterwards thermodynamics parameters associated to the system. We show numerically that one can reduce decoherence of a system by coupling this one to a driven harmonic oscillator or by carrying this system to the resonance.

Keywords

Decoherence; Propagator; Coupled oscillator

Introduction

The study of the phenomenon of decoherence has attracted increasing attention in recent years [1]. In fact, it has been recognized that decoherence is of fundamental importance in the comprehension of the nature of the boundary between the quantum and classical worlds. The nature of this boundary has been meticulously examined from both the theoretical and experimental points of view [2,3]. Decoherence is closely related to one of the greatest mysteries of physics which up to now remains unanswered, that is, the quantum measurement problem. This problem was clearly formulated by Erwin Schrodinger in 1935 in his experiment of thought known as “Schrodinger’s cat paradox” [4-6] pointing out the interaction problem between quantum systems where quantum effects dominate and classical systems which operate in the classical limit. Two linearly coupled oscillators provide an ideal test model for exploring this aspect of the quantum-classical limit interaction [7]. In this work, we use the Feynman path integral formalism which enables an evaluation of the sum over all possible paths in a consistent way. This formulation which appears to be appropriate for a correct description of the reality is exact, equivalent to the Hilbert space formalism and leads to the same results for the Schrödinger equation. It is an alternative formulation of the quantum mechanics, which introduces only functions with numerical values and does completely without operators. This paper is organized as follows: In section 2, we present the model and determine the coupled harmonic oscillator (CHO) propagator from which we deduce the driven coupled harmonic oscillator(DCHO) propagator. In section 3, we derive the thermodynamic parameters associated to the system which is entropy. In section 4, we present the numerical results and then, end with the conclusion in section 5.

The DCHO propagator

The DCHO is a system constituted of two harmonic oscillators coupled by a spring and whose extremities are subjected to external driving forces represented by equation and equation . Here, we assume that the two oscillators have equal mass m and coupling constant k .

The CHO propagator

The determination of this propagator is of great interest because it will allow us later by comparison to establish the explicit form of the DCHO propagator. The simplest version of a pair of coupled quantum mechanical simple harmonic oscillators with linear coupling consists of two identical oscillators, with equal masses, spring constants, and frequencies, plus a connecting spring with its own spring constant. Considering the system described above, the Hamiltonian reads:

equation

where x1, x2, p1, p2 are respectively the displacements and momenta of each oscillator, and ? their oscillation frequency.

From the formula equation, the Lagrangian of the system reads

equation

The differentials equations that govern the evolution of the system are given by

equation

equation

When setting f1(t)=f2(t)=0, the system of Equation(3) becomes

equation

equation

The Lagrangian is transform to

equation

Then, the action is

equation

From the definitions [9,10], the propagator of this system over a finite time interval will read [11]

equation

by making the following change of variables,

equation

the action splits into two terms

equation

that is,

equation

equation

is the classical action and yi the deviation between xi(t) and theirs classical path xci which is given by yi = xi - xci,(i = 1,2).

The propagator is thus transformed as follows:

equation

Where

equation

is a smooth function in the limit equation

By setting

equation

equation

the normalization factor F(t) of the system becomes

equation

This factor can be written as the product of the normalization factor of each oscillator as

Follows

equation

With

equation

And

equation

when expending q1 and q2 in Fourier series on eigens modes [9,10,12] one gets from Fresnel’s integral (113)

equation

And

equation

Thus, it yields that [12]

equation

Therefore, the propagator takes the form

equation

We now focus on the classical action. Performing the integration of Equation(11) and taking into account Equation(4), one finds [12]

equation

To obtain the exact form of this action, we have to solve the system of Equation(4). By setting

equation

This system becomes

equation

of which the solutions are

equation

When returning to initials variables, one gets

equation

thus, we have

equation

Therefore

equation

and

.

From the system of Equation(30) we obtain

equation

equation

and by performing the computations x1+x2 and x1-x2, we get respectively

equation

equation

By using Eqs. (27) - (33) one finds from Equation(23) that [12]

equation

substituting this result into Equation(22), we obtain the CHO propagator which is [12]

equation

equation

Deducing of the DCHO propagator: Setting f1(t) = f2(t) = 0, the DCHO is reducing to CHO. Since the Lagrangian is quadratic, then we can assume the propagator for the DCHO to have the form [13]

equation

Where a(t),b(t),c(t),d(t),g(t),h(t) are the functions of time which have to be determined.

Taking equation and equation the Hamiltonian of Equation(1) takes the form

equation

The Hamiltonian of Equation(37) satisfies the Schrödinger equation [14]

equation

that is,

equation

From Equation(37), when using Equation (35), we get

equation

From the two first relations of above system, we get, taking this relation into account

equation

When setting

equation

equation

we have from Equation(41) the following system

equation

When solving each equation of the system of Equation(43), one obtains

equation

From Equation (43) and using Equation (41), we get

equation

One can see that Equation (44) does not include the excitation forces f1(t) and f2(t). Thus setting f1(t) = f2(t) = 0 the functions a(t) and b(t) can be equalized with the coefficients of x12 and x22 of equation (34). The comparison between these two equations shows that ?1 and ?2 must be equal to zero. And so

equation

By replacing a(t) = c(t) into the forth equation of system in Equation (39) and making the following change

equation

we obtain

equation

equation

Solving these two differentials equations gives

equation

equation

Where ?1 and ?2 are the constants to be determine. When taking into account Equation (48), then, Equation(46) gives [12]

equation

When combining Eqs. (45) and (49), we find [12]

equation

Where ϑ is a constant to be determined.

