Implementation of long-range coupling

We implement long-range exchange coupling mediated by a multielectron quantum dot in a linear array of five quantum dots, as shown in Fig. 1a. The quintuple dot is defined in a GaAs two-dimensional electron gas by means of electrostatic gate electrodes deposited on top of the heterostructure (see Methods section). The middle dot is populated by a large even number of electrons, between 50 and 100 as estimated from the lithographic size of the device and the density of the two-dimensional electron gas. Its ground state is chosen to be spinless as described in ref. 4. Two two-electron double dots are tunnel-coupled4 on opposing sides of the large middle dot and are each initialized and read out using standard techniques for singlet–triplet qubits5,25.

Fig. 1 Detection of spin-exchange processes across a multielectron dot. a Scanning electron micrograph of the measured device. A multielectron dot is induced below the long segment of the horizontal gate electrode, while two two-electron double quantum dots are induced below its circular sections. Nanosecond voltage pulses applied to the blue-colored gates V j control the position of individual electrons and their mutual interactions. An external magnetic field (arrow) is applied in-plane of the device. b Operation steps. First, each double dot is initialized in a singlet state |SL/R〉, by populating the outer dots with two electrons each. Then, single electrons are moved to the inner dots, thereby turning off their exchange interaction with the outer electrons, which serve as reference spins. Next, the exchange coupling J between the inner electrons is induced, by temporarily raising V M by an amplitude parameterized by ε M (and lowering other gates to maintain constant overall charge). The exchange interaction causes flip-flops between electronic spins on the inner dots (for mechanisms, see Fig. 3a), which entangles the spin state of the left double dot with that of the right double dot. After an interaction time τ the resulting correlations in the relative alignment between inner spins and reference spins are detected by spin-to-charge conversion within each double dot, using two nearby sensor quantum dots (not shown) Full size image

The exchange interaction is induced by a sequence of sub-microsecond voltage pulses applied to the blue-colored gates in Fig. 1a, realizing the following steps (cf. Fig. 1b and further details about the pulse sequence in Supplementary Note 1). First, the outer dots are each populated by a pair of electrons. This initializes each double dot in the spin singlet state, \(|S^{{\mathrm{L/R}}}\rangle = (| \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle )/\sqrt 2\), where arrows indicate the spin state of the two electrons and the superscript L/R indicates the left and right double dot. Then, the electron pairs are rapidly separated within each double dot. Since electron wave functions effectively stop overlapping, this pulse turns off the exchange interaction within each double dot, allowing the outer dots to each store one reference spin. In the third step, V M is temporarily increased by ε M , while negative (compensation) pulses are applied to all other gates (see Supplementary Note 1). This induces an exchange interaction between the inner one-electron dots mediated by the large dot. After the interaction time τ the exchange-inducing pulse is switched off. Subsequently, spin-to-charge conversion is used to read out the relative spin alignment within each double dot5, independently and with single-shot fidelity (see Methods section).

The result of such a spin-exchange pulse sequence is shown in Fig. 2a. In the two panels, we plot the fraction of singlet outcomes (P S ) for each double dot as a function of τ and ε M . Oscillations in P S witness exchange-driven flip-flop processes between the two spins located on the inner quantum dots. The oscillation frequency increases for larger values of ε M . Consistent with complementary spin-leakage spectroscopy (see Supplementary Note 2), this indicates that positive pulses on gate V M lower the multielectron dot levels towards resonance with the inner dots, thereby increasing the rate of spin-exchange processes mediated by the multielectron dot.

Fig. 2 Exchange oscillations across the mediator and non-local correlations. a Fraction of detected singlet outcomes, P S , acquired simultaneously for the left and right double dot, as a function of interaction time τ and pulse amplitude ε M . The choice of detuning between inner dots, ε = −2 mV, corresponds to a symmetric operation point (cf. gray triangle in Fig. 3b). b Histograms of demodulated sensor voltages, when repeating a pulse cycle with τ = 2 ns many times, for three different choices of \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) as marked in panel (c). Counts bunch into four groups, each associated with a different combination of singlet (S) or triplet (T) measurement outcomes for the two double dots. Correlations within these single-shot measurement outcomes reveal the non-local nature of the interaction. c Joint probabilities of all four possible joint outcomes, as a function of the exchange-inducing pulse amplitude \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) for fixed interaction time τ = 2 ns. Here, \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\)-pulses are defined similarly to ε M -pulses, but with a different choice for the cross-compensation amplitudes (see Supplementary Note 1). Dashed line indicates the largest expected probability to detect TT for the maximally entangled state (see text) Full size image

