Abstract
Alltoall interacting, disordered quantum manybody models have a wide range of applications across disciplines, from spin glasses in condensedmatter physics over holographic duality in highenergy physics to annealing algorithms in quantum computing. Typically, these models are abstractions that do not find unambiguous physical realizations in nature. Here we realize an alltoall interacting, disordered spin system by subjecting an atomic cloud in a cavity to a controllable light shift. Adjusting the detuning between atom resonance and cavity mode, we can tune between disordered versions of a centralmode model and a Lipkin–Meshkov–Glick model. By spectroscopically probing the lowenergy excitations of the system, we explore the competition of interactions with disorder across a broad parameter range. We show how disorder in the centralmode model breaks the strong collective coupling, making the darkstate manifold cross over to a random distribution of weakly mixed light–matter, ‘grey’, states. In the Lipkin–Meshkov–Glick model, the ferromagnetic finitesized ground state evolves towards a paramagnet as disorder is increased. In that regime, semilocalized eigenstates emerge, as we observe by extracting bounds on the participation ratio. These results present substantial steps towards freely programmable cavitymediated interactions for the design of arbitrary spin Hamiltonians.
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Main
The unavoidable presence of impurities and inhomogeneities in most realworld physical systems has given a strong motivation to the study of disordered models. In such studies, important insights into the typical behaviour of a manybody system can be obtained by considering an ensemble of realizations with randomly distributed parameters^{1}. In this way, a deeper understanding of the structure of lowenergy excitations in complex quantum systems can be achieved, providing keys to interpreting transport and thermodynamics observations. Going one step further, several quantum simulation platforms, such as trapped ions^{2}, ultracold atoms^{3} and Rydberg atoms^{4,5,6}, have demonstrated the capability to implement controlled disorder into otherwise clean manybody systems. Those allowed for the investigation of nonequilibrium dynamics, revealing some of the most intriguing phenomena of random systems, such as Anderson^{7,8,9,10,11} and manybody localization^{12,13,14}.
In the last years, cavity quantum electrodynamics (QED) has emerged as a new platform for quantum simulation. By harnessing photons to tailor novel types of interaction beyond the native van der Waals and dipolar interactions between atoms, cavity QED unites the scalability of atom devices with tunable longrange interactions^{15}. Previous experiments used this platform to explore new, superradiant^{16,17,18,19} and dissipationstabilized^{20,21} phases of matter in quantum gases, as well as to demonstrate tunablerange interactions^{22} and emergent geometries using spatial and spectral addressing^{23}.
In this article, we implement random spin models on a cavity QED platform and study their lowlying excitations. Via a lightshift technique, we realize a quasirandom longitudinal field with controlled strength, which competes with an alltoall flip–flop interaction mediated by the exchange of cavity photons. Leveraging the open nature of the cavity, we observe the frequencyresolved response in the cavity field and atomic polarization channels. We exploit our setup to observe disorderdriven crossovers in two different regimes: a centralmode model where we observe a disorderinduced dressing of otherwise dark antisymmetric states with cavity photons, and a Lipkin–Meshkov–Glick (LMG) model (an instance of a Richardson–Gaudin model) where disorder competes with ferromagnetic order. As shown theoretically and experimentally, the frequencyresolved susceptibilities are sensitive to the detailed structure of excitations, providing particular insights into their localization properties. Our system is a natural starting point to investigate the spectacular consequences of strong light–matter coupling on materials properties^{24,25,26} such as transport^{27,28,29} or magnetism^{30}, where the effect of disorder due to impurities and material inhomogeneities is believed to be strongly influenced by light.
Model
Our system implements a paradigmatic model consisting of N Ising spins, mapped to internal atomic states, identically coupled to the central, bosonic photon mode of the cavity. By exposing the ith spin to a random energy shift ϵ_{i}, the model is described by the disordered Tavis–Cummingstype Hamiltonian as
Here \({\hat{a}}^{{\dagger} }\) and \(\hat{a}\) are the creation and annihilation operators of the photons in the cavity, \({\hat{\sigma }}_{i}^{r}\) are the rPauli operators acting on the Ising (pseudo)spin1/2 of the ith atom, \({\hat{S}}^{+()}=\mathop{\sum }\nolimits_{i = 1}^{N}{\hat{\sigma }}_{i}^{+()}/\sqrt{N}\) are the collective spinraising (lowering) operators and \({{{\varDelta }}}_{{{{\rm{ca}}}}}\) is the detuning between the cavity and bare atomic resonance. We set ℏ = 1 throughout the manuscript. Centralmode models^{31,32} have been used to describe a large variety of physical situations, including qubit decoherence in solidstate quantum computing platforms as well as heat and charge transport in nanostructures.
In the disorderfree instance of the Hamiltonian of equation (1) (Fig. 1a, left), the spin1/2 degrees of freedom form a manifold of N + 1 collective exchangesymmetric Dicke states coupled to light, thus called ‘bright states’, which are described by a single collective spin \(\hat{S}\). The remaining 2^{N} − (N + 1) states form a dark manifold that is decoupled from the cavity field. In the singleexcitation manifold, this structure reduces to two polaritons and N − 1 dark states. A controlled breaking of this collective spin description into macroscopic subsets that are spatially and spectrally distinguishable has recently been demonstrated by splitting atomic ensembles with the help of optical tweezers and magneticfield gradients^{23}.
In the model of equation (1), the collective spin description is broken by disorder (Fig. 1a, right). This leads to the fragmentation of the darkstate manifold into an ensemble of ‘grey eigenstates’ that are hybridizations of the delocalized photon field as well as a few localized spins with similar energies^{27}. Because the coupling to the cavity extends over the entire system, energy resonances between spins can occur at arbitrarily large distances in the presence of disorder. As a result, the grey eigenstates have wavefunctions that are neither localized nor delocalized but semilocalized over multiple, arbitrarily distant spins^{33,34}. It was recently demonstrated theoretically that for any strength of light–matter coupling, this results in a multifractal structure of the eigenstates, similar to that found at the critical points of localization–delocalization transitions^{35}. Even though they have never been directly observed, it is believed that disorderinduced grey states are responsible for the spectacular enhancement in energy and charge transport found in disordered molecular systems coupled to cavities^{27,28,36,37,38,39,40}.
Experimentally, the Hamiltonian in equation (1) is realized by an array of N = 90–800 thermal ^{6}Li atoms confined in about 160 trapping sites, positioned at the antinodes of the resonant cavity field. The spins are encoded in the \(2{S}_{1/2}^{F = 1/2}\) (g〉) and 2P_{3/2} (e〉) states of ^{6}Li atoms (Fig. 1b,c). The cavity resonance is tuned close to the 2S_{1/2}–2P_{3/2} transition at 671 nm, with the detuning given by \({{{\varDelta }}}_{{{{\rm{ca}}}}}\). Our cavity is close to concentric, leading to a singleatom cooperativity of η = (4g^{2})/(κΓ) = 6.4, where g/2π, κ/2π and Γ/2π are 2.05, 0.45 and 5.80 MHz. Due to the cloud’s temperature of 200 μK, and the reduced dipole moment for linearly polarized light at zero magnetic field, the average cavity coupling that the atoms experience is \(\bar{g}/2\uppi\) = 1.23 MHz (Methods and Extended Data Fig. 1a–c,e).
