Rotation operator approach for the dynamics of non-dissipative multi-state Landau–Zener problems: Exact solutions

Physica E 85 (2017) 74–81

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Physica E journal homepage: www.elsevier.com/locate/physe

Rotation operator approach for the dynamics of non-dissipative multi-state Landau–Zener problems: Exact solutions M.E. Ateuafack a, J.T. Diffo a,b,n, L.C. Fai a, M.N. Jipdi a a b

Mesoscopic and Multilayer Structures Laboratory, Department of Physics, Faculty of Science, University of Dschang, Cameroon Department of Physics, Higher Teachers' Training College, The University of Maroua, PO BOX 55 Maroua, Cameroon

H I G H L I G H T S








Exact solutions for the dynamics of the noisy LZ model are derived. The solutions are obtained regardless the initial configuration of the system. Stokes constants are expressed through the parabolic cylinder function. Euler angles are derived if the 3D-noise correlation function is given. Use states of an atom to store information reduce the decoherence effect.

art ic l e i nf o

a b s t r a c t

Article history: Received 1 May 2016 Received in revised form 3 August 2016 Accepted 11 August 2016 Available online 13 August 2016

The paper investigates exact time-dependent analytical solutions of the Landau–Zener (LZ) transitions for spin one-half subjected to classical noise field using rotation operator approach introduced by Zhou and co-authors. The particular case of the LZ model subjected to colored noise field is studied and extended to arbitrary spin magnitude. Transition probabilities are derived regardless of the initial configuration of the system and are found to be functions of the sort for Stokes constant. It is observed that the latter may be completely evaluated provided we have knowledge of the phase difference between noise in x− and y−directions. Transition probabilities are found to depend not only on the LZ parameter and noise frequency, but also on the states involved in the study. In particular, the coherence of the system is sustained for an exceedingly long time when many levels are considered in an atom and if in addition, the LZ parameter tends to unity and the noise' frequency is low. & 2016 Elsevier B.V. All rights reserved.

Keywords: Spin Colored noise Landau–Zener transitions Multistates system Stokes constants Coherence

1. Introduction The ability to manipulate and control spins in noisy environment has become a fundamentally challenging topic with a potentially high impact for future realization of quantum devices. This fascinating field is at the origin of new topics developed recently in some physics domains such as spintronics and molecular electronics [1–3]. These areas of modern physics aim to understand and explain the mechanisms of spin transition dynamics in static and dynamic coupling systems and transportation in quantum optics and relevant multi-level systems. For instance, Landau Zener (LZ) transitions [4–7], spin-flip in nanomagnets [8], Bose– Einstein condensates in optical lattices [9] and adiabatic n Corresponding author at: Department of Physics, Higher Teachers' Training College, The University of Maroua, PO BOX 55 Maroua, Cameroon. E-mail addresses: [email protected], [email protected] (J.T. Diffo).

http://dx.doi.org/10.1016/j.physe.2016.08.013 1386-9477/& 2016 Elsevier B.V. All rights reserved.

computing [10–20] where the environment is considered as a source of classical noise [21–23], quantum information technology [24,25] recently received great attention. In quantum information 1 processing, a spin − 2 particle is a suitable two-level system. It is the basic unit of binary quantum information [26–30] commonly known as the quantum bit or more simply the qubit. Numerous realizations of two- and multi-level LZ transitions were observed in recent quantum transport experiments [31–36]. Accordingly, large amount of literature have been devoted to the dynamic of a two-level system driven around an avoided crossing, particularly in connection with LZ physics. Asymptotical [4– 7,21,22,16,37–39] as well as time-dependent [23] LZ transition probabilities and related physical quantities were derived in different contexts using several methods. However, most results concerning the time-dependent LZ problem are obtained numerically. Few analytical expressions for the time-dependent standard LZ model [22] subjected to low-frequency colored noise

