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This present issue is an extension of the work of Y. Xiao-Zhong et al. who investigated the influence of constant external magnetic field on the decoherence of a central electron spin of atom coupled to an anti-ferromagnetic environment. We have shown in this work that the character variability of the field induces oscillations amongst the eigen modes of the environment. This observation is made via the derivation of the transition probability density of state, a manner by which critical parameters (parameters where transition occur) of the system could be obtained as it shows resonance peak. We equally observed that the two different magnons modes resulting from the frequency splitting via the application of the time-varying external B-Field, exhibit each a resonant peak of similar amplitude at different temperature ranges. This additional information shows that the probability for the central spin system to remain in its initially prepared diabatic state is enhanced for some temperature ranges for the corresponding two magnon modes. Hence, these temperature ranges where the probability density is maximum could save as decoherence free environment; an important requirement for the implementation of quantum computation and information processing in solid state circuitry. The theoretical and numerical results presented for the decoherence time and the probability density are that of a decohered central electron spin coupled to an anti-ferromagnetic spin bath. The theory is based on a spin wave approximation and on the density matrix using both transformations of Bloch, Primakov and Bogoliobuv in the adiabatic limit.

The interaction between quantum systems induced decoherence is commonly studied in the weak coupling and the strong coupling limits. Quantum decoherence is nowadays considered as the key concept in the description of the transition from the quantum to the classical world [

Ford et al. (henceforth abbreviated as FLO) in its recent publication [

The usual physical picture of decoherence [2,8] is the averaging over the unobserved degrees of freedom (the “environment”) that leads to non-unitary time evolution, with a consequent loss of information. If there is no coupling to the environment, there will be no such lost. We remark that the transition from the quantum to the classical regime due to decoherence is different from the semiclassical limit, where the classical behavior is recovered by exploiting the smallness of Planck’s constant. There are three relevant main differences to this regard: First, decoherence requires an open system, second, decoherence acts at the length-scale of the interference pattern, whereas a typical semi-classical procedure consists in evaluating a macroscopic observable on a fast oscillating probability distribution, third, decoherence is a dynamical effect; it grows with time [

In this paper we investigate the mechanism of decoherence in the simplest model of a quantum mechanically coupled oscillators, as a canonical example of the strong coupling regime characterized by frequency splitting. The conditions for the appearance of decoherence, adiabatic and non-diabatic transitions are investigated. The system of interacting oscillators is the central spin of a couple to an anti-ferromagnetic environment and subjected to a variable external magnetic field. Maintaining sufficient quantum coherence and quantum superposition properties is one of the most important requirements for applications in quantum computing and information processing, quantum teleportation [18,19], quantum cryptography [

This work is structured as follows; the model Hamiltonian of the central spin in the dual action of the antiferromagnetic bath and a parallel magnetic field is presented in Section 2. One of the most important observations in this section is frequency splitting of the antiferromagnetic spin bath due to the presence of external magnetic field, a situation where we can simulate the behavior of the system to that of a two level system a characteristic feature of the Landau-Zener scenario. The influence of the environment on the central spin dynamics is captured in the decohence factor and is evaluated in Section 3. We derive the transition probability of state in Section 4 and finally end with the conclusion.

We study a single central spin atom coupled to an antiferromagnetic spin bath environment subjected to a time dependent magnetic field. Without the external magnetic field, an anti-ferromagnetic crystal has in any elementary chain two network of spin orientation; anti-parallel spin oriented in direction and in direction and an anisotropy field,. In the presence of the external magnetic field the Hamiltonian is given for example by Equation 2.6.2 in ref. [

In an anti-ferromagnetic environment contrary to the ferromagnetic environment, the coupling factor is positive.

and the magnetic moment of each atom

where g is the gyromagnetic factor, is the Bohr magneton, the coupling constant, the exchange interaction. We consider only nearest neighbor interaction. We assumed that the spin structure of the environment may be divided into two interpenetrating sublattices and with the property that all nearest neighbor of an atom of lie on and vice versa. and represents spin operators of and atom on sublattice a and b, each sublattice contains atoms. is the applied external magnetic field in the z-direction. is the anisotropy field, assumed to be positive which approximates the effect of the crystal anisotropic energy with the property of turning for positive magnetic moment, , to align the spins on sublattice in the positive z-direction and spins on sublattice in the negative z-direction. In ref. [

and

From where we have

with the eigen value of spin Hence Equation (2.3) defines the Heisenberg Hamiltonian plus the external magnetic field in the Ising model. It is impossible to solve it exactly but conveniently when full advantage of the translational symmetry is considered. We need creation operators which create Bloch-like non localized excitations, in order to take translational symmetry into account. We consider one atom per unit cell and,’s creates localized spin deviations at a single site. We use the spin wave approximation at low temperatures and we may expect the spin deviation quantum number to be rather small. Let and to reduce Equation (2.5) and (2.6); if we neglect the product of four operators and denote by the number of nearest neighbor, the Hamiltonian of the bath follows thus

where is the free Hamiltonian of the environment and the Hamiltonian describing the excited state of the environment

