Driven dynamics of a quantum dot electron spin coupled to bath of higher-spin nuclei

Spins in self-assembled quantum dots (QDs) are under intense investigation for a variety of quantum information applications, including quantum information processing, quantum communication, and quantum transduction Ladd et al. (2010a); Liu et al. (2010); Lauk et al. (2020); Wehner et al. (2018). The relatively long coherence times, fast controllability Economou et al. (2006); Greilich et al. (2009); Makhonin et al. (2011), and good photon emission properties of these systems Ding et al. (2016); Buckley et al. (2012); Wei et al. (2014); Senellart et al. (2017) make them promising candidates for achieving high-quality spin-photon interfaces and for producing large-scale multi-photon entangled states Lindner and Rudolph (2009); Buterakos et al. (2017); Russo et al. (2018); Hilaire et al. (2020). The deterministic generation of these multi-photon entangled states has been demonstrated experimentally using the dark excitonic states of QDs Schwartz et al. (2016).

While optically controlled quantum dot spins offer a wide range of technological possibilities, hyperfine (HF) interactions between the confined spin and its surrounding nuclear spin bath have been a major impediment. This interaction is the main source of decoherence in these systems, and it also causes spectral wandering and inhomogeneities in quantum dot ensembles, aspects that have been researched extensively over the past two decades Urbaszek et al. (2013); Huang and Hu (2010, 2010); Merkulov et al. (2002); Yao et al. (2006); Stockill et al. (2016); Coish and Loss (2004); Cywiński et al. (2009a, b); Barnes et al. (2012a); Khaetskii et al. (2003); Yao et al. (2006); Bortz and Stolze (2007); Gammon et al. (2001); Chekhovich et al. (2020); Gammon et al. (1996, 1997); Chekhovich et al. (2013a); Wuest et al. (2016); Chekhovich et al. (2010a); Ladd et al. (2010b); Vink et al. (2009); Schering et al. (2019, 2020); Beugeling et al. (2016); Kleinjohann et al. (2018); Welander et al. (2014); Chekhovich et al. (2015); Kloeffel et al. (2011); Maletinsky et al. (2009); Chekhovich et al. (2012); Krebs et al. (2010); Chekhovich et al. (2013b, 2010b); Greilich et al. (2007a); Munsch et al. (2013); Prechtel et al. (2016); Schering and Uhrig (2020). However, many works have shown that the state of the bath, and consequently its deleterious effects, can be influenced by driving the electron spin. For example, several experiments have shown that driving can generate dynamic nuclear polarization (DNP), an effect that has been observed in self-assembled QDs Greilich et al. (2006, 2007b); Carter et al. (2009); Varwig et al. (2013, 2012, 2014); Xu et al. (2009); Bracker et al. (2005); Latta et al. (2009); Högele et al. (2012); Latta et al. (2011); Dominguez et al. (2020) and also in other systems such as gated QDs Foletti et al. (2009); Reilly et al. (2008); Bluhm et al. (2010); Nichol et al. (2016), quantum wires Pribiag et al. (2013) and in bulk materials Tian et al. (2017); Rej et al. (2015), findings that have been supported by a number of theory works Barnes and Economou (2011); Yang and Sham (2013); Economou and Barnes (2014); Stano and Loss (2014); Nutz et al. (2019); Sharma et al. (2017, 2019); Neder et al. (2011, 2014); Rudner and Levitov (2007); Danon et al. (2009); Oulton et al. (2007); Bulutay (2012). An important example of DNP in self-assembled QDs is the mode-locking experiment of Ref. Greilich et al. (2007b), where an ensemble of QD electron spins becomes synchronized with a periodic train of optical pulses as a consequence of DNP. Continuous-wave laser driving of the electron has been shown to create DNP in QDs as well, leading to interesting phenomena such as the line-dragging effect, i.e. the locking of an optical QD transition to the frequency of the laser Xu et al. (2009); Högele et al. (2012); Latta et al. (2009); Yang and Sham (2012); Nutz et al. (2019). Owing to the long coherence times of nuclear spins, DNP has been proposed for applications such as quantum memories Taylor et al. (2003); Denning et al. (2019), which has recently been demonstrated experimentally Gangloff et al. (2019).

