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arXiv:1810.03937 · Journal of Statistical Mechanics Theory and Experiment

The spin- s homogeneous central spin model: exact spectrum and dynamics

Rafael I. Nepomechie, Xi‐Wen Guan

Published Oct 1, 2018• 13 citations
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Abstract

Abstract We consider the problem of a central spin with arbitrary spin s that interacts pairwise and uniformly with a bath of N spins with s = 1/2. We present two approaches for determining the exact spectrum of this model, one based on properties of , and the other based on integrability. We also analyze the exact time evolution of a spin coherent state, and compute the time evolution of various quantities of physical interest, including the entanglement entropy, spin polarization and Loschmidt echo.

DOI
10.1088/1742-5468/aae2d9

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Abstract We consider the problem of a central spin with arbitrary spin s that interacts pairwise and uniformly with a bath of N spins with s = 1/2. We present two approaches for determining the exact spectrum of this model, one based on properties of , and the other based on integrability. We also analyze the exact time evolution of a spin coherent state, and compute the time evolution of various quantities of physical interest, including the entanglement entropy, spin polarization and Loschmidt echo.
The spin degree of freedom has become subject to one of the broadest and most intensively studied fields of research in physics. The present thesis is part of this field and deals with so-called central spin models. The thesis is devided into two parts. In the first one we investigate very basic properties of central spin models on an abstract level. We introduce a new, or rather enhanced, notion of quantum integrability. From this we derive an explicit and, to a certain extent, analytical relation between integrability and decoherence. Then we specify a model of two coupled Gaudin models with all coupling constants being equal to each other (``homogeneous''). We numerically study the spectral properties of the model and find the spectrum to exhibit systematically degenerate multiplets. We briefly discuss possible applications in solid state quantum information processing. The second part is devoted to concrete physical considerations in the context of solid state quantum information processing. Here we consider two exchange coupled electron spins confined in semiconductor quantum dots and evaluate the hyperfine induced spin dynamics for the case of a weak magnetic field applied to the electron spins by exact methods. In a first step we consider the two exchange coupled electron spins to interact with the same bath of nuclear spins via homogeneous couplings. In particular we derive a scaling law for the decoherence time. In a second step we try to turn vice into virtue by investigating to what extent it is possible to swap and entangle two separate nuclear spin baths. Surprisingly, it turns out that a nuclear swap is possible for weakly coupled nuclear spin baths of realistic size, but not for smaller ones. We indicate that under the same circumstances it should also be possible to fully entangle the baths. We close the thesis with a summary and an outlook on further important questions.
We calculate exactly the time-dependent reduced density matrix for the central spin in the central-spin model with homogeneous Heisenberg couplings. Therefrom, the dynamics and the entanglement entropy of the central spin are obtained. A rich variety of behaviours are found, depending on the initial state of the bath spins. For an initially unpolarized unentangled bath, the polarization of the central spin decays to zero in the thermodynamic limit, while its entanglement entropy becomes maximal. On the other hand, if the unpolarized environment is initially in an eigenstate of the total bath spin, the central spin and the entanglement entropy exhibit persistent monochromatic large-amplitude oscillations. This raises the question of to what extent entanglement of the bath spins prevents decoherence of the central spin.
Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.
We obtain analytically close forms of benchmark quantum dynamics of the collapse and revival (CR), the reduced density matrix, Von Neumann entropy, and fidelity for the XXZ central spin problem. These quantities characterize the quantum decoherence and entanglement of the system with any number of bath spins, and for a short to infinitely long time evolution. For the homogeneous central spin problem, the effective magnetic field BBB, coupling constant AAA, and longitudinal interaction $\mathrm{\ensuremath{\Delta}}$ significantly influence the time scales of the quantum dynamics of the central spin and the bath, providing a tunable resource for quantum metrology. Under the resonance condition $B=\mathrm{\ensuremath{\Delta}}=A$, the location of the mthm\mathrm{th}mth revival peak in time reaches a simple relation ${t}_{r}\ensuremath{\simeq}\frac{\ensuremath{\pi}N}{A}m$ for a large NNN. For $\mathrm{\ensuremath{\Delta}}=0$, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and a small polarization in the initial spin coherent state, our analytical result for the CR recovers the known expression found in the Jaynes-Cummings model, thus building up an exact dynamical connection between the central spin problems and the light-matter interacting systems in quantum nonlinear optics. In addition, the CR dynamics is robust to a moderate inhomogeneity of the coupling amplitudes while disappearing at strong inhomogeneity.