PaperlayerView exact revision on arXiv

Source-linked AI summary

Experimentally simulating the dynamics of quantum light and matter at ultrastrong coupling

N. K. Langford, R. Sagastizabal, M. Kounalakis, C. Dickel, A. Bruno, F. Luthi, D. J. Thoen, A. Endo, L. DiCarlo

arXiv:1610.10065v1quant-phcond-mat.supr-con

TL;DR

Experiments reaching ultrastrong coupling have largely been limited to spectroscopy, while demonstrating quantum Rabi model dynamics and its ground-state entanglement and large photon numbers remains challenging. This work uses a circuit-QED digital simulation to probe those dynamics, observing Bell-cat-like correlations and large photon populations.

  • Problem

    Demonstrating quantum Rabi model dynamics, ground-state entanglement, and large ground-state photon numbers remains an open challenge, while USC experiments have been limited to spectroscopy.

    Source: p. 1

  • Method

    A circuit-QED chip with a resonator and transmon implements a digital Rabi protocol, with a coupled probe qubit used to measure the resonator state.

    Source: pp. 1–2

  • Results

    The resonator and qubit are correlated consistently with expected Bell-cat entanglement, and the resonator builds up more than 30 photons on average in the strongest USC regions.

    Source: p. 5

  • Takeaways & Limitations

    Second-order Trotterisation and smaller Trotter steps improve performance, supporting a linear increase in step count with simulated time and progress toward quadratic scaling for universal quantum simulation.

    Source: p. 5

    For r ≳1.5, residual Kerr-type resonator nonlinearity causes some deviation from the expected photon-parity revival time.

    Source: p. 3

Abstract

from arXiv · show

The quantum Rabi model describing the fundamental interaction between light and matter is a cornerstone of quantum physics. It predicts exotic phenomena like quantum phase transitions and ground-state entanglement in the ultrastrong-coupling (USC) regime, where coupling strengths are comparable to subsystem energies. Despite progress in many experimental platforms, the few experiments reaching USC have been limited to spectroscopy: demonstrating USC dynamics remains an outstanding challenge. Here, we employ a circuit QED chip with moderate coupling between a resonator and transmon qubit to realise accurate digital quantum simulation of USC dynamics. We advance the state of the art in solid-state digital quantum simulation by using up to 90 second-order Trotter steps and probing both subsystems in a combined Hilbert space dimension $\sim80$, demonstrating the Schr\"odinger-cat like entanglement and build-up of large photon numbers characteristic of deep USC. This work opens the door to exploring extreme USC regimes, quantum phase transitions and many-body effects in the Dicke model.

METHODS

The methods use phase-controlled, second-order Trotterisation to map circuit-QED interactions onto a simulated Rabi Hamiltonian with a tunable rotating frame. They also specify the pulse implementation and resonator-state reconstruction used to analyse the dynamics.

METHODS

  • The rotating-frame detunings set the simulated frequencies, while the phase difference between successive bit-flip pulses controls the simulated resonator frequency.The methods identify ωR_r = −ω0 = −2∆φ/τ; the rotating frame allows the simulated frequency to be tuned with a weakly anharmonic transmon.Source: METHODS · pp. 7, 9

Trotter step

  • A second-order Trotter step uses three flux pulses and two single-qubit rotations, with adjacent half-duration flux pulses combined across steps.For most data, simulated τ = 20 ns and the resulting Trotter step lasts 122 ns.Source: Trotter step · p. 9

Qubit control

  • Long Trotter sequences rely on calibrated DRAG qubit rotations, with QR sequences containing up to 180 bit-flip pulses.QR drive amplitude is calibrated using 50 π-pulse pairs followed by a π/2 pulse; QW is optimised using the AllXY sequence.Source: Qubit control · p. 9

Wigner tomography reconstructions

  • Resonator states are reconstructed from direct Wigner tomography measurements using maximum-likelihood convex optimisation.The measurement operators are calculated at each phase-space position and truncated to an informationally complete set; the reconstructed density matrix yields the plotted Wigner functions.Source: Wigner tomography reconstructions · pp. 9–10

SUPPLEMENTARY INFORMATION

The supplementary information describes the cryogenic microwave and flux-control setup and reports device-fabrication choices intended to improve qubit robustness and reliability.

