Charge noise, spin-orbit coupling and dephasing of single-spin qubits
Quantum dot quantum computing architectures rely on systems in which inversion symmetry is broken, and spin-orbit coupling is present, causing even single-spin qubits to be susceptible to charge noise. We derive an effective Hamiltonian for the combined action of noise and spin-orbit coupling on a single-spin qubit, identify the mechanisms behind dephasing, and estimate the free induction decay dephasing times for common materials such as Si and GaAs. Dephasing is driven by noise matrix elements that cause relative fluctuations between orbital levels, which are dominated by screened whole charge defects and unscreened dipole defects in the substrate. Dephasing times differ markedly between materials, and can be enhanced by increasing gate fields, choosing materials with weak spin-orbit, making dots narrower, or using accumulation dots.
Developments in quantum computing hold considerable promise in the progress of modern information processing, and this has spurred a large experimental and theoretical effort investigating two-level systems that can be used as quantum bits (qubits). The need for scalability and long coherence times has led naturally to solid state spin-based devices, such as quantum dot spin systems, as ideal candidates for scalable qubits. The focus has been on single-spin Loss and DiVincenzo (1998) and singlet-triplet qubits. Petta et al. (2005) While GaAs quantum dots have been studied for many years, a substantial effort is also underway researching Si spin quantum computing architectures, Morton et al. (2011); Zwanenburg et al. (2013); Hao et al. (2014) motivated by their compatibility with Si microelectronics and long coherence times. Feher (1959); Abe et al. (2004); Tahan and Joynt (2005); Tyryshkin et al. (2006); Wang et al. (2010); Raith, Stano, and Fabian (2011); Muhonen et al. (2014) Recently, much effort has also been devoted to quantum dot systems with spin-orbit interactions, Friesen et al. (2004); Nadj-Perge et al. (2010); Pályi et al. (2012) where spin manipulation could in principle be achieved entirely by electrical means. Bulaev and Loss (2007); Szumniak et al. (2012); Budich et al. (2012)
The coherence of a solid-state spin qubit is quantified by the relaxation time and the dephasing time , both of which are determined by mechanisms that couple up spins with down spins. This coupling can either be direct, through the hyperfine interaction Merkulov, Efros, and Rosen (2002); Khaetskii, Loss, and Glazman (2003); Deng and Hu (2006); Coish and Baugh (2009) and fluctuations in the -factor, Ivchenko, Kiselev, and Willander (1997) or indirect, through the joint effect of hyperfine or spin-orbit coupling and fluctuating electric fields, such as those due to phonons Erlingsson and Nazarov (2002); Golovach, Khaetskii, and Loss (2004); Bulaev and Loss (2005); San-Jose et al. (2006); Prada, Blick, and Joynt (2008); Hu (2011); Climente, Segarra, and Planelles (2013) or charge noise. Fleetwood, Pantelides, and Schrimpf (2008); Jung et al. (2004); Huang and X.Hu (2014) Inversion symmetry breaking near an interface makes spin-orbit coupling unavoidable, even in materials such as Si in which it is weak. Wilamowski et al. (2002)
Hyperfine effects typically occur on long time scales, the nuclear bath is relatively well known and can be controlled through feedback mechanisms Bluhm et al. (2010) while in materials such as Si hyperfine coupling can be eliminated altogether through isotopic purification. Witzel, Hu, and Das Sarma (2007); Itoh and Watanabe (shed) The spin relaxation rate due to phonons is proportional to the fifth power of the magnetic field in zinc-blende materials, in which piezoelectric electron-phonon coupling is often dominant, and to the seventh power of the magnetic field in Si, in which there is no piezoelectric coupling.Prada, Blick, and Joynt (2008) Hence phonon effects become less pronounced at low magnetic fields. They also become weaker at low temperatures. Golovach, Khaetskii, and Loss (2004)
Noise is a well-known source of dephasing in charge qubits. Petersson et al. (2010); Dupont-Ferrier et al. (2013); Dial et al. (2013); Paladino et al. (2014) Experiments on quantum dots and point contacts have shown noise to be strong even at dilution refrigerator temperatures. Paladino et al. (2014); Petersson et al. (2010); Dupont-Ferrier et al. (2013); Dial et al. (2013); Takeda et al. (2013); Ribeiro et al. (2013); Buizert et al. (2008); Hitachi, Ota, and Muraki (2013); Müller et al. (2006) Noise sources include centers, which may act as traps that charge and discharge, and tunneling two-level systems, which can be modeled as fluctuating charge dipoles. Fleetwood, Pantelides, and Schrimpf (2008); Sze and Ng (2006); Zimmermann and Weber (1981); Jang, Lee, and Lee (1982); Reinisch and Heuer (2006); Biswas and Li (2006); Zimmerman et al. (2008) Noise and spin-orbit coupling give rise to nontrivial physics in 2D and 1D structures. Glazov, Sherman, and K.Dugaev (2010); Glazov and Sherman (2011); Li et al. (2013) In quantum dot spin qubits, Ref. Huang and X.Hu, 2014 has already shown that spin-orbit and noise lead to spin relaxation, and that noise and phonon effects in general become comparable at low-enough magnetic fields. Hence, at dilution refrigerator temperatures the interplay of spin-orbit and noise may set the defining bound on spin qubit coherence.
