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In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

The theory of open quantum systems has become one of the main interests of physicists during the last few decades due to their strong applications in several domains of physics, especially in solid state physics [

In fact, dissipation in an open system results from microscopic reversible interactions between the observable system and the environment. It implies irreversibility and therefore, a preferred direction in time. However, we are particularly interested in this work on the phenomenon of decoherence, which somewhat “regroups dissipation” and introduces the (partial) destruction of quantum coherence through the interaction of quantum mechanical system with its surrounding [

Indeed, the Lindblad formalism ME approach has been used to solve many concrete theoretical two modes systems and applied to various physical phenomena, for instance, in treatments of damped collective modes, in deep inelastic ions collisions, in nuclear physics [

To achieve this objective, we structure the paper as follows. In Section 2, we present a general model of the system under magnetic field effects in NC phase-space. We review the Lindblad ME approach of the density operator of the considered open system in the same section. The dynamical equation of the covariance matrix associated to the system state is derived in Section 3. We determine in Section 4 the density matrix and the Wigner function of the considered system. Section 5 is devoted to our main results where therein the decoherence time scale and the total energy of the considered system in NC phase-space are derived. Finally Section 6 presents some concluding remarks.

The model described in this work forms the standard pattern of a damped quantum harmonic oscillator in NC phase-space. Similar model was already introduced in the literature [

H = 1 2 m ( p ^ 1 − e c A ^ 1 ) 2 + 1 2 m ( p ^ 2 − e c A ^ 2 ) 2 + m 2 ( ω 1 2 q ^ 1 2 + ω 2 2 q ^ 2 2 ) + k 12 q ^ 1 q ^ 2 , (1)

where ( A ^ 1 , A ^ 2 ) are the components of the potential vector of the magnetic field, k 12 the damping parameter, ( q ^ 1 , q ^ 2 ) and ( p ^ 1 , p ^ 2 ) are respectively the noncommutative coordinates and momenta of the quantum oscillator satisfying the following commutation relations [

( q ^ 1 = q 1 − θ 2 ℏ p 2 , q ^ 2 = q 2 + θ 2 ℏ p 1 p ^ 1 = p 1 + η 2 ℏ q 2 , p ^ 2 = p 2 − η 2 ℏ q 1 (2)

where the coordinates q k and momenta p k obey the usual commutation relations [ q k , q l ] = 0 , [ p k , p l ] = 0 and [ q k , p l ] = i ℏ δ k l , k , l = 1 , 2 . By the gauge symmetric relation, the vector potentials of the magnetic field A ^ k can be defined as:

A ^ = ( − B 2 q ^ 2 , B 2 q ^ 1 ) (3)

Using the latest relation and Equation (2), the Hamiltonian (1) can be rewritten as follows:

H = 1 2 m 1 p 1 2 + 1 2 m 2 p 2 2 + m 2 ( ϖ 1 2 q 1 2 + ϖ 2 2 q 2 2 ) + k 12 ( q 1 q 2 + θ 2 ℏ ( q 1 p 1 − q 2 p 2 ) − θ 2 2 ℏ 2 p 1 p 2 ) − ( λ 1 q 1 p 2 − λ 2 q 2 p 1 ) , (4)

where

( ϖ 1 = ω 1 2 + ω c 2 4 + ω c η 2 m ℏ + η 2 4 m 2 ℏ 2 , ϖ 2 = ω 2 2 + ω c 2 4 + ω c η 2 m ℏ + η 2 4 m 2 ℏ 2 , ( λ 1 = m θ 2 ℏ ( ω 1 2 + ω c 2 4 ) + ω c 2 ( 1 + θ η 4 ℏ 2 ) + η 2 m ℏ , λ 2 = m θ 2 ℏ ( ω 1 2 + ω c 2 4 ) + ω c 2 ( 1 + θ η 4 ℏ 2 ) + η 2 m ℏ , (5)

with m 1 = m ( 1 + m ω c θ 4 ℏ ) 2 + ( m θ ω 2 2 ℏ ) 2 and m 2 = m ( 1 + m ω c θ 4 ℏ ) 2 + ( m θ ω 1 2 ℏ ) 2 . It is important to notice that the effective masses m 1 and m 2 are different due to the anisotropy of the system, and ω c = e B m c is the cyclotron frequency. The last term appearing in the Hamiltonian (4) is due to the NCty effects. This term introduces some difficulties to solve the eigenvalue equation of the corresponding Hamiltonian, thus the whole Hamiltonian requires further transformations. For this reason, let’s define a canonical transformation involving the set of coordinates-momenta ( q 1 , q 2 , p 1 , p 2 ) [

( q 1 = χ cos ( Φ ) Q 1 + 1 ρ sin ( Φ ) P 2 , q 2 = χ cos ( Φ ) Q 2 + 1 ρ sin ( Φ ) P 1 , and ( p 1 = − ρ sin ( Φ ) Q 2 + 1 χ cos ( Φ ) P 1 , p 2 = − ρ sin ( Φ ) Q 1 + 1 χ cos ( Φ ) P 2 . (6)

The new coordinates ( Q 1 , Q 2 ) and momenta ( P 1 , P 2 ) also satisfy the canonical commutation relations:

[ Q 1 , P 1 ] = [ Q 2 , P 2 ] = i ℏ , (7)

while all other permutations vanish. Using the transformations of Equation (6), the Hamiltonian (4) becomes:

