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Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method

C. Qotni, A. L. Marrakchi, S. Sayouri , Y. Achkar
Applied Mathematics and Physics. 2019, 7(1), 8-13. DOI: 10.12691/amp-7-1-2
Received August 11, 2019; Revised September 29, 2019; Accepted October 17, 2019

Abstract

The method of the invariant dynamical linear operator is very simple and may be useful to solve the Schrödinger equation, in particular in the case of the problem of a harmonic oscillator with a time dependent mass and frequency. Indeed, we have successfully used this approach to the Caldirola-Kanai oscillator. In particular, we have obtained explicit expressions of the uncertainty product and the quantum correlation coefficient. The results obtained are in good agreement with those of the literature.

1. Introduction

The harmonic oscillator is a very importantproblem in quantum mechanics, because it is one of the few problems that can be solved and that has an exact analytical solution. Furthermore, the harmonic oscillator solution and algebraic formalism haveseveral applications in modern physics (e.g. quantum electrodynamics, quantum field theory, quantum optics, etc.) 1, 2, 3, 4. This is of great importance because it means we can produce analytical solutions to systems that are not quite as idealized as the simple linear harmonic oscillator. This fact gave rise to the development of many approximate methods to find the exact quantum states for time-dependent quantum systems 5, 6, 7, 8, 9, 10, 23, 24. One of the methods which was developed several decades ago by Lewis and Riesenfeld 11, 12, 22, 25 introduces an invariant operator, known as either the Lewis-Riesenfeld invariant or the generalized invariant, for a time-dependent quantum harmonic oscillator and enables the determination of the exact quantum states of this system in terms of the eigenstates of the invariant operator.

In this work we apply the method of dynamic linear invariant for solving the problem of the harmonic oscillator with time-dependent mass and frequency. The computation of the product of the uncertainties in position and momentum as well as the correlation coefficient were carried out. Thus, the Heisenberg uncertainty principle is well verified all times 26, 27, 28, 29, 30. This remains true even when time becomes infinite; the product of the uncertainties reaches a constant value. We also compared our results with those published in the literature 27, 28, 29.

2. The Method of Dynamic Linear Invariant

We consider a harmonic oscillator with time-dependent parameters described by the following Hamiltonian:

(1)

where the momentum p and the position q are canonical conjugates, M(t) is the time dependent mass, ω(t) is the frequency, y(t) is an arbitrary parameter of coupling between the momentum and the position and F(t) is an external force.

The time evolution of the system is described by the following Schrödinger equation:

(2)

In order to study the quantum motion of this type of system, we will solve the Schrödinger equation using the method of dynamic linear invariant.

2.1. Principle and Approach

The method of dynamic linear invariant was developed by Lewis and Riesenfeld 4. It can be applied to solve the time-dependent Schrodinger equation. According to the Lewis-Riesenfeld invariant theory 4, a Hermitian operator I(t) is called invariant if it satisfies the following invariant equation:

(3)

then the general solution of the Schrodinger equation (2) can be written as:

(4)

is an eigenfunction of the linear invariant I(t)with the eigenvalue λ

is a phase function satisfying the following equation

(5)

The invariant operator I(t)is expressed in the following form:

(6)

where are time-dependent functions.

We replace I(t) in Equation 3, we obtain first-order differential equations expressed as follow:

(7a)
(7b)
(7c)

With the help of Eqs. 6 and 7, it is easy to derive the following equations :

(8)

where

(9)
(10)

The next step is to find the eigenvectors of our system which give a set of solutions of the following equation:

(11)

with

(12)

By simple integration of equation (11), we obtain the function in the following form:

(13)

On the other hand, the calculation of the matrix element in equation 5, allows to obtain the following phase function:

(14)

Therefore, the function is expressed as:

(15)

Notice that when β(t) vanishes, the phase functions μλ(t) diverge. In spite of this divergence, it can easily be shown that the wave function is always finite. Moreover, the general solution of the Schrödinger equation can be written as:

(16)

is a Gaussian function which determines the state of the system, it is given by:

(17)

with a is a real constant.

