Dephasing due to nuclear spins in large-amplitude electric dipole spin resonance
We have analyzed effects of the hyperfine interaction on electric dipole spin resonance when the amplitude of the quantum-dot motion becomes comparable or larger than the quantum dot’s size. Away from the well known small-drive regime, the important role played by transverse nuclear fluctuations leads to a gaussian decay with characteristic dependence on drive strength and detuning. A characterization of spin-flip gate fidelity, in the presence of such additional drive-dependent dephasing, shows that vanishingly small errors can still be achieved at sufficiently large amplitudes. Based on our theory, we analyze recent electric-dipole spin resonance experiments relying on spin-orbit interactions or the slanting field of a micromagnet. We find that such experiments are already in a regime with significant effects of transverse nuclear fluctuations and the form of decay of the Rabi oscillations can be reproduced well by our theory.
The interest in coherent manipulation of single electron spins has stimulated intense research efforts, leading to a great degree of control in a variety of nanostructures Hanson et al. (2007); Awschalom et al. (2013). For electrons in quantum dots, electron spin resonance (ESR) was first demonstrated in Ref. Koppens et al. (2006). However, full electric control of local spins might be a better strategy for complex architectures of many quantum dots, envisioned to realize quantum information processing Loss and DiVincenzo (1998). Thus, electric dipole spin resonance (EDSR) was developed relying on either spin-orbit couplings Golovach et al. (2006); Nowack et al. (2007) or the inhomogeneous magnetic field induced by a micromagnet Tokura et al. (2006); Pioro-Ladrière et al. (2008). The effectiveness of EDSR is highlighted by recent experiments, which could demonstrate Rabi oscillations with frequencies larger than 100 MHz for both approaches van den Berg et al. (2013); Yoneda et al. (2014). To further improve the performance of such spin manipulation schemes, it is important to characterize relevant dephasing mechanisms, and especially those which might become dominant at strong electric drive. In fact, as it will become clear in the following, a sufficiently strong drive is able to induce significant and yet unexplored modifications on how typical dephasing sources affect EDSR. For this reason, while representing the main limitation for accurate spin manipulation, dephasing is still poorly understood in large-amplitude regime of EDSR (i.e., when the amplitude of motion becomes comparable to the quantum dot’s size).
In this work we will focus on hyperfine interactions, which are well known to play an important role in the electron spin dynamics of quantum dots. In particular, the ESR dephasing was successfully interpreted in terms of a static Overhauser field, with a variance of a few mT in GaAs Koppens et al. (2006). The resulting power-law decay and a universal phase shift of the Rabi oscillations were accurately verified Koppens et al. (2007), confirming the predominance of nuclear spins over other sources of dephasing. While EDSR experiments were also generally interpreted assuming a power-law decay, the expected dependence is violated at the larger values of the drive Nadj-Perge et al. (2010); van den Berg et al. (2013); Yoneda et al. (2014). Especially, Ref. Yoneda et al. (2014) has demonstrated striking deviations from the ESR behavior, including a crossover from power-law to gaussian decay. It is also known that the electron motion, as well as the presence of the drive, can have substantial effects on spin dynamics and decoherence Laird et al. (2007); Rashba (2008); Széchenyi and Pályi (2014); Echeverría-Arrondo and Sherman (2013); Jing et al. (2014). These considerations motivate us to provide here a detailed characterization of EDSR dephasing induced by the hyperfine interaction, paying special attention to the large-amplitude regime. As a main objective behind trying to achieve faster Rabi frequencies is to decrease operation errors, we also establish the limitations on spin-flip gate fidelity imposed on EDSR by the hyperfine interaction. Finally, we compare our theory with EDSR experiments which, as we will discuss, have very recently entered the large-amplitude regime.
EDSR is induced by a driven periodic displacement of the quantum dot , which we take conventionally along . For the time-dependent wavefunction , we assume harmonic confinement along the direction of motion (which is applicable to both nanowire and lateral quantum dots):
The spin dynamics can be described with the following Hamiltonian:
where the first term is the electron Zeeman coupling, with and the Pauli matrices. The second term is the drive, for which we can generally assume while other features (e.g., the direction of ) depend on specific details of the spin-orbit coupling or magnetic gradient. The last term in Eq. (2) is the Fermi contact hyperfine interaction, where is the nuclear density. is the spin operator of nucleus , with position and coupling . The periodic time-dependence of Eq. (2) is characterized by Fourier components , of which only the static () and resonant () ones are of interest here. In fact, in a frame rotating at frequency and neglecting fast oscillating terms, the transformed Hamiltonian reads:
where longitudinal/transverse fluctuations are controlled by and , respectively. Without loss of generality, we restrict ourselves to the case rem :
where is the detuning and is defined by the second line of Eq. (3).