H = 1 2 M 1 P 1 2 + 1 2 M 2 P 2 2 + M 1 2 Ω 1 2 Q 1 2 + M 2 2 Ω 2 2 Q 2 2 + c 12 Q 1 Q 2 + μ 11 Q 1 P 2 + μ 22 Q 2 P 2 + d 12 P 1 P 2 , (8)

where

M 1 = 1 2 I 1 2 , M 2 = 1 2 I 2 2 , Ω 1 = 2 I 1 K 1 , Ω 2 = 2 I 2 K 2 , (9)

with

( I 1 2 = 1 2 ρ 2 [ ρ 2 m 1 χ 2 cos 2 ( Φ ) + m ϖ 2 2 sin 2 ( Φ ) + λ 2 ρ χ sin ( 2 Φ ) ] , I 2 2 = 1 2 ρ 2 [ ρ 2 m 2 χ 2 cos 2 ( Φ ) + m ϖ 1 2 sin 2 ( Φ ) + λ 1 ρ χ sin ( 2 Φ ) ] , K 1 2 = χ 2 2 [ ρ 2 m 2 χ 2 sin 2 ( Φ ) + m ϖ 1 2 cos 2 ( Φ ) + λ 1 ρ χ sin ( 2 Φ ) ] , K 2 2 = χ 2 [ ρ 2 m 1 χ 2 sin 2 ( Φ ) + m ϖ 2 2 cos 2 ( Φ ) + λ 2 ρ χ sin ( 2 Φ ) ] , (10)

and

( μ 11 = k 12 ρ 2 χ [ ( χ 2 ρ 2 + θ 2 4 ℏ 2 ) sin ( 2 Φ ) + θ χ ℏ ρ ] , μ 22 = k 12 ρ 2 χ [ ( χ 2 ρ 2 + θ 2 4 ℏ 2 ) sin ( 2 Φ ) − θ χ ℏ ρ ] , ( c 12 = k 12 χ 2 [ cos 2 ( Φ ) − θ 2 ρ 2 4 ℏ 2 χ 2 sin 2 ( Φ ) ] , d 12 = k 12 ρ 2 [ sin 2 ( Φ ) − θ 2 ρ 2 4 ℏ 2 χ 2 cos 2 ( Φ ) ] . (11)

The set ( ρ , χ , Φ ) satisfies the following relations:

( ρ χ = m m 1 m 2 ( λ 2 ϖ 1 2 + λ 1 ϖ 2 2 ) m 2 λ 1 + m 1 λ 2 tan ( 2 Φ ) = 2 m 1 m 2 ( m 1 λ 2 + m 2 λ 1 ) ( λ 2 ϖ 1 2 + λ 1 ϖ 2 2 ) m ( m 2 ϖ 1 2 − m 1 ϖ 2 2 ) (12)

Equation (8) is the Hamiltonian of a harmonic oscillator in two-dimensional NC phase-space, where the mass varies in terms of the NC parameters, which can provide simple solution to the eigenvalues equation.

Generally, the simplest dynamics for an open system which describes an irreversible time evolution is given by the Gorini-Kossakowski-Linblad-Sudarshan (GKLS) quantum master equation [

Parameter | Symbol | Value | Unit |
---|---|---|---|

Charge of electron | e | 1 | C |

Magnetic field | B | ∝ 10 − 2 | T |

Reduced Planck constant | ℏ | 1 | j∙s |

Frequency in the x-direction | ω 1 | 1.5 | s^{−}^{1} |

Frequency in the y-direction | ω 2 | 2.5 | s^{−}^{1} |

Effective mass | m | 1 | kg |

Light constant | c | 1 | m∙s^{−}^{1} |

Squeezing parameter | δ | 0.5 | s^{−}^{1} |

Friction coefficients | α k l , β k l , λ k l | between 0 and 1 | dimensionless |

Boltzmann constant | k | 1 | m^{2}∙kg∙s^{−2}∙K^{−1} |

Temperature | T | between 0 and 20 | K |

NC-parameter along the coordinates | θ | ∝ 10 − 38 | m^{2} |

NC-parameter along the momentum | η | ∝ 10 − 60 | kg^{2}∙m^{2}∙s^{−}^{2} |

L ( ρ ) = d ρ d t = − i ℏ [ H , ρ ] + 1 2 ℏ ∑ j ( [ V j ρ , V j ] + [ V j , ρ V j ] ) , (13)

where H denotes the open system’s Hamiltonian and the operators V k and V k † defined on the Hilbert space of H, describe the interaction of open system with the environment. Given that, we have an interest in the set of Gaussian states, we introduce such quantum dynamical semi group that preserves this set. Consequently, H is considered to be a polynomial of second degree in coordinates Q 1 , Q 2 and momenta P 1 , P 2 of the quantum oscillators. In this case, V j and V j † are taken as polynomials of first degree in these canonical observables. Then, in the linear space spanned by the coordinates and momenta, there exist only four linearly independent operators V j ( j = 1 , 2 , 3 , 4 ) [