The disorder is created by two laser beams that intersect at the position of the atoms, with frequency slightly detuned from the 2P_{3/2}–4D_{5/2} transition at 460 nm (Extended Data Fig. 1d,f), forming a lightshifting lattice with a period of 1.04 μm that is incommensurate with the trapping lattice, which has a period of 671 nm. This produces a quasirandom pattern of strong light shifts in the 2P_{3/2} state, with negligible effect on atoms in the ground state (Fig. 1b,c). These light shifts result in quasidisordered energy shifts ϵ_{i}, which translate into the spin language as random local longitudinal fields sampled from the distribution \({\rho }_{{{{\rm{a}}}}}(\epsilon )={[\uppi \sqrt{\epsilon (W\epsilon )}]}^{1}\), where W is proportional to the intensity of the control laser (Methods). We neglect the light shifts induced by the trapping light on the ground and excited states, as it is small compared with the light shift induced by the 460 nm beam.
We probe the system by weakly driving the cavity on axis with a probe beam and measuring both photon transmission proportional to \(\langle {\hat{a}}^{{\dagger} }\hat{a}\rangle\) and atomic excitations \(\langle {\hat{S}}^{z}\rangle =\langle \mathop{\sum}\nolimits_{i = 1}^{N}{\hat{\sigma}}_{i}^{z}\rangle/(2N)\) using an optical pumping technique (Fig. 2a). In the linearresponse regime, this provides us with the frequencydependent photonic and atomic (spin) susceptibilities, namely, χ_{p} and χ_{a}, respectively (equations (5), (7) and (15) provide the definitions and Extended Data Fig. 2 provides the details).
Nearresonant regime and grey states
We first investigate the regime at small \({{{\varDelta }}}_{{{{\rm{c}}}}\bar{{{{\rm{a}}}}}}\) where the cavity resonance is close to the mean atomic resonances, that is, \({{{\varDelta }}}_{{{{\rm{c}}}}\bar{{{{\rm{a}}}}}}={{{\varDelta }}}_{{{{\rm{ca}}}}}W/2\) (Fig. 2b). In the absence of disorder, we observe the canonical normalmode splitting of a width \(2g\sqrt{N}/2\pi\) of 22 MHz expected from the Tavis–Cummings model (Fig. 2c). As a result of this splitting, a Rabi gap forms at \({{{\varDelta }}}_{{{{\rm{c}}}}\bar{{{{\rm{a}}}}}}=0\), and direct atomic excitations at the bare resonance frequency are suppressed (Fig. 2c, centre). Although there are N − 1 eigenstates of the Hamiltonian lying within the gap, these are purely atomic, and the symmetry of the alltoall atom–cavity coupling prevents their excitation, rendering them completely dark.
On introducing disorder, we observe the onset of a nonzero response around zero detuning, a manifestation of the increase in photon weight of the originally dark purely atomic states. A representative spectrum of χ_{a} for W/(2π) = 26 MHz is presented in Fig. 2d. We observe that the fading out of Rabi splitting occurs via a redistribution of the spectral weight from the polaritons to a wide spectrum of midgap states. For \( {{{\varDelta }}}_{{{{\rm{c}}}}\bar{{{{\rm{a}}}}}} \gtrsim W\), a narrow, dispersively shifted cavity resonance is restored at around \({{{\varDelta }}}_{{{{\rm{pc}}}}}\) = 0 (Fig. 2d).
To further understand the evolution of the spectrum with disorder strength, we probe the photonic susceptibility at \({{{\varDelta }}}_{{{{\rm{c}}}}\bar{{{{\rm{a}}}}}}=0\,\) as a function of disorder strength W and detuning \({{{\varDelta }}}_{{{{\rm{pc}}}}}\). The results are presented in Fig. 2g,i for different mean atom numbers N. For weak disorder, photonic susceptibility χ_{p} confirms the presence of two bright polaritons, and a manifold of degenerate dark states at the centre of the Rabi gap. As the disorder becomes comparable with the collective atom–cavity coupling, that is, \(W \approx g\sqrt{N}\), we observe a smooth increase in χ_{p} at around \({{{\varDelta }}}_{{{{\rm{pc}}}}}\) = 0, signalling the onset of a finite coupling of a greystate manifold emerging from the originally dark states. Simultaneously, the polaritons’ response weakens and fades away for the largest disorder, where the spectrum consists of a resonance centred at \({{{\varDelta }}}_{{{{\rm{pc}}}}}\) = 0 strongly broadened by the disorder.
The evolution of the spectrum with disorder is driven by the fragmentation of the eigenstates, from fully delocalized bright and dark states without disorder, to randomly distributed, isolated resonances for the largest disorder. To confirm this interpretation, we compare our observation (Fig. 2g,i) with theoretical calculations (Fig. 2h,j) of the cavity transmission based on Green function techniques (see the ‘Susceptibility in the nearresonant regime’ section). The model takes into account the experimental distribution of spin energies, which is correlated and nonuniform, different from the case studied elsewhere^{35}.
Nevertheless, we have verified that the eigenfunctions are multifractal in the same way (Supplementary Section 1.1). The simulations (which take into account the measured atom number fluctuation), the effects of thermal motion on atom–cavity couplings and both losses of photons and atomic decay, are in good agreement with the observations for the lowdisorder regime. For the strongest disorder, deviations particularly appear for the upper polariton, whose signal appears moderately weaker in the experiment. We attribute this to losses induced by radiation pressure from the control laser at 460 nm, predominantly affecting the excited atoms with the largest admixture in the 4D_{5/2} manifold (Methods). For the largest disorder strength, we do not resolve the polaritons themselves but observe a clear signal from the grey states. These results are further confirmed in Fig. 2k, which presents a direct comparison of the experimental and theoretical data for photonic susceptibility χ_{p} as a function of \({{{\varDelta }}}_{{{{\rm{pc}}}}}\) for representative values of disorder strength W. The same simulation procedure also reproduces atomic susceptibility χ_{a} measured as a function of detunings (Fig. 2e,f).
We quantitatively analyse the fading out of the polariton and the emergence of grey states by comparing the photonic susceptibility in the lower (respectively middle) parts of the spectrum (Fig. 2g–j). This yields the overall photon weight of the polariton and grey states as a function of normalized disorder strength (Fig. 2l). The crossover between the light–matterinteractiondominated regime and disorderdominated regime is evident as spectral weight is smoothly transferred from the polariton to grey states, in qualitative agreement with the simulations.
Largedetuning regime and LMG model
In the centralmode model investigated so far, an essential role is played by the finite admixture of spin excitations to the delocalized photon field. For large detuning \({{{\varDelta }}}_{{{{\rm{ca}}}}}\gg g\sqrt{N}\), the cavity field is only virtually populated, giving rise to an alltoall interaction between the spins, thereby realizing an effective LMG model^{41,42,43} (Fig. 3a and the ‘Effective model and atomic susceptibility in the largedetuning regime’ section). In the presence of a longitudinal random field, the Hamiltonian for these effective dynamics reads
where J = g^{2}/\({{{\varDelta }}}_{{{{\rm{ca}}}}}\) is the strength of the spinexchange interactions. Equation (2) is a particular case of the class of exactly solvable Richardson–Gaudin models^{44,45} that are ubiquitous in quantum manybody systems^{32}.
Similar to the centralmode model, in the absence of disorder (W = 0), equation (2) describes the dynamics of a collective spin within the Hilbert subspace of symmetric states. The nonlinearity inherited from the spin–cavity coupling favours a ferromagnetic ground state, protected by a finite gap of size JN. A striking manifestation of ferromagnetism is the strong suppression of the zerofrequency magnetic response.
To realize the model of equation (2), we detune the cavity to the blue of the atomic transition by \({{{\varDelta }}}_{{{{\rm{ca}}}}}\)/2π = 92 MHz, and probe the system at frequency ω_{p} in the vicinity of the bare atomic resonance ω_{a} (Fig. 3a). In this regime, the transmission of the cavity is negligible such that χ_{p} ≈ 0, and the atomic signal χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) (equation (15) provides the definition) directly measures the transverse spin susceptibility of the system at frequency \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) = ω_{p} − (ω_{a} + 2g^{2}/\({{{\varDelta }}}_{{{{\rm{ca}}}}}\)) (equation (12)). As shown in Fig. 3b,c, in the absence of disorder, the frequency dependence of χ_{a} reveals the finite ferromagnetic gap, with magnitude \({{{\varDelta }}}_{{{{\rm{FM}}}}}\), as well as the reduced zerofrequency susceptibility at \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) = 0.