Although most of the fully quantum mechanical theoretical treatments of the hyperfine decoherence problem allow for nuclei with spin greater than 1/2 Cywiński et al. (2009b, a); Coish and Loss (2004), studies of the driven, hyperfine-induced generation of DNP have mostly focused on spin 1/2 nuclei to reduce the computational complexity of the problem Barnes and Economou (2011); Economou and Barnes (2014); Rudner and Levitov (2007); Danon and Nazarov (2008); Greilich et al. (2007b); Danon et al. (2009). The latter works typically rely on either stochastic equations or rate equations to solve for the nuclear polarization distribution. While solving the feedback problem for spin 1/2 nuclear baths can yield qualitative insights about DNP experiments, the quantitative accuracy of such models is limited by the fact that the most commonly studied semiconductor QDs are in materials such as InAs or GaAs, which contain nuclei of spin $I>1/2$. In addition to artificially reducing the size of the bath Hilbert space, assuming spin 1/2 nuclei also ignores effects such as quadrupolar interactions, which are only present for $I>1/2$. There do exist a few theoretical works that allow for $I>1/2$ Huang and Hu (2010); Yang and Sham (2013); Beugeling et al. (2016). Specifically, Huang and Hu Huang and Hu (2010) studied DNP arising from hyperfine interactions with the spin 3/2 arsenic nuclei in InGaAs by making use of Fermi’s golden rule; however, only qualitative agreement with experiment was achieved due to the need to introduce phenomenological parameters. Yang and Sham Yang and Sham (2013) presented a general framework for nuclei of arbitrary total spin by unifying the stochastic and rate-equation approaches. In this work they focused on a drift feedback loop (which allows for a possible bias in nuclear spin-flip processes) and obtained a Fokker-Planck equation for the polarization of the bath. Although this framework captures line-dragging and other DNP phenomena seen in experiments, it has only been established for continuous-wave driving, and so it is not immediately applicable to experiments with periodic driving such as the mode-locking experiment of Ref. Greilich et al. (2007b). Theoretical works that have specifically focused on mode-locking type experiments have either assumed $I=1/2$ nuclear baths Barnes and Economou (2011); Economou and Barnes (2014) or utilized semiclassical methods Schering et al. (2019, 2020). While such approaches have been successful in reproducing qualitative features seen in experiments including dynamic nuclear polarization and mode-locking, it remains an outstanding challenge to develop a more quantitatively accurate description of the driven electron-nuclear spin system. Allowing for higher spin is also important for capturing additional qualitative behavior that can arise from quadrupolar interactions.

In this paper, we develop a quantum, non-perturbative framework to solve the dynamics of an optically driven electron spin coupled to a bath of $I>1/2$ nuclear spins. We focus on DNP feedback mechanisms that arise from driving the electron with a periodic train of optical pulses while it is subject to hyperfine interactions with a nuclear spin bath, as in the mode-locking experiment Greilich et al. (2007b, 2006). Here, we also consider the effect of quadrupolar interactions. To compute DNP and its effect on the evolution of the electron spin, we use an approach based on dynamical maps and kinetic equations introduced in Refs. Barnes and Economou (2011); Economou and Barnes (2014), but, importantly, we generalize the formalism to higher nuclear spin and treat the problem non-perturbatively. Our framework provides a self-consistent description of the feedback loop between the driven electron and DNP.

We compute the nuclear spin polarization distribution and its influence on the electron spin evolution for spin 1 and spin 3/2 baths and compare the results to the $I=1/2$ case. Our approach is able to treat bath sizes of up to thousands of nuclear spins in the $I=1/2$ and $I=1$ cases, and up to several hundred spins in the $I=3/2$ case. Although evidence of mode-locking is seen in all three cases, we find that quadrupolar interactions act to suppress mode-locking for $I>1/2$, especially when the angle between the principal strain axis and the applied magnetic field is large. We also find that while HF interactions can produce a significant bath polarization that grows linearly with the number of nuclei for $I>1/2$, quadrupolar interactions work to counteract this buildup of DNP. We further show that the relative importance of quadrupolar effects grows as the magnitude of the applied magnetic field is increased. The competition between HF and quadrupolar interactions imprints clear signatures in the steady-state electron spin evolution, providing an experimental tool to measure the strength of quadrupolar couplings in a QD. Our results show that accounting for higher nuclear spin is important not only for quantitative accuracy, but also for capturing important qualitative features of the DNP process in driven QD systems.

