Transition of decoherence and entanglement induced by interaction quantum spin baths

We investigate the reduced dynamics of two qubits coupled to an interacting quantum spin bath modeled by a XXZ spin chain. By using the method of time-dependent density matrix renormalization group (t-DMRG), it is shown that when the spin bath is in an ordering state, the reduced dynamics of qubits strongly depends on the range of the spin chain that qubits coupled to. Only for ordering spin baths, both decoherence and entanglement show qualitatively different behaviors when the coupling range changes. In particular, In particular, we find that in the parametic regime where there is no ordering in the bath, the coupling range is irrelevant. On the other hand, for ordering spin baths, when the coupling range increases from local coupling to infinite range coupling, the reduced dynamics of qubits exhibits a transition in which initially entangled states no longer suffers from the entanglement sudden death when the coupling range is large. Spin baths have become a subject of interest in recent years because spin baths can induce intriguing decoherence behavior and understanding their decoherence behavior is useful in elucidating the nature of the decoherence and quantum-classical transition.[1] It is known that for some solid state qubit system the spin bath is the dominant decoherence mechanism in the low temperature regime. In contrast to bosonic baths, in real spin baths, spins are not independent . Furthermore, it is known that non-Markovian dynamics can easily emerge out of a spin bath.[2, 3] Nonetheless, the inclusion of the intra-spin interaction in the bath complicates the problem and only for some limited models with high symmetry, exact reduced dynamics can be identified. In a recent work, to overcome the difficulty associated with interacting spins, we utilize the method of time-dependent density matrix renormalization group (t-DMRG) [4, 5] to investigate the single qubit decoherence and two qubit (dis)-entanglement dynamics induced by spin bahts.[6] It is shown that both the decoherence and the entanglement strongly depend on the phase of the underlying spin bath. It is also observed that novel oscillations arise when qubits are directly coupled to the order parameter of the spin bath. The influence on the decoherence for qubits coupling the order parameter of the bath was considered by a number of groups[7, 8, 9]. It was shown that when coupling to the order parameter, the decoherence of qubits gets improved. In this work we explore the transition between weak coupling and direct coupling to the order parameter by varying the coupling profile between qubits and the spin baths. By changing the coupling range between qubits and the spin bath either in the paramagnetic or antiferromagnetic phase, it is shown that when the coupling range increases from local coupling to infinite range coupling, transitions can happen in the entanglement of the qubits so that entanglement sudden death tends to occur for smaller coupling range. Our 25th International Conference on Low Temperature Physics (LT25) IOP Publishing Journal of Physics: Conference Series 150 (2009) 042131 doi:10.1088/1742-6596/150/4/042131 c © 2009 IOP Publishing Ltd 1 results will be useful in designing spin qubits with less decoherence. We start by considering a system-bath model which is described by the total Hamiltonian H = Hsys + Hbath + Hint, where Hsys, Hbath, and Hint are respectively the Hamiltonian of the system, the bath, and the interaction between qubits and the bath. We shall set Hsys = 0. We assume that the spin bath is a spin chain characterized by the XXZ Heisenberg model Hbath = J ∑ ( S i S x i+1 + S y i S y i+1 + ∆S z i S z i+1 ) , (1) where J > 0. The system is ferromagnetic for ∆ < 1, antiferromagnetic (Ising-type) for ∆ > 1, and critical (XY-type) for −1 < ∆ < 1.[10] We Shall consider two qubits denoted as A or B. The coupling between the qubit A(B) and the bath is assumed to be linear with the form Hint = ∑ i,α 2 α i s α AS α i + ∑ i,α 2 α i s α BS α i , where α = x, y, z and i = 1, . . . , N . Here 2 α i characterizes the coupling of the spin qubit to the ith spin in the spin chain. In particular we consider the case of the Ising coupling (2i = 2 y i = 0) and isotropic Heisenberg coupling (2 x i = 2 y i = 2 z i 6= 0). The former induces pure dephasing while the laster induces both dephasing and energy relaxation. The range of the coupling is crucial for characterizing the interaction of qubits to the spin bath. For uniform coupling 2i is independent of i. In general, we shall assume that 2 α i is non-vanishing and its magnitude is uniform for i ≤ N when ths spin bath is in the ferrmagnetic phase, while for ∆ > 1 when the spin bath is in the Ising antiferromagnetic phase, 2i is staggered, proportional to (−1)i. Therefore, N is the coupling range of a single qubit. Furthermore, in order to compare entanglement for different coupling range, 2i has to be appropriately normalized so that the system has a well-defined limit in large N limit. For this purpose, we shall fix |2i | · N to be 20. In other words, for antiferromagnetic spin bath, Hint = ∑ i,α 2 α i s α AS α i = 20~sA · ∑ i(−1)~ Si/N . Therefore, for large N , the qubit ~sA couples to the Neel order parameter ∑ i(−1)~ Si/N directly so that the whole system has a well-defined large N limit. Similarly, for ferromagnetic spin bath, the qubit ~sA couples to the magnetization ∑ i ~ Si/N so that the whole system is also well defined. In this normalization, to avoid direct coupling between two qubits, N will be taken to be much less than the separation of two qubits. For a given set of parameters, we first employ static DMRG[11] to find the ground state |G〉 of the spin chain. The dimension of the truncated Hilbert space is set to be 100 when performing static DMRG and to lift degenerate ground states, a small magnetic field (staggered for antiferromagnetic case) is employed. We then use t-DRMG[4, 5] to evolve the total state starting from a product state of the form: |Φ(0)〉 = |ψsys(0)〉|G〉, where |ψsys(0)〉 is some particular system state that we are interested in. Note that the small magnetic fields used in static DMRG is turned off when performing the time evolution. However, it is checked that during the time evolution which we explore in the paper, the state of the spin chain does not flip to other denegerate ground state. Therefore, qubits only couple and creation excitations around nearby the non-degenerate ground state of the spin-chain. The detail of this procedure can be found in Ref.[6] We first explore the single qubit decoherence by calculating the Loschmidt echo L(t) ≡ |〈Ψ+(t)|Ψ−(t)〉| with |Ψ±(t)〉 being defined as the projection of |Φ(t)〉 to the qubit state |±〉: |Φ(t)〉 = C+(t)|+〉 ⊗ |Ψ+(t)〉 + C−(t)|−〉 ⊗ |Ψ−(t)〉. The decay of L(t) near t = 0 is found to be approximately Gaussian, L(t) ∼ e−αt , where α is the decay parameter.[6] Therefore, α characterizes the decoherence of a single qubit. We shall focus ourself on the antiferromagnetic phase by setting 2i = −0.3(−1)i/N . Clearly, when the number of site N that the qubit couples to the antiferromagnetic spin bath increases, effectively, the qubit couples to the order parameter of the spin bath. In Fig. 1, we plot the short time decay parameter α as a function of the number of sites the qubit coupled to. It is seen that the decay parameter decreases as the number of coupled site increases, indicating the decoherence gets improved when the qubit couples to the order parameter. The oscillation can be understood by re-examining Hint = −0.3~sA · ∑ i(−1)~ Si/N . 25th International Conference on Low Temperature Physics (LT25) IOP Publishing Journal of Physics: Conference Series 150 (2009) 042131 doi:10.1088/1742-6596/150/4/042131