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arXiv:cond-mat/0510177 · Physical Review B

Nuclear spin state narrowing via gate-controlled Rabi oscillations in a double quantum dot

D. Klauser, W. A. Coish, Daniel Loss

Published May 2, 2006• 169 citations
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Abstract

We study spin dynamics for two electrons confined to a double quantum dot under the influence of an oscillating exchange interaction. This leads to driven Rabi oscillations between the $\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩$ state and the $\ensuremath{\mid}\ensuremath{\downarrow}\ensuremath{\uparrow}⟩$ state of the two-electron system. The width of the Rabi resonance is proportional to the amplitude of the oscillating exchange. A measurement of the Rabi resonance allows one to narrow the distribution of nuclear spin states and thereby to prolong the spin decoherence time. Further, we study decoherence of the two-electron states due to the hyperfine interaction and give requirements on the parameters of the system in order to initialize in the $\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩$ state and to perform a $\sqrt{\mathrm{SWAP}}$ operation with unit fidelity.

DOI
10.1103/physrevb.73.205302

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We study spin dynamics for two electrons confined to a double quantum dot under the influence of an oscillating exchange interaction. This leads to driven Rabi oscillations between the \ensuremath∣\ensuremath↑\ensuremath↓⟩\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩\ensuremath∣\ensuremath↑\ensuremath↓⟩ state and the \ensuremath∣\ensuremath↓\ensuremath↑⟩\ensuremath{\mid}\ensuremath{\downarrow}\ensuremath{\uparrow}⟩\ensuremath∣\ensuremath↓\ensuremath↑⟩ state of the two-electron system. The width of the Rabi resonance is proportional to the amplitude of the oscillating exchange. A measurement of the Rabi resonance allows one to narrow the distribution of nuclear spin states and thereby to prolong the spin decoherence time. Further, we study decoherence of the two-electron states due to the hyperfine interaction and give requirements on the parameters of the system in order to initialize in the \ensuremath∣\ensuremath↑\ensuremath↓⟩\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩\ensuremath∣\ensuremath↑\ensuremath↓⟩ state and to perform a SWAP\sqrt{\mathrm{SWAP}}SWAP​ operation with unit fidelity.
We have evaluated hyperfine-induced electron spin dynamics for two electrons confined to a double quantum dot. Our quantum solution accounts for decay of a singlet-triplet correlator even in the presence of a fully static nuclear spin system, with no ensemble averaging over initial conditions. In contrast to an earlier semiclassical calculation, which neglects the exchange interaction, we find that the singlet-triplet correlator shows a long-time saturation value that differs from 12\frac{1}{2}21​, even in the presence of a strong magnetic field. Furthermore, we find that the form of the long-time decay undergoes a transition from a rapid Gaussian to a slow power law $(\ensuremath{\sim}1∕{t}^{3∕2})$ when the exchange interaction becomes nonzero and the singlet-triplet correlator acquires a phase shift given by a universal (parameter independent) value of $3\ensuremath{\pi}∕4$ at long times. The oscillation frequency and time-dependent phase shift of the singlet-triplet correlator can be used to perform a precision measurement of the exchange interaction and Overhauser field fluctuations in an experimentally accessible system. We also address the effect of orbital dephasing on singlet-triplet decoherence and find that there is an optimal operating point where orbital dephasing becomes negligible.
The decoherence and fidelity of spin states in a localized single-electron quantum dot in the presence of a dc magnetic field, arising either from the nuclear hyperfine interaction within the dot or due to its coupling with another localized quantum dot, are examined in detail. A general framework for determining the time evolution of the reduced density matrix. for a single dot is presented, which is exact up to the second order in interaction with any reservoir. In particular, it is applied to the problem of nuclear hyperfine coupling, and approximate estimates of coherence decay time are made when the nuclear spins are either polarized or unpolarized and the internal dynamics of nuclear spins is determined mainly by the nuclear magnetic dipole-dipole interaction. The fidelity of a pure unperturbed electronic one-qubit spin state is obtained as a function of time, which is exact even on a very short timescale of logic gate operations. The time variation of the fidelity of the same one-qubit state on the localized dot as a part of the direct product with another one-qubit state on another localized dot arising because of coupling between these quantum dots is also calculated in this paper. In this case, we include both the single-particle tunnelling between the dots as well as the direct and exchange Coulomb interactions, including on-site Coulomb repulsion. This allows for the double occupation of a single dot. It is found that the loss of fidelity of such two-qubit states due to double occupancy and additional phase errors in the presence of appreciable dot-dot coupling can become a more severe limiting factor than that due to the hyperfine interaction in individual dots.
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.
We theoretically study the interplay of spin-orbit and hyperfine interactions in dynamical nuclear polarization in two-electron semiconductor double quantum dots near the singlet-triplet ($S\ensuremath{-}{T}_{+}$) anticrossing. The goal of the scheme under study is to extend the singlet-triplet ($S\ensuremath{-}{T}_{0}$) qubit decoherence time T2∗{T}_{2}^{*}T2∗​ by dynamically transferring the polarization from the electron spins to the nuclear spins. This polarization transfer is achieved by cycling the electron spins over the $S\ensuremath{-}{T}_{+}$ anticrossing. Here, we investigate, both quantitatively and qualitatively, how this hyperfine-mediated dynamical polarization transfer is influenced by the Rashba and Dresselhaus spin-orbit interaction. In addition to T2∗{T}_{2}^{*}T2∗​, we determine the singlet return probability Ps{P}_{s}Ps​, a quantity that can be measured in experiments. Our results suggest that the spin-orbit interaction establishes a mechanism that can polarize the nuclear spins in the direction opposite to that of hyperfine-mediated nuclear spin polarization. In materials with relatively strong spin-orbit coupling, this interplay of spin-orbit and hyperfine-mediated nuclear spin polarizations prevents any notable increase in the $S\ensuremath{-}{T}_{0}$ qubit decoherence time T2∗{T}_{2}^{*}T2∗​.
In a coupled double-quantum-dot system, we present a theory for the interplay between electron and nuclear spins when the two-electron singlet state is brought into resonance with one triplet state in moderate external magnetic field. We show that the quantum interference between first-order and second-order hyperfine processes can lead to a feedback mechanism for manipulating the nuclear hyperfine fields. In a uniform external field, positive- and negative-feedback controls can be realized for the gradient of the longitudinal hyperfine field as well as the average transverse hyperfine field in the double dot. The negative feedback which suppresses fluctuations in the longitudinal nuclear field gradient can enhance the decoherence time of a singlet-triplet qubit to microsecond regime. We discuss the possibility of enhancing the decoherence time of each individual spin in a cluster of dots using the negative-feedback control on the transverse nuclear field.