The paper is structured as follows. In Sec. II, we present an overview of our framework and briefly describe each step of the calculation. In Sec. III, we lay out the approach in detail for arbitrary nuclear spin $I$ and construct the equations that govern DNP for $I=1/2$, 1, and 3/2 nuclear spin baths. We present an analytical solution for the steady-state nuclear spin polarization distribution for $I=1/2$. In Sec. IV, we numerically compute steady-state polarization distributions for $I=1$ and 3/2 and compare the results to the $I=1/2$ solution for various parameter choices. We also study the effect of DNP on the electron spin evolution. We conclude in Sec. V.

Before we describe our approach in detail, we first give an overview of the general strategy and main ingredients. This will hopefully better orient the reader for what follows and also highlight the generality of the approach, which could potentially be adapted to other time-dependent many-body problems. We still, however, frame this overview in the context of the electron-nuclear problem in QDs for the sake of concreteness and to make it readily apparent how the discussion here fits with the rest of the paper. Our framework is summarized in Fig. 1. It closely follows the approach introduced in Refs. Barnes and Economou (2011); Economou and Barnes (2014), but we outline each step in detail to keep the discussion self-contained and to make it clearer which parts must be modified to allow for higher nuclear spin.

Our focus in this work is on developing a theory that describes QD experiments in which a single electron is periodically pumped by a train of optical pulses Greilich et al. (2007b); Carter et al. (2009, 2011). Each pulse excites the electron to a trion state (a bound state of an electron and an exciton), which then decays back to the electronic ground state manifold via spontaneous emission. The underlying physical mechanism behind the formation of DNP can be understood as follows. Imagine that the electron spin starts in a pure (polarized) state and the nuclear spins are in a totally mixed (unpolarized) state. The HF interaction then transfers angular momentum from the electron onto the nuclei, creating DNP. In the absence of driving, this would lead to only a modest nuclear spin polarization, and this polarization would be short-lived because it would eventually be transferred back to the electron via the HF interaction. However, the laser pulses periodically reset the electron spin to a polarized state, enabling a net transfer of angular momentum from the laser, through the electron, and onto the nuclei. It is this transfer process that we aim to describe with our framework.

We are dealing with a system that is both open and driven. An efficient way to treat non-unitary evolution is to use dynamical maps (Nielsen and Chuang, 2010; Barnes and Economou, 2011; Economou and Barnes, 2014; Jordan and Sudarshan, 1961; Jordan et al., 2004). In this approach, the non-unitary evolution of a system from an initial state $\rho$ to a final state $\rho^{\prime}$ is implemented by applying a set of operators and summing the results:

The operators $E_{k}$ are known as Kraus operators, and they constitute a generalization of the usual unitary operators that evolve closed quantum systems to the case of non-unitary evolution in open systems. The condition $\sum_{k}E^{\dagger}_{k}E_{k}=\mathbbm{1}$ ensures that the trace of the density matrix is always unity. The advantage of Kraus operators is that they allow one to incorporate effects due to the transient occupation of excited states using operators that live purely in the ground space of the system. In the present problem, we use these operators to describe the effect of each optical pulse on the electron spin state. The entire process of optical excitation, subsequent decay, and rotation is captured by an appropriate set of Kraus operators (given in the next section) without having to explicitly include excited states or a photonic bath into the formalism. The dynamical map description works well so long as the population returns regularly to the electron spin ground states, as is the case for the periodic driving used in the mode-locking experiments.

Given a set of Kraus operators that describe the evolution of the driven electron, the next step in our theoretical framework is to find the steady-state of the electron spin. Of course, we are interested in the case where the electron spin is coupled to a nuclear spin bath through HF interactions (which are described in detail in the next section) while it is being driven. Under the condition that the electron is being pumped fast enough (which indeed is the case for the mode-locking experiments Greilich et al. (2007b)), the electron reaches its steady state on a much faster timescale compared to the electron-nuclear interaction dynamics and the electron spin decoherence time. This allows us to use a Markovian approximation in which we first solve for the driven electron steady state and then incorporate the effects due to the electron-nuclear couplings on top of this.

