{'fraction_non_alphanumeric': 0.043753329781566326, 'fraction_numerical': 0.015316995205114545, 'mean_word_length': 4.31011315417256, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dynamics of two qubit central spin under fermionic environment

In this article we derive the exact dynamics of a two qubit (spin 1 / 2) system interacting centrally with separate fermionic baths composed of qubits in thermal state. Further, each spin of a bath is coupled to every other spin of the same bath. The corresponding dynamical map is constructed. It is used to analyse the non-Markovian nature of the two qubit central spin dynamics. We further observe the evolution of quantum correlations like entanglement and discord under the inﬂuence of the environmental interaction. Moreover, we demonstrate the comparison between this exact two qubit dynamics and the locally acting fermionic central spin model. This work is a stepping stone towards the realization of non-Markovian heat engines and other quantum thermal devices.


I. INTRODUCTION
In the interactive world of quantum particles, it is almost impossible to create a physical system devoid of any external noisy influence. Quantum systems conducive for information theoretic tasks, i.e. trapped ions [1], quantum dots [2], NMR qubits [3], Josephson junctions [4] and many more, are subjected to environmental interactions. It is therefore a necessary task to investigate the dynamics of such quantum systems, which are under the influence of environmental interactions. The theory of open quantum systems [5,6], has found numerous applications, in recent times, in quantum information and its interface with various facets of quantum physics [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Over the past few decades, our understanding of such systems has stretched from the limitations of Markovian dynamics to the more intriguing and challenging domain of non-Markovian quantum systems . Even now, it is a challenging task to construct the reduced dynamics of such a system without the Born-Markov and stationary bath approximations [5].
Here we develop an exact reduced dynamics of a two qubit system subjected to fermionic baths, each of which is interacting centrally with the system of interest. Our model Hamiltonian of the two qubit system is inspired from a model of quantum thermal diode [53]. In recent times, motivated by the goal of building quantum computers, a lot of effort has been given to develop quantum versions of various thermodynamic and electronic devices like refrigerators and heat engines [54][55][56][57], thermal diodes [53,58], transistors [59][60][61][62], quantum batteries and so on [63,64]. Drawing the motivation from the long term goal of realizing experimentally feasible models of quantum thermal devices, we develop the aforementioned exact reduced dynamics of a two qubit spin system. We further analyse various thermodynamic and information theoretic properties of the system undergoing the specific open quantum evolution. We also observe the fundamentally non-Markovian behaviour of the dynamical map.
The flow of the paper is as follows: In Sec. II we introduce the model. Its reduced dynamics is developed in Sec. III. The corresponding dynamical map is discussed next, along with its operator sum representation. The dynamical map is then put to use in Sec. V. This includes analyzing the von-Neumann entropy of the reduced two-qubit system, a witness to identify inherent non-Markovianity in the dynamics, the corresponding local and global dynamical maps and finally the quantum correlations, including entanglement and discord, generated between the two spins by their open system dynamics. This is followed by the Conclusion.

II. THE MODEL
We present a model of two coupled qubits where each qubit is centrally coupled to different thermal spin baths. Further, each spin of a bath is coupled to every other spin of the same bath.
where σ i lk or σ j lk (k=x,y,z; l=1,2) are Pauli matrices corresponding to i-th or j-th spin of the l-th bath and σ 0 lk (k=x,y,z; l=1,2) are same for l-th central spin. ω 1 and ω 2 are the two central spin frequencies and δ corresponds to their coupling strength. Also, ω a and ω b are bath frequencies of two spin baths and l (l=1,2) are interaction parameters of l-th systembath. M and N are the respective number of atoms in two baths.
Using total spin angular momentum operator, J lk = MorN i=1 σ i lk (k =x,y,z,+,-; l=1 or 2 corresponding to summation upper limit M or N), we may rewrite bath Hamiltonians as, , and system-bath interaction Hamiltonians as, Following [65], we then use the Holstein-Primakoff transformation to redefine collective angular momentum operators as, for first bath and the following for the second bath Here a and a † are the bosonic annihilation and creation operators for the first spin bath having the property, [a, a † ] = 1 and b and b † represents the same for the second spin bath. After this transformation, the bath Hamiltonians appear as, and the interaction Hamiltonians of respective spin-baths as,