The Hamiltonian is made (up) of two parts

where the first part is the quantified Hamiltonian of the two magnons without interaction and the interacting Hamiltonian of the two magnons with expressions:

and

To find the total number of magnons, we do the Fourier transformation of the total Hamiltonian with the Bloch operators:

where is the vector of the primitive cell, k the wave vector. Let’s consider the restriction to the first Brillion zone and taking the inverse transformation of Equations (2.13.a) and (2.13.b)

The total number of magnons equals the total spin deviation quantum number of magnons in mode. The spin wave variable is substituted as

where is the occupation number operator for the number of magnons in mode k, is the number of particles in each sublattice and

the Fourier transform coupling constant. The operators, , , are the creation and annihilation operators of sublattices and respectively.

The Hamiltonian Equation (2.9) can then be transformed using the Bogoliubov transformation:

where the coefficients and are real and also the new operators obey the boson commutation rules:

which leads to the constraint,. The Hamiltonians respectively in Equation (2.1)-(2.3) becomes

is the energy of the free harmonic oscillator. The frequency of the two magnons at the symmetric position given by the site and the site of the system is:

In an anti-ferromagnet Cristal, the excitation of one magnon of a wave vector k lead to a change either of + 1/N, or of –1/N for the ensemble of the two anti-parallel spins of the elementary network link. There exist then two modes, and which are degenerates if the contribution of the external magnetic field is neglected. These states, up and down correspond respectively to the eigen frequencies

and

From the given expressions Equations (2.24.a) and (2.24.b), we see that the magnetic state of spin “up” or “down” depends at the same time on the excitation of the modes and and to the population and or of and. Using the kinetic energy expression of the system, the effective mass of magnon is found,

From where we have

Equation (2.26) gives the effective mass of a quasiparticle moving in the crystal with frequencies in mode where a, b, c, are the constants of system. The result of Equation (2.24.a) is not a good approximation to the situation of the spin wave [_{c} (the corresponding external field that corresponds to field with the energy equals to the energy of the environment) is evaluated; that is when the mode and at the finite oscillating time,

, and taking:

Equation (2.27) is the critical magnetic field obtained as for the constant external magnetic field in [

In Figures 1-3 are plotted the frequencies of the two oscillators. In

The dash curve corresponds to the frequency and the solid curve to the frequency.

There is a characteristic anti-crossing with a frequency splitting of

with. As, the splitting increases with the external field. Anti-crossing is a characteristic fingerprint of strong coupling.

The eigen modes of the oscillators with frequencies translate the system to that of a two level system

coupled by a constant magnetic field as analogous to the case in [

where is the transition speed and is given by the dispersion relation, where is the wave vector.

Equation (2.29) is the probability that the anti-ferromagnetic bath mode remains in the initially prepared state. The characteristic Gaussian shape of this transition is shown in

In

amplitude implying the initially prepared state of the environmental frequency mode could be tailored with precision if a certain value of the magnetic field amplitude and phase angle is used. _{c}, we have. This indicates that, this branch of magnon is no longer stable due to the externally applied magnetic field.

As a result, the anti-ferromagnetic polarization ﬂips perpendicular to the field, i.e., the magnetic field induces spin flop transition. The spin-flop transition demonstrates a significant change of the spin configuration in the anti-ferromagnetic environment. This phenomenon has been observed and investigated for many different materials [31-33]. The spin wave theory is known to describe well the low-excitation and low-temperature properties of anti-ferromagnetic materials.

Despite this low-excitation approximation, the spin wave theory also describes well the physics for and the value of the critical magnetic field of the spin-flop transition in anti-ferromagnetic materials [

for). It will be shown that an analytic expression for the decoherence time can be evaluated.

In this section, using the time evolution of the off-diagonal elements of the reduced density matrix for the central spin, we calculate the decoherence time. We assumed factorized initial state of the density matrix of the total systems, i.e.. The initial state of the central spin is described by. The density matrix of the environment is assumed to be in thermal equilibrium, that is, where Z is the partition function. We are interested in the dynamics of the off-diagonal elements of the reduced density matrix as they carry information on the phase coherence of the system. This is equivalent to calculating the time evolution of the spin-flip operator, , where and are respectively the lower and upper eigen states of. By tracing out the environmental degrees of freedom, the time evolution of can be written as:

the decoherence factor can be found using the time evolution

From Equation (2.3) the decoherence factor yields

where

The decoherence factor of the system in the volume of environment is found at low temperature; that is for, where is the temperature,

Then in the thermodynamic limit, i.e., it is obvious that as. To find the relation between and in the thermodynamic limit, we calculate:

The absolute value of the decoherence factor in the thermodynamic limit can be expressed:

where

We obtained these results analytically for the case of a variable magnetic field. It indicates that the decoherence factor displays a Gaussian decay with time (see Equation (3.37)). The factor in the exponent is different from the Markovian approximation which usually shows a linear decay in time in the exponent thus portraying nonMarkovianity a signature of strong coupling between system and environment.