The signal is broadened by the finite decay rate of the excited atomic states, which reduces to a convolution of the response with the linewidth of the atomic transition (Supplementary Section 1.2).
We now investigate this model in the presence of disorder. Similar to the centralmode model, this breaks the description in terms of a collective spin, restoring the system’s ability to explore the full Hilbert space. For a given disorder strength W, the susceptibility (Fig. 3f) shows an asymmetric peak, corresponding to a collectively enhanced response superimposed with a weak and broad background whose width traces the disorder strength (Fig. 3b–e, dashed blue line). This is a manifestation of the gradual fragmentation of the collective spin, as disorder renders the individual spins offresonant with each other. The peak is located at −\({{{\varDelta }}}_{{{{\rm{FM}}}}}\), and we denote its amplitude by \({\chi }_{{{{\rm{a}}}}}^{{{{\rm{FM}}}}}\).
Tracking the location of this peak provides a measurement of the ferromagnetic gap as a function of W. Without disorder, this gap increases linearly with atom number (Fig. 3g). With increasing disorder, it decreases smoothly towards zero (Fig. 3h), where for low enough atom numbers, the gap is zero within our error bars. This demonstrates the competition between the infiniterange cavitymediated interaction J and spectral disorder W for the dynamics of the effective model \({\hat{H}}_{{{{\rm{LMG}}}}}\).
Our results are in very good agreement with a simulation of the response χ_{a} of \({\hat{H}}_{{{{\rm{LMG}}}}}\) (see the ‘Numeric simulation of the largedetuning regime’ section), over the entire parameter regime (Fig. 3b–f): the simulated system sizes were set as the mean atom numbers N realized across all the experimental runs, and the effect of the atoms’ thermal motion on the atom–cavity coupling g has been taken into account, as in the nearresonant case. The decrease in the ferromagnetic gap (Fig. 3h) indicates a drastic change in the system properties as disorder increases. However, in the thermodynamic limit, the system is always ferromagnetic and no paramagnetic phase transition should occur. Indeed, intuitively, for any fixed disorder strength, increasing the number of atoms will always lead to an infinite number of closetoresonance spins, enforcing ferromagnetism in the thermodynamic limit for an arbitrarily large disorder strength. However, for any finite number of atoms, there exists a disorder strength large enough to bring the ferromagnetic gap close to zero, by rendering each spin essentially spectrally isolated from all the others, thus crossing the system over into a paramagnet.
More precisely, our simulations show that finite systems display a minimal gap at disorder strength W* suggestive of critical behaviour; however, the value of W* diverges with increasing atom number (Supplementary Section 1.3 and Supplementary Fig. 2).
Localization of excitations
The existence and distribution of energy resonances in disordered systems is the essence of Anderson localization. In our system, excitations can hop at arbitrarily large distances provided the spins are closely resonant. Disorder, thus, decimates the spins available for resonance by offsetting most spins from each other, but does not prevent longdistance propagation^{33,35}.
Interestingly, although our spectroscopic probe does not yield spatially resolved information, it does carry relevant insights about the localization of excitations. Indeed, general arguments based on the hierarchy of Rényi entropies (see the ‘Participation ratio and its relation to susceptibility’ section) show that a system’s magnetic response may be used to bound the participation ratio of the excitations, that is, the number of spins contributing to the wavefunction. The participation ratio PR_{1} of the first excited state obeys
at any W ≥ 0, where χ_{a,1} is the contribution of the first excited state to the atomic susceptibility when the system is probed on resonance with the transition to this state, from the global ground state (the ‘Participation ratio and its relation to susceptibility’ section provides the proof). The bound is reached for W = 0, where PR_{1} = N corresponds to a wavefunction uniformly distributed over all the spins, as well as in the limit W → ∞ in which the excitation becomes localized on a single spin (PR_{1} → 1). Our frequencyresolved measurement, thus, allows us to verify the fragmentation of the system’s collective excitations into evermore localized wavefunctions, consistent with the expectations for eigenstates of the centralmode model^{33,35,46}.
Figure 4 shows the participationratio bound deduced from our measurements, showing a decrease by more than a factor of two as disorder reaches the largest values. On normalization of PR_{1} by the mean atom number N as well as W by the corresponding zerodisorder ferromagnetic gap JN, all the data collapse onto each other and agree with the simulations. The figure also shows the theoretically predicted value of PR_{1}, which obeys the bound observed in the data.
Similar to the ferromagnetic gap, suggestive as these findings are, they do not herald a transition from delocalized to localized. For a fixed disorder strength, increasing the number of atoms leads to an infinite number of closetoresonance spins at arbitrary distances, preventing full localization but leading to a semilocalized regime similar to the critical regime of the Anderson transition^{33}.
Conclusion
Our ability to introduce controlled disorder in cavity QED offers many timely and exciting prospects for further investigations, such as the study of Bardeen–Cooper–Schrieffer superconductivity^{47,48}, where our atomic susceptibility measurements would directly map to the pairing gap. More broadly, equation (2) allows the direct simulation of Richardson–Gaudin models that are relevant to a variety of manybody systems, from superconductivity in ultrasmall grains to quark physics and neutron stars. Furthermore, the capabilities demonstrated in our experiment could also be used to study the effect of inhomogeneous broadening for quantum optics applications, particularly for superradiant laser clocks^{49}. The combination of disorder with cavitymediated interactions could further be used to study glassy phases of matter^{50,51}.
Although the finite lifetime of the employed excited state limits the current investigations to one excitation above the fully polarized state, higher excitations can be probed by encoding the spins in the groundstate manifold and coupling them via Raman transitions^{52} or through the use of atoms with longlived excited states^{43}. Last, using highresolution optics and timeresolved manipulation of the control light, it will become possible to program the otherwise homogeneous longrange cavitymediated interaction in space and time, lifting one of the most stringent restrictions for the use of cavities in quantum simulation applications. In combination with small ultracold samples of our fermionic ^{6}Li atoms, this will allow for the creation of random longrange interactions between fermionic degrees of freedom, one of the building blocks for holographic quantum matter^{53}.
Methods
Experimental apparatus
The core of our setup is a highfinesse optical resonator placed inside an ultrahigh vacuum chamber^{54}. The cavity has a finesse of 59 × 10^{3} and 13 × 10^{3} at 1,342 and 671 nm, respectively. The cavity is 25.9 mm long, 103 μm shorter than concentric, yielding a singleatom singlephoton cooperativity of η = 6.4. The 1,342 nm light is used for frequency stabilization and dipole trapping and the 671 nm light allows for resonant coupling to the D2 transition of lithium.
We use in total two lasers, a 1,342 nm diode laser (main laser) that is Raman fibre amplified and then frequency doubled to generate light at 671 nm, and a laser diode emitting at 460 nm (lightshifting laser). The main laser is used for the magnetooptical trap, absorption imaging, cavity probing and trapping of the atoms in a cavityenhanced optical dipole trap. It is stabilized to our cavity on the TEM_{04} mode at 1,342 nm. The length of the cavity itself can be controlled using piezoelectric actuators under the mirrors. We can stabilize the detuning between the D2 transition of lithium and the resonance frequency of our cavity in a large frequency range (>1 GHz), by using a sideband of the 671 nm beam sent to a saturated absorption spectroscopy cell. A feedforward scheme acting on both cavity and laser allows us to rapidly vary the cavity–atom detuning within the experimental sequence (maximum slew rate of 0.1 GHz ms^{−1}) and holding the atoms in the cavity dipole trap. The lightshifting laser is stabilized using a commercial wavemeter.