To determine the constants Λ1, Λ2 and ϑ , we make a comparison between Equations (34) and (35) for f1(t) = f2(t) = 0. From this comparison, it comes that

equation

When inserting Equations (45) and (49)-(51) into Equation(35), gathering the similar terms and replacing equation by equation we work out the DCHO propagator which reads [12]

equation

Thermodynamics Parameters for the DCHO

To determine thermodynamic parameters associated to the system, the knowledge of the time-dependent wave function is necessary as well as the wave function at initial time.

Wave function at initial time: The considered system here can be modeled as follows (Figure 1)

nano-sciences-current-research-diagram

Figure 1: Diagram of the model.

The potential energy equation of this system is:

equation

Where both k and kc are the elastic constant and kc characterize the coupling of particles at point x`=0 with each other.

The kinetic energy of this system is written as:

equation

When introduce the coordinate of the center of mass Y and the relative coordinate X.

equation

that is,

equation

Equations (53) and (54) are transformed to

equation

equation

Thus, the Hamiltonian reads

equation

From the Hamilton canonical equations, we get

equation

With

equation

Then the Schrodinger equation can be written as:

equation

By separation of variables equation while taking E= E1 + E2 this equation yields to

equation

equation

With

equation

The solutions of system in Equation(62) are looking for under the forms

equation

Where

equation

equation

Then

equation

In the fundamental state n1=0 and, n2=0, what leads to:

equation

With

equation

When returning to the initials variables, we obtain

equation

Where

equation

Time-dependent wave function: Having the propagator and Knowing the initial wave function, the solution of the time-dependent Schrodinguer equation is found by performing the integral [10,13].

equation

with equation and equation given by Equations. (52) and (69) respectively.

As a reminder, equation and x1, x2 are the initial and final coordinates of the oscillators respectively. In a much more compact form easy to manipulate, the propagator in Equation (54) can be written as follows:

equation

equation

with

equation

equation

equation

equation

equation

equation

equation

When changing I1, I2, I3 and I4 into

equation

Where

equation

this propagator can finally be expressed as follows:

equation

When putting Equation(77) into Equation(71) and doing some arrangement, we then get

equation

With

equation

equation

equation

equation

equation

When changing x1 and x2 into

equation

then, Equation(78) becomes

equation

where

equation

Equation (81) is easily integrate using Gaussian integral, then, one gets the expression of the time-dependent wave function which is

equation

Where we recall that

equation

Probability density: In term of the time-dependent wave function, the probability density is define by

equation

Carrying out this product using Gaussian integrals and taking into account Equations (83), we obtain after few arrangements the probability density of the system, which reads

equation

Entropy: By definiton, the Gibbs-Shannon entropy reads

equation

When setting

equation

equation

equation

and proceeding to the changes

equation

Equation(89) becomes

equation

Where

equation

Integrating Equation(92) using Gaussian integrals yields to the entropy of the system which is

equation

Numerical results

We notice in figure 2 that the entropy decreases with time and tends towards a limit.

nano-sciences-current-research-parameter

Figure 2: Entropy with parameter k = 0.001, kc = 0.005, ωₒ = 5.

Therefore, decoherence decrease (Figure 2).

This result is in accordance with that obtained by M.A. de Ponte et al.[14] for two coupled dissipative oscillators. Physically, this means that the whole system is organizes itself and evolves towards a coherent state in which information will be preserved.

From figure 3, we note that the entropy of the system oscillates by keeping a constant amplitude which means that the exchange between the system and its surrounding are quasi-perfect so that the system keeps a certain coherence. The system is slowly affected by external forces. Physically, this coherence can be interpreted as being due to the fact that information remains preserved in time. This result is in accordance with that obtained by Tabue et al. [15] for the pure states of a damped harmonic oscillator.

nano-sciences-current-research-entropy

Figure 3: Entropy with parameter k = 0.001, kc = k, ωₒ = 5.

Figure 4 shows that entropy increases in time, and thus decoherence increases. This means that, the whole system collects environment which accentuates decoherence and so the system disorganizes itself. One could expect such a result which is in accordance with that obtained by Tabue et al. [15] for a damped harmonic oscillator.

nano-sciences-current-research-entropy-parameter

Figure 4: Entropy with parameter k = 0.001, kc = 0, ωₒ = 5.

Conclusion

Using the Feynman path integral method we have determined the DCHO propagator after which, considering the particular case where the external driving forces are sinusoidal, we have determined the entropy associated to the system. We reveal numerically that, when two harmonic oscillators subjected to an external driving force are coupled, one compensates the effects of the other so that the whole system organizes itself and evolves towards an equilibrium position in which it will conserve certain coherence. That is explained by the decrease of the entropy which thereafter tends towards a limit. We also reveal that at resonance the system is in a coherent state which is explained by the conservation, with time of the oscillation amplitudes of the entropy of the system. Thus, we have shown that to reduce decoherence in a system one can just couple such a system with a driven harmonic oscillator or the system in question to resonance.

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