Amplified response of cavity-coupled quantum-critical systems

Abstract

A quantum critical point develops when matter undergoes a continuous transformation between distinct ground states at absolute zero. It hosts pronounced quantum fluctuations, which render the system highly susceptible to external perturbations. While light-matter coupling has rapidly moved forward as a means to probe and control quantum materials, the capacity of quantum critical fluctuations in the photon-mediated responses has been largely unexplored. Here we advance the notion that directly coupling a quantum critical mode to a quantized cavity field dramatically facilitates the realization of the elusive superradiant phase transition in equilibrium, circumventing at once the key obstacles that have prevented its attainment in spite of decades of pursuit. The superradiant phase transition develops far below the ultrastrong regime of light-matter couplings, and the transition is accompanied by the light-matter hybrid system showing strongly enhanced intrinsic squeezing and amplified quantum Fisher information. We also identify candidate cavity quantum materials platforms for validating the proposed effect. Our findings suggest a general principle by which quantum criticality amplifies the response to cavity photons. They also demonstrate that cavity coupling accesses the elevated quantum entanglement of the underlying matter at quantum criticality, thereby pointing to a pathway towards realizing the potential of highly collective quantum materials to expand the capacities of quantum information science.

Introduction

Strong correlations give rise to a rich variety of unusual physical properties ^1,2. This is especially so for systems in a quantum critical regime, where quantum fluctuations are pronounced, and physical responses are enhanced^2,3,4,5,6,7. To understand the highly collective quantum critical fluids, new means of probing them are highly desired. A defining characteristic of quantum criticality is the mixing of statics and dynamics^8,9, and indeed, singular dynamical responses have been demonstrated in the quantum critical regime^{10,11,12,13,14,15}. They not only corroborate the existence of the underlying quantum critical point (QCP), but also characterize the nature of the quantum criticality⁷. As such, external dynamical perturbations are capable of elucidating the quantum critical state¹⁶.

Here, we address how coupling to an optical cavity provides an important new means of exploring the amplified responses of quantum criticality at thermodynamic equilibrium. In a larger context, our approach is motivated by the increasing recognition that light-matter coupling can effectively interrogate and manipulate quantum materials^17,18,19,20, and engineer novel quantum states at thermodynamic equilibrium that are part light and part matter²¹. More specifically, a cavity introduces a single mode of quantized electromagnetic radiation that can be coupled to matter degrees of freedom through dipolar or Zeeman-type interactions²². It has been studied extensively in the pursuit of a superradiant phase^23,24. The latter is characterized by a macroscopic occupation of a cavity-photonic mode^25,26. The Dicke model, which describes a collection of two-level subsystems interacting with a quantized cavity mode, provides the standard setting for exploring the superradiant phase transition (SRPT)^27,28,29. The transition, taking place in the thermodynamic limit, is characterized by the development of a macroscopic occupation of the cavity mode and a spontaneous collective polarization in the matter sector. In the absence of detuning, the SRPT requires a light–matter coupling strength on the order of or exceeding 10% of the cavity mode’s energy, placing it in the ultrastrong coupling regime^24,30. Moreover, a “no-go” theorem, resulting from the requirement of gauge invariance^31,32, restricts the ability of dipolar couplings in inducing an SRPT at thermodynamic equilibrium²⁹. As a manifestation of these two challenges, the experimental realization of equilibrium SRPT in cavity-coupled systems remains elusive in spite of decades of efforts^30,33. Therefore, identifying mechanisms that ease access to the superradiant phase and the concomitant SRPT is of broad interest, as it enables controlled studies of collective quantum phenomena in light–matter interacting systems.

We focus on the effect of cavity coupling in a canonical quantum magnetic system across its QCP, as illustrated in Fig. 1a. Importantly, when the cavity mode directly (i.e., bilinearly, through a Zeeman coupling that avoids the no-go theorem, cf. Fig. 1b) couples to the degree of freedom that exhibits quantum critical fluctuations, we show here that SRPTs can be realized far below the ultrastrong coupling limit of light-matter interactions (Fig. 2a, c); thus, our work bypasses both challenges in the long-standing pursuit of equilibrium SRPT. The superradiant states, thus obtained, are highly squeezable in the vicinity of the SRPTs and support a large multipartite entanglement that can be witnessed by the quantum Fisher information. Since both aspects provide valuable metrological resources, our work indicates cavity quantum materials tuned to the vicinity of matter-QCP can potentially serve as particularly efficient quantum sensors. By analyzing the scaling behavior of intrinsic squeezing close to the SRPTs, we show that in such systems, the coherent mixing of critical matter modes and cavity photons generates a superradiant state that can be squeezed more efficiently than that in the original Dicke model. To compare and contrast our results with those for the Dicke model, we restrict our consideration to models of quantum magnetism with spin 1/2. That a direct coupling of the cavity mode to a quantum critical degree of freedom enables the underlying matter quantum criticality to sharply amplify the responses to cavity coupling represents a key new insight, which has not been recognized in previous work on such cavity-coupled systems^{34,35,36,37,38,39,40,41,42}. Conversely, our results show that cavity coupling provides a means to access the enhanced quantum entanglement of the underlying matter at quantum criticality, thereby suggesting the potential of highly collective quantum materials for quantum information science. Finally, we identify specific materials that are amenable to cavity materials engineering²¹ and provide concrete platforms for validating the effect we have advanced.