III. REDUCED DYNAMICS OF THE TWO QUBIT CENTRAL SPIN MODEL
We derive the reduced dynamics of the system of two coupled central spins evolved under the Hamiltonian H along with the two baths by tracing over the bath degrees of freedom.
By tracing out bath modes, reduced state of the system(|11 11|) becomes From the solution of Eq. (9), one can derive the values of |A 1 (m, n, t)| 2 , |B 1 (m, n, t)| 2 and |C 1 (m, n, t)| 2 following the steps as mentioned in the Appendix A.
Now the off-diagonal components of the reduced density matrix will take the following forms: The partition function in the above equations is, And from the hermiticity of the reduced density matrix, the other off-diagonal elements can be expressed as, Therefore, the reduced state of the system of two central spins after the global unitary evolution of the pair of joint system-bath state can be denoted as The elements of the density matrix are given in Appendix B.

IV. CONSTRUCTION OF THE DYNAMICAL MAP
Having constructed the density matrix of the reduced twoqubit central spin system, we now find the dynamical map of the system. To this effect, we derive the Kraus operators for the evolution of the reduced system.

A. Operator sum representation
An important facet of general quantum evolution represented by a CPTP (completely positive trace preserving) operation is the Kraus operator sum representation, given as . The Kraus operators, K i s can be constructed from the eigenvalues and eigenvectors of the Choi-Jamiołkowski (CJ) state of the corresponding dynamical map. The CJ state of the dynamic map φ(·) acting on a d−dimensional system is given by (I d ⊗ φ)|ψ ψ|, with |ψ being the maximally entangled state in d 2 dimension. For our particular case, the CJ matrix is given by, The abbreviations used in Eq. (24) are given in appendix C. One can find the eigenspectrum of the matrix in Eq. (24) numerically, and from there the Kraus operators for the evolution of the state can be derived. These Kraus operators should satisfy the relation i K † i K i = I, which is indeed the case here.

V. ANALYSIS OF THE DYNAMICAL MAP
In this section, we use the dynamical map of the reduced state of the system to shed light on the dynamics of the twoqubit central spin model.

A. von Neumann entropy
Given the reduced state of the system in Eq. (23), we can calculate the von Neumann entropy of the system as, In Fig. (2), we can see the variation of the von Neumann entropy with time for different initial states. State |11 corresponds to the condition when both central spins are initially in excited state, and state |10 corresponds to the condition when one central spin is in excited state and the other is in ground state.

B. Trace distance as a witness of non-Markovianity
The trace distance, which gives a measure of distinguishability between two quantum states, is given as D(ρ 1 , ρ 2 ) = 1 2 ||ρ 1 − ρ 2 || 1 , where ||(·)|| 1 = Tr (·) † (·). To this end, we calculate the trace distance between the state evolved with time through the dynamical map given above and the initial state, and is given by, In Fig. (3), we can see the variation of D(ρ S 1 S 2 (t), ρ S 1 S 2 (0)) with time for different initial states. The non monotonic behaviour of the trace distance between time evolved state and the initial state is a witness of non-Markovianity in the system.

C. Difference between local and global dynamical maps
In this work, we have derived the global dynamical map of two central spins (Λ 12 ). Let Λ 1 , Λ 2 be the local dynamical maps derived by solving the local Lindblad equations for each central spin [40], with the added proviso that the bath spins are interacting with each other. Λ 12 is the global map constructed here. We take the same parameter values for both the global and local maps, so that we can observe the difference between them from a common footing. For a particular initial state ρ s (0) we can calculate the trace distance as, In Fig. (4), we can see the difference between local and global dynamical maps by setting the same values for all the parameters. It is evident from the plot that though there is no interaction between the two baths in the dynamics we are evaluating, and the two baths are acting separately with individual qubits, there is a distinct difference in the dynamical behaviour of this map with the local maps acting separately with each of the qubits. Through the interaction between the qubits, bath information is passed from one environment to another and hence, despite being mutually non-interacting, the action of the bath exhibits a global trait [66]. Therefore, it is our understanding that, in those situations where baths are acting locally on a bipartite system, as presented in Eq. (1), applying local Kraus operators for each separate system-bath interaction, does not give us the complete picture.