SUPPLEMENTARY INFORMATION

Experimental setup

  • The setup routes filtered, attenuated drive and flux lines to a sample typically held near 30 mK, while AWGs and IQ mixers generate the qubit and resonator pulses.Low-noise current sources provide DC qubit-frequency tuning, combined with fast flux pulses using reactive bias tees.Source: Experimental setup · p. 12

Device fabrication

  • Fabrication changes target reduced sensitivity to two-level fluctuators, reduced coupling to flux-line voltage noise, and more reliable asymmetric-qubit junctions.The changes include rounded capacitor spacing, a centred flux-bias line, NbTiN capacitor plates, and optimised junction processing.Source: Device fabrication · p. 12

AWG AWG

The supplied section consists of the labels “1 4 3 2 pulse” and “1 4 3 2 src.”

AWG AWG

  • The cited passage records the label 1 4 3 2 src; the supplied evidence provides no further description or interpretation of that label.Source: AWG AWG · p. 13

(6GHz)

The schematic describes the cryogenic sample environment and the microwave and flux-control paths used to operate and measure the chip.

(6GHz)

  • Drive and flux lines are filtered and attenuated before reaching the sample, while AWGs and IQ mixers generate the pulses and AWG markers gate measurement pulses.The sample typically remained near 30 mK; DC bias currents were combined with fast tuning pulses through reactive bias tees.Source: Figure S1 · p. 13

Device operating parameters and qubit performance

The device operating points and coherence were chosen to support long digital simulations and photon-sensitive measurements. QR used an asymmetric two-sweet-spot design, while QW was operated below its top sweet spot.

Device operating parameters and qubit performance

  • QR was designed below RR and pulsed into resonance; its asymmetric design kept T2* typically above a few microseconds, even at steep flux gradients.The design’s reduced flux gradient relaxed the constraint on placing the resonator near QR’s top sweet spot.Source: Device operating parameters and qubit… · pp. 13–14
  • The device table reports measured qubit and resonator performance; coupling was measured by both avoided-crossing spectroscopy and vacuum-Rabi oscillations.At sweet spots, Ramsey beating was consistent with quasiparticle tunnelling; reported T2* values were fit from long, beating Ramsey signals.Source: Table S1 · p. 14
  • QW operated about 650 MHz below its top sweet spot, where T2* was typically ≳1.5 µs; it remained there because photon-sensitive operation was good.Rapidly tuning QW to the sweet spot for photon-meter measurements was therefore not used.Source: Device operating parameters and qubit… · p. 14
  • QR’s bottom sweet spot was identified by matching resonance for equal positive and negative flux pulses, locating the operating point to 1 part in 5000.Residual DC cross-talk affecting QW’s frequency was unimportant because its precise operating frequency was not critical.Source: Device operating parameters and qubit… · p. 14

Calibration of the flux distortions

Flux distortions across the pulse chain were corrected in stages rather than with one comprehensive model. Step-response measurements informed predistortion outside the fridge, while flux-pulse chevrons guided empirical optimization inside it.

Calibration of the flux distortions

  • Long trains of 10–20 ns pulses were sensitive to short- and long-time distortions from bandwidth limits and filtering in the flux line.Predistortion can compensate these effects when the system operates in a linear regime.Source: Calibration of the flux distortions · pp. 14–15
  • The calibration used a divide-and-conquer sequence: correct long-time step-response behavior first, then refine shorter-time effects, and optimize chevrons inside the fridge.The approach avoided building a single comprehensive model for all flux distortions.Source: Calibration of the flux distortions · p. 15
  • The numerical predistortion procedure samples a step response, forms its impulse response and lower-triangular transfer matrix, then inverts that matrix to obtain a kernel step response.The matrix-inversion step limits the step-response length that can be treated; sampling is chosen to cover the region of interest.Source: Calibration of the flux distortions · p. 15
  • An effectively linear AWG-amplifier step-response ramp was corrected with an exponential predistortion, followed by three exponential-approach corrections.The three added corrections used time constants between 5 µs and 500 ns.Source: Calibration of the flux distortions · p. 15
  • The bias-tee response was fit with a quadratic form rather than the naively predicted linear ramp, then used to calculate a correction.The corrected step response was measured after bias-tee compensation.Source: Calibration of the flux distortions · pp. 15–16
  • In-fridge chevrons exposed residual ramping and skin-effect asymmetry, prompting empirical bias-tee adjustment and additional skin-effect and exponential corrections.The skin-effect correction used a step-response kernel; the final chevron followed further corrections with time constants from 1500 ns to 30 ns.Source: Calibration of the flux distortions · p. 16