Fig. 1: Schematic of a cavity mode coupled to matter degrees of freedom near a QCP.

a Phase diagram as a function of a non-thermal tuning parameter x. The coupling strength (g_c) required to induce a superradiant quantum phase transition (SRPT) is minimized at the matter QCP (x_c). The minimum g_c vanishes when the cavity model couples directly to the critical mode (solid curve) but remains nonzero otherwise (dashed curve). b Schematic of the cavity magnetic field aligned parallel to the Ising spin-coupling direction. The red sinusoidal curve with arrows represents the cavity magnetic field mode. c Same setup as in b, but with the cavity magnetic field oriented perpendicular to the Ising spin-coupling direction.

Fig. 2: Phase diagram, squeezing, and entanglement for a cavity mode directly coupled to a quantum critical mode.

a Phase diagram in the large-S limit, depicting $\langle \widehat{a}\rangle$; black curve denotes continuous SRPTs that terminate at the TFIM QCP (red dot); blue bar refers to the ferromagnetic (FM) order. We set the lattice coordination number z = 2,  and spin S = 1/2. b Scaling of $\langle \widehat{a}\rangle$ with ${(g-{g}_{c})}^{\beta }$ in the large-S limit (color bar represents β): g_c = 0 for h/J ≤ (h/J)_TFIM; the Dicke model gives β = 1/2. c Phase boundary from DMRG simulations (blue points) with fit ${g}_{c}\propto {(h-{h}_{{{\rm{TFIM}}}})}^{0.65}$. d Data in c is extracted from the peaks of $\partial \langle \widehat{{a}}^{{\dagger} }\widehat{a}\rangle /\partial h$ as a function of h. We observe a reduced participation of the cavity mode in the h-tuned quantum phase transition, indicating a smooth evolution of the SRPT to the conventional TFIM quantum phase transition as g → 0. e, f Minimum and maximum variances of the quadrature ${\widehat{X}}_{\theta,\psi,\phi }$ (cf. Eq. (3)) along the phase boundary in (a). The minimum variance ${(\Delta {X}_{\min })}^{2}$ vanishes, indicating perfect intrinsic squeezing, while the maximum variance ${(\Delta {X}_{\max })}^{2}$ diverges, indicating enhanced quantum entanglement. Dashed lines show fits to numerical data as $g\to {g}_{c}^{+}$: at h = J (h > J), ${(\Delta {X}_{\min })}^{2} \sim g$ ($\sqrt{g-{g}_{c}}$), and ${(\Delta {X}_{\max })}^{2} \sim {g}^{-1}$ (${(g-{g}_{c})}^{-1/2}$). Note that (h/J)_TFIM in the large-S limit is distinct from the fully quantum solution [cf. a vs. c]; we have set ω₀ = 1 = J in (c–e).

Results

Cavity-coupled quantum critical system

A quantum spin system coupled to cavity photons contains the following ingredients. The cavity mode, denoted by the field operator $\widehat{a}$, has frequency ω₀. It couples to the α-th component (c.f. Fig. 1b, c) of the magnetization of the quantum spin system (${\widehat{S}}_{{{\bf{r}}}}^{\alpha }$), with a collective coupling constant g. The matter sector is described by the Hamiltonian H_spin, in various spatial dimensions. The overall Hamiltonian of the light-matter coupled system takes the following form:

$$\widehat{H}={\omega }_{0}{\widehat{a}}^{{\dagger} }\widehat{a}+\frac{g}{\sqrt{N}}(\widehat{a}+{\widehat{a}}^{{\dagger} }){\sum }_{{{\bf{r}}}}{{\bf{n}}}\cdot {\widehat{{{\bf{S}}}}}_{{{\bf{r}}}}+{\widehat{H}}_{{{\rm{spin}}}},$$

(1)

where n is a unit vector controlling the spin projection that couples with the photon⁴³, and N is the number of spin sites. Here, the magnetic component of light Zeeman-couples to the localized spins of the quantum spin system. Consequently, $\widehat{H}$ is gauge invariant (see “Methods”).

For concreteness, we will primarily focus on the ferromagnetic transverse field Ising model (TFIM),

$${\widehat{H}}_{{{\rm{spin}}}}=-J{\sum }_{\langle {{\bf{r}}},{{{\bf{r}}}}^{{\prime} }\rangle }{\widehat{S}}_{{{\bf{r}}}}^{x}{\widehat{S}}_{{{\bf{r}}}}^{{\prime} x}-h{\sum }_{{{\bf{r}}}}{\widehat{S}}_{{{\bf{r}}}}^{z}\,,$$

(2)

where J ≥ 0 describes the strength of the Ising spin-spin interactions and h is a transverse field that also specifies the detuning in the cavity.

Coupling to a critical degree of freedom

Our primary focus will be on the cavity photons that are Zeeman-coupled to the order parameter of the underlying Ising quantum phase transition (Fig. 1b). In this case, the cavity mode bilinearly couples to the order parameter and, as such, the singular quantum critical fluctuations of the order parameter directly affect the response of the photon field.

This corresponds to the choice ${{\bf{n}}}=\widehat{x}$, so that the photon field is linearly coupled to the magnetization ${\widehat{M}}_{x}$. In the absence of the cavity coupling, the system undergoes a continuous ferromagnetic quantum phase transition across the critical field, $h={h}_{{{\rm{TFIM}}}}$ (which equals J in the large-S limit), as the transverse field h is tuned for a fixed Ising exchange interaction J. In the ferromagnetic phase, the order parameter—the net magnetization ${m}_{x}=\left\langle \widehat{{M}_{x}}\right\rangle$—is nonzero. In the paramagnetic phase, the order parameter vanishes. The static magnetic susceptibility, χ_x, diverges upon tuning h across the critical field, ${h}_{{{\rm{TFIM}}}}$ (see “Methods”).