( V j = A j 1 P 1 + A j 2 P 2 + B j 1 Q 1 + B j 2 Q 2 V j † = A j 1 * P 1 + A j 2 * P 2 + B j 1 * Q 1 + B j 2 * Q 2 (14)

with

( A j 1 = 1 χ ( a j 1 + b j 2 θ 2 ℏ ) cos ( Φ ) + 1 ρ ( b j 2 + a j 1 η 2 ℏ ) sin ( Φ ) , A j 2 = 1 χ ( a j 1 − b j 1 θ 2 ℏ ) cos ( Φ ) + 1 ρ ( b j 1 − a j 2 η 2 ℏ ) sin ( Φ ) , B j 1 = − ρ ( a j 2 − b j 1 θ 2 ℏ ) sin ( Φ ) + χ ( b j 1 − a j 2 η 2 ℏ ) cos ( Φ ) , B j 2 = − ρ ( a j 1 + b j 2 θ 2 ℏ ) sin ( Φ ) + χ ( b j 2 + a j 1 η 2 ℏ ) cos ( Φ ) , (15)

where a j 1 , a j 2 , b j 1 , b j 2 are complex numbers. The constant term is omitted because its contribution to the generator L is equivalent to the linear terms of the Hamiltonian in P k and Q k . Then, the harmonic oscillator’s Hamiltonian (8) can be rewritten in the form:

H = H 0 + c 12 Q 1 Q 2 + μ 11 Q 1 P 1 + μ 22 Q 2 P 2 + d 12 P 1 P 2 , (16)

with

H 0 = 1 2 M 1 P 1 2 + 1 2 M 2 P 2 2 + M 1 2 Ω 1 2 Q 1 2 + M 2 2 Ω 2 2 Q 2 2 . (17)

Under the above conditions, Equation (13) becomes [

d ρ d t = − i ℏ [ H 0 , ρ ] − i 2 ℏ ( μ 11 + λ 11 ) [ Q 1 , ρ P 1 + P 1 ρ ] − i 2 ℏ ( μ 22 + λ 22 ) [ Q 2 , ρ P 2 + P 2 ρ ] − i 2 ℏ λ 12 ( [ Q 1 , ρ P 2 + P 2 ρ ] − [ P 2 , ρ Q 1 + Q 1 ρ ] ) − i 2 ℏ λ 21 ( [ Q 2 , ρ P 1 + P 1 ρ ] − [ P 1 , ρ Q 2 + Q 2 ρ ] ) − i 2 ℏ ( d 12 + α 12 ) [ P 2 , ρ P 1 + P 1 ρ ] − i 2 ℏ ( c 12 + β 12 ) [ Q 2 , ρ Q 1 + Q 1 ρ ]

+ i 2 ℏ ( λ 11 − μ 11 ) [ P 1 , ρ Q 1 + Q 1 ρ ] + i 2 ℏ ( λ 22 − μ 22 ) [ P 2 , ρ Q 2 + Q 2 ρ ] + i 2 ℏ ( α 12 − d 12 ) [ P 1 , ρ P 2 + P 2 ρ ] + i 2 ℏ ( β 12 − c 12 ) [ Q 1 , ρ Q 2 + Q 2 ρ ] − 1 ℏ 2 ∑ k , l = 1 2 [ D P k P l [ Q k , [ Q l , ρ ] ] + D Q k Q l [ P k , [ P l , ρ ] ] − D Q k P l ( [ Q k , [ P l , ρ ] ] + [ P k , [ Q l , ρ ] ] ) ] (18)

where

D Q l Q l = ℏ 2 ∑ j | A j l | 2 , D P l P l = ℏ 2 ∑ j | B j l | 2 , D Q 1 Q 2 = D Q 2 Q 1 = ℏ 2 R e [ ∑ j A j 1 * A j 2 ] ,

D P 1 P 2 = D P 2 P 1 = ℏ 2 R e [ ∑ j B j 1 * B j 2 ] , D P l Q l = D Q l P l = − ℏ 2 R e [ ∑ j A j l * B j l ] ,

D P 1 Q 2 = D Q 2 P 1 = − ℏ 2 R e [ ∑ j A j 2 * B j 1 ] , D P 2 Q 1 = D Q 1 P 2 = − ℏ 2 R e [ ∑ j A j 1 * B j 2 ] ,

α 12 = − α 21 = − I m [ ∑ j A j 1 * A j 2 ] , β 12 = − β 21 = − I m [ ∑ j B j 1 * B j 2 ] ,

λ 11 = − I m [ ∑ j A j 1 * B j 1 ] , λ 22 = − I m [ ∑ j A j 2 * B j 2 ] ,

λ 12 = − I m [ ∑ j A j 1 * B j 2 ] , λ 21 = − I m [ ∑ j A j 2 * B j 1 ] .

The quantities D P k P l , D Q k Q l , D P k Q l and D Q k P l ( k = l = 1 , 2 ) are known as the diffusion coefficients, λ k l , α 12 , α 21 , β 12 and β 21 the frictions constants. From the Cauchy-Schwarz inequality, the above coefficients satisfy the following fundamental constraints [

• The coefficients D P l P l , D Q l Q l , must be positive ∀ l .

• D P l P l D Q l Q l − D Q l P l 2 should be greater than the ratio λ l l 2 ℏ 2 4 ∀ l .

• D Q 1 Q 1 D Q 2 Q 2 − D Q 1 Q 2 2 must be greater than α 12 2 ℏ 2 4 , and

• D P 1 P 1 D P 2 P 2 − D P 1 P 2 2 must be greater than β 12 2 ℏ 2 4 .

