Quantum dynamics of a three-spin-qubit system

The most fundamental element of any quantum information (QI) processing system is the quantum bit, (qubit)—a two-state quantum system. Qubits might be two-level atoms (atom-qubit), two-level quantum dots, or spin 1/2 magnetic particles (with a non-vanishing magnetic moment) in a constant magnetic field of spin-qubits. The exchange of the quantum state between two qubits is the physical mechanism of the QI transfer or transmission from one qubit to another [1–18]. In particular, Lloyd [5] and Bose [6] have proposed to use a chain of interacting spins for transmitting QI between two spin-qubits located at two ends of this chain. The QI transmission along a spin chain has been investigated by many authors [7–15] (for a comprehensive review, see [16]). In addition to the magnetic interaction between two nearest-neighbour spin-qubits, the interaction of a spin-qubit chain with the environment causes its decoherence. The quantum dynamics of the systems of two interacting spin-qubits with decoherence has also been studied by many authors (see reviews [17, 18] and references therein), but the decoherence of the chains consisting of more than two spin-qubits was not considered. It is the purpose of the present work to study the quantum dynamics of a three-spin-qubit chain with decoherence.

In the study of the decoherence of a quantum system weakly interacting with the environment, one usually applies the Markovian approximation and has the von Neumann equation for its reduced density matrix ρ in the form

where H is the Hamiltonian of the quantum system without its interaction with the environment and L is a completely positive linear operator called the Liouvillian operator. The general form of L was established by Gorini, Kossakowskii and Sudarshan (GKS) [19]. In practice, many terms in the GKS formula give negligible contributions and one can use the Lindblad formula [20], a special case of GKS. This will be done in the present work.

In section 2, a symmetric model of three interacting identical spin-qubits is proposed and the Hamiltonian of this system is presented. We assume that there are only two main physical mechanisms of its decoherence due to the interaction of the environment: the relaxation and the dephasing of spin-qubits, and use the corresponding Lindblad formula. In section 3, the system of rate equations is established. The solution of this system of linear differential equations in the first-order approximation with respect to decoherence constants is derived in section 4. Section 5 is the conclusion.

Consider a system of three interacting identical spin-qubits with the XY Heisenberg exchange interaction of two nearest-neighbour ones, and write its Hamiltonian in the form

where

are the 2×2 unit matrix and three Pauli matrices acting on the state vector of nth spin-qubit, n=1, 2, 3. In the eight-dimensional vector space of state vectors of the system introduce the basis consisting of eight unit vectors |i₁ i₂ i₃〉, where i_n denotes the quantum state of the nth spin-qubit, i_n =0 for the state with spin projection on the z-axis and i_n =1 for that with spin projection . Eigenstates | Φ _i 〉 and eigenvalues E_i of H,

are

Suppose that there are only two physical mechanisms of the decoherence of spin-qubits due to their interaction with the environment—three separate baths or one common bath of excitations: the relaxation and the dephasing. In the case of three separate baths, each spin-qubit interacts with only one bath and two different spin-qubits interact with two different baths. The Liouvillian operator L is determined by the Lindblad formula,

with two non-negative small constants α ^_r (relaxation constant) and α ^_dp (dephasing constant). In the case when all three spin-qubits interact with one and the same bath of excitations, we have the following expression determining the Liouvillian operator L,

By using expressions (4.1)–(4.8) of eigenstates and eigenvalues of Hamiltonian H and Lindblad formula (5) or (6), one can derive the system of rate equation for the matrix elements

of the reduced density matrix ρ of the three spin-qubit-system in the Markovian approximation.

In the case of three separate baths of excitation, each spin-qubit interacts with its own bath. We have the following system of rate equations consisting of eight equations for the diagonal matrix elements :

28 equations for non-diagonal matrix elements with i<j;i,j=1,2, ... ,8,

and 28 similar equations for the matrix element with i>j, which can be obtained from equations (9.1)–(9.28) by means of suitable permutations of indices.

In the case of one common bath of excitations instead of equations (8.1)–(8.8) and (9.1)–(9.28), we have the following equations for the diagonal matrix elements:

28 equations for non-diagonal matrix elements with i<j;i,j=1,2, ... ,8:

and 28 similar equation for matrix elements with i>j, which can be obtained from equations (11.1)–(11.28) by means of suitable permutations of indices.

The system of rate equations (8.1)–(8.8), (9.1)–(9.28) and those similar to equations (9.1)–(9.28) or, the system of rate equations (10.1)–(10.8), (11.1)–(11.28) and those similar to equations (11.1)–(11.28) can be considered as the differential equations in the 64-dimensional Liouvill space [17]. By solving these equations in the first-order approximation with respect to the small constants α ^_r and α ^_dp , and then changing to basis states | ψ ₁〉=| 000 〉 , | ψ ₂〉=| 001 〉, | ψ ₃〉=| 010 〉, | ψ ₄〉=| 100 〉, | ψ ₅〉=| 011 〉, | ψ ₆〉=| 101 〉, | ψ ₇〉=| 110 〉, | ψ ₈〉=| 111 〉, with ρ _ij =〈 ψ _i ,ρ ψ _j 〉, i,j=1,2, ... ,8, we obtain explicit expressions of matrix elements ρ _ij (t) in the form

Consider first the case of three spin-qubits interacting with three distinct baths of excitations. T_ij ^kl ( t) have the following explicit expressions:

where T_ij ^kl ( t )^* is the complex conjugate of T_ij ^kl ( t). Now we consider another case: three spin-qubits interacting with one and the same bath of excitations. We have

In the special case of the three-qubit system without decoherence, the above presented expressions (13.1)–(13.73) and (14.1)–(14.83) are significantly simplified. The non-vanishing expressions in this special case are

The quantum dynamics of the symmetrical system of three interacting identical spin-qubits with the XY Heisenberg exchange interaction between two nearest-neighbour spin-qubits was studied in the Markovian approximation with respect to the weak interaction between this system and the environment. Two different cases were considered: three spin-qubits interacting with three distinct baths of excitations, or all interacting with one and the same bath of excitation. A Lindblad formula with two physical mechanisms of decoherence-relaxation and dephasing was used to take into account the influence of the environment. The systems of rate equations were solved, and explicit expressions of matrix elements of the reduced density matrix were derived in the first order approximation with respect to the small values of the decoherence constants.

The author would like to thank Academician Nguyen Van Hieu for suggesting the problem and also Dr Nguyen Bich Ha for valuable discussions.