Many-body interferometry with semiconductor spins

TL;DR

The work addresses how to access and characterize many-body spectra and chaos in semiconductor spin arrays by implementing many-body Ramsey interferometry in a gate-defined Ge/SiGe 8-spin chain. It leverages adiabatic mapping of interacting eigenstates to isolate single-qubit bases, enabling reconstruction of energy levels for 2-, 4-, and 8-spin chains and the computation of global spectral diagnostics. The study provides evidence for a localization-to-chaos crossover as exchange becomes stronger relative to Zeeman disorder, as shown by GOE-level statistics and SFF features, and demonstrates robustness against parameter fluctuations. This approach establishes gate-defined quantum-dot spin systems as scalable platforms for observing and probing many-body quantum phenomena, including localization and chaos, with potential implications for quantum simulation and annealing.

Abstract

Quantum simulators enable studies of many-body phenomena which are intractable with classical hardware. Spins in devices based on semiconductor quantum dots promise precise electrical control and scalability advantages, but accessing many-body phenomena has so far been restricted by challenges in nanofabrication and simultaneous control of multiple interactions. Here, we perform spectroscopy of up to eight interacting spins using a 2x4 array of gate-defined germanium quantum dots. The spectroscopy protocol is based on Ramsey interferometry and adiabatic mapping of many-body eigenstates to single-spin eigenstates, enabling a complete energy spectrum reconstruction. As the interaction strength exceeds magnetic disorder, we observe signatures of the crossover from localization to a chaotic phase marking a step towards the observation of many-body phenomena in quantum dot systems.

Figures (20)