In the particular case, we suppose that the asymptotic state of the considered open system is a Gibbs state ( ρ G ( ∞ ) = e − H 0 k T T r [ e − H 0 k T ] ) , corresponding to a quantum harmonic oscillator in thermal equilibrium at temperature T in two-dimensional phase-space. Under this assumption, the above quantum diffusion coefficients take the forms [

( D P l P l = ℏ M l Ω l λ l l + μ l l 2 coth ( ℏ Ω l 2 k T ) , D Q l Q l = λ l l − μ l l 2 ℏ M l Ω l coth ( ℏ Ω l 2 k T ) , (19)

other combinations being zero. In this case, the previous fundamental constraints are satisfied for example only if: λ l l > μ l l and ( λ l l − μ l l ) coth ( ℏ Ω l 2 k T ) ≥ λ l l 2 , ∀ l = 1 , 2 . Let σ A A = 〈 A 2 〉 − 〈 A 〉 2 the dispersion (variance) of a given operator A, where 〈 A 〉 = T r [ ρ A ] is the expectation value of the operator A, ρ the statistical operator (density matrix), and σ A B = 〈 A B + B A 〉 2 − 〈 A 〉 〈 B 〉 the correlation (covariance) of the operators A and B. It was demonstrated in [

σ A A σ B B − σ A B 2 ≥ 1 4 | 〈 [ A , B ] 〉 | 2 . (20)

From the master Equation (18), we obtain the following equation of motion for the expectation values of coordinates and momenta [

d n ( t ) d t = X n ( t ) , (21)

where n ( t ) = [ σ Q 1 , σ Q 2 , σ P 1 , σ P 2 ] T denotes the expectation vector and X a 4×4 matrix defined by:

X = ( − λ 11 + μ 11 − λ 21 1 M 1 − α 12 + d 12 − λ 12 − λ 22 + μ 22 α 12 + d 12 1 M 2 − M 1 Ω 1 2 β 12 − c 12 − λ 11 − μ 11 − λ 12 − β 12 − c 12 − M 2 Ω 2 2 − λ 21 − λ 22 − μ 22 ) (22)

Solving Equation (21), we obtain

n ( t ) = N ( t ) n ( 0 ) = exp ( X t ) n ( 0 ) , (23)

where n ( 0 ) denotes the initial conditions. The matrix n ( t ) must fulfill the condition

lim t → ∞ n ( t ) = 0. (24)

However, for this limit to exist, X must have only negative real eigenvalues parts. For this reason, we set k 12 = 0 , such that μ l l = d 12 = c 12 = 0 , which lead the system to uncoupled oscillators. Under these conditions, the matrix X becomes:

X = ( − λ 11 − λ 21 1 M 1 − α 12 − λ 12 − λ 22 α 12 1 M 2 − M 1 Ω 1 2 β 12 − λ 11 − λ 12 − β 12 − M 2 Ω 2 2 − λ 21 − λ 22 ) (25)

In order to evaluate the exponential matrix N ( t ) , it is important to first diagonalize the matrix X by solving the corresponding secular equation i.e.

det ( X − z I ) = 0 , (26)

where z is the eigenvalues of X and I the unit matrix. From Equation (26), one obtains an equation of fourth order with respect to the eigenvalues z, which can be easily solved. Let us for simplicity consider the particular case where β 12 = α 12 = λ 12 = λ 21 = 0 , then the secular equation obtained is:

[ ( λ 11 + z ) 2 + Ω 1 2 ] [ ( λ 22 + z ) 2 + Ω 2 2 ] = 0. (27)

The resolution of this equation provides z 1 = − λ + + i Ω + , z 2 = − λ − + i Ω − , z 3 = − λ + − i Ω + and z 4 = − λ − − i Ω − , where λ + = λ 11 , λ − = λ 22 , Ω + = Ω 1 , and Ω − = Ω 2 . To fulfill the condition of Equation (24), only the positive values of λ + and λ − are considered. Thus, using the eigenvalues z n of X, the time-dependent matrix N ( t ) can be obtained as follows:

N i j ( t ) = ∑ n M i n exp ( z n t ) M n j − 1 , (28)

where the matrix M constitutes the eigenvectors of X:

∑ j X i j M j n = z n M i n . (29)

Under the conditions N i j ( t = 0 ) = δ i j and d N i j ( t ) / d t | t = 0 = X i j . The expressions of the matrix elements N i j , can subsequently be obtained as:

N ( t ) = ( e − λ 11 t cos Ω 1 t 0 1 M 1 Ω 1 e − λ 11 t sin Ω 1 t 0 0 e − λ 22 t cos Ω 2 t 0 1 M 2 Ω 2 e − λ 22 t sin Ω 2 t − M 1 Ω 1 e − λ 11 t sin Ω 1 t 0 e − λ 11 t cos Ω 1 t 0 0 − M 2 Ω 2 e − λ 22 t sin Ω 2 t 0 e − λ 22 t cos Ω 2 t ) . (30)

Considering Equation (23) and the initial conditions defined above, one get:

{ σ Q k ( t ) = e − λ k k t [ σ Q k ( 0 ) cos ( Ω k t ) + 1 M k Ω k σ P k ( 0 ) sin ( Ω k t ) ] , σ P k ( t ) = e − λ k k t [ − M k Ω k σ Q k ( 0 ) sin ( Ω k t ) + σ P k ( 0 ) cos ( Ω k t ) ] , k = 1 , 2. (31)