Correlated measurements

Next, we demonstrate correlations between measurement outcomes for the left and right double dot. For fixed interaction time τ = 2 ns, the demodulated voltage signals for the left and right sensor (V rf,L and V rf,R ) exhibit correlations that oscillate with the amplitude of the applied pulse (Fig. 2b), confirming the non-local mechanism underlying panel 2a (in Fig. 2b, the exchange-inducing pulses parametrized by \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) are defined similarly to ε M -pulses in Fig. 2a, but employ more sophisticated cross-compensation pulses, as described in Supplementary Note 1). From these histograms, we extract the joint probabilities of detecting a singlet (S) or triplet (T) for the two double dots as a function of \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) (see Supplementary Movie 1 for animated histograms). Figure 2c clearly shows anticorrelated probabilities for detecting SS and TT, whereas the probabilities of ST and TS are small and nearly constant. The joint probabilities were extracted by fitting the histograms with four Gaussians and correcting for double-dot relaxation during the measurement pulse (see Supplementary Note 3).

The oscillatory behavior of joint probabilities results from the precession between the initialized state |SL〉|SR〉 and the maximally entangled state \(\frac{1}{2}(|S^{\mathrm{L}}\rangle |S^{\mathrm{R}}\rangle - |T_0^{\mathrm{L}}\rangle |T_0^{\mathrm{R}}\rangle + |T_ + ^{\mathrm{L}}\rangle |T_ - ^{\mathrm{R}}\rangle + |T_ - ^{\mathrm{L}}\rangle |T_ + ^{\mathrm{R}}\rangle )\). Here, the two kets denote the state of the left and right double dot, respectively, and the spin triplet states are labeled according to the standard convention, \((|T_0\rangle = (| \uparrow \downarrow \rangle + | \downarrow \uparrow \rangle )/\sqrt 2 ,|T_ + \rangle = | \uparrow \uparrow \rangle ,|T_ - \rangle = | \downarrow \downarrow \rangle )\). The coefficients associated with this entangled state explain the visibility in our measurement basis. For example, the maximum expected probability for TT is 75% in the case of perfect readout, nearly matched by the observed maxima in Fig. 2c. The observed visibility for SS and TT is further reduced by residual counts of ST and TS. We attribute this background to unintentional dynamics of the reference spins in the outer dots, arising from decoherence due to their coupling to the nuclear spin bath associated with GaAs26, and from the finite rise time of the voltage pulses.

In this experiment, we can exclude capacitive coupling as a mechanism leading to the observed oscillations, as it is small and manifests itself as a controlled phase gate between singlet–triplet qubits8,9, not as a SWAP gate between single spins.

Physical regimes of exchange interaction

In Fig. 3, we identify different mechanisms of the exchange interaction mediated by the multielectron quantum dot. For that purpose, we define a new gate voltage parameter, \(\varepsilon = (V_{{\mathrm{L2}}} - V_{{\mathrm{R1}}})/\sqrt 2 + C\) (where C is a constant; see Supplementary Note 1), which controls the relative detuning between the two inner dots, and plot the readout probabilities P S (ε, ε M ). Since τ = 6 ns is fixed for these measurements, each fringe in Fig. 3b corresponds to points of equal exchange energy J, while the density of fringes represents the gradient of J (see Supplementary Note 8 for a discussion of finite-rise-time effects). The observed exchange strength rapidly increases when ε M > 20 mV, especially for ε ≈ 0, resulting in a high density of fringes that is blurred by a combination of aliasing and decoherence. For finite |ε|, the exchange increases more slowly, resulting in a pattern that is approximately symmetric with respect to ε. While Fig. 3 presents measurements obtained from the right double dot, identical patterns are observed from the left double dot (see Supplementary Note 5). This is not the case when the occupancy of the multielectron dot is increased by one electron, as discussed in Supplementary Note 6.

Fig. 3 Physical regimes of exchange interaction. a In the Hubbard model, different spin-exchange processes dominate depending on the relative alignment of various single-particle levels (cf. Supplementary Note 7). Specifically, \(\varepsilon _{\mathrm{M}}^ \ast\) is the single-particle energy of the lowest unoccupied orbital in the multielectron dot relative to the left and right orbital, \(\varepsilon ^ \ast = (\varepsilon _{\mathrm{L}}^ \ast - \varepsilon _{\mathrm{R}}^ \ast )/2\) determines the relative detuning between the left and right orbital, and U and ε S indicate the charging energy and the level spacing of the multielectron dot. Depending on which processes are energetically allowed or suppressed, we classify different regimes as illustrated. b Measured P S (ε, ε M ) for the right double dot for fixed interaction time τ = 6 ns. Colored triangles indicate the detuning points used for Figs. 2a and 4b. Dashed lines indicate the location of independently measured charge transitions (see Supplementary Note 4). c, d Same as (b), but for reduced tunnel coupling between the multielectron dot and the inner dots. The interaction time is fixed at τ = 5 and 4 ns in (c) and (d), respectively. e Simulated \(P_{\mathrm{S}}(\varepsilon ^ \ast ,\varepsilon _{\mathrm{M}}^ \ast )\) in the Hubbard model. f Location of charge transitions (dashed lines) in the Hubbard model for the parameters used in (e). The corresponding charge configurations of the four regimes of exchange interaction are schematically indicated by dots in (a) Full size image

