Complete Readout of Two-Electron Spin States in a Double Quantum Dot

We propose and demonstrate complete spin state readout of a two-electron system in a double quantum dot probed by an electrometer. The protocol is based on repetitive single-shot measurements using Pauli spin blockade and our ability to tune on fast timescales the detuning and the interdot tunnel coupling between the GHz and sub-Hz regime. A sequence of three distinct manipulations and measurements allows establishing if the spins are in $S$, $T_0$, $T_{+}$, or $T_{-}$ state. This work points at a procedure to reduce the overhead for spin readout, an important challenge for scaling up spin-qubit platforms.

Popular Summary

Quantum information processing comes with the promise of overcoming the current computing limitations reached by classical computers on a certain class of problems. A quantum computer is not based on classical bits that can take only the 0 or 1 values but on quantum bits, or qubits, that are often encoded on two possible states of a quantum object. A potential candidate to physically implement qubits is the spin state of single electrons stored in potential traps called quantum dots.

In these systems, high-fidelity readout is usually obtained by a spin to charge, a conversion process based on the Pauli exclusion principle. The building block to implement it is a tunnel-coupled double quantum dot filled with two electrons. One acting as a qubit and one acting as an ancilla. At the end of the measurement only one bit of information is extracted from the unit cell (singlet or triplet) despite a system containing potentially two bits.

Obtaining the full spin information of the two-electron system in this simple unit cell requires, nevertheless, extra hardware such as quantum dots or electron reservoirs, that represents an important overhead for future scaling of the platform. In this article, we propose and demonstrate a strategy to perform complete spin-state readout of a two-electron system. It is based on the repetition of three single-shot measurements using Pauli spin blockade and our recent development in the control of the tunnel coupling between the GHz and the sub-Hz regime.

Figure 1

Frozen Pauli-spin-blockade measurement: protocol and fidelity. (a) Stability diagram and frozen Pauli-spin-blockade measurement scheme. The voltage $V_{B2}$ controls mostly the tunnel coupling of the DQD while $V_{B3}$ controls $ϵ$. The tunnel coupling can be controlled from the sub-Hz to GHz regime in a few hundreds of mV applied to the $B_2$ gate. The blue and red triangles indicate the positions used to initialize $S$ and mixed $S{T}_0$ spin states, respectively. The yellow circle indicates the position of the $S{T}_{+}$ avoided crossing. The PSB position is indicated by the white star, the charge-readout position by the white square while the white pentagon indicates the position used to perform selective relaxation of the $T_0$ spin state to $S$. (b) Sketch of the energy diagram of the four first spin states of two electrons in a DQD as a function of the interdot tunnel coupling when the two QD chemical potentials are equal ($ϵ=0$). At the blue triangle the exchange energy in between the two spins is high enough to preserve the singlet state. At the red triangle the exchange energy is low compared to the magnetic field gradient and mixing of the initial $S$ spin state occurs with the $T_0$. At the yellow circle, the $S$ and $T_{+}$ states are degenerated and mixing occurs via the Overhauser magnetic field and spin-orbit coupling. (c) Histogram of 50 000 single-shot measurements with an integration time of $T_{\mathrm{int}}=1.5\mathrm{ms}$. Following Connors et al. [6] we define a threshold (dashed line) to discriminate the (1,1) and (0,2) charge state minimizing the readout error. (d) Fidelity of the frozen PSB readout procedure as a function of the electrometer signal integration time. For $T_{\mathrm{int}}<0.5\mathrm{ms}$, the spin-readout error is limited by the charge-readout error $C_{\mathrm{error}}$. For longer integration times, the spin-readout error saturates and reaches $S_{\mathrm{error}}=0.43\pm 0.05\mathrm{\%}$ and $T_{\mathrm{error}}=2.7\pm 0.4\mathrm{\%}$. The uncertainty is represented by the shade around the solid line.

Figure 2

Tools for the complete readout procedure: parity readout via $T_0$ relaxation hotspot, $S$-$T_{+}$ adiabatic transformation. (a) Energy diagram of the different spin states of two electrons in a DQD as a function of $ϵ$. The fast relaxation channel of the $T_0$ is indicated via the black arrow. (b) Selective relaxation at the parity readout hotspot. The system is initialized in either a $S$-$T_0$ or $S$-$T_{+}$ mixed state and then pulsed during a variable time $T_{\mathrm{HS}}$ at the parity-readout position. (c) Localization of the $S$-$T_{+}$ avoided crossing (indicated by the yellow circle) in the voltage-gate space. (d) Singlet return probability after a ramp through the $S$-$T_{+}$ avoided crossing.

