Manybody interferometry of quantum fluids

Abstract

Characterizing strongly correlated matter is an increasingly central challenge in quantum science, where structure is often obscured by massive entanglement. It is becoming clear that in the quantum regime, state preparation and characterization should not be treated separately-entangling the two processes provides a quantum advantage in information extraction. Here, we present an approach that we term "manybody Ramsey interferometry" that combines adiabatic state preparation and Ramsey spectroscopy: Leveraging our recently developed one-to-one mapping between computational-basis states and manybody eigenstates, we prepare a superposition of manybody eigenstates controlled by the state of an ancilla qubit, allow the superposition to evolve relative phase, and then reverse the preparation protocol to disentangle the ancilla while localizing phase information back into it. Ancilla tomography then extracts information about the manybody eigenstates, the associated excitation spectrum, and thermodynamic observables. This work illustrates the potential for using quantum computers to efficiently probe quantum matter.

Fig. 1.. Preparing and interfering manybody states.

The quantum system probed in this work consists of a chain of seven capacitively coupled superconducting transmon qubits (59) [blue in (A)] connected to site-resolved readout resonators (meandering traces) and flux control (bottom traces). (B) The system is well described by the Bose-Hubbard model: particles (microwave photons) coherently tunnel between lattice sites (qubits) at a rate J, with on-site interactions U ≫ J arising from the transmon anharmonicity (1, 17, 46). Real-time flux tuning provides control of lattice site energies (δ_i for site i), allowing the deterministic manipulation of disorder that we leverage to build highly entangled states. (C) Starting in a highly disordered lattice, we initialize the system in a chosen energy N-particle eigenstate by applying π pulses to N empty sites (left) and adiabatically removing disorder to convert these states into eigenstates of the quantum fluid (right) (32). (D) To interfere superpositions of such states, we replace one of the assembly π pulses with a $\frac{\mathrm{\pi }}{2}$ pulse on the qubit with a green highlight in the figure (referred to as the ancilla qubit in the main text), producing a superposition of two (red/blue) eigenstates. Adiabatically removing disorder produces a superposition of two manybody fluid states; coherently evolving for a time T allows the eigenstates to accumulate a relative phase proportional to their energy difference; ramping back to the disordered configuration (right) relocalizes the phase difference into the single qubit that started in a superposition; a final $\frac{\mathrm{\pi }}{2}$ pulse on this qubit maps the phase information onto qubit occupancy for measurement. (E) Reinterpretation of the full manybody Ramsey sequence as a set of gates on the qubits comprising the lattice, resulting in an interference fringe versus evolution time T.

Fig. 2.. Benchmarking the manybody Ramsey protocol.

(A) To explore this protocol in the laboratory, we deterministically inject particles into a disordered lattice and remove the disorder (left), before imaging the resulting density distribution. When we inject precisely one photon (upper blue panel), adiabatic disorder removal produces the lowest-momentum particle-in-a-box state (lower blue panel); injecting two photons (upper red panel) produces the lowest-energy two-body state after disorder is removed (lower red panel). If we deterministically inject the first particle with a π pulse but $\frac{\mathrm{\pi }}{2}$ pulse the second photon, we should produce the manybody superposition state, and we observe the average of the two density distributions (green panels). (B) To demonstrate that this average density distribution corresponds to the macroscopic superposition of manybody states, we allow the superposition state to evolve on the Bloch sphere before adiabatically mapping the manybody superposition back onto a single qubit, where it can be read out via a second $\frac{\mathrm{\pi }}{2}$ pulse. (C) The resulting Ramsey fringe (versus hold time T in the manybody superposition state) evolves with a frequency given by the energy difference between the manybody states minus the frequency of the local oscillator from which the $\frac{\mathrm{\pi }}{2}$ pulses are derived, exhibiting contrast over several microseconds limited by the single-qubit T₂ (see the Supplementary Materials). In (D), we demonstrate the applicability of the approach to larger systems by applying it to superpositions of N = 2, 3 and N = 3, 4 particle fluids; the increased decay reflects the faster dephasing of states with more particles. Fits to data in (C) and (D) are plotted as solid green lines; the frequency deviation from numerics are 400 kHz, 2 kHz, and 2 MHz, respectively. Representative error bars (on first data point of each plot) reflect the SEM.

Fig. 3.. Spectroscopic signatures of adiabaticity.

The manybody Ramsey protocol relies critically on the ability to adiabatically map localized states into and out of highly entangled states (in a ramp time τ). (A) Ramping too quickly leads to diabatic excitations (purple) into other manybody states that do not interfere with the states (red/blue) in the prepared superposition (green) and thus reduce Ramsey fringe contrast. With some probability, however, these diabatic excitations are diabatically de-excited back into the initial state (red) during the backwards ramp; because these excitations evolve at different frequencies (corresponding to their energies) during the hold time T, they produce Ramsey fringes at other Fourier frequencies. (B) For the slowest ramp (τ = 1 μs), there are no diabatic excitations, producing a single Fourier feature in the Ramsey interference between N = 0 and N = 1 eigenstates. (C) For the fastest ramp (τ = 1 ns), the many diabatic excitations are reflected in additional frequencies in the Ramsey fringe beyond the dominant feature in the slow ramp. As the ramp time τ is varied over three decades, frequency components furthest from the dominant feature disappear first, with the low-offset-frequency features disappearing only for the slowest ramps, consistent with excitation rates controlled by the energy gaps of the fluid. Insets depict the time-domain Ramsey fringes for slow, intermediate, and fast ramps (top to bottom). (D) Repeating these experiments with superpositions of N = 1,2 and N = 2,3 particles demonstrates that while the proliferation of manybody states makes resolving diabatic excitations challenging, the dominant feature nonetheless appears for the slowest ramps.

Fig. 4.. Spectroscopic probes of thermodynamics.

Manybody Ramsey interferometry offers new ways to characterize synthetic quantum matter: (A to C) The chemical potential μ = E_N+1,V − E_N,V quantifies the energy required to add a particle to a manybody system, and we measure it by interfering states of different particle number. (A) is a sample dataset showing the Ramsey spectrum for the superposition of N = 1,2 particles as we vary the total system size V. For each V, the chemical potential μ is assigned to the frequency of maximal spectral density, which we plot in (B) for all fillings up to unit filling N = 0…V − 1, and all system sizes V = 1...7. Exact values calculated from numerics using our measured device’s parameters are plotted with gray dashed lines. In (C), we replot all data versus the density ρ ≡ N/V, finding a collapse onto a universal sinusoidal form (gray dashed line) consistent with a free-fermion model (50). (D to F) The pressure P = E_N,V+1 − E_N,V quantifies the energy required to change the system size, and we measure it by interfering states of different system size. We achieve controlled superpositions of system sizes using the approach shown in (D): the controlling site is U-detuned such that when it is empty, it is energetically inaccessible, reducing the system size by a single site; when it is filled, it becomes accessible and accordingly increases the system size. Using this site as the control in a manybody Ramsey experiment allows us to extract the energy difference between N particles melted into V + 1 versus V sites (see the Supplementary Materials, section B). Performing this protocol for different volumes V and particle numbers N produces the raw data in (E), which we rescale versus density in (F), again finding agreement with a free-fermion theory. Error bars, where larger than the data point, reflect the SEM (see the Supplementary Materials, section K).