Emergent Macroscopic Bistability Induced by a Single Superconducting Qubit

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Emergent Macroscopic Bistability Induced by a Single Superconducting Qubit

Riya Sett, Farid Hassani, Duc Phan, Shabir Barzanjeh, Andras Vukics, and Johannes M. Fink

PRX Quantum 5, 010327 – Published 16 February 2024

Abstract

The photon blockade breakdown in a continuously driven cavity QED system has been proposed as a prime example for a first-order driven-dissipative quantum phase transition. However, the predicted scaling from a microscopic behavior—dominated by quantum fluctuations—to a macroscopic one—characterized by stable phases—and the associated exponents and phase diagram have not been observed so far. In this work we couple a single transmon qubit with a fixed coupling strength $g$ to a superconducting cavity that is in situ bandwidth $κ$ tunable to controllably approach this thermodynamic limit. Even though the system remains microscopic, we observe its behavior becoming increasingly macroscopic as a function of $g / κ$ . For the highest realized $g / κ$ of approximately $287$ , the system switches with a characteristic timescale as long as 6 s between a bright coherent state with approximately $8 \times 10^{3}$ intracavity photons and the vacuum state. This exceeds the microscopic timescales by 6 orders of magnitude and approaches the perfect hysteresis expected between two macroscopic attractors in the thermodynamic limit. These findings and interpretation are qualitatively supported by neoclassical theory and large-scale quantum-jump Monte Carlo simulations. Besides shedding more light on driven-dissipative physics in the limit of strong light-matter coupling, this system might also find applications in quantum sensing and metrology.

Received 28 October 2022
Revised 5 October 2023
Accepted 22 December 2023

DOI:https://doi.org/10.1103/PRXQuantum.5.010327

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Riya Sett ^1,*, Farid Hassani ¹, Duc Phan ¹, Shabir Barzanjeh^1,2, Andras Vukics ^3,†, and Johannes M. Fink ^1,‡

¹Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
²Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
³HUN-REN Wigner RCP, P.O. Box 49, Budapest, 1525, Hungary

^*riya.sett@ist.ac.at
^†vukics.andras@wigner.hu
^‡jfink@ist.ac.at

Popular Summary

The recently introduced notion of macroscopic phases in microscopic quantum systems broadened the scope of phase transitions from condensed matter to quantum optics. First-order dissipative phase transitions specifically, have been predicted to be possible to realize in situations where a single qubit strongly interacts with many cavity photons in what has been called the photon blockade breakdown. The experimental hallmark for this phenomenon is the observation of stochastic bistability between a fully transparent system with thousands of intracavity photons and an opaque system where both the cavity and the qubit remain in their quantum ground state despite the same strong and continuous external drive.

In this work we extract the scaling exponents and the phase diagram of this phase transition by controllably increasing the normalized qubit-cavity coupling. We find convincing evidence that the strong-coupling limit represents the thermodynamic limit where the effect of quantum fluctuations vanishes, fast quantum jumps are replaced with hysteresis, and the average lifetime of the two phases of up to 6 s exceeds all microscopic timescales by more than 6 orders of magnitude. This is the missing evidence that justifies the observed effect being described in the language of phase transitions.

Our work sheds new light on how strongly coupled quantum systems transition into classical ones, a process that we measure to occur substantially faster than expected from numerical simulations. The need for 64 CPU years of simulation time to obtain qualitative agreement with at least parts of the presented data suggests that there may be deep connections to the complexity usually found in many-body physics but now encoded in a small, anharmonic multilevel system. This could open up new opportunities to physically encode and simulate aspects of many-body systems. Moreover, we speculate that this phase transition could also be used to detect individual excitations inside the cavity or qubit with high signal-to-noise ratio when biased close to the bistability region of the phase diagram. It could act as a quantum-jump amplifier in analogy to an avalanche photodiode.

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Vol. 5, Iss. 1 — February - April 2024