Preparation of atoms
We prepare an atomic cloud with a target atom number and size using a combination of laser cooling, spatial selection and cavityassisted feedback techniques. We start by loading the atoms from a magnetooptical trap directly into the intracavity standingwave dipole trap, with a temperature of about 200 μK and trap frequencies of ω_{⊥}/2π = 22 kHz and ω_{∥}/2π = 1.4 MHz in the transverse and longitudinal directions, respectively.
At this point, the cavity resonance frequency is set 1 GHz reddetuned with respect to the D2 transition. We then start an optical molasses phase using the magnetooptical trap beams and probing the cavity using light detuned by a fixed amount with respect to the resonance of the empty cavity. The dispersive shift in the cavity is reduced as atoms are lost during the molasses, until the probe becomes resonant with the cavity, leading to an increased transmitted photon flux detected by a singlephoton counter. The molasses is stopped when the target atom number set by the predefined dispersive shift is reached and the sequence can continue. When turning off the optical molasses beams, we make sure that all the atoms are optically depumped into the \(2{S}_{1/2}^{F = 1/2}\) manifold.
At this point of the sequence, the atomic cloud measures a length of 0.5 mm, populating about 750 pancakes, each containing between 0.4 and 4.0 atoms on average. We empty all but the central 180 sites using radiation pressure, by imaging an opaque mask on the centre of the cloud with a laser resonant on the D2 transitions (Extended Data Fig. 1a). We then shift the cavity on resonance within 30 ms with the \(2{S}_{1/2}^{F = 3/2}\)–2P_{3/2} transition, leading to a detuning of 228 MHz (hyperfine splitting of ^{6}Li) with respect to the \(2{S}_{1/2}^{F = 1/2}\)–2P_{3/2} transition resonant with the atoms.
We then perform fast cavity transmission spectroscopy by sweeping a weak probe over the cavity resonance. The dispersive shift in the cavity is used to extract the initial number of atoms in the F = 1/2 state. A similar sweep is performed after the interrogation of the disordered system. Together, they allow for the characterization of probeinduced atom losses on a shottoshot basis.
The deterministic preparation suppresses drifts in the atom number and allows for longterm averages of the experimental results. However, we still observe shottoshot fluctuations in the atom number that stem from the Poissonian nature of the trap loading and losses and the atom removal procedure described above. The standard deviation of these fluctuations is about 35 and 60 atoms for the data presented in Figs. 2 and 3, respectively.
Implementation of disorder
We encode the twolevel system using the \(2{S}_{1/2}^{F = 1/2}\) (g〉) and 2P_{3/2} (e〉) states of our ^{6}Li atoms. The transition frequency of the atoms can be tuned by light shifting the excited state e〉. In particular, this is achieved by dressing the 2P_{3/2} state with the higherlying 4D_{5/2} manifold using a control laser at 460 nm detuned from resonance (Fig. 1c) by \({{{\varDelta }}}_{{{{\rm{blue}}}}}\). We first calibrated the light shift of the excited state—due to a single Gaussian beam with a waist of 120 μm at \({{{\varDelta }}}_{{{{\rm{blue}}}}}\) = 50 MHz—by performing absorption spectroscopy of the D2 transition, similar to another work^{55}. Taking the absorption images of the cloud at different imaging frequencies, we reconstructed the spatial distribution of the light shift of one of the two identical beams that generate the lightshifting lattice when sent together (Extended Data Fig. 1d). We performed this spectroscopy both in situ and after releasing the atoms from the cavity dipole trap, allowing us to measure the traprelated shift of the 2P_{3/2}–4D_{5/2} transition to be 90 MHz.
Furthermore, we characterized the dependence of the cavity transmission spectrum on the detuning of the lightshifting laser, showing an avoided crossing for both states of the Autler–Townes doublet, particularly the lightshifted singlephoton 2S_{1/2}–2P_{3/2} transition and twophoton 2S_{1/2}–4D_{5/2} transition (Extended Data Fig. 1e). We observed increased atom losses for small detunings of the lightshifting laser, pointing towards radiationpressureinduced atom losses, occurring when atoms are promoted to the 2P_{3/2} state during the spectroscopic measurements. We minimized this effect by choosing the maximal detuning (400 MHz blue detuned from the 2P_{3/2}–4D_{5/2} transition), allowing us to go up to W = 26 MHz for the maximum available power of the laser of 7.3 mW per lattice beam.
The lightshifting laser produces a dipole potential on the atoms in the ground state of about 5 × 10^{−6} smaller than the light shift of the 2P_{3/2} state, negligible compared with the intracavity trapping potential.
Both lightshifting lattice beams are linearly polarized perpendicular to the cavity axis, and set the direction of the quantization axis. We then probe the cavity using πpolarized light, to avoid any vector lightshift effect of the lightshifting beam. Because atoms reside in the F = 1/2 hyperfine manifold, the π transition used for cavity interrogation is free of tensor lightshift effects. As a result, even though our sample comprises an incoherent mixture of the two magnetic sublevels of the F = 1/2 manifold, the two components experience a strictly identical light shift and probe beam, contributing equally to the signal without further broadening effects. Crossoptical pumping between the two does not deteriorate the signal in the linearresponse regime explored in this work.
Interrogation
Once the preparation phase is completed, we tune the cavity to the desired length and illuminate the cloud with the lightshifting lattice. We send a cavity probe pulse with a duration of 5 or 60 μs for the measurements presented in Figs. 2 and 3, respectively. During this measurement, we monitor the photons leaking out of the cavity using a singlephoton counter, to infer the optical response. At zero magnetic field, the transition between g〉 and e〉 is not closed, and an atom in the 2P_{3/2} state can decay into the F = 3/2 groundstate manifold, denoted as an auxiliary state a〉. This state is not coupled to the cavity field, owing to the large hyperfine splitting. Since the decay can only happen from state e〉, the population accumulated in the F = 3/2 state is directly proportional to the excitedstate population \(\langle {\hat{S}}_{z}\rangle\) integrated over the probepulse duration.
The population of the F = 3/2 state is measured after the interrogation of the disordered system using a cavity transmission spectroscopy, with the cavity tuned on resonance with the F = 3/2 to 2P_{3/2} transition (Extended Data Fig. 2b, right). In this configuration, the power of the cavity transmission is suppressed by 1/(1 + η)^{2} in the presence of a single atom in the F = 3/2 state, yielding a singleatomlevel sensitivity for the detection of atomic response.
In practice, we implement the detection by sweeping the frequency of the onaxis probe over the cavity resonance, yielding an average photon count of four photons per sweep for the empty a〉 manifold (Extended Data Fig. 2c (inset), green histogram). The frequency sweep is essential since it removes the systematic effects of dispersive shifts on the depumping detection resulting from the presence of atoms in the g〉 state. Extended Data Fig. 2c shows the dependence of the number of transmitted photons on the laser power during interrogation, showing the expected exponential trend (see the ‘Measurement of atomic susceptibility’ section), allowing for the characterization of atomic susceptibility. At large probe powers, we observe a deviation from the exponential model that is due to saturation effects and atom losses. The data presented in this work were measured at different probe powers, and the measurements with an average photon count below 1.5 photons per sweep were neglected (Extended Data Fig. 2c, dashed line), ensuring that no additional broadening of the resonances is introduced.
Susceptibility in the nearresonant regime
In this section, we provide some details on the calculations of the susceptibility in the nearresonant regime, whose results are presented in the ‘Nearresonant regime and grey states’ section.