M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

limit are now available [23]. It is instructive to note that all these results were obtained via some initial conditions imposed to the quantum system. Experimentally, this is not an easy task because accurate control of spin is still a challenge for both theoreticians and engineers. Here, we found a way out of solving the problem without preparing the system from some initial conditions. This paper is another contribution to the analysis and understanding of the time-dependent non dissipative LZ problem that is generalized for the multi-level problem. With the help of the rotation operator approach detailed in Ref. [40] we derive for the first time the exact time-dependent transition probabilities for the two and multi-level LZ model in random noise field. This is independent of the noise regime and the initial configuration of the quantum system. Our discussion is reported in terms of relaxation and dephasing. The motivation behind the mathematical developments implemented in this paper resides also on the derivation of the Stokes constant and phase [41] which help to tackle inelastic collision and give consistent results for non-adiabatic probabilities. The present paper is organized as follows: in Section 2, considering the LZ model subjected to arbitrary transverse noise field, we derive and solve the master equation governing the dynamics of Eulers' angles. The resolution of this equation in the special case of colored noise is considered in Section 3. We also investigate the 1 transition dynamics of spin − 2 system. Section 4 presents the case where an arbitrary spin is subjected to a classical transverse noise with Gaussian realizations. Section 5 is the conclusion.

2. Model Hamiltonian and general equation The model describes a quantum system consisting of two states where level crossings are induced by a classical noise having both diagonal and off-diagonal fluctuating matrix elements. The total Hamiltonian of the system reads:

^ H = 2ξtσz +

∑ gℓ(t )σℓ,

ℓ = (x, y).

ℓ

(1)

Here σℓ are Pauli matrices, ξ > 0 is the constant sweep velocity and gℓ (t ) the noise fields. The relevance of this model comes from the fact that its Hamiltonian (1) contains only terms linear in the spin operator components and thus, obeys a SU(2) algebra. This implies that the system may be treated as some rotations in a three-dimensional space; hence, the propagator of the system can be written in a form described by the Wigner rotation operator in which all the dynamic information is included [42–45]:

⎛ γ (t ) ⎞ ⎛ β(t ) ⎞ ⎛ α(t ) ⎞ ^ ^ σ ⎟exp⎜ i σ ⎟exp⎜ i σ⎟ U (t , t0) ≡ U (α, β, γ ) = exp⎜ i ⎝ 2 z⎠ ⎝ 2 y⎠ ⎝ 2 x⎠

(2)

where α(t ), β(t ), γ(t ) are the three time-dependent Euler angles. Applying the rotation operator approach, Zhou and coauthors [40,45] cast this problem into three differential equations:

⎧ γ (̇ t ) + α̇(t )cosβ(t ) + g (t ) = 0 z ⎪ ⎪ ̇ ⎨ β (t ) + gx(t )sinγ (t ) + gy(t )cosγ (t ) = 0 ⎪ ⎪ ⎩ α̇(t )sinβ(t ) − gx(t )cosγ (t ) + gy(t )sinγ (t ) = 0.

ḟ +

75

i i g (t )f 2 + igz (t )f − g−(t ) = 0 2 + 2

(4)

where the dot stands for g±(t ) = gx(t ) ± igy(t ) , gz (t ) = 2ξt and

time

⎛ β(t ) ⎞ ⎟exp(iγ ( t )). f (t ) = − tan⎜ ⎝ 2 ⎠

The most important feature of this approach is that it leads to a straightforward calculation of accumulated phases, probabilities of spin transitions and coherence evolutions of the system. Following suitable recombinations and transformations, the Euler kinematic equations (3) can be recast into the Riccati equation [46]:

and

(5)

We introduce the transformations

⎧ f (t ) = f˜ (t )exp⎨ −i ⎩

∫t

t 0

⎫ gz (t′)dt′⎬ ⎭

(6)

and

⎧ Ω±(t ) = g±(t )exp⎨ ±i ⎩

∫t

t 0

⎫ gz (t′)dt′⎬ ⎭

(7)

and write the differential equation (4) in the form 2 i i f˜ ̇ + Ω+(t )f˜ − Ω−(t ) = 0. 2 2

