# Coherence Characteristics of a GaAs Single Heavy-Hole Spin Qubit Using a Modified Single-Shot Latching Readout Technique

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Sample and Single-Shot Readout

_{x}Ga

_{1−x}As (x = 0.5) undoped heterostructure using fine surface Schottky gates fabricated by e-beam lithography. A global accumulation gate is deposited over a 110 nm Al

_{2}O

_{3}dielectric layer and is used to generate 2D holes at the GaAs/AlGaAs interface located 65 nm below the surface [14]. Gates labeled as V

_{L}and V

_{R}are used to tune the energy levels of the dots and to perform the spin manipulations and readout protocols described below. Voltage detuning pulses and microwave (MW) bursts of variable amplitude durations are applied to the left control gate, V

_{L}. Current through the nearby charge sensor labeled in Figure 1a as I

_{CS}is used to detect single-hole charges and map out the charge occupation of the DQD device as a function of V

_{L}and V

_{R}gate voltages in the form of a stability diagram shown in Figure 1b. The charge sensor current was amplified by a room temperature current-voltage converter (Basel Precision Instruments SP983C) and sent to a digital voltmeter set to 1 NPLC = 16 ms averaging time. More details on the device have been presented in our previous publications reviewed in ref. [7]. The device is capable of reaching the last hole regime in each dot. Figure 1b shows a stability diagram covering the four charge configurations of this regime. The vertical dash line accompanied by points is the detuning line used in the manipulation and readout protocols that will be described below. In this study, a single-hole spin qubit is formed in the left dot. This state is manipulated using MW pulses applied to the left control gate, V

_{L}, while the right QD is used for fast spin readout and as a memory register to store the information.

_{L}< 20 ns). By contrast, the tunneling barrier from the right dot to the right lead was very slow (t

_{R}>> 16 ms). The tunneling time between the two dots on resonance was set to ${t}_{C}\approx 100\mathrm{ns}$. This satisfies the condition for the technique to work which is ${t}_{L}\ll {t}_{C}\ll {t}_{R}$. The detailed protocol used in the measurements is shown in Figure 2. The pulse sequence is illustrated in Figure 2a together with times and labels relevant to the positions in the stability diagram. In Figure 2b, the critical locations labelled in Figure 2a resulting from the pulse are indicated on the stability diagram (M—measure, T—transfer, I—initialization, and D—drive). In Figure 2c, schematics of the energy levels and the occupations during the whole sequence (stages (1) to (7)) are presented. To begin describing the sequence, we consider the occupation possibilities resulting from the previous cycle since this is required to perform the necessary initialization. At the end of any cycle (at position M), the two charge occupation possibilities are either a single hole in the right dot (0, 1) or an empty system (0, 0). For each new cycle, it is first necessary to initialize the system in the ground state with a spin-up hole in the left dot. This is accomplished by first ensuring the system is empty by transferring the hole in the right dot via the left dot to the left lead, i.e., (0, 1) to (1, 0) to (0, 0) (steps 1 through 3 in the sequence), and then adding a single hole from the left lead to the left dot. By waiting sufficiently longer than the T

_{1}spin relaxation time, we can ensure the system is initialized in the spin-up state in the left dot (Step 4 in the sequence). In this work, we used an initialization wait time of 10 ms, which is much longer than the ${T}_{1}$. In Figure 2c, the schematics illustrate both readout situations i.e., where the system is in either of (0, 0) or (0, 1) at the end of the previous cycle. The single hole in the left dot is then manipulated with microwave pulses to perform specific measurements using Rabi, Ramsey, Hahn echo, or CPMG pulse sequences. This is step 5 in the diagram shown in Figure 2. At the conclusion of the microwave sequence, the readout is initiated by aligning the upper spin level in the left dot with the lower spin level in the right dot. If the upper spin level of the left dot is occupied at the end of the microwave pulses, it will have a 50% occupation probability of transferring to the right dot at the end of the alignment step (i.e., creating a (0, 1) charge occupation). If the lower spin state is occupied at the end of the manipulation, the system will remain in the (1, 0) state at the end of the transfer process. An alignment time of ~200 ns was found to be optimal (step 6 in the sequence) for the hole transfer step. At the conclusion of the transfer step, the left dot, if occupied, is emptied into the left lead (step 7 in the sequence). Thus at the end of the whole sequence there are two charge occupation possibilities (which are easily distinguished by the charge detector as described above). If the hole occupied the lower spin state (up) at the end of the manipulation, the device charge state would be (0, 0). If the upper (spin up) state were occupied, then each of the (0, 1) and (0, 0) charge configurations would be occupied with a 50% probability. The measurements involve monitoring the charge occupation of the system after each manipulation pulse.

## 3. Rabi Oscillations

_{0}= 18.91 GHz.

