Relaxation and readout visibility of a singlet-triplet qubit in an Overhauser field gradient

Using single-shot charge detection in a GaAs double quantum dot, we investigate spin relaxation time ($T_1$) and readout visibility of a two-electron singlet-triplet qubit following single-electron dynamic nuclear polarization (DNP). For magnetic fields up to 2 T, the DNP cycle is in all cases found to increase Overhauser field gradients, which in turn decrease $T_1$ and, consequently, reduce readout visibility. This effect was previously attributed to a suppression of singlet-triplet dephasing under a similar DNP cycle. A model describing relaxation after singlet-triplet mixing agrees well with experiment. Effects of pulse bandwidth on visibility are also investigated.

Figure 1

(a) Micrograph of device lithographically identical to the one measured. Gate voltages, $V_L$ and $V_R$, set the electrostatic energy of left and right dot. A sensor quantum dot on the right allows fast measurement of the double-dot charge state via rf reflectometry. The direction of applied magnetic field, $B$, is indicated, as well as the GaAs crystal axes. (b) Change of sensor dc conductance $\Delta g$, with double-dot charge state $(N_L,N_R)$, constrained to (1,1) and (0,2) in this work. The qubit state is controlled by the (1,1)–(0,2) energy detuning, $\epsilon$, set by gate voltages $V_L$ and $V_R$ along the diagonal axis through the markers S and M. The scaling of detuning is $\left|\epsilon \right|=\eta \sqrt{{\displaystyle \Delta V_L^2+\Delta V_R^2}}$, with a lever arm $\eta =40$ $\mu$eV/mV and voltage detunings, $\Delta V_L$ and $\Delta V_R$, from the (1,1)–(0,2) charge degeneracy.[24] Markers indicate gate voltages during pump and probe cycles. Singlet preparation at point P. Pump: $S$-$T_{+}$ mixing at point I; see text. Probe: Separation of singlet for $S-T_0$ mixing at point S and measurement at point M at variable detuning, $80\mu \mathrm{eV}<{\epsilon }_M<260\mu \mathrm{eV}$.

Figure 2

(a) Energy level diagram as function of detuning, $\epsilon$, for the (1,1) charge state except where noted. Pulse-cycle detunings, ${\epsilon }_P$ of singlet preparation, ${\epsilon }_I$ of $S$-$T_{+}$ resonance, and ${\epsilon }_S$ of $S$-$T_0$ precession, are labeled. The relaxation channels of triplet state, $T_0$, during measurement at ${\epsilon }_M$ are indicated: Charge relaxation, ${\Gamma }_S$, after $S$-$T_0$ mixing by nuclear field difference, $\Delta B_z$, and processes not involving $\Delta B_z$, at rate ${\Gamma }_T$. The $S$-$T_0$ mixing is suppressed by the exchange energy splitting, $J$, due to two anticrossings, at $\epsilon =0$ between singlet states of (1,1)–(0,2), and at $\epsilon ={\epsilon }_T\sim 300$ $\mu$eV between triplet states of (1,1)–(0,2). (Inset) Pump cycle around $S$-$T_{+}$ anticrossing at ${\epsilon }_I$. (b) Ramping detuning to ${\epsilon }_S$ over a time ${\tau }_R$ converts an initial $S$ into an admixture of $S$ and $\mid \uparrow \downarrow \rangle$ with amplitudes that depend on ${\tau }_R$. (c) Charge relaxation rate, ${\Gamma }_S$, of metastable $S$ state [Eq. (9)]; singlet probability, $P_{T-S}$, from admixture [Eq. (5)]; exchange $J$ [Eq. (6)], as functions of detuning, $\epsilon$ using experimental parameters (see text).

Figure 3

(a) Probability, $P_S$, of singlet measurement outcome (gray scale) as function of $S$-$T_0$ mixing time, ${\tau }_S$, and time, $\Delta t$, after a 60 s, $\sim 4$ MHz pump cycle; taken at $B=200$ mT. Note the near-unity singlet measurement probability with low-visibility, high-frequency oscillations at small $\Delta t$. (b) Vertical cuts through (a), showing $P_S\left({\tau }_S\right)$ curves, from which visibilities, $V$, and nuclear field differences, $\Delta B_z$, are extracted via fits (solid curves) to Eq. (10), for $\Delta t=150$ and 300 s. (c) Normalized Fourier amplitudes, $A_{\mathrm{FFT}}$ (gray scale), of data in (a). (d) Vertical cuts through (c) for $\Delta t=150$ s and 300 s following pumping cycle. (e) Overhauser field difference, $\Delta B_z$ at $B_{\mathrm{ext}}=100$ mT (black) and 20 mT (red) as function of time, $\Delta t$, after pumping, using Eq. (1), which gives $\Delta B_z\sim f_S$ mT/(6.2 MHz). $\Delta B_z$ decays faster at lower fields, as expected for spin diffusion.[3, 16] (f) Visibility, $V$, of $S$-$T_0$ oscillations as function of $\Delta t$ for data in (a) (black) and similar data at $B_{\mathrm{ext}}=20$ mT (red).

