Quantum Simulation of Antiferromagnetic Heisenberg Chain with Gate-Defined Quantum Dots

Quantum-mechanical correlations of interacting fermions result in the emergence of exotic phases. Magnetic phases naturally arise in the Mott-insulator regime of the Fermi-Hubbard model, where charges are localized and the spin degree of freedom remains. In this regime, the occurrence of phenomena such as resonating valence bonds, frustrated magnetism, and spin liquids is predicted. Quantum systems with engineered Hamiltonians can be used as simulators of such spin physics to provide insights beyond the capabilities of analytical methods and classical computers. To be useful, methods for the preparation of intricate many-body spin states and access to relevant observables are required. Here, we show the quantum simulation of magnetism in the Mott-insulator regime with a linear quantum-dot array. We characterize the energy spectrum for a Heisenberg spin chain, from which we can identify when the conditions for homogeneous exchange couplings are met. Next, we study the multispin coherence with global exchange oscillations in both the singlet and triplet subspace of the Heisenberg Hamiltonian. Last, we adiabatically prepare the low-energy global singlet of the homogeneous spin chain and probe it with two-spin singlet-triplet measurements on each nearest-neighbor pair and the correlations therein. The methods and control presented here open new opportunities for the simulation of quantum magnetism benefiting from the flexibility in tuning and layout of gate-defined quantum-dot arrays.

Popular Summary

Quantum simulators can offer insights into properties of matter with underlying quantum-mechanical correlations. Interactions between spins induce such correlations and can result in complex magnetic behavior. Here, we compose artificial quantum matter out of electrons confined in a semiconductor and demonstrate quantum simulation of magnetism when only the spin degree of freedom remains.

The electrons are electrostatically confined in a linear array of quantum dots, with each site occupied with only one electron. The electrons then form an antiferromagnetic spin chain with exchange interactions arising from wave function overlaps, which can be controlled with voltages on the gate electrodes. We then extract properties of the spin chain using several techniques. First, we study the energy spectrum as a function of interaction strengths and an external magnetic field. Then, we induce global coherent oscillations of the spin chain. Finally, we prepare the antiferromagnetic ground state of the spin chain with homogenous interactions, which we characterize with correlation measurements on all nearest-neighbor pairs of spins.

By leveraging the in situ control of interaction strengths between spins, the diversity of techniques to extract information, and the design flexibility of lithographically defined quantum dot lattices, the quantum-dot platform offers a broad range of opportunities to study quantum magnetism.

Figure 1

Device and spin-chain operation. (a) False-colored scanning electron micrograph of a device nominally identical to the one used for the experiments. The resistance meter, indicated with $\mathrm{\Omega }$, shows the location of the sensing dot. Plunger gates for the dots are colored in red and labeled with $P_i$, and the plunger for the sensing dot is labeled with SDP. (b) Schematic illustration of the experimental sequence for the spin chain, consisting of five stages: initialization of local singlets or triplets, separation of singlets, manipulation of exchange couplings and time evolution, isolation of either the left and the right pair or the middle pair by switching off specific exchange couplings, and pairwise singlet-triplet readout.

Figure 2

Energy spectroscopy. Left- and right-pair correlated singlet-triplet probabilities from independent single-shot Pauli spin blockade readout as a function of (a) left- and right-pair detunings and (d) magnetic field and middle-pair detuning. A decrease in singlet-singlet probability and an increase in one of the other probabilities correspond to an anticrossing between the low-energy global singlet state and a polarized state. In (d), the right axis shows the calculated Zeeman splitting taking a Landé $g$ factor of $-0.44$. (b) Numerical simulation of the left- and right-pair detuning for which the low-energy global singlet state is degenerate with a polarized state. The parameters for the numerical simulation are obtained from separate spin funnel measurements for the left and right exchange coupling (see Appendix pp3-s5) and from the Fourier transform in Fig. 3. (c) Numerical simulation of the middle-pair detuning and Zeeman splitting for which the low-energy global singlet state is degenerate with a polarized state. The diagram shows only the energies for the polarized states for which the magnetic field lowers the energy. The energy of the low-energy global singlet state is set to zero as a reference. Points in (b) and (c) with the same detunings and magnetic field are indicated with a “$+$”. The legend for (c) is the same as for (b).

Figure 3

Global coherent exchange oscillations. Coherent oscillations (a),(c) within the triplet subspaces and (e) within the singlet subspace. In (a), the difference in left- and right-pair detuning is varied, and in (c),(e), the middle-pair detuning is varied (with $J_{12}=J_{34}$). (b) Fourier transform of the data in (a). The frequency minimum corresponds to $J_{12}=J_{34}$. (d) Fourier transform of the data in (c). At $J_{12}=J_{23}=J_{34}=J_{\mathrm{hom}}$, indicated by the white dotted line, the most visible frequency is equal to $J_{\mathrm{hom}}/\sqrt{2}$. (f) Fourier transform of the data in (e). The frequency minimum corresponds to $J_{12}=J_{23}=J_{34}=J_{\mathrm{hom}}$ and is equal to $\sqrt{3}{J}_{\mathrm{hom}}$. Insets in all panels show numerical simulations of the experiment.