Calibration of the photon meters

Ramsey measurements on QW served as photon meters through its dispersive dependence on resonator photon number. Calibration set the parity condition and effective delay, while pulse design traded sensitivity against photon-number range.

Calibration of the photon meters

  • The photon meter uses QW’s dispersive frequency shift with resonator photon number: increasing Ramsey delay τ increases sensitivity but narrows the range where the response remains linear.The small-phase condition is required for the approximate linear mapping from measured probability to average photon number.Source: Calibration of the photon meters · pp. 16, 17, 19
  • For the 0–8-photon meter, a 4 ns Ramsey separation yielded an effective τ of ∼19 ns; higher photon-number dynamic range requires frequency refocusing and is ultimately limited by drive-pulse bandwidth.Source: Figure S5 · p. 18, Calibration of the photon meters · p. 19
  • The parity condition was calibrated at the 0.5-excitation crossing after loading one resonator photon; a 383 ns wait was robust to photon decay and QW dephasing.The principal accuracy limitation is calibration of QW’s zero-photon frequency, which sets how accurately the crossing identifies the pulse delay.Source: Calibration of the photon meters · p. 19
  • A 383 ns pulse separation corresponded to an effective delay τ of about 398 ns and a dispersive shift 2χ/2π = −1.26 MHz.The effective delay calibration accounts for finite pulse duration.Source: Calibration of the photon meters · p. 19
  • Photon-meter cross-talk was calibrated without loading resonator excitations, then compensated by adjusting the phase of QW’s second Ramsey pulse.This procedure could be adapted to flux-pulse sequences of interest.Source: Calibration of the photon meters · p. 19

Calibration of Wigner tomography

Wigner tomography used a coherent resonator displacement followed by photon-parity readout. Calibration against zero- and one-photon states showed radial symmetry and a small displacement-scale offset.

Calibration of Wigner tomography

  • A 50 ns square pulse displaced the resonator state before parity readout; drive frequency and amplitude were calibrated using the photon meter.The amplitude fit used the centre of the meter’s linear range, where its mapping is most accurate.Source: Calibration of Wigner tomography · pp. 19–20, Figure S6 · p. 20
  • Zero- and one-photon tomograms showed radial symmetry, while their maximum visibilities fell short because full parity-meter calibrations were not applied.The one-photon state was prepared by a calibrated SWAP pulse between QR and RR.Source: Calibration of Wigner tomography · p. 20
  • Fits to zero- and one-photon tomogram slices indicated a ∼5% displacement overestimate, consistent with a measured Gaussian width of 0.526 ± 0.003 versus 0.5 expected.The fits used a classical mixture of zero- and one-photon Wigner-function cross-sections with a free x-axis scaling parameter.Source: Calibration of Wigner tomography · p. 20

Analog vs Digital Jaynes-Cummings Dynamics

Digital Jaynes–Cummings interactions initially produce satellite resonances rather than the natural-resonance chevron, but a compensation pulse brings their dynamics into very good agreement with the analog interaction. The same digital Rabi protocol also shows revival behavior with ground-state initialization.