In this case, the ferromagnetic and the superradiant phases mutually cooperate because both the light-matter (g) and Ising (J) terms weaken the field-polarized state while commuting with each other. Since in the ${{\bf{n}}}=\widehat{x}$ limit the model is not exactly solvable, in order to demonstrate this cooperation and explore its consequences, we will first obtain the zero-temperature phase diagram supported by $\widehat{H}$ in the large-S limit, and, subsequently, verify these predictions for d = 1 in the spin-1/2 case through density matrix renormalization group (DMRG) calculations.

We isolate the k = 0 magnon mode (henceforth, represented by ${\widehat{{\mathfrak{b}}}}_{0}$; see Methods), and solve for the polaritonic normal modes. The vanishing of the dispersion at a critical coupling, g_c, triggers a Bose-Einstein condensation (BEC) in the corresponding polaritonic mode, which amounts to an SRPT. In the standard Dicke model, corresponding to (J, S) → (0, 1/2), ${g}_{c}^{2}={\omega }_{0}h$, which indicates the need for an ultrastrong light–matter coupling at weak detunings²⁴. A nonzero J reduces g_c and favors the nucleation of a superradiant state.

A key result of our work is that the critical cavity-spin coupling for the SRPT vanishes at the TFIM QCP. Consider a fixed J, as $h\to {h}_{{{\rm{TFIM}}}}^{+}$, g_c vanishes (as seen from computations in the large-S limit in Methods, Eq. (8)). For a fixed J and ω₀, superradiant states are present in the entire region bounded from below by the curve $g={g}_{c}(h/J)\Theta \left((h/J)-{(h/J)}_{{{\rm{TFIM}}}}\right)$ on the (h, g) plane. In Fig. 2a we identify this region by plotting 〈a〉 for the one-dimensional cavity-TFIM. As the phase boundary is approached from the g > g_c side, 〈a〉 vanishes continuously as g, g², and $\sqrt{g-{g}_{c}}$ for $h < {h}_{{{\rm{TFIM}}}}$, $h={h}_{{{\rm{TFIM}}}}$, and $h > {h}_{{{\rm{TFIM}}}}$, respectively, as depicted in Fig. 2b and described in detail in Supplementary Note 1. This variation in the scaling of 〈a〉 indicates the presence of distinct scaling regimes in the superradiant phase that reflect the phase diagram of the underlying matter sector.

An important question is what happens in the extreme quantum limit. To address this issue, we have performed DMRG simulations for spin-$\frac{1}{2}$ (i.e., $S=\frac{1}{2}$) TFIM coupled to a cavity mode (see Supplementary Note 2 for details). Figure. 2c, d shows a line of continuous quantum phase transitions between the Ising-paramagnetic normal phase and a superradiant phase on the $h > {h}_{{{\rm{TFIM}}}}$ side of the phase diagram. The numerically obtained phase boundary is such that ${g}_{c}\propto {(h-{h}_{{{\rm{TFIM}}}})}^{\zeta }$ with ζ = 0.65 ≈ 2/3. Not surprisingly, the scaling exponents obtained by DMRG simulations deviate from the large-S result. Importantly, though, our result shows that the phase diagram is robust when the quantum fluctuations are fully accounted for.

Intrinsic squeezing and quantum entanglement

The extreme propensity of the fully quantum system towards an SRPT, as revealed by our DMRG simulations, sets the stage for us to determine the metrological and quantum entanglement implications of our findings. The intermixing between the cavity mode and the critical spin degree of freedom captures the coherence between the light and matter sectors, which is described in terms of an intrinsic two-mode squeezing^44,45,46. Specifically, in the large-S limit, the variance of the polaritonic operator,

$${\widehat{X}}_{\theta,\phi,\psi }(h/J)=\frac{1}{2}[{e}^{i\phi }(\cos \theta \,\delta \widehat{a}+{e}^{i\psi }\sin \theta \,\delta \widehat{b})+\,{{\rm{h.c.}}}]\,,$$

(3)

with $\delta \widehat{a}$ and $\delta \widehat{b}$ representing fluctuations about $\langle \widehat{a}\rangle$ and $\langle {\widehat{{\mathfrak{b}}}}_{0}\rangle$, respectively, and (ϕ, ψ, θ) being optimization parameters, is minimized to zero at the SRPT (see “Methods”). The intrinsic squeezing in the limit $g\to {g}_{c}^{+}(h/J)$ is sensitive to the three superradiant regimes identified above, and the minimum variance, $\Delta {X}_{\min }^{2}(h/J)$, scales as ${(g-{g}_{c})}^{0}$, (g − g_c), and ${(g-{g}_{c})}^{1/2}$ for $h < {h}_{{{\rm{TFIM}}}}$, $h={h}_{{{\rm{TFIM}}}}$, and $h > {h}_{{{\rm{TFIM}}}}$, respectively; the last two cases for S = 1/2 are shown in Fig. 2e. We note that the vanishing of $\Delta {X}_{\min }^{2}(h/J)$ with g → g_c for $h > {h}_{{{\rm{TFIM}}}}$ can be alternatively viewed as the existence of a perfect intrinsic squeezing at a fixed g as h tunes the system across an SRPT (c.f. Fig. 2a). This is a remarkable outcome when contrasted with the weak squeezing found in pure TFIM (g = 0) as h is tuned across ${h}_{{{\rm{TFIM}}}}$⁴⁷.