Figure 3

Complete readout procedure and description of the three-step measurement. (a) Typical measured current across the SET during the three different steps of the readout for the four possible spin states. Blue shaded areas correspond to a pulse to the PSB position, the green shaded one is when the system is at charge-readout position. The PSB, parity and $S$-$T_{+}$ shaded area are artificially extended in the chronogram and the SET traces are slightly offset for clarity. The white area corresponds to the time needed for the system to stabilize for readout after a manipulation. (b)–(d) Output of the complete readout procedure for singlet state, $S$-$T_0$ mixed state, and $T_{+}$ initialization. The inset corresponds to the expected result for an errorless complete readout procedure.

Figure 4

Spin-state operations observed via the complete readout procedure. (a) Pulse sequence performed to obtain (b). The system is initialized in a singlet state at a large value of $t_c$ ($V_{B2}=-0.6\mathrm{V}$) in the (1,1) charge-state region ($V_{B3}=-1.06\mathrm{V}$), the interdot gate $B_2$ is pulsed during 500 at a variable amplitude $V_{B2}$. A complete readout procedure is then performed, the output is classified using the methodology described in Fig. 3. (b) Singlet transformation with respect to the interdot tunnel coupling. (c) Pulse sequence used to perform $S$-$T_0$ exchange oscillations. The initial $S$ spin state is conserved via a nonadiabatic pulse across the $S$-$T_{+}$ avoided crossing from the blue triangle position to the green triangle one. It is then transformed adiabatically to $|\uparrow \downarrow ⟩$ or $|\downarrow \uparrow ⟩$ (depending on the sign of $\mathrm{\Delta }{B}_z$) via a ramp from the green triangle to the red one. From there, the exchange coupling $J$ is increased via an interdot tunnel coupling pulse $V_{B2}^{\mathrm{exch}}$ during a fixed time ${\tau }_{\mathrm{exch}}=1.6$. During this time the system evolves at a frequency $J$ between $|\uparrow \downarrow ⟩$ and $|\downarrow \uparrow ⟩$, and finally the mirrored sequence is applied to stop the oscillations and map the two states to $S$ and $T_0$. (d) Spin-state probability at the end of the exchange pulse sequence. As expected the $S$ and $T_0$ probability are oscillating out of phase. We observe a negligible leakage to the $T_{+}$ and $T_{-}$ state proving the efficiency of the procedure to perform a swap operation.

Figure 5

Electron micrograph of a sample similar to the one measured. The three QDs (lower dashed circles) are probed via a local electrometer (top dashed circle). The green gates mainly control the dot-dot and dot-reservoir tunnel rates. The red gates are biased with negative voltages to prevent leakage between the gates. Four electron reservoirs are located at the white crossed squares and are used to either load electrons in the QD array for the bottom ones or apply a bias through the SET for the top ones.

Figure 6

Calibration of the Pauli-spin-blockade position in the voltage-gate space. (a) Pulse sequence performed to calibrate the PSB position. The system is initialized in a $S$ or mixed $S$-$T_0$ state via a 500 ns voltage pulse at the blue or red triangle, respectively. It is then pulsed to the PSB position (white star) during 20 ns to perform a spin-to-charge conversion and the configuration of the DQD is read at the white square position during 5 ms. (b) PSB calibration experiment. The pulse sequence described in (a) is performed for a varied position of the PSB along the detuning axis. We plot the (0,2) charge-state probability as a function of the PSB position along the dotted line $V_{B3}^{\mathrm{PSB}}$ for the two initializations possible. The readout position is chosen at the white star position where the mixed-state probability reaches 50%.

Figure 7

Calibration of the relaxation hotspot in the voltage-gate space. (a) Experimental workflow used to identify the relaxation hotspot where parity readout is performed. The system is initialized in a singlet state and pulsed in the (1,1) charge region at a low value of tunnel coupling in order to prepare an equipartition of $S$ and $T_0$ population. The system is then pulsed during $5\text{μ}\mathrm{s}$ at coordinates ($V_{B2}^{\mathrm{PR}}$,$V_{B3}^{\mathrm{PR}}$), finally a FPSB procedure is applied to determine the proportion of $S$ at the end of the relaxation pulse. (b) Singlet probability at the end of the pulse sequence presented in (a) as a function of the pulse coordinates ($V_{B2}^{\mathrm{PR}}$,$V_{B3}^{\mathrm{PR}}$). We observe lobes of high singlet probability indicating that the initialized $T_0$ state has relaxed to the $S$ state during the $5\text{μ}\mathrm{s}$ pulse. The position C where the highest singlet probability is recorded, is selected as the parity-readout position. (c) Energy diagram of the relevant spin states and identification of the region where $|T_0(1,1)⟩$ and $|S(1,1)⟩$ mixing can occur. Once the $|T_0(1,1)⟩$ state is transformed it relaxes rapidly through phonon emission to $|S_1(0,2)⟩$.