_{L}. In the experiment presented in Figure 3b, the MW frequency was kept at the peak frequency while the pulse duration and the output amplitude of the MW generator were varied. As expected, the Rabi frequency increased with the MW amplitude. It is also qualitatively evident that the Rabi coherence time ${T}_{Rabi}$ is longer at larger MW driving amplitudes. This is consistent with Figure 3a, where the apparent coherence of the Rabi oscillations diminishes with detuning from the central frequency, which is equivalent to reducing the effective driving force.

_{Rabi}saturated at about 50 MHz. It indicates that the hole spin $\pi $ rotation can be performed in 12 ns, which is very close to the reported values of Rabi rotations in electronic GaAs DQD devices equipped with micromagnets [16]. his is an impressive result as it is expected to be more challenging to manipulate heavy hole spins due to their large g-factor anisotropy [4].

_{MW}= f

_{0}, and fitted each trace with an exponentially decaying sinusoid $P\left(\uparrow \right)\propto \mathrm{exp}\left(-t/{T}_{Rabi}\right)\mathrm{sin}\left(2\pi \left(t-{t}_{0}\right)/w\right)$, where $t$ is the MW pulse duration, ${T}_{Rabi}$ is the Rabi coherence time, ${t}_{0}$ is an instrumental phase factor, and $w$ = 1/f

_{Rabi}is the Rabi period. From this procedure, we determined two fitting parameters, T

_{Rabi}and f

_{Rabi}, both of which vary with the MW amplitude. Since the MW amplitude reaching the device is less well calibrated, it is instructive to examine the dependence of ${T}_{Rabi}$ as a function of ${f}_{Rabi}$, which is plotted in Figure 3d. Note, because such a dependence of ${T}_{Rabi}\left({f}_{Rabi}\right)$ has not yet been discussed in the literature, we approximate it with a linear dependence (dashed line). Qualitatively, we can conclude that ${T}_{Rabi}$ increases with the Rabi frequency, corresponding to an increasing driving amplitude. This behavior is expected if ${T}_{Rabi}$ is limited by the hyperfine interaction [6]. However, decoherence that occurs within a single trace is not explicitly taken into account in Ref. [6], rather, the decay is attributed to averages resulting from different hyperfine fields for each individual trace. In the data presented in Figure 3d, the linear dependence, when extrapolated to zero, intersects the Y-axis at ${T}_{Rabi}$ = 40 ns, not at zero. We speculate that this indicates that ${T}_{2}^{*}$ free evolution decoherence processes need to be taken into account to simulate the dependence ${T}_{Rabi}\left({f}_{Rabi}\right)$ presented in Figure 3d. Further theoretical and experimental work is required to fully understand decoherence in Rabi measurements. We note that decoherence that occurs within a single trace is not explicitly taken into account in the theoretical treatment in Ref. [6]; rather, the decay is totally attributed to averages resulting from different hyperfine fields for each individual trace.

_{0}leads to a reduced excitation strength, effectively to a reduced power. Therefore, the two phenomena presented in Figure 3d,c should be considered together to understand the underlying microscopic mechanisms affecting the Rabi coherence time.

## 4. Ramsey and CPMG Results and Discussion

_{1}measurements in Figure 4e. The spin relaxation ${T}_{1}$ measurements agree closely with the previous results confirming the very good stability of the device for spin relaxation as well as the independence of ${T}_{1}$ on specific gate voltages (i.e., fine details of the dot potential). This is not trivial, for example, the ${T}_{1}$ spin relaxation time may be affected by heavy hole–light (HH-LH) hole interactions which may be expected to depend upon details of the confining potential as defined by the gate voltages. In our device, the confining potential is very shallow, resulting in the light-hole states lying above the confining DQD barrier resulting in an extended spatial distribution of the light-hole wave function. This, on one hand, minimizes the HH-LH interaction, while on the other hand, it makes certain properties of the device more predictable, including the absolute value of the effective g*-factor and it large anisotropy with respect to the magnetic field direction [4].