(8)

Letting

f˜ (t ) =

2i dZ , Z ( t )Ω+( t ) dt

(9)

then

dZ 1 =− 4 dt

∫t

t 0

g+(t )g−(t′)exp

{ − iξ( t

2

− t′2

)}Z(t′)dt′

(10)

which can be interpreted as the general equation governing the time variation of Euler's angles for a two-level system subjected to a classical random field noise. For a well-defined noise correlation function, Eq. (10) is computed and the three Euler angles are obtained provided the complex function f (t ) derived from the aforementioned relationships.

3. Dynamics of the Euler angles for a two-level system subjected to a classical noise with Gaussian realizations In this section, we suppose the system described by the Hamiltonian (1) subjected to a classical noise that obeys the Markovian Gaussian process specified by the time correlation functions

gℓ (t )gℓ (t′) = J2 δℓℓ exp( − Υ |t − t′|). ′ ′

(11)

Here ⋯ denotes the statistical average, J is the averaged amplitude of the diagonal and off-diagonal coupling and Υ is the decay constant. An alternative way to solve the master equation derived above consists on averaging Eq. (10) over all possible realizations of the considered system. Hence we show that

⎧ i⎛ Υ 2 ⎞⎫ ⎟⎬ Z (t ) = A− exp⎨ − ⎜ ξt 2 − 2iΥt − 4 4ξ ⎠⎭ ⎝ ⎩ ⎪

⎪

⎪

⎪

⎛ ⎧ 5π ⎫ Υ ⎞ ⎟⎟exp⎨ i ⎬ × D−iλ⎜⎜ t ξ − i ξ⎠ ⎝ ⎩ 4 ⎭ (3)

derivative

⎪

⎪

⎪

⎪

(12)

and

⎧ i⎛ Υ 2 ⎞⎫ ⎟⎬ Z ̇(t ) = iλA− ξ exp⎨ − ⎜ ξt 2 − 2iΥt − 4ξ ⎠⎭ ⎩ 4⎝ ⎪

⎪

⎪

⎪

⎛ ⎧ 5π ⎫ Υ ⎞ ⎟⎟exp⎨ i ⎬ . × D−iλ − 1⎜⎜ t ξ − i ξ⎠ ⎝ ⎩ 4 ⎭ ⎪

⎪

⎪

⎪

(13)

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M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

Fig. 1. Time evolution of the Euler angle β in a two-level LZ system in the presence of Gaussian colored noise. From left to right, the columns represent the diabatic, intermediate and adiabatic regimes respectively. From top to bottom, the rows respectively indicate the slow, the intermediate and the fast noise regimes.

Fig. 2. Time evolution of the Euler angle γ in a two-level LZ system in the presence of Gaussian colored noise. From left to right, the columns represent the diabatic, intermediate and adiabatic regimes respectively. From top to bottom, the rows respectively indicate the slow, the intermediate and the fast noise regimes.

Here λ = J 2 /2ξ is the renormalized noise intensity and Dn( z ), the parabolic cylinder function [47]. The parameter λ is introduced hereafter to distinguish between the sudden ( λ⪡1), the intermediate ( λ → 1) and the adiabatic ( λ⪢1) transition limits. In Eq. (13), A− is the normalization factor which will not explicitly be evaluated. Hence the method developed in this paper does not require the knowledge of the initial conditions of the system.

