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Journal of Physics Condensed Matter

Decoherence and fidelity of single-electron spin states in quantum dots: effects of nuclear hyperfine coupling and double occupancy

A. K. Rajagopal, S. D. Mahanti, Sudhanshu Shekhar Jha

Published Nov 13, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/18/47/013

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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 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 study the dynamics of entanglement of two electron spins in two quantum dots, in which each electron is interacting with its nuclear spin environment. Focusing on the case of uncoupled dots, and starting from either Bell or Werner states of two qubits, we calculate the decay of entanglement due to the hyperfine interaction with the nuclei. We mostly focus on the regime of magnetic fields in which the bath-induced electron spin flips play a role, for example, their presence leads to the appearance of entanglement sudden death at finite time for two qubits initialized in a Bell state. For these fields, the intrabath dipolar interactions and spatial inhomogeneity of hyperfine couplings are irrelevant on the time scale of coherence (and entanglement) decay, and most of the presented calculations are performed using the uniform-coupling approximation to the exact hyperfine Hamiltonian. We provide a comprehensive overview of entanglement decay in this regime, considering both free evolution of the qubits, and an echo protocol with simultaneous application of $\ensuremath{\pi}$ pulses to the two spins. All the currently relevant for experiments bath states are considered: the thermal state, narrowed states (characterized by diminished uncertainty of one of the components of the Overhauser field) of two uncorrelated baths, and a correlated narrowed state with a well-defined value of the zzz component of the Overhauser field interdot gradient. While we mostly use concurrence to quantify the amount of entanglement in a mixed state of the two electron spins, we also show that their entanglement dynamics can be reconstructed from measurements of the currently relevant for experiments entanglement witnesses and the fidelity of quantum teleportation, performed using a partially disentangled state as a resource.
We consider two qubits interacting by means of an optical cavity, where each qubit is represented by a single-electron spin confined to a quantum dot. It is known that electron spins in III-V semiconductor quantum dots are affected by the decoherence due to the hyperfine interaction with nuclear spins. Here we show that the interaction between two qubits is influenced by the Overhauser field as well. Starting from an unpolarized nuclear ensemble, we investigate the dependance of the fidelities for two-qubit gates on the Overhauser field. We include the hyperfine interaction pertubatively to second order in our analytical results and to arbitrary precision numerically.
The motivating background for this thesis lies in the recent development in the coherent optical control of electron spin states in semiconductor quan- tum dots. One possible and attractive application for these achievements is a physical implementation of a quantum computer based on spin qubits (quan- tum bit) in quantum dots. However, one of the obstacles in building such quantum computer is decoherence caused by the interaction of the electron spins with nuclear spins of the host material. In this thesis we theoretically investigate the effects caused by the hyperfine interaction in optically active semiconductor quantum dots. In Chapter 2 we assume optically induced rotations of single electron spins in semiconductor quantum dots. The optical control of the electron spin states is considered to be conducted by means of Raman type optical transitions between electron spin states. We investigate the influence of nuclear spins on the performance of the single-qubit gates by incorporating the additional effect of the Overhauser field into the electron spin dynamics. To calculate the errors caused by the hyperfine interaction, we determine average fidelities of rotations around characteristic axes in the Bloch sphere in the presence of nuclear spins analytically with perturbation theory up to second order in the Overhauser field. By applying numerical averaging over the nuclear field distribution we find the average fidelity to the arbitrary orders of the hyper- fine interaction. In Chapter 3 we investigate the dynamics of spin qubits interacting by means of an optical cavity. The qubits are represented by a single electron spin each confined to a quantum dot. Knowing that electron spins in III-V semiconduc- tor quantum dots are affected by the decoherence due to the hyperfine inter- action with nuclear spins, we find that the interaction between two qubits is consequently influenced by the Overhauser field. Assuming an unpolarizied nuclear ensemble, we investigate the fidelities for two-qubit gates depending on the Overhauser field. We include the hyperfine interaction perturbatively to second order in our analytical results, and to arbitrary precision numeri cally. Quantum dots can be used not only as physical systems for qubits, but also as sources for entangled photons. In chapter 4 we consider an emission of entangled photons by spontaneous decay of the biexciton state to the ground state. Due to electron-hole exchange interaction, the intermediate excitonic states have a finite fine structure splitting. The electron spins of the excitons also interact via hyperfine interaction with the nuclear spins. We study the temporal and fine structure splitting dependance of the coherence measures considering the effect of the nuclear magnetic field. We find that the hy perfine interaction contributes considerably to the reduction of the photon coherence. One of the possibilities to minimize the decoherence effects of the hyperfine interaction is to polarize the nuclear spins to 100%. However, up to date it was not possible to reach such degree of the nuclear polarization in experi- ments. In a recent experiment by Chekhovich et al. it was shown that high nuclear spin polarization can be achieved in self-assembled quantum dots by exploiting an optically forbidden transition between a heavy hole and a trion state. A fully polarized state in this case was predicted by a classical rate equation, but could not be obtained experimentally. Therefore we theoret- ically investigate this problem with the help of a quantum master equation in Chapter 5, and we show that a fully polarized state cannot be achieved due to formation of a nuclear dark state. Moreover, we demonstrate that the maximal degree of polarization depends on the form of the electron wave function inside of the quantum dot.
We investigate in detail, using both analytical and numerical tools, the decoherence of electron spins in quantum dots (QDs) coupled to a bath of nuclear spins in magnetic fields or with various initial bath polarizations, focusing on the longitudinal relaxation in low and moderate field and polarization regimes. An increase of the initial polarization of nuclear-spin bath has the same effect on the decoherence process as an increase of the external magnetic field, namely, the decoherence dynamics changes from smooth decay to damped oscillations. This change can be observed experimentally for a single QD and for a double-QD setup. Our results indicate that substantial increase of the decoherence time requires very large bath polarizations, and the use of other methods (dynamical decoupling or control of the nuclear spins distribution) may be more practical for suppressing decoherence of QD-based qubits.