In our calculations, we account for fluctuations in both atom number N and atom–cavity couplings g. Specifically, we average the susceptibility over 1,000 different values of N randomly sampled from a normal distribution. The mean and standard deviation of the N distribution have been determined by fitting the experimental data at W = 0. For each value of N, we consider a generalized version of the Tavis–Cummings Hamiltonian in equation (1), namely,
where couplings g_{i} are randomly generated to account for the finite temperature of atoms and the polarization of probe light. Specifically, the g_{i} values are proportional to the square root of the cavityfield intensity at the atom positions, which are extracted from a thermal distribution (the ‘Preparation of atoms’ section provides the parameters). We also account for the fact that the N atoms are randomly distributed across ~100 pancakes: to this end, we randomly select N site energies among the set \({\epsilon }_{i}\in \{\frac{W}{2}\cos (2\uppi {\lambda }_{\mathrm{l}}j/{\lambda }_{\mathrm{s}})\,,\,j=1,\ldots ,100\}\), where λ_{l} = 671 nm is the lattice wavelength and λ_{s} = 1,040 nm is the lightshift wavelength.
For each value of N, as per other work^{35,56}, we employ a Green function formalism in the linearresponse regime. In such a situation, the cavity susceptibility (cavity transmission) at a given probe–cavity detuning \({{{\varDelta }}}_{{{{\rm{pc}}}}}\) is
where G〉 is the ground state. In equation (5), we introduced the nonHermitian Hamiltonian as
which includes the generalized Tavis–Cummings Hamiltonian (equation (4)) and two terms describing cavity losses and atom decay, respectively. Similarly, the atomic susceptibility is computed by summing the transition probabilities to all the atomic states \({\hat{\sigma }}_{i}^{+}\left\vert G\right\rangle\), namely,
Measurement of atomic susceptibility
We now show that the atomic susceptibility χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) (see the ‘Largedetuning regime and LMG model’ section), can be extracted from the measurements of atomic population P_{A}(t) of the auxiliary state a〉 at a given point in time t. Intuitively, it is plausible that χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) and P_{A}(t) should be connected: on one hand, χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) is simply a rescaling of the absorptive part of the dynamic susceptibility χ″(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) (equations (13)–(15)) of the effective model described by equation (11) and thus quantifies the timeaveraged energy absorbed by this system when subjected to a perturbation at frequency \({{{\varDelta }}}_{{{{\rm{pa}}}}}\). On the other hand, the system can absorb energy from the probe beam only via coherent excitations of the atomic population from state g〉 (\(2{S}_{1/2}^{F = 1/2}\)) to e〉 (2P_{3/2}). The population of state a〉 (\(2{S}_{1/2}^{F = 3/2}\)) can then change only via spontaneous decay from state e〉 at rate Γ_{a}. Therefore, detecting P_{A}(t) > 0 implies that the system has absorbed energy via atomic excitations. Furthermore, the probability to excite the system into a collective state containing an atomic excitation on probing is maximized when the probe frequency \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) is resonant with transitions from the system’s collective ground state. It follows that the total atomic population P_{A}(t_{meas}) found in state a〉 after the interrogation time is a measure of how susceptible the effective model was to excitations introduced by the probe at frequency \({{{\varDelta }}}_{{{{\rm{pa}}}}}\).
We give the above intuition an analytic foundation by modelling the experimental sequence (see the ‘Preparation of atoms’, ‘Implementation of disorder’ and ‘Interrogation’ sections) via a Lindblad master equation (Supplementary Section 1.2 provides details of the derivation). We derive an equation of motion for P_{A}(t) in terms of χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) using the fact that the probe beam’s amplitude Ω_{p} is much weaker than the natural linewidth Γ of ^{6}Li, that is, that atomic excitations decay at a rate much faster than the rate at which they are introduced by the probe beam, namely, Ω_{p} ≪ Γ. This yields the relation
evaluated here at the measurement time t = t_{meas}. This result confirms the monotonic relation between P_{A}(t) and χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)). It is obtained with respect to the experiment’s initial conditions p_{G}(0) = 1 and P_{A}(0) = 0, and is valid for times t ≫ (Γ/2)^{−1}.
The saturation of P_{A}(t_{meas}) as a function of probe power Ω_{p}^{2} (Extended Data Fig. 2c) is captured by equation (8). Further, for a given probe power, the saturation rate is maximal at those probe frequencies \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) at which χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) is the largest: since population transfer from G〉 to state m〉 of the singleexcitation manifold (SEM) (see the ‘Effective model and atomic susceptibility in the largedetuning regime’ section) is maximized when the probe frequency is resonant with the transition frequency E_{mG} (that is, resonant with a frequency at which the system is most susceptible to perturbations, as quantified by χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\))), the concomitant accumulation of population in the auxiliary state is also maximized. Conversely, for a fixed measurement time t_{meas}, saturation of the signal P_{A}(t_{meas}) can be suppressed by reducing the probe’s power. This is crucial for the precision of the experimental data presented in Fig. 3 (see the ‘Interrogation’ section) where the experimental technique for measuring P_{A}(t_{meas}) is discussed.
Effective model and atomic susceptibility in the largedetuning regime
In this section, we demonstrate that the dynamics of our system are described by the effective Hamiltonian of equation (2) when the cavity is tuned far into the dispersive regime, such that \({{{\varDelta }}}_{{{{\rm{ca}}}}}\) ≡ ω_{c} − ω_{a} is the dominant energy scale.
Our starting point is the disordered Tavis–Cummings Hamiltonian \({\hat{H}}_{{{{\rm{TC}}}}}\), which is expressed in equation (1) relative to the rotating frame of the bare atomic resonance frequency ω_{a}. Within this rotating frame, the probe beam is described by the perturbation \(\hat{V}(t)={{{\varOmega }}}_{{{{\rm{p}}}}}{\mathrm{e}}^{\mathrm{i}({\omega }_{{{{\rm{p}}}}}{\omega }_{{{{\rm{a}}}}})t}{\hat{a}}^{{\dagger} }+{{{\rm{h.c.}}}}\), with probelaser and Rabi frequency ω_{p} and Ω_{p}, respectively.
The total Hamiltonian is \({\hat{H}}_{{{{\rm{TC}}}}}+\hat{V}(t)\), and thus, the equation of motion of the (Heisenberg picture) photonic operator \(\hat{a}(t)\) is
where the last term takes into account cavity losses. Using the fact that these are subdominant, that is, \({{{\varDelta }}}_{{{{\rm{ca}}}}}\) ≫ κ (see the ‘Effective model and atomic susceptibility in the largedetuning regime’ and ‘Largedetuning regime and LMG model’ sections), the cavity mode adiabatically follows the evolution of the spin degrees of freedom as
Substituting this expression into \({\hat{H}}_{{{{\rm{TC}}}}}+\hat{V}(t)\,\) eliminates the cavity mode, and one obtains (up to an irrelevant constant term) the effective spin Hamiltonian as
with \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) ≡ ω_{p} − (ω_{a} + 2g^{2}/\({{{\varDelta }}}_{{{{\rm{ca}}}}}\)) and \({\hat{H}}_{{{{\rm{LMG}}}}}\) is as per equation (2). We note that the above equations are obtained after performing an additional rotatingframe transformation, which serves only to remove an otherwise constant contribution of \((2{g}^{2}N/{{{\varDelta }}}_{{{{\rm{ca}}}}}){\hat{S}}^{z}\) to equation (2).