Thus, we can observe that, the expectation values of the coordinates and momenta decay quickly due to the exponential factors e − λ 11 t and e − λ 22 t and vanish when t → ∞ . Let us now determine the variance and covariance matrices. For this reason, let σ ( t ) and D be respectively the covariance and the diffusion coefficients matrices where the elements are defined as follows:

D = ( D Q 1 Q 1 D Q 1 Q 2 D Q 1 P 1 D Q 1 P 2 D Q 2 Q 1 D Q 2 Q 2 D Q 2 P 1 D Q 2 P 2 D P 1 Q 1 D P 1 Q 2 D P 1 P 1 D P 1 P 2 D P 2 Q 1 D P 2 Q 2 D P 2 P 1 D P 2 P 2 ) and σ ( t ) = ( σ Q 1 Q 1 σ Q 1 Q 2 σ Q 1 P 1 σ Q 1 P 2 σ Q 2 Q 1 σ Q 2 Q 2 σ Q 2 P 1 σ Q 2 P 2 σ P 1 Q 1 σ P 1 Q 2 σ P 1 P 1 σ P 1 P 2 σ P 2 Q 1 σ P 2 Q 2 σ P 2 P 1 σ P 2 P 2 ) , (32)

Thus, from the master Equation (18), the equations of motion corresponding to the quantum correlations of canonical observables Q 1 , Q 2 and P 1 , P 2 are the following [

d σ d t = X σ + σ X T + 2 D , (33)

where the matrix X is defined in Equation (25), with X T its transposed matrix. The time-dependent solution of (33) is given by:

σ ( t ) = N ( t ) ( σ ( 0 ) − σ ( ∞ ) ) N T ( t ) + σ ( ∞ ) , (34)

where N ( t ) is defined in Equation (30). The matrix σ ( ∞ ) is time independent and solves the static problem Equation (33) (i.e. d σ d t = 0 ). Assuming that the limit of σ ( t ) exists as t goes to infinity (i.e. l i m t → ∞ σ ( t ) exists), then one has σ ( ∞ ) = l i m t → ∞ σ ( t ) . We can therefore, obtain the solutions of Equation (33) giving the elements of σ ( t ) . For illustration, the elements σ Q k Q k , σ P k P k and σ P k Q k ( k = 1 , 2 ) are given by:

{ σ Q k Q k ( t ) = e − 2 λ k k t ( cos 2 ( Ω k t ) ( σ Q k Q k ( 0 ) − σ Q k Q k ( ∞ ) ) + 1 M k Ω k sin ( 2 Ω k t ) ( σ Q k P k ( 0 ) − σ Q k P k ( ∞ ) ) + 1 M k 2 Ω k 2 ( σ P k P k ( 0 ) − σ P k P k ( ∞ ) ) sin 2 ( Ω k t ) ) + σ Q k Q k ( ∞ ) ; σ P k P k ( t ) = e − 2 λ k k t ( M k 2 Ω k 2 sin 2 ( Ω k t ) ( σ Q k Q k ( 0 ) − σ Q k Q k ( ∞ ) ) − M k Ω k sin ( 2 Ω k t ) ( σ q k p k ( 0 ) − σ q k p k ( ∞ ) ) + ( σ P k P k ( 0 ) − σ P k P k ( ∞ ) ) cos 2 ( Ω k t ) ) + σ P k P k ( ∞ ) ; σ P k Q k ( t ) = e − 2 λ k k t ( − M k Ω k 2 sin ( 2 Ω k t ) ( σ Q k Q k ( 0 ) − σ Q k Q k ( ∞ ) ) + cos ( 2 Ω k t ) ( σ P k Q k ( 0 ) − σ P k Q k ( ∞ ) ) + 1 2 M k Ω k ( σ P k P k ( 0 ) − σ P k P k ( ∞ ) ) sin ( 2 Ω k t ) ) + σ P k Q k ( ∞ ) , (35)

where the remaining elements can be easily found following the same approach. It can be observed that the matrix elements σ i j decay rapidly to zero due to the exponential terms exp ( − 2 λ 11 t ) , exp ( − 2 λ 22 t ) and exp ( − ( λ 11 + λ 22 ) t ) . The matrix elements of σ ( ∞ ) depend on X and D, thus should be evaluated using the relation [

σ ( ∞ ) = 2 ∫ 0 ∞ N ( t ′ ) D N ( t ′ ) T d t ′ . (36)

This relation is remarkably important, since it gives a very simple connection between the asymptotic values ( t → ∞ ) of σ ( t ) and the diffusion coefficients D. One observe that the asymptotic values σ Q k Q k ( ∞ ) , σ P k P k ( ∞ ) and σ P k Q l ( ∞ ) do not depend on initial values σ Q k Q k ( 0 ) , σ P k P k ( 0 ) and σ P k Q l ( 0 ) . The initial state represents a correlated coherent state with variance and covariance of coordinates and momenta defined as [

σ Q k Q k ( 0 ) = ℏ δ 2 M k Ω k , σ P k P k ( 0 ) = ℏ M k Ω k 2 δ ( 1 − r k 2 ) and σ P k Q k ( 0 ) = ℏ r k 2 δ 1 − r k 2 . (37)