2.2. Gaussian Wave Packet: Fluctuation and Correlation

We express the wave function as follows:

(18)

with

(18a)
(18b)
(18c)

Thus, we obtain the general wave function of the gaussian wave packet as:

(19)

where are such as:

(19a)
(19b)
(19c)
(19d)

The computed average quadratic deviations in the state have the following expressions:

(20a)
(20b)

Using the definition of the following quantum correlation coefficients

(21)

where is an anticommutator.

We succeeded in writing these coefficients in a new form

(22)

The uncertainty relation of Heisenberg can be written as:

(23)

This last equation suggests that the product of uncertainties is minimal when the quantum correlation coefficient tends to zero. Therefore, to satisfy the Heisenberg uncertainty principle, this coefficient has to converge to a constant when time tends to infinity.

3. Application: Study of the Harmonic Oscillator with Time-dependent Mass and Frequency

In this section, we consider an example, which allows us to apply the theoretical approach developed in previous sections; The example we choose is the Caldirola-Kanai (CK) oscillator 4, 13 with time-dependent mass and frequency. We apply the method of dynamic linear invariant for studying the (CK) oscillator. The aim is to test our method of resolution in the case of this type of oscillator.

The CK Hamiltonian has the following expression

(24)
3.1. Solution for the Differential Equation of β(t)

The second-order differential equation obtained in β(t)(Eq. 8)is given by

(25)

where M(t) is a time-dependent mass and is a time-dependent frequency.

To solve this equation, we proceed to use the following variable change, which allows a simple computation:

(26)

with the initial condition

(27)

Substitution of Y(t) of equation (26) into equation (2) yields:

(28)

In order to solve this differential equation and to obtain systematic solutions without making explicit choices of mass or frequency variation, we have considered the simplest situation in which we have considered that the Damping term is a real constantσ such that:

(29)

Integration of equation (29) gives:

(30)

Thus equations (28) become

(31)

It is a differential equation which admits three solutions under the three following conditions or

Time variation of the mass has been chosen to satisfy the expression M(t) such as:

(32)

By a simple use of equation (30) we can obtain the frequency function as:

(33)

Now, we use the expressions (32) and (33)of M(t) and (t) to calculate the solutions of the second differential equation (31) corresponding to the three domains which the constant σ could respect.

a)

In this case, the solution of equation (31) can be written in the form:

(34)

where

(35)

Which leads to the function f1k.

b)

The solution obtained in this case is the follows:

(36)

And the function is given as:

(37)

c)

For the latter case, we determined the solution in the form:

(38)

and the function in this case can be written as:

(39)
3.2. Quantum Correlation Coefficient

From equations (19a) and (22) we can calculate the quantum correlations coefficients for the different domains of σ:

a)

In this case the quantum correlation coefficient obtained is:

(40)

It is a non-time-dependent coefficient.

b)

The quantum correlation coefficient obtained is:

(41)

Notice that this coefficient is expressed in the form of a linear combination of real exponential time functions.

c)

For the last case, we obtain:

(42)

We notice that this coefficient is expressed in the form of a linear combination of complex exponential time functions.

3.3. Heisenberg Uncertainty Relation

In this part we will use the expression given in equation (23) to calculate the Heisenberg uncertainty product following the three domains of σ.

a)

In this case the Heisenberg product is given as:

(43)

b)

We obtained the Heisenberg product as follows:

(44)

where n is a real constant as:

(45)

c)

In this case we obtain:

(46)

4. Discussion

The different values of the uncertainty product obtained from the computation of the quantum correlation coefficient for Kanai-Caldirola harmonic oscillator, show that the Heisenberg uncertainty principle is well satisfied for any value of time t, and that this product depends mainly on the value of σ. The uncertainty relations obtained are closely related to the concept of coherent states, in the sense that the product ∆q(t)∆p(t) turns out to be not minimum. This result has been interpreted by Pedrosa using the concept of squeezed states 15, 17, 18. Moreover, the comparison of our results with those obtained by Achkar et al. 16 allows the usefulness and the efficiency of this method. The approach used by Achkar et al. is based on the Floquet theorem, which transforms the time-dependent Schrödinger equation to the time-independent one, the set of results is shown in Table 1.

The rewriting of our expressions of uncertainty products is intended to make easier the comparison of the two results. Thus, these expressions have the same form as that obtained by Achkar et al. for each domain of σ.