Having obtained the above effective model, we now derive the form of atomic susceptibility χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) in the dispersive regime. In particular, χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) is obtained from the absorptive part χ″(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) of the dynamic susceptibility of effective model \({\hat{H}}_{{{{\rm{LMG}}}}}\) of equation (2), when the latter is initialized in its ground state \(\left\vert G\right\rangle {\equiv \bigotimes }_{i = 1}^{N}{\left\vert g\right\rangle }_{i}\,\) and subsequently subjected to the probe via the interaction \(\hat{{{{\mathcal{V}}}}}(t)\) of equation (12). Studying the dynamic susceptibility is motivated by the fact that the probe beam is weak (Ω_{p} ≪ Γ), so that one may treat \(\hat{{{{\mathcal{V}}}}}(t)\) as a perturbation within the regime of linear response^{57}. In particular, Ω_{p} ≪ Γ implies that atomic excitations decay much faster than the rate at which they are introduced into the system, so that one may study the limit in which there is at most a single excitation present in the system. That is, one needs to only consider the eigenstates G〉 and \({\{\left\vert m\right\rangle \}}_{m = 1}^{N}\), where the latter set of states forms the SEM of \({\hat{H}}_{{{{\rm{LMG}}}}}\). We denote the respective eigenenergies as \({{{{\mathcal{E}}}}}_{G},{{{{\mathcal{E}}}}}_{m}\), and the spectral gaps as \({E}_{mG}\equiv {{{{\mathcal{E}}}}}_{m}{{{{\mathcal{E}}}}}_{G}\), for m = 1, …, N. With respect to this basis, we then have
In what follows, we approximate the Dirac delta functions in equation (13) as Lorentzian responses:
Here γ is the linewidth of the (normalized) resonance, which according to the Wiener–Khintchine theorem^{58,59} corresponds to a finite experimental measurement time 1/γ. The dimensionless atomic susceptibility χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) (Fig. 3) is then finally obtained from χ″(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\)) as
At zero disorder, only the first excited state \(\left\vert m=1\right\rangle ={\hat{S}}^{+}\left\vert G\right\rangle\,\) contributes, such that on resonance, χ_{a}(\({{{\varDelta }}}_{{{{\rm{pa}}}}}\) = E_{1G}) = χ_{a,1}(E_{1G}) = N (Fig. 4, inset).
Participation ratio and its relation to susceptibility
Here we prove the relation, given in inequality (3), between the atomic susceptibility and participation ratio. The participation ratio quantifies the extent to which a given state is (de)localized over a basis of interest. In our context, we wish to study the (de)localization of an SEM eigenstate \(\left\vert m\right\rangle \equiv \mathop{\sum }\nolimits_{i = 1}^{N}{c}_{mi}{\hat{\sigma }}_{i}^{+}\left\vert G\right\rangle\) of the LMG Hamiltonian of equation (2) over the spins i of the system. This is quantified by the participation ratio as
of which the limiting values 1 and N are respectively obtained at \({c}_{mi}={\delta }_{i,{i}^{* }}\) (full localization at some site i*, achieved at W → ∞), and \({c}_{mi}=\frac{1}{\sqrt{N}},\,\forall \,i\) (full delocalization over all N sites, achieved at W → 0).
Our proof of inequality (3) relies on the identification of PR_{1} and χ_{a,1}(E_{1G}) (equation (17)) as monotonic functions of different Rényi entropies \({H}_{\alpha }(\overrightarrow{p})=\frac{1}{1\alpha }\log ({\sum }_{i}{p}_{i}^{\alpha })\), and then exploiting the hierarchy \({H}_{{\alpha }_{1}}(\overrightarrow{p})\ge {H}_{{\alpha }_{2}}(\overrightarrow{p})\) for any real numbers α_{2} ≥ α_{1} ≥ 0 (ref. ^{60}). To this end, we note that (1) on resonance \({{{\varDelta }}}_{{{{\rm{pa}}}}}\) = E_{mG}, the mth summand of the atomic susceptibility defined in equation (15) reduces to
whose limiting values are obtained with the same distributions of c_{mi} as for PR_{m} (equation (16)). (2) Using Perron–Frobenius theory^{61}, one can show that the lowest SEM eigenstate m = 1〉 of \({\hat{H}}_{{{{\rm{LMG}}}}}\) (as defined in equation (2)) satisfies c_{1i} ≥ 0, ∀ i = 1, …, N. Hence, one has that \({c}_{1i}=+\sqrt{{p}_{1i}}\), where p_{mi} ≡ c_{mi}^{2} are the probabilities associated to the amplitudes c_{mi}.
Now, for the identification with Rényi entropies, we expand both sides of inequality (3) and employ point (2) above. This yields
where α_{1} = 1/2, α_{2} = 2 and \({\overrightarrow{p}}_{1}\equiv ({p}_{11},{p}_{12},\ldots ,{p}_{1N})\,\). Since exp(x) is monotonic, the hierarchy of Rényi entropies is preserved and thus
This concludes the proof.
We briefly comment on how the participationratio bound of inequality (3) may be measured: due to the finite atomic linewidth (see the ‘Measurement of atomic susceptibility’ section), extracting only the m = 1 summand of the atomic susceptibility is not feasible, as nearby resonances will add to the measured signal. What can be feasibly extracted is the amplitude \({\chi }_{{{{\rm{a}}}}}^{{{{\rm{FM}}}}}\) of the full susceptibility of equation (15), which satisfies \({\chi }_{{{{\rm{a}}}}}^{{{{\rm{FM}}}}}\equiv {\chi }_{{{{\rm{a}}}}}({E}_{1G})\ge {\chi }_{{{{\rm{a}}}},1}({E}_{1G})\), by definition. This is the data shown in Fig. 4.
In closing, we note that inequality (3) (as well as its looser form in terms of the full χ_{a}(E_{1G})) becomes an equality in both limits of m = 1〉 being fully (de)localized. This, too, follows from the above expression in terms of Rényi entropies: for all α ≥ 0, \({H}_{\alpha }(\overrightarrow{p})=\log (N)\) if p_{i} = 1/N, ∀ i = 1, …, N (maximal uncertainty), and \({H}_{\alpha }(\overrightarrow{p})=0\) if \({p}_{i}={\delta }_{i,{i}^{* }}\) for some i* = 1, …, N (maximal certainty).
The above discussion exemplifies that the participation ratio is an entropic measure, quantifying the degree of (un)certainty ((de)localization)—obtained from some state’s expansion coefficients—as to its spread over a chosen set of degrees of freedom (basis). In fact, for any basis {i〉}, the generalized inverse participation ratio IPR_{q}(ψ〉) is related to Rényi entropies via its multifractal dimension D_{q}: combining equations (2) and (3) in Supplementary Information, one has \({D}_{q}=\frac{d}{1q}\frac{\log ({{{{\rm{IPR}}}}}_{q}(\left\vert \psi \right\rangle ))}{\log (N)}\). As in the above discussion, this can be expressed in terms of Rényi entropies as \({D}_{q}=d\frac{{H}_{\alpha = q}(\overrightarrow{p})}{\log (N)}\), where \(\overrightarrow{p}=({\left\vert \langle i = 1 \psi \rangle \right\vert }^{2},\ldots ,{\left\vert \langle i = N \psi \rangle \right\vert }^{2})\). This relation exemplifies the intimate link between entropy and quantifiers of a state’s (de)localization properties, and has as an immediate consequence that D_{q} decays monotonically with q ≥ 0.
Numeric simulation of the largedetuning regime
We compute χ_{a} and participation ratios by diagonalizing the random LMG Hamiltonian of equation (2) for system sizes N = 303 and 610. These system sizes correspond to the mean atom numbers realized in the experiment, which were determined from the dispersive shift JN = g^{2}N/\({{{\varDelta }}}_{{{{\rm{ca}}}}}\) measured at zero disorder (W = 0), for each iteration of the measurement sequence (see the ‘Preparation of atoms’ section). The effect of the atoms’ thermal motion on the value of g was taken into account for the conversion of the dispersive shifts into atom numbers, as well as for the matrix elements of the Hamiltonian. Taking the mean atom number across all the experimental runs yields the system sizes quoted above.
