Abstract
Quantum emitters coupled to optical resonators are quintessential systems for exploring fundamental phenomena in cavity quantum electrodynamics (cQED)1 and are commonly used in quantum devices acting as qubits, memories and transducers2. Many previous experimental cQED studies have focused on regimes in which a small number of identical emitters interact with a weak external drive3,4,5,6, such that the system can be described with simple, effective models. However, the dynamics of a disordered, many-body quantum system subject to a strong drive have not been fully explored, despite its importance and potential in quantum applications7,8,9,10. Here we study how a large, inhomogeneously broadened ensemble of solid-state emitters coupled with high cooperativity to a nanophotonic resonator behaves under strong excitation. We discover a sharp, collectively induced transparency (CIT) in the cavity reflection spectrum, resulting from quantum interference and collective response induced by the interplay between driven inhomogeneous emitters and cavity photons. Furthermore, coherent excitation within the CIT window leads to highly nonlinear optical emission, spanning from fast superradiance to slow subradiance11. These phenomena in the many-body cQED regime enable new mechanisms for achieving slow light12 and frequency referencing, pave a way towards solid-state superradiant lasers13 and inform the development of ensemble-based quantum interconnects9,10.
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Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank A. Ruskuc, T. Xie, C.-J. Wu, O. Vendrell and R. Finkelstein for discussion. This work was supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (contract number DE-SC0012704), Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY-1733907) with support from the Moore Foundation and by the Office of Naval Research awards no. N00014-19-1-2182 and N00014-22-1-2422 and the Army Research Office MURI programme (W911NF2010136). The device nanofabrication was performed in the Kavli Nanoscience Institute at the California Institute of Technology. M.L. acknowledges the support from the Eddleman Graduate Fellowship. R.F. acknowledges the support from the JASSO Graduate Scholarship. J.R. acknowledges the support from the Natural Sciences and Engineering Research Council of Canada (NSERC) (PGSD3-502844-2017). J.C. acknowledges support from the IQIM Postdoctoral Fellowship.
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A.F. conceived the experiment. M.L. and R.F. built the experimental set-up, performed the measurements and analysed the data. J.R. fabricated the device. M.L., R.F., B.Z., M.E., J.C. and A.F. interpreted the results. M.L., R.F., J.C. and A.F. wrote the manuscript, with input from all authors. All work was supervised by J.C. and A.F.
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Extended data figures and tables
Extended Data Fig. 1 Experimental setup.
a, Laser A addresses the A transition. Optical pulses are generated using AOMs, which are driven with gated radiofrequency sources. Part of it can be split off for use as a local oscillator for heterodyne measurements. A second laser, laser C, can be used to perform optical hole burning on the C transition. The combined light is sent through a circulator, to the device and the reflected light is sent to a SNSPD for time-resolved photon counting. b, Scanning electron microscopy image of the device.
Extended Data Fig. 2 Cavity–ion coupling.
a, Energy-level spectrum of 171Yb3+:YVO4 at zero external magnetic field. The optical transitions whose polarizations are along the cavity mode are shown in blue (A, E, I). Because the three transitions are separated by an amount on the order of GHz, larger than the inhomogeneous broadening, each transition is spectrally well resolved and can be regarded as an isolated, effective two-level system; for example, the levels labelled |e⟩ and |g⟩ form a two-level system for the A transition. b, Schematic showing the relative population of ions in each transition, in which the I transition has double the population owing to a doubly degenerate ground state. c, Cavity reflection spectrum at weak laser power shows three DIR peaks corresponding to the A, E and I transitions. The peak corresponding to A is marked with orange, as we focus on this transition in the main text.
Extended Data Fig. 3 Numerical simulation under the mean-field approximation of cavity reflection for a system with higher cooperativity.
a, Simulated cavity reflection showing narrow CIT for Δinh/(2πC) = 1 kHz and γ/2π = 0.3 kHz. b, CIT width with varying γ in this high-cooperativity system, showing a strong dependence on γ. See Supplementary Information for simulation details.
Extended Data Fig. 4 Decay rates between Dicke states in a bad cavity regime.
Here Dicke states are formed by six identical two-level systems. a, Collective decay (red) governed by Γc = 4g2/κ decays vertically, preserving total spin J. b,d,e, Individual decay (green) governed by spontaneous emission γs decays diagonally. c,f, Individual dephasing (beige) γd couples neighbouring J states with the same M. With higher M, the diagonal decay rates are faster towards larger J and slower towards smaller J.