• Figure 1: Many-body Ramsey interferometry protocol. (A) Eigenenergies of the Heisenberg Hamiltonian with on-site disorder as a function of time. The eigenstates in the isolated and interacting regimes are connected by adiabatic transformations (AT). Waiting for a time $\tau$ in the interacting regime allows for the superposition states to acquire a relative phase. (B) In the isolated regime, one spin can be initialized in a superposition state of $\uparrow$ and $\downarrow$. When interactions are turned on, $\ket{\downarrow}$ ($\ket{\uparrow}$) is adiabatically transformed to $\ket{\psi_g}$ ($\ket{\psi_e}$) and the initial spin excitation can delocalize over the array: the orange shades encode the probability of finding the spin excitation on a specific site. After a time $\tau$ and returning to the isolated basis, the states will have acquired a phase $E_g\tau/h$ and $E_e\tau/h$, respectively. (C) Bloch-sphere representation of the many-body Ramsey protocol. The relevant phase accumulation occurs in the interacting regime. At the end, the accumulated phase is projected back onto the readout basis. (D) The $\uparrow$-probability of the qubit initialized in a superposition will oscillate with a frequency $hf = E_e-E_g$ allowing extraction of the many-body energy level spacings. (E) False-colored scanning electron microscope image of a nominally identical device to the one measured. Four charge sensors (S) in the corners of the device are used for RF-reflectometry charge sensing. The plunger gates (orange) $p_i$ and the barrier gates (blue and teal) $b_{ij}$ confine eight unpaired charges in a 2x4 arrangement of quantum dots. An external magnetic field $B = 10mT$ is applied as indicated. The scale bar is 200 nm.
• Figure 2: Interferometry implementation. (A) Top: Energy level diagram of a two-spin system as a function of the lab time. Bottom: the corresponding pulse on the detuning $\epsilon$. The grey dashed lines indicate the state trajectories resulting from the ramp in detuning $\epsilon$, passing through the spin-orbit induced avoided crossing (green dashed circles). After letting the system acquire relative phase $\phi$ for a time $\tau = t_f-t_0$, another ramp back towards the readout point closes the interference loop of the two trajectories. We can read the resulting triplet probability, which will oscillate at a frequency $hf = h\phi(\tau)/\tau=E_{\downarrow\uparrow}-E_{\downarrow\downarrow}$. Any phase accumulated before $t_0$ and after $t_f$ only adds a constant offset to $\phi$. (B) Probability of finding the final state in a triplet as a function of ramptime $\tau_{ramp}$ and free evolution time $\tau$ for spin pair 5-6. Two sets of oscillations can be seen which are further evidenced by the fast Fourier transform (FFT) in C. (C) Squared norm of FFT of B. The inset shows the FFT amplitude at $f_6$ as a function of $\tau_{in,out}$. The red dashed line marks the ramp time corresponding to $P_{LZ} = 0.5$. (D) Extracted Larmor frequencies $f_L$ defining the magnetic disorder landscape. (E) Energy spectrum as a function of time and barrier voltage (as sketched in the bottom panel). After the initialization through the avoided crossing we can choose to apply a local SWAP operation. If no SWAP (a SWAP) is applied, we follow the lower (higher) branch. After adiabatically activating the exchange interaction, we let the system acquire a relative phase. The adiabatic turn-off is followed by no (a) SWAP in case we selected the lower (higher) branch. Subsequently, we close the interference loop like in A. (F) Sum of FFTs of oscillations as a function of the inter-qubit barrier $b_{56}$, using 200 ns ramps. When turning interactions on and off without SWAPs applied, we observe the lower frequency. In this case, spin 6 is prepared in a superposition and we label the frequency $f_6$. When we apply a SWAP before and after the phase acquisition and adiabatic exchange turn-on and off, we observe the upper frequency. Here spin 5 is initialized in a superposition and we call this frequency $f_5$. The brown and purple dashed lines are the fitted model (Eq. \ref{['eq:4spinHam']}) from which we extract the exchange interaction strength at each point. The minimum of $f_6$ corresponds to $2\Delta_{SO}$.
• Figure 3: Interferometry of extended spin chains. (A) FFTs of oscillations for spins in a four-spin chain as a function of global barrier gate voltage, in percent, where $b = 0\%$ corresponds to $J\ll \Delta E_Z$ and $b = 100\%$ to $J> \Delta E_Z$. The panels are ordered to reflect the position of the spin initialized in a superposition in the array. The four frequencies $f_i$ allow reconstruction of the first spin manifold. (B) FFTs of oscillations for the initial states in the inset. On the left (right), we initialize spin 2 (4) in a superposition, while spin 4 (2) is initialized in $\ket{\uparrow}$. (C) To reconstruct the energy levels of the second manifold we add $f_4$ ($f_2$) measured in A, represented by the solid gray lines, to the corresponding frequencies measured in B. We see that the data points lie almost perfectly on top of each other, and return the frequency of the state $\ket{\downarrow\uparrow\downarrow\uparrow}$ from two complementary measurements. (D) Reconstructed spectrum of the four-spin chain 1-2-3-4. When redundant frequencies have been measured, we chose the one with the fewest artifacts. The gray lines show the spectrum obtained from the fitted model.
• Figure 4: Inferring global system parameters. (A) FFTs of oscillations for a chain of eight spins. The panels are ordered to reflect the position of the measured spin in the array. The barrier gates are all swept together from low exchange $J\ll \Delta E_Z$ ($b = 0\%$) to high exchange $J>\Delta E_z$ ($b = 100\%$). Except for spin 7, we can always clearly identify the main frequency. (B) Extracted frequencies from A with overlaid fitted model in gray. At the colored dashed lines we reconstruct higher spin manifolds. (C) Energy levels of the first and second spin manifold extracted for three different values of $\frac{J}{\Delta E_Z}$ highlighted in B. We observe clearly separated manifolds for $\frac{J}{\Delta E_Z}\approx 0$. For $\frac{J}{\Delta E_Z}\approx 2$ the energy levels tend to repel each other signaling correlations and, consequently, a tendency for chaotic behavior. (D) Visualization of the definition of $s_i$. (E) Energy level uniformity distribution for the three exchange configurations calculated from the energy levels extracted in C. The gray dashed (black dashed dotted) line represents the expected theoretical distribution for localized (chaotic) systems. The data displays a trend to more chaotic behavior as interactions increase, signaled by $P(r\to0)\to 0$ for $\frac{J}{\Delta E_Z}\approx 2$. (F) Spectral form factor calculated for the energy levels extracted in C. We observe a systematic trend to reach a deeper early-time minimum smaller than the thermalization plateau (black dashed line at a value of $1/N$) as we increase the interaction strength. For longer times, oscillatory behavior masks the universal features of the SFF, since we do not average over disorder realizations.
• Figure S1: Different ways to capture eigenvalue statistics for a matrix of size $2^L\times 2^L$ with $L=8$: averaged spectral form factor (left), energy spacing distribution (middle), and energy spacing ratio distribution (right). In the averaged spectral form factor, we also plot a single instance in transparent font to illustrate the non-self-averaging of the SFF Prange1997. In orange, we show the statistics for the uniformly distributed eigenvalues, and in blue, the ones from a Gaussian unitary ensemble. Similarly, for the GOE, we plot the spectral measures in red. For all quantities, we draw the eigenvalues from their respective distribution and average over $N_\text{sim}=100$ samples. For the histograms, we choose a bin size of 60.