Substituting relations (12) and (13) in (9) yields

⎛⎛ ⎧ 5π ⎫ ⎞ Υ ⎞ −2λ ξ D−iλ − 1⎜⎜ ⎜ t ξ − i ⎟exp⎨ i ⎬⎟⎟ ξ ⎝ ⎠ ⎩ 4 ⎭⎠ ⎝ . f (t ) = ⎛⎛ ⎞ ⎧ 5π ⎫ ⎞ Υ g+(t )D−iλ⎜⎜ ⎜ t ξ − i ⎟exp⎨ i ⎬⎟⎟ ξ⎠ ⎩ 4 ⎭⎠ ⎝⎝

(14)

M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

Considering Eq. (5) then Euler's angles: 2⎞ ⎛ ⎜ 1 − f ( t) ⎟ β(t ) = arccos⎜ , ⎜ 1 + f t 2 ⎟⎟ () ⎠ ⎝

(

)

γ (t ) = π + Arg f ( t )

(15)

(16)

and

α 2(t ) =

∫t

t

dt1 0

∫t

t1 0

dt′ [〈g (t1)gx(t′)〉cosγ (t1)cosγ (t′) sinβ(t1)sinβ(t′) x

+ 〈gy(t1)gy(t′)〉sinγ (t1)sinγ (t′)].

(17)

In the following figures, we perform the time-scaled transformation τ = t ξ and introduce Υ˜ = Υ / ξ , a dimensionless parameter that relates the environment relaxation and oscillatory times of the two-level system. The latter is introduced hereafter to distinguish between the low-( Υ˜ ⪡1), intermediate-( Υ˜ → 1) and high( Υ˜ ⪢1) frequency noise limits. Figs. 1 and 2 show the time-dependence of the Euler angles β and γ. It is observed that both noise frequency and LZ parameter significantly contribute to the variation of these Euler angles. Some oscillations are also observed for sudden (intermediate) and for low- (intermediate-) frequency noise regimes suggesting that adiabatic and high-frequency noise limits contribute to the destruction of these oscillations after the crossing point. It follows that the coherence of the system is sustained for an exceedingly long time and consequently the system exhibits long term memory effect within these regimes. Thus the coherence of the system can entirely be described by the Euler angles. Interestingly, the β-angle evolves around its minimum value β = 0 and rises at the crossing point to a maximum value that

77

depends on λ and Υ˜ values (Fig. 1). However, before the crossing, the β-angle is slightly different from zero in proportion as λ increases. The γ-angle is completely defined by virtue of the argument of the complex function f ( t ) which in turn is evaluated provided the phase difference between noise in the x− and y− directions is given. In Fig. 2, we assume that this dephase is equal to π corre4

sponding to the case where there is no noise' correlation between the x− and y−directions. Thus, the γ-angle cannot be accurately evaluated (and thus the phase accumulation) for a two level quantum system subjected to Gaussian colored noise if the only information about the noise is given by relation (11). However, the assumption is applicable for short time-correlation. In this perspective, the Euler angle γ can be approximated to a Gaussian function centered at the crossing point τ = 0 considering adiabatic regime.

4. Dynamics of the Landau–Zener transition in multi-state systems In this section, we consider again Eq. (1) in the following form:

^ H = 2ξ t

j

∑

j

m|j, m〉〈j, m| + g+(t )

m =−j

∑ ( j − m + 1)( j + m) m =−j

j

|j, m〉〈j, m − 1| + g−(t )

∑ ( j − m)( j + m + 1) |j, m〉 m =−j

〈j, m + 1|

(18)

where g±(t ) = gx(t ) ± igy(t ) are the noise fields. In the multistate Hamiltonian (18), we suppose the function ψjm ≡ |j, m〉(m = − j, − j + 1, … , j − 1, j ) described in time t0 and

Fig. 3. Time evolution of the adiabatic survival transition probability in a two-level LZ system in the presence of Gaussian colored noise. From left to right, the columns represent the sudden, intermediate and adiabatic regimes respectively. From top to bottom, the rows respectively indicate the slow, the intermediate and the fast noise regimes.