D. Quantum Correlations
Quantum correlations are a very useful resource in quantum information processing. To this end, we investigate the quantum correlations in the reduced state of the system given in Eq. (23). Concurrence [67] is a measure of quantum entanglement in the system given by, where λ i s are the eigenvalue of the matrix √ ρ S 1 S 2ρ S 1 S 2 √ ρ S 1 S 2 in decreasing order, and ρ S 1 S 2 = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ). Another popular candidate for the measurement of quantum correlations is quantum discord [68]. It includes the quantum correlations due to quantum effects in the system which may not necessarily be due to quantum entanglement. For the reduced state given in Eq. (23), quantum discord can be given as, where S (ρ S 2 ) and S (ρ S 1 S 2 ) are the von Neumann entropy of the reduced subsystem T r S 1 [ρ S 1 S 2 ] and joint von Neumann entropy of the reduced system ρ S 1 S 2 , respectively. S (ρ S 1 |S 2 ) is the quantum conditional entropy given by, where ρ S 1 |Π k is the state of the reduced system when measurement operator Π k is operated on subsystem S 2 , such that ρ S 1 |Π k = 1 p k T r S 2 (Π k ρ S 1 S 2 Π † k ). p k is the probability associated with the measurement operators Π k given by p k = T r(Π k ρ S 1 S 2 Π † k ). The generalized measurement operators Π k in 2 qubits, are given by Π 1 = I S 1 ⊗ |u u| S 2 and Π 2 = I S 1 ⊗ |v v| S 2 , where |u = cos(θ) |1 + e iφ sin(θ) |0 and |v = sin(θ) |1 − e iφ cos(θ) |0 . The parameters θ and φ vary in the range 0 ≤ θ ≤ π/2 and 0 ≤ φ ≤ 2π. We consider the two-qubit maximally entangled state, |ψ(0) S = 1 √ 2 (|00 +|11 ) as the initial state of the central spin system and study the variation of quantum correlations with time, depicted in Fig. (5). The profile of both the quantum discord and the concurrence is similar. There exist time intervals where entanglement is zero, but discord is non-zero. Revivals in the quantum correlations depicting the information backflow nature of non-Markovianity of the system are observed.

VI. CONCLUSION
In this article, we have derived the exact reduced dynamics of two coupled central spins where each spin is centrally coupled to different thermal baths. We develop the corresponding dynamical map by constructing the relevant Kraus operators. This helps to shed light into the reduced dynamics of the coupled central spin model. In particular, we show the evolution of von Neumann entropy, and an evidence of the non-Markovian evolution of the system in the form of trace distance. We also calculate the difference between the local and global dynamical maps of evolution and show that there is a distinct difference in the exact dynamics of the two qubit system interacting separately with two non-interacting environments and that of the phenomenological application of local Kraus operations of the same physical picture. It is our assertion that the later does not give us the complete picture of the exact dynamical behaviour of such interactions, where we overlook the information flow between the baths via the interaction between the qubit systems. Moreover, we have also seen the effect on the quantum correlations, in particular entanglement and quantum discord, between the central spins as the dynamics of the system evolves in time. Revival of the quantum correlations benchmarks the non-Markovian behaviour in the central spin system.
Appendix A: Solution of simultaneous linear differential equations from boundary conditions: Matrix method Suppose, three simultaneous linear differential equations are written as: And we have to find the solution using the given boundary conditions: X(0) = 1, Y(0) = 0 and Z(0) = 0.
The problem can be solved by plugging in the well-known Matrix method in this context. According to this, one can write Eq. (A1) using column and square matrices of dimension 3 in a following way, For simplicity, we denote, M= Any of the three eigenvalues, say λ j ( j=1,2,3) of matrix, M can be found by solving the characteristic Eq, And corresponding to each λ j , eigenvectors can be expressed as, , except for the case when λ j = b, c. Then according to the Matrix method, the general solution of Eq. (A1) in terms of column matrix will take the form, where the co-efficients v j can be computed from three boundary conditions given as, , We see that, we are able to come up with the solution (A4) only when λ j is calculated from the Eq. (A3). For solving the three roots of such cubic equation, we follow the procedure given below. Let us say, a cubic equation is in the form: Then we define the following quantities, where, ∆ is called the discriminant of the cubic Eq. (A7). Next, denoting a complex number, − 1 2 + √ 3 2 i = ϕ, we may write the roots of Eq. (A7) as, Note that, the root x 1 is always real and the roots x 2 and x 3 are complex and conjugate to each other only when ∆ < 0. Otherwise, all the roots are real and they become equal when ∆ = 0.
In our case, expanding the Left Hand Side of Eq. (A3) in a power series of λ j for all j, we identify, a 1 = 1, b 1 = −(a+b+c), c 1 = ab+bc+ac− f d−eg and d 1 = f dc+egb−abc. As described above, λ j 's( j=1,2,3) can be determined and the solution (A4) can be attained accordingly.