Importantly, in the vicinity of the SRPT, the squeezing is stronger at the QCP compared to the case of the pure Dicke model. In particular, comparing what happens at the QCP $(h={h}_{{{\rm{TFIM}}}})$ with that in the disordered regime $(h > {h}_{{{\rm{TFIM}}}})$, for a fixed distance from the respective SRPTs, δg ≡ g − g_c, $\Delta {X}_{\min }^{2}(h={h}_{{{\rm{TFIM}}}})/\Delta {X}_{\min }^{2}(h > {h}_{{{\rm{TFIM}}}}) \sim \sqrt{\delta g}$, which vanishes as δg → 0. This reflects the interplay between the approach to the SRPT and the underlying quantum criticality of the TFIM. The reduction of $\Delta {X}_{\min }^{2}$ at the QCP from that in the disordered regime reflects the increased precision with which ${\widehat{X}}_{\min }$ can be measured in principle at the QCP. By contrast, for the ordered regime $(h < {h}_{{{\rm{TFIM}}}})$ and at sufficiently weak g, the spin-sector possesses a long-range order, which is not conducive to squeezing; here, the only meaningfully squeezable quadrature comes solely from the photon sector, which does not exhibit a perfect squeezing⁴⁸ (see Supplementary Note 3).

The elevated intrinsic squeezing at the QCP ($h={h}_{{{\rm{TFIM}}}}$) indicates the enhancement of light-matter quantum entanglement⁴⁵. The latter can be described in terms of the variance of the variable conjugate to ${\widehat{X}}_{\min }$. The procedure for identifying this conjugate variable, ${\widehat{X}}_{\max }$, is presented in the “Methods”. The variance, $\Delta {X}_{\max }^{2}$, becomes large, as shown in Fig. 2f; near the QCP, $\Delta {X}_{\max }^{2}$ diverges $\sim \frac{1}{g-{g}_{{{\rm{c}}}}}$, which is stronger than the $\sim \frac{1}{{(g-{g}_{{{\rm{c}}}})}^{1/2}}$ form arising in the Dicke model as well as in the disordered regime ($h > {h}_{{{\rm{TFIM}}}}$). For the pure state we are considering, this variance is proportional to (is equal to 1/4 of) the polaritonic quantum Fisher information⁴⁹, capturing the degree of light-matter quantum entanglement.

Cavity coupling to a non-critical mode

For comparison, we now turn to the case where the light-matter coupling is orthogonal to the Ising order parameter, corresponding to ${{\bf{n}}}=\widehat{y}$ (i.e., Fig. 1c). In this case, the light-matter and Ising terms no longer commute. Consequently, the ferromagnetism competes with the superradiant state, and their respective fluctuations mutually frustrate each other. This competition results in a complex phase diagram^34,35 as shown in Fig. 3a. The SRPT boundary reaches a minimum in the vicinity of the TFIM QCP, which underscores the role of the matter QCP in facilitating the superradiant phase (c.f., Fig. 1a). Moreover, this minimum corresponds to a tricritical point that generates an anomalous scaling for $\langle \widehat{a}\rangle \sim {(g-{g}_{c})}^{\beta }$ with β ≈ 0.25 in its vicinity and supports a rich set of crossover behaviors, as portrayed in Fig. 3b (also, see Methods and Supplementary Note 4).

Fig. 3: Phase diagram for a cavity mode coupled to non-critical degrees of freedom.

a Analytically obtained exact phase diagram in the full quantum limit of S = 1/2, where the color bar indicates $\langle \widehat{a}\rangle$; red solid line marks the QCP of the one-dimensional TFIM; dashed (solid) black lines denote discontinuous (continuous) SRPTs; red dot represents the tricritical point. b Scaling of $\langle \widehat{a}\rangle$ with ${(g-{g}_{{{\rm{c}}}})}^{\beta }$ in the vicinity of the tricritical point at h/J ≈ 0.55. The crossover behavior is dictated by Eq. (10).

Other models and robustness

We now address the robustness of the SRPT facilitated by the matter quantum criticality by considering a different cavity-coupled model with the spin Hamiltonian describing the 1D ferromagnetic XY model,

$${\widehat{H}}_{{{\rm{spin}}}}=-\frac{J}{2}{\sum }_{i}\left[(1+\Delta ){\widehat{S}}_{i}^{x}{\widehat{S}}_{i+1}^{x}+(1-\Delta ){\widehat{S}}_{i}^{y}{\widehat{S}}_{i+1}^{y}\right]\,.$$

(4)

Since ${\widehat{H}}_{{{\rm{spin}}}}$ supports distinct types of orderings (see “Methods”), a fixed light-matter vertex can represent coupling to either critical or non-critical matter-modes, depending on the location of the model parameters in its phase diagram. Here, the choice ${{\bf{n}}}=\widehat{x}$ or $\widehat{y}$ (${{\bf{n}}}=\widehat{z}$) in Eq. (1) corresponds to coupling the cavity mode to a critical (non-critical) matter mode. The main contrast with the TFIM lies in the fact that as Δ → 0⁻ (Δ → 0⁺), for ${{\bf{n}}}=\widehat{x}$ (${{\bf{n}}}=\widehat{y}$), the diverging correlation length of the fluctuations in the ${\widehat{S}}^{x}$ (${\widehat{S}}^{y}$) channel continuously suppresses the g_c, even though magnetic order persists in the ${\widehat{S}}^{y}$ (${\widehat{S}}^{x}$) channel (see Section V of the SI). For any choice of n, however, g_c is minimized in the vicinity of Δ = 0 (c.f. Supplementary Note 5), consistent with our earlier analysis of the cavity-TFIM variants.