To bring the nuclei into the framework, we first solve for the joint evolution of one nuclear spin coupled to the driven electron spin. Although the HF interaction generates unitary dynamics, this is disrupted periodically by the pulses, and this in turn leads to an effective non-unitary dynamical map for the nuclear spin that depends on the electron steady state under the Markovian approximation. We extract nuclear spin-flip rates from this effective nuclear spin evolution operator; these rates provide information about the movement of population between the different nuclear spin levels.

We calculate the steady state of the entire nuclear spin bath using a rate equation that depends on the spin-flip rates obtained from the single-nucleus solution. A critical step is that we build in self-consistent system-environment feedback by modifying the flip rates. To understand this, we first need to describe the Overhauser effect Overhauser (1953), which is the main feedback mechanism between the electron and nuclei. A polarized nuclear spin bath acts as an effective magnetic field and therefore shifts the Zeeman frequency of the electron. However, the interaction between the electron and the nuclear spin bath is reciprocal; not only will the state of the electron change under the Overhauser field, but the nuclear spins will also be affected by the Knight field Knight (1949), i.e., the effective magnetic field due to polarization of the electron. The Knight field is given by the electron steady state spin vector, and so it enters into the nuclear spin flip rates, as explained above. The electron steady state (and hence the Knight field) in turn depends on the total magnetic field, which includes the Overhauser field due to nuclear polarization. These interdependencies constitute a complete feedback loop that must be treated self-consistently. We do this by making the nuclear spin-flip rates depend on the net nuclear polarization of the bath. The steady-state of the rate equation then gives the polarization distribution of the nuclear spin bath with feedback included. Finally, we use this nuclear polarization distribution to perform the Overhauser shift on the Zeeman frequency of the electron and update the nuclear-bath-averaged electron spin steady-state self-consistently.

The framework we have just outlined can be thought of as a self-consistent dynamical mean-field approach. In the following section, we describe each step of our formalism as it applies to the periodically driven electron-nuclear problem in full detail. Our method is quite general and can be applied to baths of any nuclear spin. We focus on the cases $I=1/2$, 1, and 3/2 to illustrate the various steps.

The full Hamiltonian of the nuclear spin bath and the driven electron is given by

Here, $H_{0,e}$ describes the electronic degrees of freedom in the QD in the absence of driving:

where $\omega_{e}$ is the electron spin Zeeman frequency, $\hat{S}_{z}$ is the spin operator in the electronic ground space, and $\omega_{\bar{T}}$ is the energy of the trion state $|\bar{T}\rangle$. We take the magnetic field to be oriented along the $z$ direction, while the optical axis lies in the $x$ direction (see Fig. 2). We neglect the second trion level $\ket{T}$ in $H_{0,e}$ because it is not excited by the laser polarization we are considering. This driving is described by the Hamiltonian

where we assume the drive laser is left-circularly polarized (red arrow in Fig. 2) with periodic temporal profile $\Omega(t+T_{R})=\Omega(t)$, so that each pulse couples the electron spin state $\ket{\bar{x}}$ to the trion state $\ket{\bar{T}}$. The latter decays via spontaneous emission with rate $\gamma_{e}$. This process arises from interactions with a photonic bath, which is represented by the term $H_{res}$. We do not give an explicit expression for this term as it is not explicitly considered in what follows. The Zeeman splitting of the nuclear spins is given by $H_{0,n}=\omega_{n}\sum_{i}\hat{I}_{z}^{i}$.

The HF interaction is given by the contact term:

where $N$ is the number of nuclei that interact appreciably with the electron. The first term is referred to as the Overhauser term, and it gives rise to an effective magnetic field seen by the electron spin in the case of nonzero nuclear spin polarization. The second term generates flip-flop interactions under which the electron spin flips with a nuclear spin. These terms are responsible for transferring angular momentum from the electron onto the nuclei, while the Overhauser term is the primary mechanism for feedback between the nuclear spin polarization and the electron spin evolution. The HF couplings $A_{i}$ are determined by the magnitude of the electronic wave function at the location of the nuclear spin $I^{i}$. However, on timescales short compared to $N/\mathcal{A}\sim\mu$s, where $\mathcal{A}$ is the total HF interaction energy, the variations in these couplings do not significantly affect the electron spin evolution Barnes et al. (2012b). Here, we focus on fast optical driving where the electron reaches a steady state over a timescale of about 100 ns Economou and Barnes (2014), which allows us to make the “box model” approximation in which all the HF couplings are taken equal: $A_{i}=A\equiv\mathcal{A}/N$ Barnes et al. (2011, 2012b).