5. Conclusion

In this paper, we presented a new theoretical approach for a direct resolution of this problem of the quantum harmonic oscillator with time-dependent mass and frequency using a simple analytical approach based on the choice of a suitable variable change giving a symmetrical shape to the equation (Eq.28). Then we have introduced - as a first approximation - a constant, σ, combining the variation of M(t) and (t) (Eq.29), which allows us to have a differential equation with constant coefficients. Thus, the resolution becames systematic without any explicit choice of M(t) or (t). The expression of β(t) has been determined for three domains of values of σ, we have shown that the solution β(t) is in the form of linear combinations of cos and sin functions, which tend towards the usual oscillating solution of the harmonic oscillator independent of time. Forms of solutions have been determined for various choices of explicit expressions of M(t) or (t) already used by other authors 1, 2, 4, 5, 11, 12, 14, 16, and proved to be in agreement with the results of these authors deduced from different methods.

The results obtained show that the Heisenberg uncertainty principle is satisfied for any value of time. The comparison of our results with those obtained by other authors using other methods proves the efficiency of this method as a calculation tool.

Nomenclature

H(t) Hamiltonien

P momentum

M(t) mass function

(t) frequency

Q position

ν parameter

β(t) solution of the motion equation

(t) time dependentparameter

γ(t) time dependent coupling function

I(t) linear invariant

time dependentparameter

time dependentparameter

B(t) time dependentparameter

time dependentparameter

time dependentparamete

eigenfunction

state of the system

g(t) gausiun function

(t) time dependentparameter

k arbitrary parameter

(t) time dependentparameter

ρ probability density

wave function

λ eigenvalue

Acknowledgments

One of us, C. Qotni, would like to thank H. Laribou for helpful discussion.

References

[1]  D. Bohm. Prentice Hall Quantum theory, 1951 New York. ISBN : 0-486-65969-0.
In article      
 
[2]  E. Fradkin. Field Theories of condensed matter physics. Second edition 2013 ISBN 978-0-521-76444-5.
In article      View Article
 
[3]  M. Suhail Zubairy. Marlan O Quantum Optics. Scully, American Journal of Physics 67, 648 New York 1997.
In article      View Article
 
[4]  M. Gell-Mann and F. E. Low Quantum Electrodynamics at Small Distances. Phys. Rev. 95, 1300 – Published 1 September 1954.
In article      View Article
 
[5]  R. Kosloff Time-Dependent Quantum-Mechanical Methods for Molecular Dynamics.J. Phys. Chem. 1988, 92, 2087-2100.
In article      View Article
 
[6]  F. Cooper, S. Pi, and P. Stancioff. Quantum dynamics in a time-dependent variational approximation. Phys. Rev. D 34, 3831-Published 15 December 1986.
In article      View Article  PubMed
 
[7]  D Kosloff. A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics. Journal of Computational Physics. Volume 52, Issue 1, October 1983, Pages 35-53.
In article      View Article
 
[8]  M. S. Abdalla Canonical treatment of harmonic oscillator with variable mass, , Phys. Rev. A 33, 2870 – Published 1 May 1986.
In article      View Article  PubMed
 
[9]  I. A. Pedrosa, and B. Baseia Phys. Harmonic oscillator with time-dependent mass and frequency and a perturbative potential. C. M. A. Dantas, Rev. A 45, 1320 – Published 1 February 1992.
In article      View Article  PubMed
 
[10]  C M Cheng and P C W Fung The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator. 1988 J. Phys. A: Math. Gen. 21, 4115.
In article      View Article
 
[11]  H. R. Lewis, Jr. and W. B.Riesenfeld,An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field. J. Math. Phys. 10, 1458 (1969).
In article      View Article
 
[12]  H. R. Lewis Jr. Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators. Journal of Mathematical Physics 9, 1976 (1968).
In article      View Article
 
[13]  Piero Caldirola. Forze non conservative nella meccanica quantistica. Il Nuovo Cimento, 18(9): 393-400, 1941.
In article      View Article
 
[14]  E. Kanai. On the Quantization of the Dissipative Systems. Progress of Theoretical Physics, 3(4): 440-442, 1950.
In article      View Article
 
[15]  I. A. Pedrosa Comment on “Coherent states for the time-dependent harmonic oscillator”., Phys. Rev. D 36, 1279-Published 15 August 1987..
In article      View Article  PubMed
 