Here, δ represents the squeezing parameter and r k the correlation coefficient at t = 0 with | r k | < 1 , where r k = σ P k Q k σ Q k Q k σ P k P k . In the case of a thermal bath where μ l l = 0 , with the coefficients (19) one can deduce that

σ Q k Q k ( ∞ ) = ℏ 2 M k Ω k coth ( ℏ Ω k 2 k T ) , σ P k P k ( ∞ ) = M k Ω k ℏ 2 coth ( ℏ Ω k 2 k T ) and σ P k Q l ( ∞ ) = 0. (38)

The semigroup dynamics density operator which must hold for a quantum Markovian process is valid only for the weak-coupling regime, with the damping λ k k obeying the inequality λ k k ≪ Ω k . Moreover, the diffusion coefficients D Q 1 Q 2 , D P 2 Q 1 , D P 1 Q 2 and D P 1 P 2 are generally zero for uncoupled oscillators interacting with an usual environment. Consequently, the expectation values σ Q 1 Q 2 , σ P 2 Q 1 , σ P 1 Q 2 and σ P 1 P 2 vanish when t → ∞ . It is a very interesting point that the general theory of Lindblad allows coupling via the environment between uncoupled oscillators ( d 12 = 0 , c 12 = 0 , μ l l = 0 , ( l = 1 ; 2 )). Furthermore, according to the definition of the parameters in terms of the vectors A j k and B j k , the diffusion coefficients above can be different from zero. In this case, a structure of the environment is reflected in the motion of the oscillators.

In this section, the NC anisotropic oscillator interacting with the usual environment is considered in the same idea presented in the previous section, but with k 12 = 0 , α 12 = 0 , β 12 = 0 , λ 12 = 0 , λ 21 = 0 . Considering these conditions and following the method developed by Lampo et al. [

d ρ d t = ∑ k = 1 2 { [ i ℏ M k ( ∂ 2 ∂ Q k 2 − ∂ 2 ∂ Q ′ k 2 ) − i M k Ω k 2 2 ℏ ( Q k 2 − Q ′ k 2 ) ] ρ − λ k k 2 ( Q k − Q ′ k ) ( ∂ ∂ Q k − ∂ ∂ Q ′ k ) ρ } + ∑ k = 1 2 [ λ k k 2 [ ( Q k + Q ′ k ) ( ∂ ∂ Q k + ∂ ∂ Q ′ k ) + 2 ] ρ + D Q k Q k ( ∂ ∂ Q k + ∂ ∂ Q ′ l ) 2 ρ − D P k P k ℏ 2 ( Q k − Q ′ k ) 2 ρ ]

+ ∑ k ≠ l = 1 2 [ − i D Q k P l ℏ ( Q k − Q ′ k ) ( ∂ ∂ Q l + ∂ ∂ Q ′ l ) ρ + D Q k Q l ( ∂ ∂ Q k + ∂ ∂ Q ′ k ) ( ∂ ∂ Q l + ∂ ∂ Q ′ l ) ρ ] − ∑ k ≠ l = 1 2 D P k P l ℏ 2 ( Q k − Q ′ k ) ( Q l − Q ′ l ) ρ − ∑ k = 1 2 2 i D Q k P k ℏ ( Q k − Q ′ k ) ( ∂ ∂ Q k + ∂ ∂ Q ′ k ) ρ (39)

Following [

∂ W ∂ t = ∑ k = 1 2 { [ − P k M k ∂ W ∂ Q k + M k Ω k 2 ∂ W ∂ P k ] + λ k k ( ∂ ( P k W ) ∂ P k + ∂ ( Q k W ) ∂ Q k ) } + ∑ k = 1 2 [ D Q k Q k ∂ 2 W ∂ 2 Q k + D P k P k ∂ 2 W ∂ 2 P k + 2 D Q k P k ∂ 2 W ∂ Q k ∂ P k ] + ∑ k ≠ l = 1 2 [ 2 D Q k Q l ∂ 2 W ∂ Q k ∂ Q l + 2 D P k P l ∂ 2 W ∂ P k ∂ P l + 2 D Q k P l ∂ 2 W ∂ Q k ∂ P l ] . (40)

In Equation (40), the first four terms on the right-hand side give a purely unitary evolutions, the second four terms are the dissipative terms and have a damping effect (exchange of energy with environment), the last twelve terms are diffusive (noise) terms and produce fluctuation effects in the evaluation of the system. Moreover D P k P k promotes diffusion in momentum and generates decoherence in coordinate Q k . It reduces the off-diagonal terms, responsible for correlations between spatially separated pieces of the wave packet. Analogically, D Q k Q k promotes diffusion in coordinate and generates decoherence in momentum P k . The terms D Q k Q l , D Q k P l , D Q k P k , and D P k P l ( k ≠ l = 1 , 2 ) are the so-called “anomalous diffusion” terms. They promote diffusion in the variables P k P l + P l P k , Q k P l + P l Q k , Q k P k + P k Q k , and Q k Q l + Q l Q k respectively. Just like the other diffusion terms but they do not generate decoherence [