In this paper we build on previous decoherence work de Sousa and Das Sarma (2003); Hu and Das Sarma (2006); Chirolli and Burkard (2008); Culcer, Hu, and Das Sarma (2009); de Sousa (2009); Ramon and Hu (2010); Culcer and Zimmerman (2013) and devise a theory of dephasing due to the combined effect of charge noise and spin-orbit interactions, with two aims in mind. The first is to understand conceptually how spin-orbit and noise cause dephasing. For example, noise can give relative fluctuations between levels, virtual transitions between levels, as well as fluctuations in spin-orbit constants. We wish to isolate the terms that are responsible for dephasing. The second aim is to study the sensitivity to spin-orbit coupling across common materials with similar noise profiles. We study a sample qubit with the same specifications in different materials, we determine sample s due to common noise sources, discuss the variation in across materials, and seek methods to improve generally.
We consider a single-spin qubit implemented in a symmetric, gate-defined quantum dot, located at a sharp flat interface (Fig. 1) in a dilution refrigerator at 100mK. The qubit is described by the Hamiltonian . The kinetic energy and confinement term
where is the effective dot radius and the effective mass. The eigenstates of are the Fock-Darwin states, with the ground and first excited states given by
These have energies for the orbital ground state and for the twofold degenerate first orbital excited state. The orbital level splitting is assumed to be the dominant scale, so that only the ground and first excited states are considered. The Zeeman Hamiltonian , with the vector of Pauli spin matrices. Since is constant, the orbital effect of can be absorbed into the effective dot radius . We have also not taken into account multi valley effects in Si. For a certain interaction to couple valley states appreciably, it must be sufficiently sharp in real space. Neither the spin-orbit coupling due to the interface field nor the electric field of the defect satisfy this requirement – even though these interactions are important in relaxation in particular around hot spots. Huang and Hu (2014)
The spin-orbit term . The Rashba term , stems from structure inversion asymmetry, where here is an operator in real space, is the unit vector perpendicular to the interface, and is determined by a material specific parameter as well as the interface electric field . Winkler (2003) Thus is also sensitive to stray electric fields and fluctuates in time, thus we let where . For a quantum dot on a (001) surface the linear Dresselhaus term is usually the dominant bulk inversion asymmetry contribution,Winkler (2003) where , with a material-specific parameter and the width of the -confinement perpendicular to the interface. Since can be obtained from by a spin rotation, they give rise to qualitatively similar physics. In Si due to inversion symmetry, whereas Rashba spin-orbit coupling is expected generally in a 2D electron gas near an interface, and should be present in all gate-defined dots. In zincblende structures and may comparable in magnitude in certain parameter regimes, though for Vm ( 0.1 V/10 nm), the Rashba term is expected to be the dominant spin-orbit contribution.