[16]  Y. Achkar, S. Sayouri S., A.L. Marrakchi. A Simple Approach For The Problem Of The Harmonic Oscillator With Time Dependent Mass And Frequency.., Physical & Chemical News. ISSN 1114-3800. 2008, No. 39, pp. 19-24.
In article      
 
[17]  M. Cariglia, C. Duval, G. W. Gibbons, P. A. Horváthy Eisenhart lifts and symmetries of time-dependent systems Annals of Physics, Volume 373, October 2016, Pages 631-654
In article      View Article
 
[18]  Chia-Chun Chou. Dissipative quantum trajectories in complex space: Dampedharmonic oscillator Annals of Physics, Volume 373, October 2016, Pages 325-345
In article      View Article
 
[19]  Lin Zhang, Weiping Zhang. Lietransformation method on quantum state evolution of a general time dependent driven and damped parametric oscillator Annals of Physics, Volume 373, October 2016, Pages 424-45.
In article      View Article
 
[20]  Roberto Garra, Giorgio S. Taverna, Delfim F. M. Torres. Fractional Herglotz variational principles with generalized Caputo derivatives Chaos, Solitons & Fractals, Volume 102, September 2017, Pages 94-98
In article      View Article
 
[21]  Meng-Yun Lai, Xiao-Yin Pan, Yu-Qi Li. ]Wave function for dissipative harmonically confined electrons in a time dependent electric field Physica A: Statistical Mechanics and its Applications, Volume 453, 1 July 2016, Pages 305-315
In article      View Article
 
[22]  C. Qotni, A. L. Marrakchi, S. Sayouri, Y. Achkar, Analytical study of the Forced Harmonic oscillator with time dependent parameters by Means of the Invariant dynamical linear operator, Internatinnal Review of Physics (I.RE.PHY.), vol.7, N.2 ISSN 1971-68X, April 20.
In article      
 
[23]  F. A. Brito, F. F. Santos, J. R. L. Santos, Harmonic oscillators from displacement operators and thermodynamics,Physica A: Statistical Mechanics and its Applications, Volume 516, 15 February 2019, Pages 78-89.
In article      View Article
 
[24]  H. de la Cruz, J. C. Jimenez, R. J. Biscay,On the oscillatory behavior of coupled stochastic harmonic oscillators driven by random forcesStatistics & Probability Letters, Volume 146, March 2019, Pages 85-89.
In article      View Article
 
[25]  T. Iwai and B. Zhilinskii, The 2D Kramers-Dirac oscillator Physics Letters A, In press, corrected proof, Available online, 1 February 2019.
In article      View Article
 
[26]  Jaume Giné and Claudia Valls On the dynamics of the Rayleigh–Duffing oscillator Nonlinear Analysis: Real World Applications, Volume 45, February 2019, Pages 309-319.
In article      View Article
 
[27]  Shinichi Deguchi, Yuki Fujiwara and Kunihiko Nakano, Two quantization approaches to the Bateman oscillator model, Annals of Physics, In press, accepted manuscript, Available online, 8 February 2019.
In article      View Article
 
[28]  P. Facchi, S. Pascazio, F. V Pepe and K. Yuasa, ”Long-lived entanglement of two multilevel atoms in awaveguide,” J. Phys. Commun 8, 3 (2018)..
In article      View Article
 
[29]  T. M. El-Shahat, M. Kh. Ismail, and A. F. Al Naim, ”Damping in the Interaction of a Field and Two Three-Level Atoms Through Quantized Caldirola–Kanai Hamiltonian,” J. Russ. Laser Res 39, 231(2018).
In article      View Article
 
[30]  Fardin Kheirandish, Exact density matrix of an oscillator-bath system: Alternative derivation Physics Letters A, Volume 382, Issue 46, 23 November 2018, Pages 3339-3346.
In article      View Article
 
[31]  M. Ghorbani, M. Javad Faghihi and H. Safari, ”Wigner function and entanglement dynamics of a twoatom two-mode nonlinear Jaynes–Cummings model,” J. Opt. Soc. Am. B 34, 1884 (2017)..
In article      View Article
 
[32]  R. Daneshm and and M. K. Tavassoly, ”Description of Atom-Field Interaction via Quantized Caldirola-Kanai Hamiltonian,” Int. J. Theor. Phys 56, 10773 (2017)..
In article      View Article
 
