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arXiv:0801.0992 · Physical Review B

Long-time electron spin storage via dynamical suppression of hyperfine-induced decoherence in a quantum dot

Wenxian Zhang, Nikolaos P. Konstantinidis, V. V. Dobrovitski, B. N. Harmon, Léa Ferreira dos Santos, Lorenza Viola

Published Mar 27, 2008• 61 citations
arXivHTMLPDF

Abstract

The coherence time of an electron spin decohered by the nuclear spin environment in a quantum dot can be substantially increased by subjecting the electron to suitable dynamical decoupling sequences. We analyze the performance of high-level decoupling protocols by using a combination of analytical and exact numerical methods, and by paying special attention to the regimes of large interpulse delays and long-time dynamics, which are outside the reach of standard average Hamiltonian theory descriptions. We demonstrate that dynamical decoupling can remain efficient far beyond its formal domain of applicability, and find that a protocol exploiting concatenated design provides best performance for this system in the relevant parameter range. In situations where the initial electron state is known, protocols able to completely freeze decoherence at long times are constructed and characterized. The impact of system and control nonidealities is also assessed, including the effect of intrabath dipolar interaction, magnetic field bias and bath polarization, as well as systematic pulse imperfections. While small bias field and small bath polarization degrade the decoupling fidelity, enhanced performance and temporal modulation result from strong applied fields and high polarizations. Overall, we find that if the relative errors of the control pulse flip angles do not exceed 3%, decoupling protocols can still prolong the coherence time by up to 2 orders of magnitude.