The noise Hamiltonian is a random function of time. We do not include gate noise in our model, and we first consider random telegraph noise (RTN). In the simplest case, in which the qubit is only sensitive to one defect, represents a fluctuating Coulomb potential, screened by the nearby 2D electron gas. The 2D screened Coulomb potential is written in terms of its Fourier transform, which is a function of momentum Davies (1998)
with the relative permittivity, the Tomas-Fermi wave vector, and the Fermi wave vector (the contribution from is negligible Culcer and Zimmerman (2013)). The matrix elements entering are , , and . For RTN we can write for , and is a Poisson random variable with switching time . 111The effect of fluctuators with s can be eliminated in experiment through dynamical decoupling. Hence, on physical grounds, we impose 1 s as a cutoff for the switching time.
Additional (extrinsic) spin-orbit coupling arises from the electric field of the defect itself. Yet for a charge defect located 40 nm away from the dot this field is several orders of magnitude smaller than the interface electric field . Because the matrix element involved is second order in , the contribution this makes to dephasing is many orders of magnitude smaller than the Rashba interaction due to , and will not be considered further.
In the basis , with representing up and down spins, the Hamiltonian reads
where and are the Zeeman-split orbital levels including the noise terms, the Zeeman energy , and the spin-orbit terms and , with (not a function of time).
The qubit subspace is simply the Zeeman-split orbital ground state , which has been singled out in the top left hand corner of Eq. 4. These two states are coupled by to spin-aligned orbital excited states and by to orbital excited states with anti-aligned spin. By projecting onto this subspace we encapsulate the combined effect of spin-orbit coupling and noise in an effective qubit Hamiltonian . To achieve this, we carry out a Schrieffer-Wolff transformation, eliminating higher orbital excited states. Winkler (2003); Aleiner and FalÕko (2001); Golovach, Khaetskii, and Loss (2004); Stano and Fabian (2006) Keeping terms up to the second order in this transformation,
where (not a function of time) and . We retain only terms of first order in and . Equation (5) implies that, in addition to , there exists an effective Zeeman term , where represents an effective fluctuating effective magnetic field due to the combined action of spin-orbit and noise. For convenience has units of energy and, for RTN, . We will also use for the magnitude of . Since the Rashba and Dresselhaus contributions are added in quadrature, there is no sweet spot for dephasing.
The noise matrix elements appearing in may be divided into two categories. The diagonal elements cause different orbital levels to fluctuate by different amounts, while the off-diagonal element causes transitions between different orbital levels. If the qubit is initialized in an off-diagonal state, the diagonal elements () in give dephasing. These terms involve the intraband matrix elements of the defect potentials. An additional contribution comes from fluctuations in , which lead to fluctuations in itself. These fluctuations can be interpreted as a modulation of the -factor, and are expected to come from defects in the substrate right above the dot, which modify . Since the dot region is depleted, whole charge defects cannot fluctuate, except in the very special case in which the defect lies right above the dot. Hence defects contributing to are expected to be mostly charge dipoles, stemming for example from passivated traps. Although these are weaker than whole charge defects, they are unscreened, leading to a subtle competition. Thus, generally, dephasing stems from noise matrix elements that cause relative fluctuations between orbital levels. In contrast, if the qubit is initialized in the spin-up state, the off-diagonal elements () in give relaxation ( processes), which was studied in detail in Ref. Huang and X.Hu, 2014. These elements are of first order in and involve the interband defect matrix element . 222We expect whole charge defect potentials to be dominant in relaxation since they are much stronger than dipole potentials Culcer and Zimmerman (2013).
In order to study dephasing further and obtain quantitative estimates of , we focus on a single-spin qubit described by a spin density matrix . The spin density matrix satisfies the quantum Liouville equation
The spin density matrix . Any spin component can be found as , with tr the matrix trace. We restrict our attention to RTN for the time being. Using the time evolution operator , we obtain the general time evolution of the spin as
where we have defined and, for RTN, , with . The two components of have exactly the same time evolution. Since , if is initialised, . Averaging over noise realisations de Sousa and Das Sarma (2003); Culcer, Hu, and Das Sarma (2009); Culcer and Zimmerman (2013)
where . All systems of interest in this work satisfy , in which case we may approximate . When the denominator of the is expanded in , only the leading term in the expansion may be retained. Physically, in this case, the time dynamics of are a random walk in time, and the spread in leads to motional narrowing. As a result, the initial spin decays exponentially as , where
For whole charge defects, where dephasing is dominated by fluctuations in the orbital energy, we may set and retain . For dipole charge defects we have .