Discussion

Experimental implications

The ferromagnetic-TFIM quantum phase transition can be studied in the quasi-one-dimensional materials CoNb₂O₆⁵⁰, as well as higher-dimensional systems, such as LiHoF₄⁵¹ and CrI₃⁵². These materials can be directly coupled to a quantized cavity mode to access the propensity for SRPT, and elevated photon–matter squeezing and entanglement in the vicinity of the magnetic QCP, as presented in this work. The feasibility of coupling magnetic materials to cavity modes has been demonstrated in various cavity-magnonic systems^53,54,55. Thus, there is a good prospect for potentially realizing our proposal on existing experimental platforms^19,56 with the externally applied magnetic field h serving as a practical tuning parameter³³. Finally, also of interest in the present context are strange metals that exhibit strong quantum fluctuations⁵⁷. Our findings suggest that cavity coupling provides a means to access the elevated multipartite entanglement^13,14 of such systems.

Outlook and summary

Our framework extends to driven-dissipative systems. In particular, cavity quantum systems have emerged as attractive platforms for realizing nonequilibrium phenomena where light plays a key role^{17,18,19,58,59,60,61,62}. Both features discussed here—the suppression of g_c and enhanced squeezing and entanglement—can be generalized to nonequilibrium settings, which may also provide pathways for accessing the critically enhanced squeezing and QFI identified in this work⁶³. In closing the paragraph, we note that modeling a cavity as a single mode is an idealization, and realistic cavities generally require multi-mode modeling⁶⁴. Such a generalization is also necessary to systematically connect with the details of cavity design and the formal notion of a thermodynamic limit (see Supplementary Note 4). Generalizing to multi-mode settings, appropriate for cavity magnonics platforms, it can be shown that the conclusions based on the single-mode model remain unchanged (see “Methods” and Supplementary Note 4).

To summarize, we have theoretically demonstrated that quantum critical fluctuations in the matter sector greatly amplify the response to the cavity-photon coupling and, especially, promote the formation of a superradiant state. This tendency is particularly striking when the cavity photons directly couple to the critical matter degree of freedom. Here, in the quantum critical regime, the superradiant phase becomes accessible far below the ultrastrong coupling limit of cavity–matter interactions. Moreover, the system shows intrinsic squeezing and enhanced quantum Fisher information. In other words, coupling to the cavity photons provides a way to access the elevated quantum entanglement of the underlying matter at quantum criticality, a finding that points a way towards realizing the potential of highly collective quantum materials for expanding the capacities of quantum information science. In this way, our work identifies a general principle for harnessing matter quantum criticality in cavity quantum materials to realize SRPTs at thermodynamic equilibrium and opens a new route of investigation for designing cavity quantum materials and generating metrologically useful quantum states.

Methods

Cavity-coupled TFIM—additional properties

For ${{\bf{n}}}\cdot \widehat{z}=0$, $\widehat{H}$ in Eqs. ((1) and (2)) has a ${{\mathbb{Z}}}_{2}$ symmetry associated with $(a,{{\bf{n}}}\cdot {\widehat{{{\bf{S}}}}}_{{{\bf{r}}}})\to -(a,{{\bf{n}}}\cdot {\widehat{{{\bf{S}}}}}_{{{\bf{r}}}})$²⁹. (For ${{\bf{n}}}\cdot \widehat{z}=1$, the model lacks the ${{\mathbb{Z}}}_{2}$-symmetry due to the applied magnetic field, and the ground state supports a photon condensate at any non-vanishing model parameters.) In the J = 0 limit, $\widehat{H}$ reduces to the well-known Dicke model, and the system undergoes a spontaneous ${{\mathbb{Z}}}_{2}$-symmetry breaking as g exceeds ${g}_{c}(J=0)=\sqrt{{\omega }_{0}h}$. The resultant superradiant phase is characterized by a macroscopic occupation of the bosonic mode, $\langle \widehat{a}\rangle \ne 0$, and a non-trivial spin-polarization, $\langle {{\bf{n}}}\cdot \widehat{{{\bf{S}}}}\rangle \ne 0$. In the opposite limit, g = 0, light and matter sectors are decoupled, and the ground state is the product state of the zero photon occupation state and the ground state of the TFIM. Notably, the spin sector undergoes a ferromagnet to paramagnet (field-polarized state) quantum phase transition as the ratio h/J is tuned across a critical value, ${(h/J)}_{{{\rm{TFIM}}}}$.

Here, we couple the cavity mode to the α-th component of the net magnetization, ${m}_{\alpha }=\langle {\widehat{M}}_{\alpha }\rangle$ where ${\widehat{M}}_{\alpha }\equiv {\sum }_{{{\bf{r}}}}{\widehat{S}}_{{{\bf{r}}}}^{\alpha }/N$ with N being the total number of sites. In the ferromagnetic (paramagnetic) phase m_x ≠ 0 (m_x = 0). At a fixed J, the static susceptibility, χ_x, diverges as ${\chi }_{x} \sim | h-{h}_{{{\rm{TFIM}}}}{| }^{-\gamma }$ upon tuning h across h_TFIM, with the critical exponent γ being dimension-dependent. This divergent susceptibility identifies ${\widehat{M}}_{x}$ as the critical mode that is associated with the quantum phase transition in the TFIM. The cavity mode directly couples to ${\widehat{M}}_{x}$ for ${{\bf{n}}}=\widehat{x}$.