Figure 4

(a) Parametric plot of extracted visibility, $V$, of $S$-$T_0$ precession and nuclear field gradient, $\Delta B_z$, for three values of measurement-point detuning, ${\epsilon }_M$. (b) rf voltage amplitude, $\langle v_{\mathrm{rf}}\rangle$, as function of time at ${\epsilon }_M$ following separation, for two values of $\Delta B_z$, averaged over an ensemble of 8000 point with separation times ${\tau }_S$ ranging from 1 to 100 ns. Exponential fits with offsets (solid curves, formula in graph) give $T_1\sim 13$ $\mu$s for $\Delta B_z=3$ mT and $T_1\sim 0.8$ $\mu$s for $\Delta B_z=15$ mT. (c) Parametric plot of relaxation time, $T_1$, of $T_0$ state versus Overhauser gradient, $\Delta B_z$. Solid lines are a best fit of Eq. (4) over all values of ${\epsilon }_M$, with a single set of fit parameters ${\Gamma }_S\left({\epsilon }_M\right)$ [see (e)], ${\Gamma }_T\sim (40$ ${\mu \mathrm{s})}^{-1}$, and tunnel coupling, $t_T=12$ $\mu$eV, which determines $J\left({\epsilon }_M\right)$ via Eq. (6). (d) $T_1$ as function of detuning ${\epsilon }_M$, for three different $\Delta B_z$. Model (curves) based on Eq. (9) for ${\Gamma }_S\left({\epsilon }_M\right)$ with parameters from (e). (e) Singlet relaxation rate ${\Gamma }_S$ from fit of Eq. (4) to data in [(c) and (d)] along with fit to Eq. (9) (solid line) with fit parameters $\alpha \sim 11$ $\mu \mathrm{eV}{\mathrm{ns}}^{-1}$ and $\beta \sim 1600$ ${\left(\mu \mathrm{eV}\right)}^2{\mathrm{ns}}^{-1}$. The functional form is consistent with rate contributions from 3D ($\alpha$) and 2D ($\beta$) piezoelectric phonons; see Ref. 33. (f) Visibility, $V$, from fits to $S$-$T_0$ precession data [Fig. 3b], for ${\epsilon }_M=240$ $\mu$eV, along with model visibilities, with $V_T$ based on Eq. (11). Single-shot measurement visibility, $V_M$, is calculated from the measured $T_1$ and measurement signal-to-noise ratio. The pure singlet precession visibility, $V_J$, Eq. (2), reflects finite exchange, $J_S\sim 10\mathrm{neV}\sim 0.5$ mT, at point S.[29]

Figure 5

(a) Parametric plot of measured visibility, $V$ [Fig. 3f], versus nuclear field difference, $\Delta B_z$ [Fig. 4e] for applied fields of 200 and 20 mT. Model of total visibility, $V_T$, using measured relaxation time, $T_1$ [Fig. 3f]. (b) Triplet relaxation time, $T_1$, as function of nuclear field difference, $\Delta B_z$, along with the model, using Eq. (4). The $\Delta B_z$ dependence of $V$ and $T_1$ does not depend on the applied magnetic fields.

Figure 6

(a) Parametric plot of visibility, $V$, of $S$-$T_0$ precession as a function of field gradient, $\Delta B_z$, for ramp times ${\tau }_R$ [legend in (b) and (c)] from initial singlet to separation point S and back to measurement point M, along with model total visibility, $V_T$ (solid curve) based on measured $T_1$ values. (b) Relaxation time, $T_1$, as function of nuclear field difference, $\Delta B_z$, shows no dependence on ramp time, ${\tau }_R$. (c) Ramp-rate visibility factor $V_R=V/V_T$ as function of $\Delta B_z$ for different ${\tau }_R$, along with exponential fits, $V_R=e^{-\Delta B_z/B_{\mathrm{W}}}+V_0$[36] (d) $B_{90\%}$, the nuclear field difference for which $V_R=0.9$, as function of inverse ramp duration, $1/{\tau }_R$. Without an intentional ramp, ${\tau }_R\sim 3$ ns.