We choose the random energy shifts ϵ_{i} in two different ways: (1) from the incommensurate lightshift potential generating correlated quasirandom disorder (as discussed in the main text) and (2) independent and identically distributed ρ_{a}. For both cases, we find quantitative agreement of χ_{a} as well as the participation ratio, within numerical accuracy.
The Hamiltonian matrix is constructed with respect to the basis states \(\left\vert i\right\rangle ={\hat{\sigma }}_{i}^{+}\left\vert G\right\rangle\) of the SEM, and diagonalized exactly. In the absence of disorder, that is, ϵ_{i} = 0 ∀ i, the diagonalization is analytically tractable, and the eigenenergies are \({{{{\mathcal{E}}}}}_{1}=NJ/2\) and \({{{{\mathcal{E}}}}}_{m}=NJ/2\) for m = 2,…, N. Consequently, the zerodisorder ferromagnetic gap \({{{\varDelta }}}_{{{{\rm{FM}}}}}\equiv {{{{\mathcal{E}}}}}_{2}{{{{\mathcal{E}}}}}_{1}=JN\), as mentioned in the main text. However, the presence of disorder mixes the Hamiltonian’s zerodisorder eigenstates, necessitating the analysis through numerical diagonalization. Using the numerically determined eigenenergies and eigenstates, we compute the atomic susceptibility and participation ratio using equations (15) and (16), respectively. We average these quantities with respect to 2,000 disorder realizations of the Hamiltonian, the results of which are illustrated in Figs. 3 and 4. The corresponding variances are strongly suppressed, falling within the linewidths of the simulated data.
Data availability
The datasets generated and analysed in the current study are available via Zenodo at https://doi.org/10.5281/zenodo.7074544 (ref. ^{62}).
Code availability
The codes used for the analysis of the experimental data and for simulations are available from the corresponding author upon reasonable request.
References
Parisi, G. Field Theory, Disorder and Simulations Vol. 49 (World Scientific, 1992).
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).
Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).
Lippe, C. et al. Experimental realization of a 3D random hopping model. Nat. Commun. 12, 6976 (2021).
Signoles, A. et al. Glassy dynamics in a disordered Heisenberg quantum spin system. Phys. Rev. X 11, 011011 (2021).
Marcuzzi, M. et al. Facilitation dynamics and localization phenomena in Rydberg lattice gases with position disorder. Phys. Rev. Lett. 118, 063606 (2017).
Roati, G. et al. Anderson localization of a noninteracting Bose–Einstein condensate. Nature 453, 895–898 (2008).
Billy, J. et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453, 891–894 (2008).
Jendrzejewski, F. et al. Threedimensional localization of ultracold atoms in an optical disordered potential. Nat. Phys. 8, 398–403 (2012).
Kondov, S. S., McGehee, W. R., Zirbel, J. J. & DeMarco, B. Threedimensional Anderson localization of ultracold matter. Science 334, 66–68 (2011).
Maier, C. et al. Environmentassisted quantum transport in a 10qubit network. Phys. Rev. Lett. 122, 050501 (2019).
Schreiber, M. et al. Observation of manybody localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
Smith, J. et al. Manybody localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907–911 (2016).
Lukin, A. et al. Probing entanglement in a manybody localized system. Science 364, 256–260 (2019).
Mivehvar, F., Piazza, F., Donner, T. & Ritsch, H. Cavity QED with quantum gases: new paradigms in manybody physics. Adv. Phys. 70, 1–153 (2021).
Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).
Guo, Y. et al. An optical lattice with sound. Nature 599, 211–215 (2021).
Zhang, X. et al. Observation of a superradiant quantum phase transition in an intracavity degenerate Fermi gas. Science 373, 1359–1362 (2021).
Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2017).
Dogra, N. et al. Dissipationinduced structural instability and chiral dynamics in a quantum gas. Science 366, 1496–1499 (2019).
Kongkhambut, P. et al. Observation of a continuous time crystal. Science 377, 670–673 (2022).
Vaidya, V. D. et al. Tunablerange, photonmediated atomic interactions in multimode cavity QED. Phys. Rev. X 8, 011002 (2018).
Periwal, A. et al. Programmable interactions and emergent geometry in an array of atom clouds. Nature 600, 630–635 (2021).
Ebbesen, T. W. Hybrid light–matter states in a molecular and material science perspective. Acc. Chem. Res. 49, 2403–2412 (2016).
GarciaVidal, F. J., Ciuti, C. & Ebbesen, T. W. Manipulating matter by strong coupling to vacuum fields. Science 373 eabd0336 (2021).
Bloch, J., Cavalleri, A., Galitski, V., Hafezi, M. & Rubio, A. Strongly correlated electron–photon systems. Nature 606, 41–48 (2022).
Orgiu, E. et al. Conductivity in organic semiconductors hybridized with the vacuum field. Nat. Mater. 14, 1123–1129 (2015).
Lerario, G. et al. Highspeed flow of interacting organic polaritons. Light Sci. App. 6, e16212 (2017).
Appugliese, F. et al. Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. Science 375, 1030–1034 (2022).
Thomas, A. et al. Large enhancement of ferromagnetism under a collective strong coupling of YBCO nanoparticles. Nano Lett. 21, 4365–4370 (2021).
Prokof’ev, N. V. & Stamp, P. C. E. Theory of the spin bath. Rep. Prog. Phys. 63, 669–726 (2000).
Dukelsky, J., Pittel, S. & Sierra, G. Colloquium: exactly solvable RichardsonGaudin models for manybody quantum systems. Rev. Mod. Phys. 76, 643–662 (2004).
Botzung, T. et al. Dark state semilocalization of quantum emitters in a cavity. Phys. Rev. B 102, 144202 (2020).
Scholes, G. D. Polaritons and excitons: Hamiltonian design for enhanced coherence. Proc. R. Soc. A 476, 20200278 (2020).
Dubail, J., Botzung, T., Schachenmayer, J., Pupillo, G. & Hagenmüller, D. Large random arrowhead matrices: multifractality, semilocalization, and protected transport in disordered quantum spins coupled to a cavity. Phys. Rev. A 105, 023714 (2022).
Schachenmayer, J., Genes, C., Tignone, E. & Pupillo, G. Cavityenhanced transport of excitons. Phys. Rev. Lett. 114, 196403 (2015).
Feist, J. & GarciaVidal, F. J. Extraordinary exciton conductance induced by strong coupling. Phys. Rev. Lett. 114, 196402 (2015).
GonzalezBallestero, C., Feist, J., Gonzalo Badía, E., Moreno, E. & GarciaVidal, F. J. Uncoupled dark states can inherit polaritonic properties. Phys. Rev. Lett. 117, 156402 (2016).
Chávez, N. C., Mattiotti, F., MéndezBermúdez, J. A., Borgonovi, F. & Celardo, G. L. Disorderenhanced and disorderindependent transport with longrange hopping: application to molecular chains in optical cavities. Phys. Rev. Lett. 126, 153201 (2021).
Balasubrahmaniyam, M. et al. From enhanced diffusion to ultrafast ballistic motion of hybrid light–matter excitations. Nat. Mater. 22, 338–344 (2023).
Lipkin, H. J., Meshkov, N. & Glick, A. J. Validity of manybody approximation methods for a solvable model: (I). Exact solutions and perturbation theory. Nucl. Phys. 62, 188–198 (1965).
Makhalov, V. et al. Probing quantum criticality and symmetry breaking at the microscopic level. Phys. Rev. Lett. 123, 120601 (2019).
Muniz, J. A. et al. Exploring dynamical phase transitions with cold atoms in an optical cavity. Nature 580, 602–607 (2020).