The cavity coupling, gauge invariance, and absence of the no-go theorem

The spins are Zeeman-coupled to a fluctuating magnetic field generated by the electromagnetic field inside the cavity,

$${\widehat{H}}_{\widehat{a}-\widehat{S}}={\mu }_{B}{g}_{B}{{\bf{B}}}(t)\cdot {\sum }_{{{\bf{r}}}}{\widehat{{{\bf{S}}}}}_{{{\bf{r}}}}\,.$$

(5)

Upon quantization of the electromagnetic field, the nth component of the total spin couples to the photon momentum quadrature $i(\widehat{a}-{\widehat{a}}^{{\dagger} })$: ${\mu }_{B}{g}_{B}{{\bf{B}}}(t)\to ig(\widehat{a}-{\widehat{a}}^{{\dagger} }){{\bf{n}}}$, where g = Cμ_Bg_B is the effective light-matter coupling strength with C being a parameter that depends on the details of the cavity^53,64,65. After a canonical transformation $\widehat{a}\to -i\widehat{a}$ that swaps the photon position and momentum operators, Eq. (1) provides an equivalent description of the coupled system.

Unlike the electric-dipole interaction, which involves the vector potential A and requires a compensating diamagnetic term ( ∝ A ⋅ A) to preserve gauge invariance, the Zeeman coupling depends on the magnetic field B = ∇ × A, which is itself gauge invariant under A → A + ∇ χ. Consequently, ${\widehat{H}}_{\widehat{a}-\widehat{S}}$ is gauge invariant, and no additional A²-type term operates. This is in sharp contrast to the case of mobile two-level emitters or atoms with kinetic Hamiltonian ${\widehat{H}}_{{{\rm{spin}}}}={\widehat{{{\bf{p}}}}}^{2}/(2m)$^32,64, where gauging via $\widehat{{{\bf{p}}}}\to \widehat{{{\bf{p}}}}-e\widehat{{{\bf{A}}}}$ inevitably generates the diamagnetic term that leads to a no-go theorem against an SRPT at equilibrium³¹ that continues to be discussed²⁹. The manifestly gauge-invariant form of the Zeeman coupling means that our case is not subject to any no-go theorem. A distinct microscopic perspective that reaches the same conclusion is presented in Supplementary Note 6.

Cavity coupled to a critical degree of freedom

We start from a large-S analysis by introducing Holstein-Primakoff bosons with the Ising paramagnetic state as the reference, ${\widehat{S}}_{{{\bf{r}}}}^{z}=S-{\widehat{b}}_{{{\bf{r}}}}^{{\dagger} }{\widehat{b}}_{{{\bf{r}}}}$ and ${\widehat{S}}_{{{\bf{r}}}}^{-}=\widehat{b}_{{{\bf{r}}}}^{{\dagger} }\sqrt{2S-\widehat{b}_{{{\bf{r}}}}^{{\dagger} }{\widehat{b}}_{{{\bf{r}}}}}$. Here, ${\widehat{b}}_{{{\bf{r}}}}$ destroys the quantum of spin-fluctuations transverse to the field-polarization direction—a “magnon”—at site r. The effective Hamiltonian governing the resultant system of coupled photons and magnons is obtained from Eq. (1) by expanding about the large-S saddle point and retaining terms up to order S⁰,

$${\widehat{H}}_{{{\rm{eff}}}} = {\omega }_{0}{\widehat{a}}^{{\dagger} }\widehat{a}+\sqrt{\frac{S}{2N}}\,g(\widehat{a}+{\widehat{a}}^{{\dagger} }){\sum }_{{{\bf{r}}}}(\widehat{b}_{{{\bf{r}}}}+\widehat{b}_{{{\bf{r}}}}^{{\dagger} })\\ -\frac{S}{2}J{\sum }_{\langle {{\bf{r}}},{{{\bf{r}}}}^{{\prime} }\rangle }({\widehat{b}}_{{{\bf{r}}}}^{{\dagger} }{\widehat{b}}_{{{{\bf{r}}}}^{{\prime} }}+{\widehat{b}}_{{{\bf{r}}}}{\widehat{b}}_{{{{\bf{r}}}}^{{\prime} }}+\,{{\rm{h.c.}}})+h{\sum }_{{{\bf{r}}}}{\widehat{b}}_{{{\bf{r}}}}^{{\dagger} }{\widehat{b}}_{{{\bf{r}}}}.$$

(6)

We note two key features. First, the photons couple only to a global magnon operator. This implies that, in the large-S limit, only the k = 0 mode in the magnon sector is sensitive to the cavity coupling. Second, as shown below, the magnon modes whose BECs lead to the superradiant and ferromagnetic phases, respectively, are in fact identical. This underscores the cooperation between the two phases.