[33]  R. Daneshmand and M. K. Tavassoly, ”Damping effect in the interaction of a -type three-level atom with a single-mode field: Caldirola–Kanai approach,” Laser Phys 26, 065204 (2016).
In article      View Article
 

Published with license by Science and Education Publishing, Copyright © 2019 C. Qotni, A. L. Marrakchi, S. Sayouri and Y. Achkar

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Normal Style
C. Qotni, A. L. Marrakchi, S. Sayouri, Y. Achkar. Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method. Applied Mathematics and Physics. Vol. 7, No. 1, 2019, pp 8-13. http://pubs.sciepub.com/amp/7/1/2
MLA Style
Qotni, C., et al. "Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method." Applied Mathematics and Physics 7.1 (2019): 8-13.
APA Style
Qotni, C. , Marrakchi, A. L. , Sayouri, S. , & Achkar, Y. (2019). Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method. Applied Mathematics and Physics, 7(1), 8-13.
Chicago Style
Qotni, C., A. L. Marrakchi, S. Sayouri, and Y. Achkar. "Analytical Computation of the Caldirola-Kanai Oscillator Parameters by the Dynamic Invariant Method." Applied Mathematics and Physics 7, no. 1 (2019): 8-13.
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[1]  D. Bohm. Prentice Hall Quantum theory, 1951 New York. ISBN : 0-486-65969-0.
In article      
 
[2]  E. Fradkin. Field Theories of condensed matter physics. Second edition 2013 ISBN 978-0-521-76444-5.
In article      View Article
 
[3]  M. Suhail Zubairy. Marlan O Quantum Optics. Scully, American Journal of Physics 67, 648 New York 1997.
In article      View Article
 
[4]  M. Gell-Mann and F. E. Low Quantum Electrodynamics at Small Distances. Phys. Rev. 95, 1300 – Published 1 September 1954.
In article      View Article
 
[5]  R. Kosloff Time-Dependent Quantum-Mechanical Methods for Molecular Dynamics.J. Phys. Chem. 1988, 92, 2087-2100.
In article      View Article
 
[6]  F. Cooper, S. Pi, and P. Stancioff. Quantum dynamics in a time-dependent variational approximation. Phys. Rev. D 34, 3831-Published 15 December 1986.
In article      View Article  PubMed
 
[7]  D Kosloff. A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics. Journal of Computational Physics. Volume 52, Issue 1, October 1983, Pages 35-53.
In article      View Article
 
[8]  M. S. Abdalla Canonical treatment of harmonic oscillator with variable mass, , Phys. Rev. A 33, 2870 – Published 1 May 1986.
In article      View Article  PubMed
 
[9]  I. A. Pedrosa, and B. Baseia Phys. Harmonic oscillator with time-dependent mass and frequency and a perturbative potential. C. M. A. Dantas, Rev. A 45, 1320 – Published 1 February 1992.
In article      View Article  PubMed
 
[10]  C M Cheng and P C W Fung The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator. 1988 J. Phys. A: Math. Gen. 21, 4115.
In article      View Article
 
[11]  H. R. Lewis, Jr. and W. B.Riesenfeld,An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field. J. Math. Phys. 10, 1458 (1969).
In article      View Article
 
[12]  H. R. Lewis Jr. Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators. Journal of Mathematical Physics 9, 1976 (1968).
In article      View Article
 
[13]  Piero Caldirola. Forze non conservative nella meccanica quantistica. Il Nuovo Cimento, 18(9): 393-400, 1941.
In article      View Article
 
[14]  E. Kanai. On the Quantization of the Dissipative Systems. Progress of Theoretical Physics, 3(4): 440-442, 1950.
In article      View Article
 
[15]  I. A. Pedrosa Comment on “Coherent states for the time-dependent harmonic oscillator”., Phys. Rev. D 36, 1279-Published 15 August 1987..
In article      View Article  PubMed
 
[16]  Y. Achkar, S. Sayouri S., A.L. Marrakchi. A Simple Approach For The Problem Of The Harmonic Oscillator With Time Dependent Mass And Frequency.., Physical & Chemical News. ISSN 1114-3800. 2008, No. 39, pp. 19-24.
In article      
 
