Spin of a Multielectron Quantum Dot and Its Interaction with a Neighboring Electron

We investigate the spin of a multielectron GaAs quantum dot in a sequence of nine charge occupancies, by exchange coupling the multielectron dot to a neighboring two-electron double quantum dot. For all nine occupancies, we make use of a leakage spectroscopy technique to reconstruct the spectrum of spin states in the vicinity of the interdot charge transition between a single- and a multielectron quantum dot. In the same regime we also perform time-resolved measurements of coherent exchange oscillations between the single- and multielectron quantum dot. With these measurements, we identify distinct characteristics of the multielectron spin state, depending on whether the dot’s occupancy is even or odd. For three out of four even occupancies, we do not observe any exchange interaction with the single quantum dot, indicating a spin-0 ground state. For the one remaining even occupancy, we observe an exchange interaction that we associate with a spin-1 multielectron quantum dot ground state. For all five of the odd occupancies, we observe an exchange interaction associated with a spin-$1/2$ ground state. For three of these odd occupancies, we clearly demonstrate that the exchange interaction changes sign in the vicinity of the charge transition. For one of these, the exchange interaction is negative (i.e., triplet preferring) beyond the interdot charge transition, consistent with the observed spin-1 for the next (even) occupancy. Our experimental results are interpreted through the use of a Hubbard model involving two orbitals of the multielectron quantum dot. Allowing for the spin correlation energy (i.e., including a term favoring Hund’s rules) and different tunnel coupling to different orbitals, we qualitatively reproduce the measured exchange profiles for all occupancies.

Popular Summary

The manipulation of electron spins is one way to encode and process algorithms in a quantum computer. While researchers have thoroughly studied the use of spins in a single qubit (the quantum equivalent of a digital bit), there has not been as much research on two-qubit devices, a necessary step for practical quantum computers. For two qubits to efficiently exchange information, they must sit very close to each other, which presents a serious fabrication challenge. One proposed workaround makes use of a third quantum party to act as both a mediator and a way to reliably separate adjacent qubits. Here, we experimentally demonstrate how a multielectron quantum dot (a nanometer-sized semiconductor speck) can meet many of the necessary requirements of a mediator.

We embed a quantum dot (containing up to 100 electrons) in an array of conventional one-electron spin qubits, and we determine systematically how its interaction with the qubits depends on electron number and electrical tuning. We report novel features of coherent spin dynamics that are particularly relevant for the implementation of long-distance coupling of spin qubits, coherent shuttling of spin states, and high-fidelity two-qubit gates in arrays of spin qubits. These new functionalities can be switched simply by changing the occupation number by one, and they can be tuned accurately with small changes in gate voltages.

The large size of the quantum dot might allow the coherent operation of complex arrays of spin qubits with minimal crosstalk and in geometries that are not restricted to one-dimensional chains. Large semiconducting dots as mediators in larger spin-qubit arrays could lead to new functionalities arising from the spin-exchange properties.

Figure 1

(a) Scanning-electron micrograph of the device, colored in gray and red to indicate metallic gate electrodes that deplete the 2DEG below the surface (negative voltages). Accumulation gates (colored in green, positive voltages) steepen the resulting confining potential of the quantum dots that form underneath. Voltage pulses applied to the gates $V_{L,M,R}$ control individual electrons in the triple-dot array on a nanosecond time scale. (b) Illustration of the electron configuration in the resulting triple quantum dot. We refer to the left and middle dot as the double dot, and to the right dot as the multielectron dot. The precise single-particle level structure and occupation number of the right dot determine the spin properties of the multielectron dot, and are the focus of this experiment. (c) Charge diagrams of the triple quantum dot in the absence of voltage pulses. The left-hand panel shows the interdot charge transition of the two-electron double quantum dot. The right-hand panel presents the charge transition at which the middle electron transfers to the multielectron quantum dot, for different initial occupations of the multielectron dot ($K-1$, $K$, and $K+1$) depending on $V_R$. Labels $P_K$, $R_K$, and $S_K$ indicate positions in gate-voltage space at which the electron pair is, respectively, prepared, read out, and separated, if appropriate time-dependent voltage pulses are applied. In particular, arrows labeled ${\zeta }_K$ and ${ϵ}_K$ indicate axes in gate-voltage space used to define the voltage pulses in this experiment. For example, gate voltages associated with the interaction step ($I_K$) are varied systematically (see Sec. 2), but always remain on the ${\zeta }_K$ or ${ϵ}_K$ axis. (d) Operating principle of probing the multielectron spin state by the two-electron double dot. First, a pair of electrons is prepared in a singlet state on the left dot. Next, one of these electrons is transferred to the middle dot, allowing a spin-sensitive interaction ($J$) with the multielectron dot. In the last step, the spin of the middle electron is measured relative to the reference electron in the left dot by means of Pauli blockade. (e) Implementation of (d) by voltage pulses along ${\zeta }_K$ and ${ϵ}_K$. The outcome of each interaction cycle depends on pulse amplitude ($ϵ$) and duration ($\tau$), and, crucially, on the occupation and spin of the multielectron dot.