We isolate the k = 0 magnon mode, and solve for the polaritonic normal modes supported by ${\widehat{H}}_{{{\rm{eff}}}}$. We observe that the ferromagnetic exchange interaction, J, serves as an additional detuning parameter, such that the resonant regime is renormalized to ω₀ = h − zSJ, where z is the coordination number of the lattice on which ${\widehat{H}}_{{{\rm{spin}}}}$ is defined. Using the Nambu basis $\widehat{\Phi }=(\widehat{a}\,{\widehat{{\mathfrak{b}}}}_{0}\,\widehat{a}^{{\dagger} }\,{\widehat{{\mathfrak{b}}}}_{0}^{{\dagger} })$, where ${\widehat{{\mathfrak{b}}}}_{{{\bf{k}}}}$ is the k-th Fourier mode of ${\widehat{b}}_{{{\bf{r}}}}$, we find two branches in the Bogoliubov spectrum,

$${E}_{\pm }=\frac{1}{2\sqrt{2}}\sqrt{{\Omega }_{+}\pm \sqrt{{\Omega }_{-}^{2}+8{g}^{2}hS{\omega }_{0}}}\,,$$

(7)

where ${\Omega }_{\pm }={\omega }_{0}^{2}\pm h(h-zSJ)$. The vanishing of the dispersion of the ‘−’ branch triggers a BEC in the corresponding polaritonic mode. The condition for the vanishing of E₋ determines the critical cavity-coupling for an SRPT,

$${g}_{c}(h/J)=\sqrt{\frac{{\omega }_{0}h}{2S}\left[1-\frac{{(h/J)}_{{{\rm{TFIM}}}}}{h/J}\right]}\,,$$

(8)

where ${(h/J)}_{{{\rm{TFIM}}}}=zS$. The phase boundary is shown in Fig. 2a. We note that one would arrive at the same conclusion by studying the pole structure of the dressed photon/cavity-mode propagator obtained by integrating our the collective spin mode, as discussed in Section IB of the SI. At a fixed J, as $h\to {h}_{{{\rm{TFIM}}}}^{+}$, g_c vanishes with a mean-field exponent, ${g}_{c} \sim {(h-{h}_{{{\rm{TFIM}}}})}^{1/2}$. We note that g_c = 0 for $h\le {h}_{{{\rm{TFIM}}}}$ because the ${{\mathbb{Z}}}_{2}$ symmetry of $\widehat{H}$ is already broken by the ferromagnetic order.

We proceed to carry out DMRG calculations for the extreme quantum case, with $S=\frac{1}{2}$ and in one dimension. Details of the calculation are given in Supplementary Note 2.

Quadrature-squeezing and quantum Fisher information

We semi-classically determine the photon-magnon quadrature that is most squeezed in the vicinity of the SRPT. For this purpose, we derive (see Supplementary Note 3) an effective Hamiltonian that governs the excitations above a mean-field state specified by $\left(\langle \widehat{a}\rangle,\langle {\widehat{{\mathfrak{b}}}}_{0}\rangle \right)=\sqrt{2SN}(\alpha,-\beta )$ with α, β ≥ 0,

$$\delta \widehat{H}^{\prime}= {\omega }_{0}\delta {\widehat{a}}^{{\dagger} }\delta \widehat{a}+{h}_{{{\rm{eff}}}}(g,h,J)\delta {\widehat{b}}^{{\dagger} }\delta \widehat{b}+{g}_{{{\rm{eff}}}}(g,h,J)(\delta \widehat{a}+\delta {\widehat{a}}^{{\dagger} })(\delta \widehat{b}+\delta {\widehat{b}}^{{\dagger} })\\ \,+{\Delta }_{{{\rm{pair}}}}(g,h,J)\left(\delta \widehat{b}\delta \widehat{b}+\delta {\widehat{b}}^{{\dagger} }\delta {\widehat{b}}^{{\dagger} }\right)\,,$$

(9)

where $\delta \widehat{a}$ ($\delta \widehat{b}$) is the fluctuation about $\langle \widehat{a}\rangle$ ($\langle {\widehat{{\mathfrak{b}}}}_{0}\rangle$) and the effective parameters are defined in Section III of the SI. The quadrature in Eq. (3) is the general linear combination of $\delta \widehat{a}$ and $\delta \widehat{b}$, which results in a hermitian operator and it is analogous to a generalized position operator in simple harmonic oscillators⁶⁶ (see Supplementary Note 3).

We use the Optim.jl package in Julia to numerically determine the set of angles ${\left.(\theta,\psi,\phi )\right|}_{\min }$ for which the variance of ${\widehat{X}}_{\theta,\psi,\phi }$ is lowest. We refer to this operator as ${\widehat{X}}_{\min }$.

Because of Heisenberg’s uncertainty relations, there must exist an operator ${\widehat{X}}_{\max }$ that is conjugate to ${\widehat{X}}_{\min }$: $\left[{\widehat{X}}_{\min },{\widehat{X}}_{\max }\right]=-\frac{i}{2}$, and whose variance is maximized. The same computational method leads us to the needed set of angles ${\left.(\theta,\psi,\phi )\right|}_{\max }$ for which the variance of ${\widehat{X}}_{\theta,\psi,\phi }$ is the highest. This operator is the desired ${\widehat{X}}_{\max }$. It can be checked that the product of the two variances equals 1/16, satisfying the lower bound of the uncertainty relation for variances of bosonic mode operators⁶⁷.