Figure 4

Adiabaticity of state preparation and readout. (a) Singlet probability for middle-pair readout as a function of ramp-in time and evolution time. (b) Triplet-triplet probability for left- and right-pair readout as a function of ramp-in time and evolution time. Insets in (a) and (b) show numerical simulations of the experiment. (c) Numerical simulation of the overlap, $P_{\mathrm{gnd}}=|⟨\psi |S_0⟩{|}^2$, between the state at the manipulation stage, $|\psi ⟩$, and the low-energy singlet eigenstate $|S_0⟩$, as a function of ramp-in time. The overlap is an indication for the success of the state preparation. (d) Middle-pair singlet and triplet probabilities from experiment (solid lines) and numerical simulations (dashed lines) as a function of middle-pair detuning for the isolation stage after the manipulation. The experimentally obtained probabilities are corrected for relaxation during the readout time (see Appendix pp3-s6).

Figure 5

Probing the low-energy singlet eigenstate of the homogeneous spin chain. (a) Singlet and triplet probability for middle-pair readout and (b) singlet-triplet correlated probabilities from left- and right-pair readout as a function of middle-pair detuning and with $J_{12}=J_{34}$. Experimental data are shown as solid lines and the results from numerical simulations as dashed-dotted lines. Vertical dashed lines indicate the middle-pair detuning for which the exchange couplings are calibrated to be homogeneous. The experimentally obtained probabilities are corrected for relaxation during the readout time (see Appendix pp3-s6).

Figure 6

Triplet energies. (a) Energies of global triplet states as a function of $J_{12}-J_{34}$, with $J_{23}=0.2\mu \mathrm{eV}$ and $J_{12}+J_{34}=2\mu \mathrm{eV}$, revealing a minimum in energy difference at $J_{12}=J_{34}$ for $T_0^{+}$ and $T_1^{+}$. (b) Triplet energies as a function of $J_{23}$ with $J_{12}=J_{34}=1\mu \mathrm{eV}$, showing that at $J_{12}=J_{23}=J_{34}$ the energy spacings are equal. The legend for (b) is the same as for (a). An external magnetic field results in an overall offset and, thus, does not change the triplet energy differences.

Figure 7

Charge-stability diagrams for nearest-neighbor pairs, with (a) left, (b) middle, and (c) right. Pulse positions are indicated with small white circles and are labeled with a letter corresponding to the pulse stage in Table 2. Primes are added to characters for which the indicated voltage position is a projection, because the true position is in a different charge occupation, which is outside the respective two-dimensional plane in the four-dimensional charge-stability space. Solid arrows represent the part of the pulse sequence for which only one variant is used, though the operation voltage position $O$ is varied throughout the experiments. Dashed arrows correspond to sequences for initialization of either a singlet-singlet $L_{SS}$ or triplet-singlet $L_{TS}$. Dotted arrows correspond to sequences for readout (and isolation) of either the middle pair, $R_M$ ($I_M$), or readout (and isolation) of the left pair, $R_L$ ($S_L^{\prime }$), and the right pair, $R_R$ ($S_R^{\prime }$).

Figure 8

Histograms of correlated left- and right-pair readout directly after initialization of (a) the singlet-singlet product state and (b) the triplet-singlet product state. For both the left- and the right-pair readout, the integration time is $8\mu \mathrm{s}$, and the bin width is [1.9, 1.9] a.u.

Figure 9

Separate exchange coupling measurements. The dashed blue lines are fits to the data. (a) Spin funnel on left pair, (b) Fourier transform of triplet-subspace coherent oscillations shown in Fig. 3(c), (c) spin funnel on right pair.

Figure 10

Readout characteristics (a)–(c) show relaxation curves for the Pauli spin blockade readout of the left, middle, and right pair, respectively. The corresponding $T_1$ values are 80.9, 25.0, and $114.8\mu \mathrm{s}$. (d) Histogram of middle-pair readout with $8\mu \mathrm{s}$ integration time and bin width 4.375 a.u., and the counts are in thousands. (e) Histogram of correlated left- and right-pair readout. For both readouts, the integration time is $8\mu \mathrm{s}$, and the bin width is [2.0, 2.0] a.u.

Figure 11

Automatic calibration of pulse voltage offsets. (a) Sensing dot calibration, with the red marker corresponding to the top of the Coulomb peak, and the green marker is the half-height on the rising flank, (b) interdot transition between (0202) and (1102), (c) dot-reservoir transition between (0202) and (1202), (d) interdot transition between (0202) and (0211), (e) dot-reservoir transition between (0202) and (0212). The traces are not centered at the same voltages, but voltage offsets are added for each scan based on the previous calibration.