[17]  M. Cariglia, C. Duval, G. W. Gibbons, P. A. Horváthy Eisenhart lifts and symmetries of time-dependent systems Annals of Physics, Volume 373, October 2016, Pages 631-654
In article      View Article
 
[18]  Chia-Chun Chou. Dissipative quantum trajectories in complex space: Dampedharmonic oscillator Annals of Physics, Volume 373, October 2016, Pages 325-345
In article      View Article
 
[19]  Lin Zhang, Weiping Zhang. Lietransformation method on quantum state evolution of a general time dependent driven and damped parametric oscillator Annals of Physics, Volume 373, October 2016, Pages 424-45.
In article      View Article
 
[20]  Roberto Garra, Giorgio S. Taverna, Delfim F. M. Torres. Fractional Herglotz variational principles with generalized Caputo derivatives Chaos, Solitons & Fractals, Volume 102, September 2017, Pages 94-98
In article      View Article
 
[21]  Meng-Yun Lai, Xiao-Yin Pan, Yu-Qi Li. ]Wave function for dissipative harmonically confined electrons in a time dependent electric field Physica A: Statistical Mechanics and its Applications, Volume 453, 1 July 2016, Pages 305-315
In article      View Article
 
[22]  C. Qotni, A. L. Marrakchi, S. Sayouri, Y. Achkar, Analytical study of the Forced Harmonic oscillator with time dependent parameters by Means of the Invariant dynamical linear operator, Internatinnal Review of Physics (I.RE.PHY.), vol.7, N.2 ISSN 1971-68X, April 20.
In article      
 
[23]  F. A. Brito, F. F. Santos, J. R. L. Santos, Harmonic oscillators from displacement operators and thermodynamics,Physica A: Statistical Mechanics and its Applications, Volume 516, 15 February 2019, Pages 78-89.
In article      View Article
 
[24]  H. de la Cruz, J. C. Jimenez, R. J. Biscay,On the oscillatory behavior of coupled stochastic harmonic oscillators driven by random forcesStatistics & Probability Letters, Volume 146, March 2019, Pages 85-89.
In article      View Article
 
[25]  T. Iwai and B. Zhilinskii, The 2D Kramers-Dirac oscillator Physics Letters A, In press, corrected proof, Available online, 1 February 2019.
In article      View Article
 
[26]  Jaume Giné and Claudia Valls On the dynamics of the Rayleigh–Duffing oscillator Nonlinear Analysis: Real World Applications, Volume 45, February 2019, Pages 309-319.
In article      View Article
 
[27]  Shinichi Deguchi, Yuki Fujiwara and Kunihiko Nakano, Two quantization approaches to the Bateman oscillator model, Annals of Physics, In press, accepted manuscript, Available online, 8 February 2019.
In article      View Article
 
[28]  P. Facchi, S. Pascazio, F. V Pepe and K. Yuasa, ”Long-lived entanglement of two multilevel atoms in awaveguide,” J. Phys. Commun 8, 3 (2018)..
In article      View Article
 
[29]  T. M. El-Shahat, M. Kh. Ismail, and A. F. Al Naim, ”Damping in the Interaction of a Field and Two Three-Level Atoms Through Quantized Caldirola–Kanai Hamiltonian,” J. Russ. Laser Res 39, 231(2018).
In article      View Article
 
[30]  Fardin Kheirandish, Exact density matrix of an oscillator-bath system: Alternative derivation Physics Letters A, Volume 382, Issue 46, 23 November 2018, Pages 3339-3346.
In article      View Article
 
[31]  M. Ghorbani, M. Javad Faghihi and H. Safari, ”Wigner function and entanglement dynamics of a twoatom two-mode nonlinear Jaynes–Cummings model,” J. Opt. Soc. Am. B 34, 1884 (2017)..
In article      View Article
 
[32]  R. Daneshm and and M. K. Tavassoly, ”Description of Atom-Field Interaction via Quantized Caldirola-Kanai Hamiltonian,” Int. J. Theor. Phys 56, 10773 (2017)..
In article      View Article
 
[33]  R. Daneshmand and M. K. Tavassoly, ”Damping effect in the interaction of a -type three-level atom with a single-mode field: Caldirola–Kanai approach,” Laser Phys 26, 065204 (2016).
In article      View Article