On the relatively short time scales of the EDSR experiments, it is appropriate to describe with a static random classical magnetic field. In the lab frame and for infinite-temperature nuclear spins, the variance of the Overhauser field is given by However, is for a reference frame moving with the dot and rotating at frequency . As a consequence, its statistical properties differ from the ones in the lab frame. We still have , but Eq. (3) implies that , have an interesting dependence on the strength of the drive. For finite and as in Eq. (1) we can evaluate , as follows, in terms of Hypergeometric functions:
In the above formulas, the only dependence is on , i.e., the amplitude of motion relative to the width of the electron wavefunction. and are plotted in Fig. 1(a), showing that for only the longitudinal fluctuations survive. Therefore, in this limit one recovers the same behaviour of ESR. At finite , the value of is a decreasing function of amplitude while the transverse fluctuations become non-zero and have a non-monotonic dependence on . Such transverse fluctuations can serve at as a driving term Laird et al. (2007), and in this context were previously discussed through an expansion at small Rashba (2008) or numerical evaluation Széchenyi and Pályi (2014). However, both scenarios of EDSR (i.e., based on a micromagnet or spin-orbit coupling) are in a physical regime distinct from Refs. Laird et al. (2007); Rashba (2008); Széchenyi and Pályi (2014), because is typically much smaller than the drive . To see this, we notice that Eq. (6) implies , thus:
In Eq. (7) we defined the useful parameter . is approximately constant (since ) and is typically small, according to our later estimates. Therefore, transverse nuclear fluctuations provide an additional dephasing mechanism which becomes progressively more important, until the maximum effect is reached at . We will discuss how the effect of becomes dominant over in a regime of sufficiently strong EDSR drive, which was already realized by recent experiments Yoneda et al. (2014).
Although we cannot provide a general closed-form result, several relevant features can be explicitly characterized. In particular, at sufficiently large drive and detuning we can neglect the components of perpendicular to [see Eq. (4)], to obtain:
The Rabi decay time is:
with the typical inhomogeneous dephasing time associated with nuclear spins. Equation (9) implies a crossover between the ESR power-law decay at weak drive to the gaussian decay of the strong-drive regime.
To exemplify this behavior, we first consider the resonant condition () when, besides , the form of the decay is determined by and the coefficient . An example of numerical results for is shown in Fig. 1(b), assuming . We confirm that the ESR power-law decay Koppens et al. (2006, 2007) is recovered when but significant deviations from this known dependence appear at larger strength of the drive. In the large-drive limit, we find a gaussian decay with no universal phase shift, as Eq. (9) becomes an excellent approximation. This crossover to the strong-drive regime occurs when the effect of in Eq. (8) becomes dominant over , i.e., . We estimate the typical values of using the limit of at small (which is justified if ), i.e., and . This yields the condition:
in good agreement with the numerical results of Fig. 1(b).
Of special interest is the decay timescale:
which follows simply from Eq. (10) and is plotted in Fig. 1(c). The increasing strength of with the drive leads to a significantly faster decay, as also seen in the time domain results of Fig. 1(b). However, it is important to note that the fidelity of the rotation grows monotonically with , as shown in Fig. 1(d). For a quantitative analysis, we set in Eq. (8) and perform an expansion up to second order in . After statistical averaging, we have:
Equation (13) includes a contribution proportional to but the additional dephasing from transverse fluctuations is more than compensated by the decrease of and the faster Rabi frequency. Thus, Eq. (13) shows that it is always advantageous to apply a stronger drive and the effect of the hyperfine interaction on -rotation error can be reduced below any desired threshold with a sufficiently large Yoneda et al. (2014).
Decay at finite detuning.
Considering a finite detuning yields further insight on the role of longitudinal/transverse nuclear fluctuations. The effect of is illustrated in Fig. 2, where panel (c) confirms that Eq. (9) provides an excellent approximation in the strong drive regime. A first consequence is that the Rabi oscillations approach the “chevron” pattern of Fig. 2(a). Furthermore the decay time gets reduced at finite , which is illustrated in Figs. 2(c) and (d).
The dependence of on has a simple physical explanation, as it can be traced to the difference in strength between transverse and longitudinal nuclear fluctuations shown in Fig. 1(a). Since Eq. (4) implies that a finite detuning corresponds to a field along in the rotating frame, the relevant component of the nuclear fluctuations (i.e., along the total driving field) becomes a weighted average of and . Since , the nuclear fluctuations gets enhanced by a finite detuning. As shown in Fig. 2(d), the dependence of on is particularly pronounced at smaller values of . This is natural, as the ratio is large in this regime, see Fig. 1(a), while the nuclear fluctuations become more isotropic at larger . Thus, studying the dependence of on allows one to explore how the relative strength of longitudinal/transverse nuclear fluctuations evolves with .
We conclude our analysis of the Rabi oscillations by commenting briefly on the stationary limit , which is a useful quantity to estimate . Common methods rely either on the drive dependence at resonance (as done in the ESR experiment of Ref. Koppens et al. (2007)) or the linewidth (considering finite detunings). We find that, even if the Rabi oscillations are sensitively modified by transverse nuclear fluctuations, the effect on is negligible in the current experimental regime . A significant difference between ESR and EDSR only appears when (for more detais, see sup ). Therefore, methods to extract from are still valid for large-amplitude EDSR.