The observed pattern can be understood by monitoring the charge distribution during the interaction step (Supplementary Note 4), and overlaying the fringes in Fig. 3b with the observed charge transitions (dashed lines). Consistent with simulations from the Hubbard model discussed below, shown in Fig. 3e, f, we identify each region with a different electron configuration, as illustrated by dots in Fig. 3a. In region I, the inner dots remain singly occupied, and the multielectron dot keeps its initial charge state. This corresponds to an indirect exchange interaction, where virtual tunneling through the multielectron quantum dot mediates superexchange19. In regions IIa and IIb, one of the electrons has relocated onto the multielectron dot, forming an effective spin-1/2 many-body state which directly exchange-couples to the other electron spin. The mirror symmetry of IIa and IIb with respect to ε* = 0 reflects the left–right symmetry of the device, with minor deviations in the experimental data arising from a slight inequality in the tunneling barriers between the multielectron dot and the inner dots. In region III, the chemical potential of the multielectron dot is sufficiently low such that both electrons relocate onto the multielectron dot. Depending on their relative spin alignment, singlet-like or triplet-like, both electrons occupy either the lowest orbital, or the lowest and second lowest orbital, respectively. The energy difference between these spin configurations sets the coupling strength of this (rapid) onsite exchange interaction. It is related to two mesoscopic parameters, namely the single-particle level spacing of the two orbitals, and the spin correlation energy4,27,28.

The different mechanisms of the exchange interaction can also be distinguished by reducing the tunnel couplings between the multielectron and the inner dots (Fig. 3c, d). We observe P S ≈ 1 throughout region I, indicating that the reduced coupling effectively turns off the indirect exchange regime. Region I now appears separated by a sharp boundary, i.e., charge transition, from the direct exchange regions IIa,b. Similarly, in Fig. 3c, a sharp boundary between clear and blurred oscillation fringes is observed, indicating the transition between direct and onsite exchange regimes. Upon further reduction of tunnel coupling (Fig. 3d), an arc pattern is observed in region III. This behavior results from a sweet spot and a sign reversal of the exchange interaction4,23, as argued below.

Model and sweet-spot behavior

To verify all four regimes, we evaluate a Hubbard model of the two inner quantum dots coupled to the multielectron quantum dot (see level structure in Fig. 3a and Supplementary Note 7). Using realistic parameters, this model qualitatively reproduces data in Fig. 3b, including the fringe pattern and the charge stability diagram (Fig. 3e, f).

Quantitative insight into the fast dynamics of onsite exchange can be obtained by reducing τ to 2 ns. This circumvents blurring and aliasing effects, revealing a characteristic arc pattern at the transition between direct and onsite exchange regimes (Fig. 4a). This pattern (similar to the one observed in Fig. 3d) is in fact a fingerprint of the exchange profile J(ε, ε M )22: Retaking any pixel, say along a cut at ε = −8 mV (blue triangle in Fig. 3b), as a function of τ results in oscillations with frequency f = J/h21. Extracting f for the cut shown in Fig. 4b reveals a non-monotonic behavior of the exchange coupling with respect to ε M (Fig. 4d). The presence of a maximum in frequency followed by a zero crossing is similar to exchange profiles studied in refs. 4,23, and arises if direct exchange (which depends on orbital-specific tunnel matrix elements) competes with onsite exchange (which depends on electron correlation effects and, for relatively small orbital level spacing, can be negative). Accordingly, to qualitatively reproduce the chevron pattern of Fig. 4b, we must include two unoccupied orbitals of the multielectron quantum dot4,24, as well as a finite rise time of the applied voltage pulses (see Methods and Supplementary Note 7). The results of such a simulation are shown in Fig. 4d and are in good agreement with experimental data.

Fig. 4 Sweet-spot behavior and competition between direct and onsite processes. a P S (ε, ε M ) for reduced interaction time τ = 2 ns. A fingerprint pattern, related to a sweet spot in the exchange profile J(ε, ε M ), emerges at the crossover from direct to onsite regimes. Measured (b) and simulated (c) time-dependent exchange oscillations for fixed ε = 8 mV (marked by blue triangle in Fig. 3b). Enhanced oscillation visibility along the chevron pattern indicates that operation at the sweet spot prolongs coherence. d Coupling strength of the quantum mediator, demonstrating high speed (GHz), sweet spot (maximum), and sign reversal, controlled by small voltage changes in ε M . Data points represent the oscillation frequency extracted from rows in panel (b), which we identify with the exchange coupling strength J = hf. Solid line is a guide to the eye Full size image