ρ ( Q 1 , Q 2 , Q ′ 1 , Q ′ 2 , t ) = 1 2 π det G exp { − 1 2 { ( ( R − Q ¯ ) T , r T ) [ G − 1 − i ℏ H − i ℏ H T β ℏ 2 ] ( R − Q ¯ r ) } + i ℏ ∑ k = 1 2 σ P k r k } , (41)

and

W ( Q 1 , Q 2 , P 1 , P 2 , t ) = 1 det ( 2 π σ ( t ) ) exp [ − 1 2 ( y − n ( t ) ) σ ( t ) − 1 ( y − n ( t ) ) T ] , (42)

where R = ( Q 1 + Q ′ 1 2 Q 2 + Q ′ 2 2 ) , Q ¯ = ( σ Q 1 σ Q 2 ) , r = ( Q 1 − Q ′ 1 Q 2 − Q ′ 2 ) , H = G − 1 F T and β = E − F G − 1 F T , with E = [ σ P 1 P 1 0 0 σ P 2 P 2 ] , F = [ σ Q 1 P 1 0 0 σ Q 2 P 2 ] , and G = [ σ Q 1 Q 1 0 0 σ Q 2 Q 2 ] , y = ( Q 1 , Q 2 , P 1 , P 2 ) . n ( t ) = ( σ Q 1 , σ Q 2 , σ P 1 , σ P 2 ) denotes the expectation vector, and σ ( t ) is the dispersion (correlation) matrix defined by:

σ ( t ) = ( G F F E ) . (43)

Thus, in the case of a thermal bath and when the asymptotic state is a Gibbs state and considering Equation (38), we obtain the following steady state solution:

〈 Q 1 Q 2 | ρ ( ∞ ) | Q ′ 1 Q ′ 2 〉 = ρ 1 ρ 2 = [ M 1 Ω 1 π ℏ coth ε 1 exp { − M 1 Ω 1 4 ℏ [ ( Q 1 + Q ′ 1 ) 2 coth ε 1 + coth ε 1 ( Q 1 − Q ′ 1 ) 2 ] } ] × [ M 2 Ω 2 π ℏ coth ε 2 exp { − M 2 Ω 2 4 ℏ [ ( Q 2 + Q ′ 2 ) 2 coth ε 2 + coth ε 2 ( Q 2 − Q ′ 2 ) 2 ] } ] (44)

and

W ∞ = W 1 W 2 = [ 1 π ℏ coth ε 1 exp { − 1 ℏ coth ε 1 [ M 1 Ω 1 Q 1 2 + 1 M 1 Ω 1 P 1 2 ] } ] × [ 1 π ℏ coth ε 2 exp { − 1 ℏ coth ε 2 [ M 2 Ω 2 Q 2 2 + 1 M 2 Ω 2 P 2 2 ] } ] , (45)

with ε k = ℏ Ω k k T . We observe that, all stationary solutions to the evolution equations obtained in the long time limit are possible as a result of a balance between the wave packet spreading induced by the Hamiltonian and the localizing effect of the Lindblad operators.

In this section, we mainly focus on the decoherence time scale and the total energy of a two-dimensional harmonic oscillator in an NC phase-space. For the reasons of simplicity, let θ = α × 10 − 38 m 2 and η = β × 10 − 60 kg 2 ⋅ m 2 ⋅ s − 2 , where θ and η are the NC parameters, whose values are selected following the works of refs. [

As previously mentioned, the diffusion coefficients in momentum D P k P k produces decoherence along coordinates Q k . Mathematically, classical limit is often associated with the size of Planck’s constant. In the macroscopic limit, this

Planck constant ℏ is negligible compared to other quantities with dimension of action, such as D P 1 P 1 〈 ( Q 1 − Q ′ 1 ) 2 〉 + D P 2 P 2 〈 ( Q 2 − Q ′ 2 ) 2 〉 . Thus, in this condition, the terms of Equation (39) containing the quantity − D P k P k ℏ 2 dominate on others. To kindly appreciate its effects, let us rewrite an explicit solution of the master Equation (Equation (39)) approximating the right hand side by the dominating terms. In that case, the following time depend evolution of the density matrix is obtained:

〈 Q 1 Q 2 | ρ ( t ) | Q ′ 1 Q ′ 2 〉 = 〈 Q 1 Q 2 | ρ ( 0 ) | Q ′ 1 Q ′ 2 〉 exp { − [ D P 1 P 1 ℏ 2 ( Q 1 − Q ′ 1 ) 2 + D P 2 P 2 ℏ 2 ( Q 2 − Q ′ 2 ) 2 ] t } = 〈 Q 1 Q 2 | ρ ( 0 ) | Q ′ 1 Q ′ 2 〉 exp { − 1 t d e c o t } . (46)

We can thus, easily deduce the decoherence time scale as:

t d e c o = ℏ 2 D P 1 P 1 ( Q 1 − Q ′ 1 ) 2 + D P 2 P 2 ( Q 2 − Q ′ 2 ) 2 , (47)

which reduces to:

t d e c o = 2 ℏ λ 11 M 1 Ω 1 σ Q 1 Q 1 ( 0 ) coth ε 1 + λ 22 M 2 Ω 2 σ Q 2 Q 2 ( 0 ) coth ε 2 , (48)

where the quantities ( Q k − Q ′ k ) 2 are in order of the initial dispersion in coordinates σ Q k Q k ( 0 ) , in the particular case of thermal bath (i.e. at zero temperature). Considering Equation (37), Equation (48) becomes:

t d e c o = 4 λ 11 δ coth ε 1 + λ 22 δ coth ε 2 , (49)

with τ 1 = 2 k T ℏ Ω 1 = 1 ε 1 and τ 2 = 2 k T ℏ Ω 2 = 1 ε 2 . The quantity t d e c o defines the so-called decoherence time scale which as we previously mentioned defines the particular time that the system initially quantum, starts behaving classically (i.e. the particular time that the system starts losing its quantum properties). Based on the above analytical expressions, we have numerically simulated the decoherence time scale as function of the temperature on one hand and as function of the cutting frequency (magnetic field) on the other hand. One can easily observe on