We turn our attention next to noise. In semiconductors noise is Gaussian Kogan (2008) and is fully described by its spectral density . The Fourier transform of this spectral density has the form , where is a parameter typically inferred from experiment. Based on our estimates for RTN above we expect whole charge defects to dominate dephasing. Hence, for the effective fluctuating magnetic field acting in the qubit subspace, we may write approximately . To study dephasing, we write , where
The low-frequency cut-off is usually taken to be the inverse of the measurement time. At times such as we consider here, we can approximate
where the dephasing time is estimated by
Since this definition of is approximate, we plot the full time evolution of in Fig. 2.
We consider a sample dot with radius nm located at , and as calculated in Refs. Tahan and Joynt, 2005; Winkler, 2003. For a defect in the plane of the dot with nm, eV, eV and eV. Next we estimate the change in due to a dipole defect right above the dot () and nm away from it. Sze and Ng (2006); Fleetwood, Pantelides, and Schrimpf (2008) The potential of an unscreened charge dipole located a distance away from the dot, is . The charge dipole has dipole moment , where . We take the expectation value of using , and compare it with the matrix element of , yielding . We use this figure in all our estimates since for all materials considered are of very similar magnitudes. For noise we extract from experiment. For Si/SiGe we use Ref. Takeda et al., 2013, and for GaAs Ref. Petersson et al., 2010, while for InAs and InSb, in the absence of experimental data, we use the same as for GaAs.
The results are listed in Table 1, which is the central result of this paper. For all materials, whole charge defects dominate dephasing. Table 1 shows that terms of second-order in spin-orbit are effective in causing dephasing, and the dependence on causes vast differences in dephasing times between materials. Hence, using materials with a small such as Si can improve coherence enormously. If spin-orbit coupling is needed for electric dipole spin resonance, increasing will align the charge dipoles. Although that increases and with it dephasing, it also reduces the gate time by an equal amount. Moreover, for noise, , so by halving the dot radius the dephasing time can be increased by an order of magnitude (Fig. 2; for RTN, ). One can also use pulse sequences, Ribeiro et al. (2013) lower the temperature to reduce , use accumulation dots, in which there is no nearby 2DEG, or focus on reducing charge noise. Buizert et al. (2008); Takeda et al. (2013); Hitachi, Ota, and Muraki (2013)
Following existing calculations of , Tahan and Joynt (2005) we have taken the -confinement in the form of a square well, whereas semiconductor interfaces are more accurately described by a triangular well. Nevertheless, since the form of and is dictated by symmetry, they will be identical in structure for triangular confinement, thus we may simply treat and as phenomenological parameters. Finally, fluctuations in affect . Although this effect, likewise driven by fluctuating dipoles, can be calculated in the same way as the renormalization of by , we expect its contribution to be minor, in exact analogy with .
In summary, we have shown that spin-orbit coupling and charge noise are an effective source of dephasing in single-spin qubits even in materials such as GaAs in which spin-orbit coupling is weak. Based on realistic experimental parameters vast differences in spin dephasing times exist between common materials. In the future we will devise a full model of noise Martin and Galperin (2006) as an ensemble of incoherent RTNs, Müller et al. (2006) where qubit dynamics is nontrivial. Burkard (2009) Dephasing of hole spin qubits, in which spin-orbit interactions are also strong but the heavy hole-light hole coupling cannot be ignored, will likewise be studied in a future publication.
Acknowledgements.We thank R. Winkler, Sven Rogge, Joe Salfi, Andrea Morello, K. Takeda, Amir Yacoby, L. Vandersypen, Neil Zimmerman, S. Das Sarma, Alex Hamilton, Xuedong Hu, Guido Burkard, Mark Friesen, Andrew Dzurak, Menno Veldhorst, Floris Zwanenburg, J. R. Petta, and Matt House for enlightening discussions.
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