## 5. Conclusions

_{CPMG}= 0, 1 corresponding to Ramsey and Hahn echo experiments, correspondingly. Despite the p-type nature of the hole wave function, which is predicted to lead to a reduced decoherence dueto hyperfine interactions, unexpectedly, we observe very similar ${T}_{2}^{*}$ magnitudes to the electronic spin qubits in GaAs quantum dots. We explain this observation by the approximately 10-fold smaller size of the hole quantum dots due to larger effective mass leading to larger fluctuations $\propto 1/\sqrt{N}$ of the nuclear field bath, where N is number of nuclei being in contact interaction with the single hole spin. We also explored the coherence properties of a single hole spin characterizing it by various time constants, ${T}_{Rabi},{T}_{2}^{Ramsey}$ and ${T}_{2}^{CPMG}$ as a function of the MW detuning frequency, excitation amplitude, and ${N}_{CPMG}$. Even though the measured coherence time in our single hole GaAs device is smaller than ones reported in germanium and silicon quantum dots [25,35,36,37,38], GaAs remains a promising material for spin optoelectronic devices and applications, such as photon to spin transducers in long-range quantum repeater architectures [2,3]. The discussed microscopic physical mechanisms limiting the coherence time may also be relevant and useful for other material systems. The geometry of our dot electrostatic potential is very anisotropic [4], which is known to strongly affect coherence times, as is found in Ref. [39]. It is possible that the relaxation parameters of hole qubits in GaAs can be extended by making the QD confining potentials more circular isotropic.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) An SEM image of the double quantum dot (DQD) device similar to the one used in this work but without the global gate laid over the fine surface gates. The yellow and red arrows show the direction of the charge sensing and inter-dot currents. The blue circles indicate the approximate locations of the two quantum dots. The labeled V

_{L}and V

_{R}gates are used to control the dot occupation and the detuning energy during the spin qubit manipulations. (

**b**) An example of the charge stability diagram measured via the charge sensor current, dI

_{CS}/dV

_{L}. Each region is labeled with the left and right dot hole occupations (n

_{L}, n

_{R}). The detuning trajectory is indicated as the dashed white line with the detuning points of the control pulse sequence during the manipulation and readout protocol labeled as measurement (M), transfer (T), initialization (I), and (D) Drive.

**Figure 2.**(

**a**) The control pulse sequence used in the spin manipulation and read-out experiments. The steps are enumerated in brackets (N) in correspondence with the energy diagrams in (

**c**). The left axis is given in millivolts of the control pulse stages labeled by a letter as measurement (M), transfer (T), initialization (I), and (D) Drive. The same letters are used in the diagrams (

**b**,

**c**). (

**b**) A schematic of the stability diagram is shown in Figure 1b nb. The gate detuning voltages $\mathsf{\Delta}{V}_{L}$ and $\mathsf{\Delta}{V}_{R}$ are presented with respect to the drive point D. The vertical position of the stage points (D, I, T, M) match those marked in the Y-axis of (

**a**). (

**c**) The sequence of DQD energy diagrams with indicated occupations and transitions at each detuning point produced by the control pulse in (

**a**). (For details see text).

**Figure 3.**(

**a**) Average spin-up probability to map out the Rabi oscillations as a function of the MW burst duration (Y-axis) and the MW frequency (X-axis); (

**b**) the 2D map of the Rabi oscillations measured as the average spin-up probability in the plane of the MW Pulse Duration and MW Amplitude; (

**c**) Rabi frequency as a function of MW pulse amplitude; (

**d**) Period of Rabi oscillations as a function of Rabi frequency tuned by the MW burst amplitude; (

**e**) An example of Rabi oscillations as a function of the MW burst duration; (

**f**) Period of Rabi oscillations as a function of magnetic field stepped around the resonance frequency. The top scale shows effective frequency detuning. (For more details see text).

**Figure 4.**(

**a**) Ramsey fringes measured by the spin-up probability using a sequence of two $\frac{\pi}{2}$ MW pulses of different frequencies (X-axis) separated by the wait time (Y-axis). Dotted lines indicate the expected position of the fringes (for more details see text); (

**b**) Spin-up probability measured in the Ramsey experiment (points) fitted by an exponential decay (solid line); (

**c**) spin up probability measured in Hahn echo experiment (points) fitted with an exponential step function (solid line); (

**d**) coherence time for different number of refocusing CPMG pulses for two values of magnetic field (for details see text); (

**e**) longitudinal spin relaxation time ${T}_{1}$ combined with Ramsey and Hahn echo results.

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**MDPI and ACS Style**

Marton, V.; Sachrajda, A.; Korkusinski, M.; Bogan, A.; Studenikin, S.
Coherence Characteristics of a GaAs Single Heavy-Hole Spin Qubit Using a Modified Single-Shot Latching Readout Technique. *Nanomaterials* **2023**, *13*, 950.
https://doi.org/10.3390/nano13050950

**AMA Style**

Marton V, Sachrajda A, Korkusinski M, Bogan A, Studenikin S.
Coherence Characteristics of a GaAs Single Heavy-Hole Spin Qubit Using a Modified Single-Shot Latching Readout Technique. *Nanomaterials*. 2023; 13(5):950.
https://doi.org/10.3390/nano13050950

**Chicago/Turabian Style**

Marton, Victor, Andrew Sachrajda, Marek Korkusinski, Alex Bogan, and Sergei Studenikin.
2023. "Coherence Characteristics of a GaAs Single Heavy-Hole Spin Qubit Using a Modified Single-Shot Latching Readout Technique" *Nanomaterials* 13, no. 5: 950.
https://doi.org/10.3390/nano13050950