The total energy of the anisotropic oscillator in a NC phase space can be derived from the mean value of the Hamiltonian as:

〈 H 〉 = σ P 1 P 1 2 M 1 ( t ) + σ P 2 P 2 2 M 2 ( t ) + 1 2 M 1 Ω 1 2 σ Q 1 Q 1 ( t ) + 1 2 M 2 Ω 2 2 σ Q 2 Q 2 ( t ) + μ 11 σ P 1 Q 1 ( t ) + μ 22 σ P 2 Q 2 ( t ) + d 12 σ P 1 P 2 ( t ) + c 12 σ Q 1 Q 2 ( t ) + 1 2 M 1 σ P 1 2 ( t ) + 1 2 M 2 σ P 2 2 ( t ) + 1 2 M 1 Ω 1 2 σ Q 1 2 ( t ) + 1 2 M 2 Ω 2 2 σ Q 2 2 ( t ) + μ 11 σ P 1 ( t ) σ Q 1 ( t ) + μ 22 σ P 2 ( t ) σ Q 2 ( t ) + d 12 σ P 1 ( t ) σ P 2 ( t ) + c 12 σ Q 1 ( t ) σ Q 2 ( t ) . (50)

Since the expectation and variances values of the coordinates and momenta decrease exponentially in time, the energy is dissipated to a minimum value. Assuming that the conditions (38) hold, the minimum value of the total energy of the system is given by the following expression:

E min = E ( ∞ ) = ℏ ( Ω 1 2 coth ℏ Ω 1 2 k T + Ω 2 2 coth ℏ Ω 2 2 k T ) . (51)

At zero temperature, i.e. T = 0 , the minimal energy becomes

E ¯ min = ℏ ( Ω 1 2 + Ω 2 2 ) , which corresponds to the energy of the system in its ground state. In the particular case where ω 1 = ω 2 implying Ω 1 = Ω 2 = Ω and θ = η = 0 , the minimum energy of the ground state can be reduced to E ¯ ′ min = ℏ Ω , and we recover the ground state energy of the usual harmonic oscillator in the commutative case.

It can be observed that, for fixed values of the temperature T (the thermal bath temperature), the total energy increases quickly with the magnetic field (cutting frequency), and that under lower magnetic field effects, the total energy for

different NC parameter values coincides. But under intense magnetic field effects, the NCty structure of the phase-space considerably affects the system since the energy increase with these effects. In addition, it is observed that, for extremely low temperature, the total energy of the system increase significantly with the NCty effects, however as the temperature increases, the energy increases too and for high temperature, the energy of the system in NC phase-space coincides with that of the system in commutative space (blue solid curve). This represents the fact that, when the temperature increases in the system, the coupling between the particle and the bath becomes considerably strong, the whole system takes a large dimension and then, the effects of quantum gravity manifest in NC phases-space are not more detectable. Similar results were found by Tchoffo et al. [

In this paper, we studied decoherence of a damped anisotropic harmonic oscillator under magnetic field effects in two-dimensional NC phase-space. For this reason, the evolution of the system was studied within the framework of the Lindblad ME theory for open quantum systems, considering the general case of an environment consisting of a thermal bath at arbitrary temperature. Based on the above mentioned theory, the damping of the expectation values of coordinates and momenta was evaluated as functions of time. Then, from the ME of an NC damped anisotropic oscillator approach, the time evolution of the density matrix and the Wigner function were derived systematically. It turned out by solving these equations that:

• The solutions follow Gaussian distribution. Moreover, from the expectation values and the variances of coordinate and momenta, the total energy of the system was evaluated.

• The total energy increases significantly with the NCty effects, and as the temperature increases, the energy increases too. However, for high temperature, the energy of the system in NC phase-space coincides with that of the system in commutative space. Similar effects were observed with the magnetic field effects.

• In addition, the decoherence time scale was analytically derived, and its simulation proved that, the NCty effects present significant impact on this time scale, and thus, considerably affect decoherence in the system in the sense that, the decoherence time scale increases as the NC parameters increase. It turned out therefore that, the decoherence time scale was improved by NCty effects.

• The decoherence time was found to have a similar scale as the time after which statistic fluctuations become comparable with quantum fluctuations as expected, and the values of the scales become closer with the growth of temperature, magnetic field, and NCty effects.

The authors declare no conflicts of interest regarding the publication of this paper.

Tcoffo, M., Yinde Deuto, G., Nsangou, I., Azangue Koumetio, A., Yonya Tchapda, L.S. and Tene, A.G. (2020) Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space. Journal of Applied Mathematics and Physics, 8, 2801-2823. https://doi.org/10.4236/jamp.2020.812207