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M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

Fig. 4. Time evolution of the adiabatic survival transition probability in a three-level LZ system in the presence of Gaussian colored noise. Transition probability for a system with spin j¼ 1 to end up in state −1 From left to right, the columns represent the sudden, intermediate and adiabatic regimes respectively. From top to bottom, the rows respectively indicate the slow, the intermediate and the fast noise regimes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 5. Time evolution of the adiabatic survival transition probability in a four-level LZ system in the presence of Gaussian colored noise. Transition probability for a system 3 3 with spin j = to end up in state − . From left to right, the columns represent the sudden, intermediate and adiabatic regimes respectively. From top to bottom, the rows 2 2 respectively indicate the slow, the intermediate and the fast noise regimes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

79

angular momentum j. Under the influence of the time-dependent noise field, the system evolves non-adiabatically in time t to a certain configuration described by the state ψjm ≡ |j, m′〉. The two ′ ^ states are connected via the relation |j, m′〉 = U (t0, t )|j, m〉. The ^ matrix elements of the evolution operator U (t0, t ) read:

environmental parameters. Thus in LZ scenario, the time dependent diabatic and adiabatic transition probabilities are given by

^ ^ (j ) Um m(t0, t ) = j, m′ U ( t0, t ) j, m ′

and

(19)

⎛⎛ ⎧ 5π ⎫ ⎞ Υ ⎞ Pd(t ) = D−iλ⎜⎜ ⎜ t ξ − i ⎟exp⎨ i ⎬⎟⎟ ξ⎠ ⎩ 4 ⎭⎠ ⎝⎝

2

×Θ

where m ( m′) is a projection of j on a given axis of the basis j, m ( j, m′ ). It follows that the system has N = ( 2j + 1) accessible or

⎛⎛ ⎧ 5π ⎫ ⎞ Υ ⎞ Pa(t ) = λ D−iλ − 1⎜⎜ ⎜ t ξ − i ⎟exp⎨ i ⎬⎟⎟ ξ⎠ ⎩ 4 ⎭⎠ ⎝⎝

^ (j ) non accessible states and Um m(t0, t ) is a ( 2j + 1) × ( 2j + 1) matrix ′ with respect to m′ and m. Following [42], these matrix elements are evaluated with the help of the following relation:

respectively. Here, we let

^ ( j) ^ ( j) j Umm (t0, t ) ≡ Umm (α, β, γ ) = eim ′ γ ( t )dm( ) m(β(t ))eimα( t ) ′ ′ ′

(20)

⎛⎛ ⎧ 5π ⎫ ⎞ Υ ⎞ ⎟exp⎨ i ⎬⎟⎟ Θ−1 = D−iλ⎜⎜ ⎜ t ξ − i ξ ⎝ ⎠ ⎩ 4 ⎭⎠ ⎝

2

×Θ (29)

2

2

⎛⎛ ⎧ 5π ⎫ ⎞ Υ ⎞ ⎟exp⎨ i ⎬⎟⎟ . + λ D−iλ − 1⎜⎜ ⎜ t ξ − i ξ ⎝ ⎠ ⎩ 4 ⎭⎠ ⎝

where 1

m +m ⎡ ( j + m′)!( j − m′)! ⎤ 2 ⎛ β(t ) ⎟⎞ ′ j ⎥ ⎜ cos dm( ) m(β(t )) = ⎢ ⎠ ′ ⎢⎣ ( j + m)!( j − m)! ⎥⎦ ⎝ 2 m −m ⎛ β(t ) ⎟⎞ ′ m − m, m + m) ′ (cosβ(t )) h(j − m′ × ⎜ sin ⎝ ′ 2 ⎠

(28)

It is instructive to note that for a relative long time interval, and considering the expansion

(21)

is the general expression of the pure probability density for a multi-level system subjected to a classical random noise and, 2

