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Central spin models: Quantum integrability andhyperfine induced spin dynamics

B. Erbe

Published Aug 16, 2011• 0 citations
OpenAlex

Abstract

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.

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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 consider a system of two strongly coupled electron spins in zero magnetic field, each of which is interacting with an individual bath of nuclear spins via the hyperfine interaction. Applying the long-spin approximation that we introduced in a previous paper [Europhys. Lett. 95, 47009 (2011)] (here each bath is replaced by a single long spin), we numerically study the electron spin and entanglement dynamics. We demonstrate that the decoherence time is scaling with the bath size according to a power law. As expected, the decaying part of the dynamics decreases with increasing bath polarization. However, surprisingly it turns out that, under certain circumstances, combining quantum dots of different geometry to the double dot setup has a very similar effect on the magnitude of the spin decay. Finally, we show that even for a comparatively weak exchange coupling the electron spins can be fully entangled.
We numerically study the hyperfine induced nuclear spin dynamics in a system of two coupled quantum dots in zero magnetic field. Each of the electron spins is considered to interact with an individual bath of nuclear spins via homogeneous coupling constants (all coupling coefficients being equal). In order to lower the dimension of the problem, the two baths are approximated by two single long spins. We demonstrate that the hyperfine interaction allows to utilize the nuclear baths for quantum information purposes. In particular, we show that it is possible to swap the nuclear ensembles on time scales of seconds and indicate that it might even be possible to fully entangle them. As a key result, it turns out that the larger the baths are, the more useful they become as a resource of quantum information. Interestingly, the nuclear spin dynamics strongly benefits from combining two quantum dots of different geometry to a double-dot setup.
We numerically study the hyperfine induced nuclear spin dynamics in a system of two coupled quantum dots in zero magnetic field. Each of the electron spins is considered to interact with an individual bath of nuclear spins via homogeneous coupling constants (all coupling coefficients being equal). In order to lower the dimension of the problem, the two baths are approximated by two single long spins. We demonstrate that the hyperfine interaction allows to utilize the nuclear baths for quantum information purposes. In particular, we show that it is possible to swap the nuclear ensembles on time scales of seconds and indicate that it might even be possible to fully entangle them. As a key result, it turns out that the larger the baths are, the more useful they become as a resource of quantum information. Interestingly, the nuclear spin dynamics strongly benefits from combining two quantum dots of different geometry to a double-dot setup.
We investigate hyperfine induced electron spin and entanglement dynamics in a system of two quantum dot spin qubits. We focus on the situation of zero external magnetic field and concentrate on approximation-free theoretical methods. We give an exact solution of the model for homogeneous hyperfine coupling constants (with all coupling coefficients being equal) and varying exchange coupling, and we derive the dynamics therefrom. After describing and explaining the basic dynamical properties, the decoherence time is calculated from the results of a detailed investigation of the short-time electron-spin dynamics. The result turns out to be in good agreement with experimental data.
The dynamics and decoherence of an electronic spin-1/2 qubit coupled to a bath of nuclear spins via hyperfine interactions in a quantum dot is studied. We show how exact results from the integrable solution can be used to understand the dynamic behavior of the qubit. It is possible to predict the main frequency contributions and their broadening for relatively general initial states analytically, leading to an estimate of the corresponding decay times, which are related to T1{T}_{1}T1​ of the electron. Furthermore, for a small bath polarization, a distinct low-frequency time scale is observed.
The time evolution of spin states of two electrons interacting with a nuclear spin bath in a quantum dot system is studied. The hyperfine interaction between the electrons and the nuclear spins is modeled by an isotropic Heisenberg interaction, and the interaction between the electron spins by Heisenberg exchange. Depending on the extent of the overlap between the spatial wave functions of the electrons, there are two physically different cases, namely, the two qubits either interact with the same set of nuclear spins or they see different nuclear spin environments. In the two cases, the decoherence of the two-qubit state is studied analytically. We have identified a class of two-qubit states that have a rich dynamics when the exchange interaction between the qubits becomes large in comparison to the hyperfine interaction strengths. The decoherence time scale is determined as a function of the bath-spin distribution and the polarizations of the initial two-qubit state. States with large decoherence times are identified by performing a minimization over all the two-qubit pure states.