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

^{2}Laboratory 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.

**Visit for more related articles at** Journal of Nanosciences: Current Research

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.

Decoherence; Propagator; Coupled **oscillator**

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 and . 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:

where x_{1}, x_{2}, p_{1}, p_{2} are respectively the displacements and momenta
of each oscillator, and ? their oscillation frequency.

From the formula , the Lagrangian of the system reads

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

When setting *f*_{1}(t)=f_{2}(t)=0, the system of Equation(3) becomes

The Lagrangian is transform to

Then, the action is

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

by making the following change of variables,

the action splits into two terms

that is,

is the classical action and yi the deviation between x_{i}(t) and theirs classical path x_{ci} which is given by y_{i} = x_{i} - x_{ci},(i = 1,2).

The propagator is thus transformed as follows:

Where

is a smooth function in the limit

By setting

the normalization factor F(t) of the system becomes

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

Follows

With

And

when expending *q*_{1} and *q*_{2} in Fourier series on eigens modes [9,10,12] one gets from Fresnel’s integral (113)

And

Thus, it yields that [12]

Therefore, the propagator takes the form

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

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

This system becomes

of which the solutions are

When returning to initials variables, one gets

thus, we have

Therefore

and

From the system of Equation(30) we obtain

and by performing the computations x_{1}+x_{2} and x_{1}-x_{2}, we get
respectively

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

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

Deducing of the DCHO propagator: Setting *f*_{1}(t) = *f*_{2}(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]

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

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

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

that is,

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

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

When setting

we have from Equation(41) the following system

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

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

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

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

we obtain

Solving these two differentials equations gives

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

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

Where ϑ is a constant to be determined.

To determine the constants Λ_{1}, Λ_{2} and ϑ , we make a comparison between Equations (34) and (35) for *f*_{1}(t) = f_{2}(t) = 0. From this comparison,
it comes that

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

**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)**

The potential energy of this system is:

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

The kinetic energy of this system is written as:

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

that is,

Equations (53) and (54) are transformed to

Thus, the Hamiltonian reads

From the Hamilton canonical equations, we get

With

Then the Schrodinger equation can be written as:

By separation of variables while taking E= E_{1} + E_{2} this equation yields to

With

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

Where

Then

In the fundamental state n_{1}=0 and, n_{2}=0, what leads to:

With

When returning to the initials variables, we obtain

Where

**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].

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

As a reminder, and x_{1}, x_{2} 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:

with

When changing I_{1}, I_{2}, I_{3} and I_{4} into

Where

this propagator can finally be expressed as follows:

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

With

When changing x_{1} and x_{2} into

then, Equation(78) becomes

where

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

Where we recall that

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

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

**Entropy:** By definiton, the Gibbs-Shannon entropy reads

When setting

and proceeding to the changes

Equation(89) becomes

Where

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

**Numerical results**

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

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.

**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.

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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