( − 1) −r −t r, t hn( )(cosβ(t )) = n ( 1 − cosβ(t )) ( 1 + cosβ(t )) 2 n! ⎛ ⎞n⎡ d r+n t + n⎤ ×⎜ ⎟ ( 1 − cosβ(t )) ( 1 + cosβ(t )) ⎦ ⎝ dcosβ(t ) ⎠ ⎣ (22) are the Jacobi polynomials. If the spin system is initially in the eigenstate ( j, m′ ) the probability for the transition from eigen⎡ ( j) ⎤2 state ( j, m′ ) to ( j, m ) is ⎢⎣ dm (β (t ))⎥⎦ . ′m

⎤ ⎛ z 2 ⎞⎡ n ( n − 1) n( n − 1)( n − 2)( n − 3) Dn(z ) = z nexp⎜⎜ − ⎟⎟⎢ 1 − + − ⋯⎥ 2 4 ⎢ ⎥⎦ 2z 2·4·z ⎝ 4 ⎠⎣ ⎛ ( n + 1)( n + 2) + ⋯⎤⎥, z 2 ⎞⎡ 2π − n − 1 z exp⎜⎜ iπn + ⎟⎟⎢ 1 + − ⎢ ⎥⎦ 4 ⎠⎣ Γ ( − n) 2. z 2 ⎝ we achieve in the limit of zero-frequency noise ( Υ˜ = 0) the traditional Landau–Zener formulas [4–7]:

Pd( ∞) = exp{ −2πλ}

(30)

and

Pa( ∞) = 1 − exp{ −2πλ}.

(31)

4.1. Transition probabilities for a two-level system subjected to colored noise 1

4.2. Transition probabilities for N-level ( N > 2) system subjected to colored noise

Consider j = 2 , then

⎡ β(t ) β(t ) ⎤ −sin ⎥ ⎢ cos 1/2) 2 2 ⎥ ( dm m = ⎢ . ′ t t β ( ) β ( ) ⎥ ⎢ sin cos ⎢⎣ 2 2 ⎥⎦

(23)

It follows that

P − 1 →− 1 = P 1 → 1 = cos2 2

2

2

2

β(t ) 2

(24)

β(t ) 2

(25)

and

P − 1 → 1 = P 1 →− 1 = sin2 2

2

2

2

where

β(t ) 1 = 2 1 + f (t ) 2

(26)

β(t ) f (t ) 2 = . 2 1 + f (t ) 2

(27)

cos2 and

sin2

Considering Eqs. (23) and (26) and in accordance with [41], f (t ) and γ(t ) are respectively the time-dependent Stokes' constant and phase. The novelty here is that these constants are functions of

Interaction between a quantum system and its environment results in loss of quantum coherence. The bottleneck for implementing quantum computers is the maintenance of a macroscopically coherent state of the quantum system [48–52]. An alternative way to overcome this problem is to reduce the environment degrees of freedom interacting with the quantum system. This can be achieved by using more than two levels in each ion in the linear ion trap scheme, thus reducing the number of ions needed to be stored in the trap and accordingly the number of atoms involved in the process [53]. This is an advantage because of the greater information capacity of multilevel atomic systems and thus, the main motivation for making the transition from binary to multi-valued quantum logic. In this section, we present the pure 1

density probability matrix for some values of j ( j > 2 ): For j¼1,

⎡ a2 − 2 ab ⎢ 1) ( dm m = ⎢ − 2 ab a2 − b2 ′ ⎢ ⎣ b2 2 ab 3

For j = 2 ,

b2 ⎤ ⎥ 2 ab⎥. ⎥ a2 ⎦

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M.E. Ateuafack et al. / Physica E 85 (2017) 74–81

⎡ a3 3 a2b ⎢ 2 2 ⎢ − 3 a b a a − 2b2 3/2 dm( m) = ⎢ ′ ⎢ 3 ab2 b b2 − 2a2 ⎢ ⎢⎣ −b3 3 ab2