Figure 2

Schematic of the even-occupied spinless multielectron dot, coupled to the two-electron double dot. Symbols ${ϵ}_{L/M/R}$ indicate the single-particle energies of the lowest orbitals in the double dot and the lowest unoccupied orbital in the multielectron dot. Arrows indicate tunnel couplings within the double dot ($t_{\mathrm{DD}}$) and between the middle and the multielectron dot ($t$). For simulations, the detuning of the right dot relative to the left dot is varied (${ϵ}^{*}$), which allows the generation of energy spectra [as in Fig. 3] and comparison to experimental leakage spectroscopy data [see discussion of Figs. 3 and 3].

Figure 3

Spin-0 ground-state behavior of the $2N$ occupied multielectron dot. (a) Schematic representation of the electron configuration for even-occupied multielectron dot with spin-0 ground state. Electrons below the Fermi energy in the multielectron dot are assumed to pair up into spin singlets. (b) Eigenstates of a two-electron triple dot system for fixed applied magnetic field, calculated as a function of ${ϵ}^{*}$. For suitable input parameters the resulting spin states (labeled) are a useful model for comparison with experimental data obtained from multielectron-dot charge states $(2,0,2N)$, $(1,1,2N)$ and $(1,0,2N+1)$ (see main text). (c) Experimental leakage spectroscopy for $K=2N$, revealing strong exchange coupling in $(2,0,2N)$, i.e., within the double dot, and vanishing exchange interaction in $(1,0,2N+1)$. We associate the sharp feature of suppressed $P_S$ (white triangle) with the $S-T_{+}$ crossing and the overall suppression of $P_S$ above $ϵ\approx 45\mathrm{mV}$ with $S-T_0$ oscillations arising from Overhauser gradients. (d) Leakage spectrum expected from the $S-T_{+}$ crossing within the Hubbard model (white triangle in panel (b)). (e) Time-resolved measurement of coherent oscillations between $|S⟩$ and $|T_0⟩$ two-electron spin states in $(1,0,2N+1)$ charge state, for $B_{\parallel }=700\mathrm{mT}$ and $ϵ=60\mathrm{mV}$. The observed fluctuations of precession frequencies with laboratory time $\mathcal{T}$ are characteristic of fluctuating Overhauser field gradients arising from nuclear spin dynamics within the GaAs sample.

Figure 4

Leakage spectroscopy of the double dot coupled to the multielectron dot at the transition between (a) $(2,0,2N-2)$, $(1,1,2N-2)$, and $(1,0,2N-1)$ and (b) $(2,0,2N-4)$, $(1,1,2N-4)$, and $(1,0,2N-3)$.

Figure 5

(a) Schematic of the three-electron triple dot, which serves as a reference for discussing the odd-occupied multielectron dot tunnel coupled to the double dot. (b) Energy diagram of the three-electron triple-dot spin states for a finite external magnetic field. (c) Measured leakage spectrum for the three-electron triple dot. Markers indicate leakage features attributed to the level crossings marked in (b). (d) Measured exchange oscillations reveal a monotonically increasing frequency, corresponding to monotonically increasing $J_R(ϵ)$ in (b).

Figure 6

Schematic of a two-electron double quantum dot coupled to an odd-occupied spin-$1/2$ multielectron dot. Symbols ${ϵ}_{L/M/R1/R2}$ indicate single-particle energies of the lowest orbitals in the double dot and the two lowest orbitals above the effective vacuum in the multielectron dot. Arrows indicate tunnel couplings between the left and middle orbital ($t_{\mathrm{DD}}$) and between the middle orbital and each of the two orbitals in the multielectron dot ($t_{1/2}$). The single-particle energy difference between the two orbitals on the multielectron dot is indicated by $\mathrm{\Delta }E={ϵ}_{R2}-{ϵ}_{R1}$. Detuning ${ϵ}^{*}=({ϵ}_L-{ϵ}_{R1})/2$ is varied when calculating the energy diagram and leakage spectrum presented in Figs. 7, 7, 9, and 9. Within the range of relevant parameters, low-lying orbitals in the right dot remain doubly occupied (i.e., spinless) and can be ignored when solving Eq. (3), thereby yielding a three-electron Hubbard model. [Similarly, a four-electron Hubbard model is solved to model the spin-1 case in Figs. 13 and 13].