Richardson, R. W. A restricted class of exact eigenstates of the pairingforce Hamiltonian. Phys. Lett. 3, 277–279 (1963).
Gaudin, M. Diagonalisation d’une classe d’hamiltoniens de spin. J. Phys. 37, 1087–1098 (1976).
Buccheri, F., De Luca, A. & Scardicchio, A. Structure of typical states of a disordered Richardson model and manybody localization. Phys. Rev. B 84, 094203 (2011).
LewisSwan, R. J. et al. CavityQED quantum simulator of dynamical phases of a BardeenCooperSchrieffer superconductor. Phys. Rev. Lett. 126, 173601 (2021).
Kelly, S. P., Thompson, J. K., Rey, AnaMaria & Marino, J. Resonant light enhances phase coherence in a cavity QED simulator of fermionic superfluidity. Phys. Rev. Research 4, L042032 (2022).
Bychek, A., Hotter, C., Plankensteiner, D. & Ritsch, H. Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling. Open Res. Europe 1, 73 (2021).
Gopalakrishnan, S., Lev, B. L. & Goldbart, P. M. Frustration and glassiness in spin models with cavitymediated interactions. Phys. Rev. Lett. 107, 277201 (2011).
Habibian, H., Winter, A., Paganelli, S., Rieger, H. & Morigi, G. Boseglass phases of ultracold atoms due to cavity backaction. Phys. Rev. Lett. 110, 075304 (2013).
Davis, E. J., Bentsen, G., Homeier, L., Li, T. & SchleierSmith, M. H. Photonmediated spinexchange dynamics of spin1 atoms. Phys. Rev. Lett. 122, 010405 (2019).
Chowdhury, D., Georges, A., Parcollet, O., & Sachdev, S. SachdevYeKitaev models and beyond: window into nonFermi liquids. Rev. Mod. Phys. 94, 035004 (2022).
Sauerwein, N., CantatMoltrecht, T., Grigoras, I. T. & Brantut, J.P. Vibration damping platform for cavity quantumelectrodynamics experiments. Rev. Sci. Instrum. 93, 033203 (2022).
Brantut, J. P. et al. Lightshift tomography in an opticaldipole trap for neutral atoms. Phys. Rev. A 78, 031401 (2008).
Leslie, S., Shenvi, N., Brown, K. R., StamperKurn, D. M. & Whaley, K. B. Transmission spectrum of an optical cavity containing N atoms. Phys. Rev. A 69, 043805 (2004).
Jensen, J. & Mackintosh, A. R. Rare Earth Magnetism. The International Series of Monographs on Physics (Clarendon Press, 1991).
Wiener, N. Generalized harmonic analysis. Acta Math. 55, 117–258 (1930).
Khintchine, A. Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109, 604–615 (1934).
Beck, C. & Schögl, F. Thermodynamics of Chaotic Systems: An Introduction (Cambridge Univ. Press, 1993).
Meyer, C. D. Matrix Analysis and Applied Linear Algebra (SIAM, 2000).
Sauerwein, N. et al. Engineering random spin models with atoms in a highfinesse cavity. Zenodo https://doi.org/10.5281/zenodo.7074544 (2022).
Acknowledgements
N.S., F.O., T.C.M. and J.P.B. acknowledge funding from the Swiss National Science Foundation (grant no. 184654), the Sandoz Family Foundation and EPFL. G.P. acknowledges support from the Institut Universitaire de France (IUF) and the University of Strasbourg Institute of Advanced Studies (USIAS), the ANR via CLIMAQS and aQCess ANR21ESRE0032. P.U., S.B. and P.H. acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 804305); Provincia Autonoma di Trento; and Q@TN, the joint lab among University of Trento, FBKFondazione Bruno Kessler, INFN—National Institute for Nuclear Physics and CNR—National Research Council. S.B. acknowledges CINECA for the use of highperformance computing resources under ISCRAC project ISSYK2 (HP10CP8XXF). F.M. acknowledges the highperformance computing centre of the University of Strasbourg for providing the computing time.
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N.S., F.O. and T.C.M. designed and assembled the experimental setup. N.S. and F.O. performed the measurements and analysed the data. P.U., S.B. and F.M. performed the calculations and numerical simulations. J.P.B., P.H. and G.P. planned and supervised the experimental (J.P.B.) and theoretical (P.H. and G.P.) work. All the authors contributed to the interpretation of the results and writing of the manuscript.
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Extended data
Extended Data Fig. 1 Characterization of lightshift of excited state.
a, Absorption picture of an atomic cloud in the cavity dipole trap. The grey needle in the middle is used to shield central atoms from the absorption light, making it possible to empty the outer pancake traps using the radiation pressure force of the absorption imaging. b and c, Absorption image of the cloud at maximal shift (green arrow in d) and zero shift (red arrow in d). d, Spatiallyresolved absorption spectroscopy of the D2 transition in the presence of a single lattice beam (Gaussian beam centred on x = 0 mm with waist of 120μm). Each horizontal line of plot d is the yintegrated optical density. The light shifting laser was 50 MHz blue detuned from the 2P_{3/2}–4D_{5/2} transition. We can observe both resonances of the Autler–Townes doublet, the strong lightshifted single photon transition (top) and the faint twophoton transition (bottom). e, Cavity transmission spectroscopy for cloud illuminated with homogenous lightshifting beam. The dashed blue lines indicate the expected resonance frequencies of the Autler–Townes doublet. f, 2D map of cavity transmission spectra for different detunings of the lightshifting laser from the 2P_{3/2}–4D_{5/2} transition. Each horizontal line is a cavity transmission spectrum. In the colorbar, red indicates a large number of photons transmitted through the cavity. The black horizontal dashed line marks the configuration of panel e.
Extended Data Fig. 2 Experimental sequence and measurement of atomic susceptibility.
2c: data are presented as mean values +/ SEM, Averages run over 371 measurements. a, Timeline of the experimental sequence. The core elements are the interrogation of the disordered cavity–atom system (red) and the subsequent detection of depumped atoms in the \(\left\vert a\right\rangle\)(F = 3/2) state. b, Illustration of the probe configurations during interrogation and depumping detection. I, When probing the disordered system, the excited state of the atoms is dressed with the lightshifting laser, indicated by the blue colour of the level and the shift ϵ. The photons entering the cavity from the probe have two decay channels. Either they leak out of the cavity on the other side (red wiggled arrow) where they will be detected by a single photon counter, or they can be lost by freespace spontaneous emission of an atomic excitation (orange wiggled line). At zeromagnetic field, the transition from \(\left\vert g\right\rangle\) to \(\left\vert e\right\rangle\) is not closed, therefore spontaneous emission events can depump the atom into the \(\left\vert a\right\rangle\) state. II, These depumped atoms can be detected by measuring the cavity transmission. If the cavity is on resonance with the \(\left\vert a\right\rangle\)–\(\left\vert e\right\rangle\) transition and only a single atoms is in state \(\left\vert a\right\rangle\) the transmission gets strongly suppressed. c, Calibration of depumping signal. The mean number of transmitted photons during the depumping detection is plotted against the probe power during the interrogation of the disordered system. The error bars of the data represent statistical fluctuations. The orange line shows a fit of the theoretically expected relation [see Eq. (8)]. The inset shows a histogram of detected photons for low (green) and high (red) probe powers (see arrows on xaxis for configuration).
Supplementary information
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Supplementary Section 1, Figs. 1 and 2 and references.
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Sauerwein, N., Orsi, F., Uhrich, P. et al. Engineering random spin models with atoms in a highfinesse cavity. Nat. Phys. 19, 1128–1134 (2023). https://doi.org/10.1038/s41567023020333
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DOI: https://doi.org/10.1038/s41567023020333
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