( (

3 ab2

) b( 2a − b ) ) a( a − 2 b ) 2

2

− 3 a2b

2

2

b3 ⎤ ⎥ 3 ab2 ⎥ ⎥ 3 a2b ⎥ ⎥ a3 ⎥⎦

where

⎛ β(t ) ⎞ ⎛ β(t ) ⎞ ⎟ and b = sin⎜ ⎟. a = cos⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ The time dependent survival transition probabilities for two-, three- and four-levels are illustrated in Figs. 3–5 for different values of the LZ parameter and noise frequency. It is observed that for low- (intermediate-) frequency noise and sudden (intermediate) regimes, before the avoided crossing, all the probabilities are smooth monotonic functions of time testifying that the environment is pumping energy to the system. This stage is the so called dephasing. At the vicinity of the crossing point, the transition probability rises (due to dephasing) from zero and achieves a maximum value. From here two situations can occur:


the transition probability asymptotically achieves a saturation




point with fast (Fig. 3a–d) or slow (see the blue curve in Fig. 4a– d and green curve in Fig. 5a–d) decaying oscillations. Here there is complete transfer of population. the transition probability decreases and asymptotically tends to zero (see the red curve in Fig. 4a–d and red and blue curves in Fig. 5a–d). Here there is no complete transfer of population and after a relatively long time, only the initial state is occupied.

The oscillations observed reveal the fact that energy is being transferred from the system to environment (relaxation) via the coupling. Hence the competition between dephasing and relaxation begins within the rising or after the maximum value of the probability is achieved. Both LZ parameter and noise frequency are found to play a crucial role in the coherence of the system. This coherence is sustained for an exceedingly long time interval when more than two levels are considered in the system and if in addition, we impose an intermediate regime ( λ → 1) and low-frequency noise. To complete our picture, we note that independent of the number of quantum levels involved in the physical system, a complete destruction of the interference is observed in the adiabatic and high-frequency noise regimes. Here all the probabilities are smooth monotonic (see Fig. 3) or non-monotonic (see Figs. 4 and 5) functions of time. The nonmonotonic character however depends on the initial state of the system and with gradual changing conditions allow the system to adapt to its configuration. On the other hand, high-frequency noise induced transitions permits the quenching of the relative phase of the states by the noise factor before and after the crossing. It is also observed that the transition occurring before, on or after the crossing depends on the state in which the system is initially prepared and on the LZ parameter as well as the noise frequency.

5. Conclusion We have investigated the dynamics of the two-level Landau– Zener transitions induced by time-linearly varying energy levels of the system and (or) classical Gaussian noise with both diagonal and off-diagonal fluctuating matrix elements applying the rotation operator approach developed in [40]. The same technique is extended to the cases of transitions in multi-states system. Rather than solving the Von Neumann equation for the density matrix

[23], this method leads to solving the general equation (10). Eq. (10) is solved in the Gaussian colored noise environment and the Euler angles are derived. It is important to note that relevant solutions of Eq. (10) are not related to the initial configuration of the system. The results obtained are general since they take into account the entire noise regime (slow-, intermediateand fast-frequency noise) and LZ parameter. In this wise, the Bloch sphere is demonstrated to be a representation of a qubit with the Eulerian angles being essential parameters tailoring information. The approach developed in this paper gives clear identification of Stokes phases during each population transfer and is suitable for the study of coherence evolution of rigid bodies. Eulerian angles are found to depend on the Stokes' constants which are expressed through the parabolic cylinder function. This may be useful in quantum interferometers and coherent preparation of states. We note that α− and γ−angles (thus the phase accumulation mα + m′γ ) are completely determined provided the dephasing between the x− and y−axis is given. It is also observed that the coherence of the system is more pronounced when many states are considered in the system. This feature is more perceptible in intermediate regime and gradually disappears for low- to high-frequency noise limits (see the middle column in Figs. 3–5). This is an indication that in quantum information processing, rather than consider many atoms that may interact and give rise to noise, it will be gainful to use the states of the atom to store information.

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