The quadrupolar interaction is given by (Abragam, 1970; Slichter, 1996)

This interaction occurs due to the coupling of the nuclear quadrupole moment to electric field gradients caused by strain in the semiconductor lattice, and it is only present for $I>1/2$. The presence of quadrupolar interactions has lead to striking phenomena in various types of experiments conducted in QDs. A few examples include the anomalous Hanle effect Krebs et al. (2010) and suppression of spin diffusion (Maletinsky et al., 2009). Line-dragging phenomena have also been associated with the presence of quadrupolar interactions (Högele et al., 2012; Latta et al., 2009; Yang and Sham, 2013). The coupling strength $\nu_{Q}$ is referred to as the nuclear quadrupole resonance frequency, which is estimated to be around 2.8 MHz for As Maletinsky et al. (2009). The quadrupole resonance frequency generally depends on the local strain in the vicinity of each nuclear spin, and so it generally varies across the material. Here, we assume that the strain remains roughly constant over the QD, and so we take all the frequencies to be equal: $\nu_{Q}^{i}=\nu_{Q}$. The operator $\hat{I}_{z^{\prime}}$ in Eq. (6) is the component of the nuclear spin operator along the principal axis of the electric field gradient. Our focus will be on the case of QDs with cylindrical symmetry in which the electric field gradient makes an angle $\theta$ with the magnetic field. Therefore, we have $\hat{I}_{z^{\prime}}=\hat{I}_{z}\cos\theta+\hat{I}_{x}\sin\theta$, which then gives Abragam (1970):

When $\theta=0$, $H_{Q}$ creates non-uniform energy spacings between the nuclear spin levels. For $\theta\neq 0$, $H_{Q}$ has the additional effect of driving $\Delta m_{I}=\pm 1$ and $\Delta m_{I}=\pm 2$ nuclear spin-flip transitions, where $m_{I}$ is the eigenvalue of $\hat{I}_{z}$. Notice that the rate for $\Delta m_{I}=\pm 1$ transitions is maximal at $\theta=\pi/4$, while the rate for $\Delta m_{I}=\pm 2$ transitions is largest for $\theta=\pi/2$, which is also the value of $\theta$ where the non-uniformity in the energy level spacings is zero. Thus, we see that the role of $H_{Q}$ changes as $\theta$ varies from 0 to $\pi/4$, and from $\pi/4$ to $\pi/2$. Because $H_{Q}$ is $\pi$-periodic in $\theta$, it suffices to focus on the range $0\leq\theta\leq\pi/2$.

In the following subsections, we carry out the various steps of the formalism as outlined above in Sec. II. In Sec. III.1, we introduce the Kraus operators that describe the optical pumping process and subsequent spontaneous emission generated by each pulse. In Sec. III.2, we then combine these with the Larmor precession between pulses to construct a dynamical map that evolves the electron spin vector over one complete driving cycle (with duration $T_{R}$). This dynamical map is used to obtain the steady state of the electron spin vector. In Sec. III.3, we derive a dynamical map that describes the effective evolution of a single nuclear spin hyperfine-coupled to the electron and subject to quadrupolar effects, where we use the Markovian approximation to freeze the electron in its steady state as described above. We use this dynamical map to compute the nuclear spin-flip rates in Sec. III.4. We obtain non-perturbative analytical expressions for these flip rates for an $I=1/2$ bath and for $I=1$ and $I=3/2$ baths in the case of no quadrupolar interactions. Results with the quadrupolar interaction included are obtained numerically. We take into account multinuclear effects in Sec. III.5 by constructing kinetic equations for $I=1/2$, 1, and 3/2 that govern the time evolution of the nuclear spin polarization distribution of the entire bath. In these equations, we include Overhauser shifts in the flip rates to incorporate dynamical feedback effects between the electron and nuclei. The steady states of these equations describe the DNP that is created through the combination of optical pumping of the electron and hyperfine flip-flops. These equations are then solved in various cases in Sec. IV, where we also describe how the resulting polarization distributions in turn modify the evolution of the electron spin.