Extended Data Fig. 5 Master equation simulation of dissipative many-body dynamics.
a, Simulation of six identical ions excited with a long (50-μs) pulse. The peak of total squared atomic polarization ⟨J+J−⟩ (red) is plotted as a function of excitation power along with individual (blue) and correlation (orange) terms from equation (17), in which the peak amplitudes are calculated from the values immediately after the excitation pulse, to emulate the peak counts measurements. b, Simulated population dynamics of different subspaces in the Dicke basis as a function of excitation power. Note that the superradiant ladder here refers to all of the states in the J = 3 manifold except the ground state. We find that the evolution of the superradiant population with power aligns with the correlation term in a. c, Simulated Dicke state population distribution for different powers. The size of the black circles represents the relative population weights at the end of the excitation pulse. With low power (power = 0.5 AU, left), primarily the lower excitation superradiant ladder (orange bars) is populated. With increased power (power = 1 AU, middle), the subradiant subspace (blue bars) begins to populate, including the long-lived dark subradiant states. At high power (power = 2 AU, right), the system approaches a completely mixed state. Here the population distribution seems to be unequal among the Dicke states owing to the varying degeneracies of the states in the subradiant subspace.
Extended Data Fig. 6 Comparison of simulated uncoupled and coupled ensembles.
Master equation simulation of the peak counts with power for a single homogeneous ensemble of five ions (red), two coupled subensembles of five ions at 0 MHz and two ions detuned by 5 MHz (blue) and two uncoupled subensembles of five ions at 0 MHz and two ions detuned by 5 MHz (orange). Here uncoupled refers to the fact that the peak emission is simulated separately for the subensembles of five and two ions and later added together. Note that the peak emission reflects the cavity population ⟨a†a⟩ (Methods). The qualitatively similar behaviour of the uncoupled and coupled subensemble cases motivates the simulation of an inhomogeneous ensemble by means of incoherent addition of many uncoupled smaller subensembles in Fig. 3e.
Extended Data Fig. 7 Frequency and power dependence of peak counts.
a, Peak counts with excitation power and laser detuning. We repeat the pulsed excitation measurement at different frequencies along the inhomogeneous line in the A transition and observe that the S-curve shifts to higher power. b, Horizontal cuts of a at different detunings. Dashed red line indicates the maximum peak counts for different detunings. c, Extracted local maximum of peak counts as a function of laser detuning. The maximum of peak counts decreases with increased detuning from the centre. We note that this behaviour differs from Fig. 4b, in which the S-curve also shifted towards higher powers and the maximum of peak counts increased with laser detuning. Here the S-curve shifts towards higher powers because of the CIT profile; as the laser is detuned from the centre, less power enters the cavity, resulting in effectively more power being required to excite the ions. At the same time, the decrease of the maximum of peak counts indicates that there are less ions resonant with the laser when detuned from the centre.
Extended Data Fig. 8 CIT width and depth comparison between the A and I transitions.
Measured CIT width (a) and depth (b) with power and corresponding fits for the A (red) and I (blue) transitions. We expect the I transition to have different width and depth values, as the cooperativity is twice as high, but the dephasing rate is expected to be more than 100 times larger than A. Although we find similar depth values, the width differs between the two. In particular, the I transition starts out much broader than A, as expected from the worse coherence properties. Despite this, the minimum width is slightly narrower for the I transition, indicating that indeed the cooperativity is larger for the I transition. Further discussion of the fit parameters is in Methods.
Extended Data Fig. 9 Numerical simulation of the cavity phase across the CIT.
The simulated cavity phase arg(a) (red) across the CIT for a laser power of 0.5 nW, exhibiting a relative π phase shift. The corresponding cavity reflection spectrum (blue) as a function of laser frequency for reference, identical to Fig. 2c, purple line.
Extended Data Fig. 10 Optical switch based on CIT.
a, CIT response time. We park the laser at the centre of the CIT and measure the reflection as a function of time after turning the pump on. The reflection signal decreases as the CIT is created in sub-μs timescales. b, Schematic of an ideal two-port optical switch with an integrated filter with CIT. A transmission port and a transverse pump port are added to realize this application. For the best extinction ratio, the cavity should be two-sided and critically coupled such that κ1/κ = κ2/κ = 0.5, in which κ1 and κ2 are the coupling rates from ports 1 and 2, respectively. The top schematic shows that when the pump is off and DIR is formed, the signal is entirely reflected (port 1). The bottom schematic shows that, when the pump is on and CIT is created, the signal is transmitted (port 2) within the CIT window (spectral filter). See Supplementary Information for a more detailed discussion.
Supplementary information
Supplementary Information
This file contains Supplementary Sections 1–7 and the Supplementary References. Sections 1–7 also include Supplementary Table 1 and Supplementary Figs. 1–10.
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Lei, M., Fukumori, R., Rochman, J. et al. Many-body cavity quantum electrodynamics with driven inhomogeneous emitters. Nature 617, 271–276 (2023). https://doi.org/10.1038/s41586-023-05884-1
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DOI: https://doi.org/10.1038/s41586-023-05884-1
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