In

decoherence time increases exponentially with the anisotropy field. This behavior shows the fact that the anisotropy gives rise to stronger polarization of the environmental spin and reduces the effect of the external field on the decoherence of the central spin.

As discussed in[

An alternative way to understand the field-dependent decoherence time may be in terms of quantum correlations. There is a kind of trade-off between the external magnetic field and the anisotropy field. The anisotropy field renders the anti-ferromagnetic environment stable. On the other hand, the external magnetic field tends to reduce the anti-ferromagnetic order of the environment. Therefore the stronger the external magnetic field is, the smaller the anti-ferromagnetic order. On the contrary, the larger the anisotropy field is, the stronger the correlation of the anti-ferromagnetic environment. If the constituents (spins) of the environment maintain appreciable correlations or entanglement between themselves, then there is a restriction on the entanglement between the central spin and the environment [34,35]. As a consequence, this sets a restriction on the amount that the central spin may decohered [32,33,36]. Thus as far as the decoherence of the central spin is concerned, the anisotropy field has a similar effect on the exchange interaction strength between the constituents (spins) of the anti-ferromagnetic environment. Strong intra-environmental interaction results in a strong anti-ferromagnetic correlation, thus an effective decoupling of the central spin from the environment and a suppression of decoherence [32,33]. Therefore the decoherence time increases with the increase of the anisotropy field but decreases with the increase of the strength of the external magnetic field.

In the subsequent section we find the transition probability of state in the system, considering that the density matrix of the environment is in thermal equilibrium.

At thermodynamic equilibrium, the density state of the system is expressed as

H is the total Hamiltonian and T the Boltzmann temperature. Let’s evaluate the partition function of the system, at the thermodynamic equilibrium

Here, g is the gyromagnetic factor, E_{0} the energy of the free harmonic oscillator.

If we let, the number of magnon for the different creation operator and respectively

Then we have the mean value of the probability density of state.

where

The solid curve represents the vibration mode with frequency and the dot curve the mode with frequency. Here, is the gap between the two vibrational modes where looking at Equation (2.28), it results that and may be interpreted as the energy necessary for spin transition from low spin energy state to high energy spin state.

Figures 7-9 show plots of the probability density of state as a function of temperature. The plots demonstrate a resonance peak within some temperature range. This provides us with additional information on the range of values of the temperature, anisotropy field, magnetic field and other system parameters for which the central spin system is sensitive to and possibly undergoes transition. Transforming temperature into frequency via the Matsubara relation, we could talk of triple resonance comprising of the driving field frequency, anisotropy field frequency and the environmental eigen modes frequency. The resonance peak in the plot of the probability density of state for the two vibrational modes corresponds to minimum decoherence effect of the environment and the driving field on the central spin. In

(2.29), it results that and may be interpreted as the energy necessary for spin to make transition from its low energy state (with frequency mode) to its high energy state (with frequency mode) and vice versa. In

That is, the stronger the anisotropy field, the stronger the decoherence of the central spin system.

The solid and dash curves are the plot considering the external magnetic field constant whereas the dotted curve is the plot of the probability density of state subjected to a variable magnetic field.

We observed from

We have studied the decoherence of a central spin coupled to an anti-ferromagnetic environment in the presence of a variable external magnetic field. The results, obtained using the spin wave approximation in the thermodynamic limit, show that the decoherence factor displays a Gaussian decay with time. It is shown that the probability density of state occurs at some critical values of magnetic field and temperatures. The probability density as a function of temperature is characterized by a resonant peak corresponding to some critical parameters of the system which here are the critical external magnetic field, anisotropy field and temperatures. The probability of the central spin to remain in the initially prepared state is maximum at these values and spin-flop transition is suppressed. Out of these parameter range spin-flop transition occurs, consistent with the QPT as studied in [37-39]. It is equally seen that strong anisotropy field enhances the probability density and reduces decoherence of the anti-ferromagnetic environment of the central spin. Therefore, in order to reduce the loss of coherence of the central spin, we could decrease the environmental temperature, choose variable magnetic field with high amplitude (which) could lead to dark state of the environment, and choose the anti-ferromagnetic surrounding or underlying anti-ferromagnetic materials with a strong crystal anisotropy field. The frequency eigen mode dependence on the phase angle as shown in (