Analog vs Digital Jaynes-Cummings Dynamics

  • The digital chevron showed satellite resonances absent from analog measurements and lacked a chevron at the natural resonance near 2.45 Vpp.The satellite spacing was around 50 MHz, the inverse of the pulse duration.Source: Analog vs Digital Jaynes-Cummings… · p. 21, Figure S7 · p. 22
  • Extra interaction-off times cause relative qubit–resonator phase accumulation that can prevent successive pulses from continuing the interaction at natural resonance.Satellite peaks arise when phase accumulation allows the interaction to build constructively from pulse to pulse.Source: Analog vs Digital Jaynes-Cummings… · p. 21
  • A 5 ns compensation flux pulse corrected the digital JC interaction, yielding very good agreement with the analog version, with slightly lower visibility and effective coupling.The measured effective coupling was about 1.8 MHz, versus about 1.95 MHz for the analog interaction.Source: Analog vs Digital Jaynes-Cummings… · p. 21
  • Ground-state-initialized simulations also showed qubit revivals, and the reported comparison found very good agreement between ground- and excited-state initialization.The paper contrasts this with the Jaynes–Cummings interaction, where |g, 0⟩ is decoupled and does not undergo nontrivial dynamics.Source: Analog vs Digital Jaynes-Cummings… · pp. 21–22

Trotterisation performance vs Trotter order

Second-order Trotterisation preserves the characteristic USC plateaus and revivals at coupling ratios where first-order Trotterisation produces qualitatively different dynamics. Its symmetric sequence cancels first-order error terms, while low coupling and adequate coherence remain important practical constraints.

Trotterisation performance vs Trotter order

  • The excited- versus ground-state comparison directly verifies the simulated Rabi model’s symmetrical behavior.The figure caption also identifies revival timing in line slices.Source: Figure S8 · p. 23
  • Low qubit–resonator coupling was required for reasonable simulation fidelity given flux-pulsing bandwidth limits, while longer runs constrained coherence.Shorter-than-anticipated resonator coherence proved the biggest limitation.Source: Trotterisation performance vs Trotter… · pp. 22–23
  • Second-order Trotterisation cancels first-order error terms, pushing the largest errors to third order in simulation time.For two-part Hamiltonians, it modifies pulses only in the first and last Trotter steps.Source: Trotterisation performance vs Trotter… · p. 23
  • Only second-order Trotterisation showed the characteristic USC plateaus and revivals at r ≳ 0.5; first-order dynamics became qualitatively different.The two orders agreed reasonably well below r = 0.5.Source: Trotterisation performance vs Trotter… · p. 23

Trotterisation performance vs Trotter step size

Increasing the number of Trotter steps from 24 to 60 reduced Trotter error, with the measured results agreeing closely with simulations that include photon decay. Error is most evident at high coupling, and only the smallest measured step size removes the described effects.

Trotterisation performance vs Trotter step size

  • Trotter error is most visible at high r, where central features depart from expected plateaus and the dynamical landscape can break apart into lower-coupling regimes.The paper relates the weaker low-r error to the Rabi model’s approximation by the Jaynes–Cummings model in that regime.Source: Trotterisation performance vs Trotter… · p. 23
  • Increasing Trotter steps from 24 to 60 over 1.2 µs significantly reduced error; only the smallest measured step size left the described effects absent from measurements.In ideal simulations, central features remained even at that step size and disappeared only with a still smaller 10 ns step.Source: Trotterisation performance vs Trotter… · pp. 23–24, Figure S9 · p. 24
  • Measured results closely matched numerical Trotter dynamics including photon decay, which the paper identifies as the primary experimental limitation.With decay included, fine details were less prominent and oscillation dynamics washed out more rapidly.Source: Trotterisation performance vs Trotter… · pp. 23–24

Qubit entropy dynamics

Tomography of the reduced qubit state revealed entropy and purity revivals consistent with the reported qubit-parity revivals, including out to r > 0.8. The revival is treated as a signature of surviving entanglement with the resonator when its states re-coalesce.

Qubit entropy dynamics

  • A purity revival signals resonator entanglement when the resonator states re-coalesce at the origin in phase space.A revival occurs only if the underlying entanglement remains present at re-coalescence.Source: Qubit entropy dynamics · pp. 24–25
  • Qubit-purity revivals were observed out to r > 0.8, consistent with the qubit-parity revivals.The reduced qubit state was reconstructed after each Trotter step using maximum-likelihood tomography.Source: Qubit entropy dynamics · p. 25, Figure S11 · p. 26
  • Entropy revivals appeared smaller than parity revivals because purity is quadratic in the qubit population difference; an appropriate square root made them consistent.The inset indicates low background noise and distinct revivals.Source: Qubit entropy dynamics · p. 25
Loading 1610.10065v1…