Comparison to experiments.
We now discuss the application of our theory to available experimental data. In Fig. 3 we show an analysis of the data shown in Fig. 2(c) of Ref. van den Berg et al. (2013), obtained from InSb nanowire dots with strong spin-orbit interaction. As seen, our theory is able to reproduce well the Rabi oscillations and the fit yields values and consistent with . For EDSR driven by the spin-orbit coupling we have:
where is the spin-orbit length and the angle between the spin-orbit field and . Using nm, nm Nadj-Perge et al. (2012), , and mT van den Berg et al. (2013) gives . Figure 3(b) implies , which is consistent with this estimate.
While Fig. 3(b) has , we estimate that larger values of were achieved in a recent experiment on GaAs quantum dots Yoneda et al. (2014). There, the drive is based on a micromagnet for which numerical simulations give (1 mT/nm) Yoneda et al. (2014). Using the largest achieved Rabi frequency MHz, , and nm (corresponding to orbital energies meV), we obtain which is relatively close to the condition at which fluctuations are most effective. On the other hand, the regime of motional narrowing com (b) does not appear to be within reach of current experiments. For GaAs quantum dots Petta et al. (2005); Koppens et al. (2006, 2007); Laird et al. (2007), which allows us to estimate a typical range .
Several findings of Ref. Yoneda et al. (2014) are in good agreement with our discussion, including the transition to a “chevron” pattern in the strong drive regime Yoneda et al. (2014). As a result, our panels (a) and (b) of Fig. 2 are remarkably similar to the corresponding panels of Ref. Yoneda et al. (2014). Furthermore, a crossover from power-law decay (for 5 mT) to gaussian decay (for 15 mT) was observed. The strength of the drive for such crossover is compatible with Eq. (11), which can be rewritten as , using the above estimates of , and . We also show in Fig. 1(c) that Eq. (12) is able to reproduce the dependence of on the drive strength, with reasonable fit parameters for GaAs quantum dots ( ns, ).
In conclusion, we have characterized the dephasing induced by the hypefine interaction in large-amplitude EDSR, and we showed that transverse fluctuations of the Overhauser field are likely to play an important role in this regime, recently achieved experimentally. It should be mentioned that also other dephasing sources were suggested for EDSR, such as paramagnetic impurities, charge noise, and photon-assisted tunneling Nadj-Perge et al. (2010); van den Berg et al. (2013); Yoneda et al. (2014). In the absence of clear evidence (or specific predictions) about these alternative mechanisms, our theory offers further means to test if nuclear spins are indeed the dominant effect, e.g., through a detailed analysis of as function of both drive and detuning. In fact, it is unlikely that other types of dephasing would induce the same type of sensitive dependence on discussed in relation to Eq. (10) and Fig. 2.
From a more general point of view, our study is helpful to assess the limitations to spin manipulation due to the quantum-dot motion and the nuclear-spin bath. These two aspects are unavoidable for EDSR based on III-V semiconductors, in contrast to ESR or spin manipulation based on group-IV materials Kawakami et al. (2014); Veldhorst et al. (2014). Nuclear fluctuations, on the other hand, do not represent a fundamental obstacle to EDSR, since high-fidelity gates can be achieved at sufficiently large amplitude.
We thank W. A. Coish, T. Otsuka, P. Stano, and J. Yoneda for useful discussions. S. C. acknowledges funding from NSFC (grant No. 11574025). D. L. acknowledges support from the Swiss NSF and NCCR QSIT.
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Supplemental material for “Dephasing due to nuclear spins in large-amplitude electric dipole spin resonance”
We discuss here here the behavior of , i.e., the stationary value of the spin-flip probability under a continuous drive. As a function of detuning, displays a peak around and, like in regular ESR (the limit), a stronger drive leads to a general increase of , as well as broadening in . Representative examples are shown if Fig. S1(a).
We first examine the behavior of the peak value as a function of the drive strength. Taking in Eq. (9) of the main text gives in the large-drive regime, thus the interesting drive dependence of occurs when are smaller than . By neglecting in Eq. (8) such transverse nuclear fluctuations, we obtain the following expression:
which is similar to the ESR result, except that the nuclear fluctuations are replaced here by a reduced value . We have checked that Eq. (S1) is in agreement with direct numerical evaluation. As seen in Fig. S1(b), deviations from the ESR result exist in general but in the current experimental regime can be safely neglected, because they only become important when .
By considering the dependence of on detuning, we can reach a similar conclusions about the EDSR linewidth. Figure S1(c) shows that transverse fluctuations have a significant effect for and , when the linewidth is significantly narrower than for ESR. However, very small deviations from the Voigt profile occur for currently more typical values . As stated in the main text, we can thus conclude that is much less sensiteve to the physical origin of the drive than the decay of Rabi oscillations. One can see, contrasting Fig. S1(b) with Fig. 1(b) of the main text, that significant deviations from the ESR decay appear at relatively small values of , when has not yet saturated to 1/2.