Figure 7

(a) Schematic of the odd-occupied multielectron quantum dot with spin-$1/2$ ground state, tunnel coupled to the two-electron double quantum dot. (b) The inferred energy diagram at the transition between $(2,0,2N-3)$, $(1,1,2N-3)$, and $(1,0,2N-2)$ charge configurations, for a finite magnetic field $B_{\parallel }=330\mathrm{mT}$. The markers indicate crossings revealed by the leakage spectroscopy measurement presented in (c). Energies are measured relative to the energy of the state $|Q⟩\propto |\uparrow \uparrow \downarrow ⟩+|\uparrow \downarrow \uparrow ⟩+|\downarrow \uparrow \uparrow ⟩$. (d) Time-resolved measurement of exchange oscillations between the $2N-3$ occupied spin-$1/2$ multielectron dot and the middle electron, obtained at zero magnetic field. The dashed line indicates pulse parameters for which the resulting quantum state is to first order insensitive to fluctuations in $ϵ$. (e) Leakage spectrum expected from the Hubbard model. (f) Exchange profile $J(ϵ)$ extracted from the pattern of exchange oscillations in (d).

Figure 8

Leakage spectroscopy (a) and time-resolved measurement of exchange oscillations (b) for the multielectron dot occupied by $2N-1$ electrons.

Figure 9

(a) Schematic of the odd-occupied multielectron quantum dot with spin-$1/2$ ground state, tunnel coupled to the two-electron double quantum dot. (b) Calculated energy diagram at the transition between $(2,0,2N+1)$, $(1,1,2N+1)$, and $(1,0,2N+2)$ charge configurations, for a finite magnetic field $B_{\parallel }=900\mathrm{mT}$ and input parameters motivated by the observed spectrum in (c). Markers indicate crossings revealed by the leakage spectroscopy measurement presented in (c). Energies are measured relative to the energy of the state $|Q⟩\propto |\uparrow \uparrow \downarrow ⟩+|\uparrow \downarrow \uparrow ⟩+|\downarrow \uparrow \uparrow ⟩$. (d) Time-resolved measurement of exchange oscillations between the $2N+1$ occupied spin-$1/2$ multielectron quantum dot and the middle electron, obtained at zero magnetic field. Color scale as in (c). The dashed line indicates pulse parameters for which the resulting quantum state is to first order insensitive to fluctuations in $ϵ$. (e) Leakage spectrum expected from the Hubbard model. (f) Exchange profile $J(ϵ)$ extracted from the pattern of exchange oscillations in (d).

Figure 10

Leakage spectroscopy (a) and time-resolved measurement of exchange oscillations (b) for the multielectron dot occupied by $2N-5$ electrons.

Figure 11

Leakage spectroscopy (a) and time-resolved measurement of exchange oscillations (b) for the multielectron dot occupied by $2N+3$ electrons.

Figure 12

Illustration of qualitatively different exchange profiles arising from the interplay between the level spacing in the multielectron quantum dot $\mathrm{\Delta }E$, spin correlation energy $\xi$, and tunnel couplings between a single-electron dot and two lowest orbitals of the multielectron quantum dot $t_{1/2}$. Colored lines in the insets I–IV represent the energies of the three-spin states with $S=1/2$, $S_z=-1/2$ as a function of detuning ${ϵ}^{*}$.

Figure 13

(a) Schematic of the even-occupied multielectron quantum dot with spin-1 ground state, tunnel coupled to the two-electron double quantum dot. (b) The inferred energy diagram at the transition between $(2,0,2N+2)$, $(1,1,2N+2)$, and $(1,0,2N+3)$ electronic configurations, for a finite magnetic field. The markers indicate the crossings revealed by leakage spectroscopy presented in (c). (d) Time-resolved measurement of exchange oscillations between the $2N+2$ occupied spin-$1/2$ multielectron quantum dot and the middle electron. (e) Calculated leakage spectrum, extracted from the calculated energy diagram as described in Sec. 4. (f) Dependence of the exchange energy extracted from the leakage spectroscopy in (c). (g) Leakage spectroscopy in a magnetic field applied perpendicular to the 2DEG plane, $B_{\perp }$.