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Figure 1
Experimental realization. (a) Schematics of the experimental device consisting of a superconducting transmon qubit fabricated on a silicon substrate that is placed at the antinode of the fundamental mode of a 3D copper cavity. The cavity has a fixed-length port (red) and an in situ variable-length pin coupler port (blue). (b) Measured cavity transmission spectra with the qubit far detuned for different coupler positions (color coded) together with a fit to Eq. (5) (dashed lines) and the extracted $κ / 2 π$ .
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Figure 2
Photon blockade breakdown measurements at $g / κ \approx 39.1$ . (a) The Jaynes-Cummings ladder for a three-level atom illustrating the PBB effect in the frequency domain: single-photon (blue) and multiphoton transitions (red) are indicated according to the measured spectrum at the Rabi-splitting frequencies and near resonance, respectively. As indicated, the third atomic state causes a third subladder in the spectrum starting from the second rung, whose spacing is close to the bare-system frequency. This is a possible explanation for the PBB bistability persisting in the case of resonant driving. (b) Measured cavity transmission spectra for $ω_{A} = ω_{R}$ for two applied external drive strengths: $1.05 MHz$ , revealing a typical vacuum Rabi spectrum, and $167.01 MHz$ , where a sharp peak at $ω_{R}$ is observed. (c) Measured cavity output bistability at $ω = ω_{R}$ in the time domain indicating the dwell times of the bright state ( $t_{on}$ ) and the dim state ( $t_{off}$ ). (d)–(f) Measured quadrature histograms (proportional to cavity $Q$ functions convolved with amplifier noise) for the dim phase at $η / 2 π = 105 MHz$ , for the bistable region with equal probability at $167 MHz$ , and for the bright phase at $210 MHz$ , respectively. (g) Measured histograms of the output power (arranged vertically and with probability color coded) as a function of input power. The maxima indicated by circles trace out a typical bistability curve; see Fig. 4. (h) Extracted average dwell times in the dim and bright states [Eq. (6)] as a function of $η$ . The error bars represent the standard error that is extracted from five sections of the full dataset. The dwell time and drive strength corresponding to half-filling where $t_{dwell}^{\dim} = t_{dwell}^{bright}$ are indicated with an asterisk. The inset shows the measured slow-switching eigenvalue $1 / τ$ , from Eq. (7), in the unit of $κ$ as a function of drive amplitude $η$ close to half-filling [65].
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Figure 3
Finite-size scaling towards the thermodynamic limit. (a) Measured dwell times in the dim state (blue) and the bright state (red) are shown as a function of $η / g$ for different values of $κ$ . Black stars denote half-filling values, where the probabilities of being in the dim state or the bright state are equal. The dashed line is a power law fit. The error bars represent the standard error that is extracted from at least three sections of the full dataset. (b)–(d) Measured scaling laws as a function of control parameter $g / κ$ at half-filling (asterisks). Fitting the experimental data (dashed lines) yields the exponents $t_{dwell}^{*} \propto (g / κ)^{5.4 \pm 0.2}$ for the dwell time, $⟨ n ⟩^{*} \propto (g / κ)^{0.96 \pm 0.05}$ for the average resonator photon number, and $η^{*} / g \propto (g / κ)^{- 0.52 \pm 0.03}$ for the input drive strength. We compare the experimental values (stars) with those obtained from QJMC simulations (polygons with color code) for computationally manageable $κ = 11.5, 14.3, and 18.1 MHz$ . The simulations include three, five, or seven transmon levels both without transmon dephasing (light blue) and with dephasing increasing linearly with the level number (light red) with $γ_{ϕ, 1} / 2 π = 50$ kHz and $γ_{1} = 0$ . Details of the simulation are given in the main text and in Appendix pp1.
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Figure 4
Photon blockade breakdown phase diagram. (a) Phase diagram on the $Δ$ - $η$ plane obtained from the neoclassical theory for two transmon levels. (b) Typical multistability curves for the transmitted intensity as a function of the drive amplitude obtained from the approximate three-level neoclassical theory detailed in Appendix pp2. The curves are different for $Δ = 5 MHz$ and $Δ = - 5 MHz$ , proving the asymmetry of the phase boundaries. For $Δ = 0$ , the curve obtained from the two-level theory is plotted as a dotted line for comparison, revealing the role of the third level in the appearance of a bistability on resonance. The solution of the three-level theory contains very high order polynomials that lead to numerical problems when the experimental parameters are used. Therefore, here and in (a), model parameters were used: $g_{1} / κ = 30$ , $g_{2} / κ = 2.5$ , and $Δ_{an} / κ = - 40$ (see Appendix pp1 for the definition of all the parameters of the full model). (c) Experimental PBB phase diagram at $g / κ \approx 132$ with boundaries (points) and the half-filling drive strength (dotted line) obtained from the experimental trajectories that exhibit the temporal bistable signal [(e)–(h)]. The color scale of the density plot depicts the ratio $r = (h_{1} - h_{2}) / (h_{1} + h_{2})$ , where $h_{1}$ and $h_{2}$ are the amplitudes of the peaks of the measured quadrature histograms corresponding to the dim and bright states, respectively; see Figs. 2–2. (d) Measured dwell times at half-filling as a function of drive detuning. Error bars are extracted from the mean and standard error of the measured dim and bright dwell times. (e)–(h) Experimentally observed telegraph signals that define the phase boundaries at different $η$ values at $Δ = 0$ in the range of $η / 2 π$ from 85.6 to 115.5 MHz as indicated by a double arrow in (c).
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Figure 5
Simulated example trajectories for the three possibilities of modeling dephasing of higher transmon levels for otherwise identical parameters (five transmon levels, $γ_{1} = 0$ , $γ_{ϕ, 1} / 2 π = 50$ kHz, $η / 2 π = 100$ MHz, and $κ / 2 π = 14.3$ MHz): no dephasing (top, blue), flux noise model (top, red), and charge dispersion model (bottom, blue). The last case leads to qualitatively incompatible results with very large noise levels and only partially stabilized attractors.
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