DOI
10.1103/physrevb.77.125336

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The coherence time of an electron spin decohered by the nuclear spin environment in a quantum dot can be substantially increased by subjecting the electron to suitable dynamical decoupling sequences. We analyze the performance of high-level decoupling protocols by using a combination of analytical and exact numerical methods, and by paying special attention to the regimes of large interpulse delays and long-time dynamics, which are outside the reach of standard average Hamiltonian theory descriptions. We demonstrate that dynamical decoupling can remain efficient far beyond its formal domain of applicability, and find that a protocol exploiting concatenated design provides best performance for this system in the relevant parameter range. In situations where the initial electron state is known, protocols able to completely freeze decoherence at long times are constructed and characterized. The impact of system and control nonidealities is also assessed, including the effect of intrabath dipolar interaction, magnetic field bias and bath polarization, as well as systematic pulse imperfections. While small bias field and small bath polarization degrade the decoupling fidelity, enhanced performance and temporal modulation result from strong applied fields and high polarizations. Overall, we find that if the relative errors of the control pulse flip angles do not exceed 3%, decoupling protocols can still prolong the coherence time by up to 2 orders of magnitude.
The dominant source of decoherence for an electron spin in a quantum dot is the hyperfine interaction with the surrounding bath of nuclear spins. The decoherence process may be slowed down by subjecting the electron spin to suitable sequences of external control pulses. We investigate the performance of a variety of dynamical decoupling protocols using exact numerical simulation. Emphasis is given to realistic pulse delays and the long-time limit, beyond the domain where available analytical approaches are guaranteed to work. Our results show that both deterministic and randomized protocols are capable to significantly prolong the electron coherence time, even when using control pulse separations substantially larger than what expected from the upper cutoff frequency of the coupling spectrum between the electron and the nuclear spins. In a realistic parameter range, the total width of such a coupling spectrum appears to be the physically relevant frequency scale affecting the overall quality of the decoupling. 1
The dominant source of decoherence for an electron spin in a quantum dot is the hyperfine interaction with the surrounding bath of nuclear spins. The decoherence process may be slowed down by subjecting the electron spin to suitable sequences of external control pulses. We investigate the performance of a variety of dynamical decoupling protocols using exact numerical simulation. Emphasis is given to realistic pulse delays and the long-time limit, beyond the domain where available analytical approaches are guaranteed to work. Our results show that both deterministic and randomized protocols are capable to significantly prolong the electron coherence time, even when using control pulse separations substantially larger than what expected from the upper cutoff frequency of the coupling spectrum between the electron and the nuclear spins. In a realistic parameter range, the total width of such a coupling spectrum appears to be the physically relevant frequency scale affecting the overall quality of the decoupling.
We investigate the performance of dynamical decoupling methods at suppressing electron spin decoherence from a low-temperature nuclear spin reservoir in a quantum dot. The controlled dynamics is studied through exact numerical simulation, with emphasis on realistic pulse delays and the long-time limit. Our results show that optimal performance for this system is attained by a periodic protocol exploiting concatenated design, with control rates substantially slower than expected from the upper spectral cutoff of the bath. For a known initial electron spin state, coherence can saturate at long times, signaling the creation of a stable ``spin-locked'' decoherence-free subspace. Analytical insight into saturation is obtained for a simple echo protocol, in good agreement with numerical results.
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.
This dissertation is a study of the decoherence of a solid-state spin qubit, either that of a localized electron spin or a donor nucleus, caused by a nuclear spin bath relevant to semiconductor quantum computer architectures. In the presence of an external magnetic field and at low temperatures, the dominant decoherence mechanism is the spectral diffusion of the qubit spin resonance frequency due to the temporally fluctuating random magnetic field associated with the dipolar interaction induced flip-flops of nuclear spin pairs. The qubit spin dephasing due to this random magnetic field depends intricately on the quantum dynamics of the nuclear spin bath, making the coupled decoherence problem difficult to solve. We provide a formally exact solution of this non-Markovian quantum decoherence problem which numerically calculates accurate spin decoherence at short times, of particular relevance in solid-state spin quantum computer architectures. A quantum cluster expansion method is motivated, developed, and tested for the spectral diffusion problem. The method is applicable to any ideal pulse sequence applied to the qubit. Dynamical decoupling sequences, which aim to prolong qubit coherence, are analyzed. In particular, concatenated dynamical decoupling sequences are shown to prolong not only the coherence time over the entire sequence but also the length of time between pulses necessary to maintain coherence. This is shown to result from successive low-order cancellations in applicable perturbative expansions with each level of concatenation. Each cancellation, however, will require the inclusion, in the cluster expansion, of increasingly large clusters to obtain the lowest-order results. These larger clusters in the lowest order often dominate decoherence and therefore invalidate, as being overly optimistic, the pair approximation as a means to study the effect of concatenated dynamical decoupling. We present numerical results from our cluster expansion technique for echoes of single (Hahn), concatenated, and periodic pulse sequences using realistic models of a localized electron in phosphorus doped Si and in a GaAs quantum dot and of a P donor nucleus in Si or GaAs. In the Si:P electron spin decoherence problem, we consider, along with spectral diffusion, the effects of anisotropic hyperfine (AHF) interactions and suggest a technique to suppress electron spin echo envelope modulations (ESEEM), an additional source of decoherence resulting from the AHF interactions. Our calculations for the Si:P Hahn echoes, including the effects of both anisotropic hyperfine interactions and spectral diffusion, are in excellent agreement with experimental results. Our calculations of concatenated pulse sequence echoes offer important predictions for the effectiveness of a promising strategy to preserve qubit coherence in semiconductor quantum computer architectures.
We investigate the effectiveness of different dynamical decoupling protocols for storage of a single qubit in the presence of a purely dephasing bosonic bath, with emphasis on comparing quantum coherence preservation under uniform versus nonuniform delay times between pulses. In the limit of instantaneous bit-flip pulses, this is accomplished by establishing a different representation of the controlled qubit evolution, where the decoherence behavior after an arbitrary number of pulses is directly expressed in terms of the uncontrolled decoherence function. In particular, analytical expressions are obtained for approximation of the long- and short-term coherence behavior for both Ohmic and supra-Ohmic environments. By focusing on the realistic case of pure dephasing in an excitonic qubit, we quantitatively assess the impact of physical constraints on achievable pulse separations, and show that little advantage of high-level decoupling schemes based on concatenated or optimal design may be expected if pulses cannot be applied sufficiently fast. In such constrained scenarios, we demonstrate how simple modifications of repeated periodic-echo protocols can offer significantly improved coherence preservation in realistic parameter regimes. We expect similar conclusions to be relevant to other constrained qubit devices exposed to quantum or classical phase noise.