In principle, the large variance of ${\widehat{X}}_{\max }$ can be utilized as a resource for high-precision parameter estimation⁴⁹. In particular, we consider unitarily imprinting the parameter ϑ as $\left|{\psi }_{0}\right\rangle \to \left|\psi (\vartheta )\right\rangle=\exp -i\vartheta {\widehat{X}}_{\max }\left|{\psi }_{0}\right\rangle$. The Cramérs-Rao bound dictates that for m independent measurements of ϑ, its variance $\Delta {\vartheta }^{2}\ge 1/(m{F}_{Q}({\widehat{X}}_{\max }))$, where ${F}_{Q}(\widehat{X})$ is the quantum Fisher information associated with the operator ${\widehat{X}}_{\max }$ in the state $\left|{\psi }_{0}\right\rangle$⁶⁸. Assuming $\left|{\psi }_{0}\right\rangle$ is a pure state, one obtains ${F}_{Q}({\widehat{X}}_{\max })=4\Delta {\widehat{X}}_{\max }^{2}$ (see Supplementary Note 3), which implies, in principle, Δϑ² can be reduced to zero.

Cavity coupling to a non-critical mode

As described in the main text, we also consider the case of cavity coupling to a non-critical model, corresponding to ${{\bf{n}}}=\widehat{y}$. From the large-S limit, we can see that the cavity mode directly couples to ${\sum }_{{{\bf{r}}}}(\widehat{b}_{{{\bf{r}}}}-\widehat{b}_{{{\bf{r}}}}^{{\dagger} })$. Therefore, the k = 0 magnon mode that must condense to produce a superradiant phase, $({\widehat{\bar{b}}}_{{{\bf{0}}}}-{\widehat{\bar{b}}}_{{{\bf{0}}}}^{{\dagger} })/i$, is distinct from the ferromagnetic order parameter of the pure matter sector, $({\widehat{\bar{b}}}_{{{\bf{0}}}}+{\widehat{\bar{b}}}_{{{\bf{0}}}}^{{\dagger} })$. Because these two spin modes are orthogonal, phase transitions in the two sectors remain decoupled at the leading order in the large-S limit.

Instead of pursuing higher order corrections in 1/S, here, we focus on d = 1 with S = 1/2 and derive an analytically exact free energy in terms of 〈a〉 ^34,35. In this approach, 〈a〉 is treated as a real-valued order parameter, $\langle a\rangle \equiv \sqrt{N}\phi /2$, and the spin degrees of freedom are integrated out to obtain the ground state energy density, ${{{\mathcal{E}}}}_{g}(\phi )$ (see Supplementary Note 4). Minimizing ${{{\mathcal{E}}}}_{g}$ with respect to ϕ yields the phase diagram shown in Fig. 3a.

For h/J < 1/2, the superradiant transition is discontinuous; at the TFIM critical point, ${(h/J)}_{{{\rm{TFIM}}}}=1/2$, the Ising order vanishes while the superradiant transition remains discontinuous with a reduced critical coupling ${g}_{{{\rm{c}}}}\approx 0.87\sqrt{J{\omega }_{0}}$ (vs. ${g}_{{{\rm{c}}}}\approx 0.92\sqrt{J{\omega }_{0}}$ at h = 0). For h/J > 1/2, the ground-state energy expands as ${{{\mathcal{E}}}}_{g}/J=r{\phi }^{2}+u{\phi }^{4}+v{\phi }^{6}$, with $r=({g}_{{{\rm{c}}}}^{2}-{g}^{2})/4{J}^{2}$ and ${g}_{{{\rm{c}}}}=\sqrt{J{\omega }_{0}/f(h/J)}$, where the coefficients f, u, v are shown in Sec IV of the SI. In the range (h/J)_TFIM < h/J < (h/J)_tri ≈ 0.55, we find u < 0, v > 0, yielding a discontinuous SRPT that extends the first-order transition line from h/J≤(h/J)_TFIM. For h/J > (h/J)_tri, u > 0 and the SRPT becomes continuous, smoothly connecting to the Dicke limit ($h/J,g/\sqrt{{\omega }_{0}J}\to \infty$ at fixed $g/\sqrt{{\omega }_{0}h} \sim 1$). The intersection between the two types of QPTs at h/J = (h/J)_tri defines the tricritical point, whose scaling we analyze below.

In the vicinity of the tricritical point with a fixed J and ω₀, u ∝ (h − h_tri)/J and the number of photons in the condensate obtains the scaling form

$${{\mathcal{N}}}={\phi }^{2}={\left(g/{g}_{c}-1\right)}^{\frac{1}{2}}{f}_{{{\mathcal{N}}}}\left(\frac{u/\sqrt{v}}{\sqrt{g/{g}_{c}-1}}\right)\,.$$

(10)

Here, the dimensionless function ${f}_{{{\mathcal{N}}}}(x)={c}_{1}/\left[{c}_{2}x+\sqrt{1+{({c}_{2}x)}^{2}}\right]$ with c_n’s being dimensionless parameters, and it has the limiting behaviors, ${lim}_{x\to 0}{f}_{{{\mathcal{N}}}}(x) \sim 1$ and ${lim}_{x\to \infty }{f}_{{{\mathcal{N}}}}(x) \sim 1/x$. Therefore, in the superradiant phase at h = h_tri, ${{\mathcal{N}}} \sim {(g/{g}_{c}-1)}^{\frac{1}{2}}$, while ${{\mathcal{N}}} \sim (g/{g}_{c}-1)$ for h > h_tri. While the latter is the standard mean-field result, the former is a peculiarity of tricritical points, which was also observed in a variant of the pure Dicke model⁶⁹. As shown in Fig. 3b, ${f}_{{{\mathcal{N}}}}$ controls the crossover between the two scaling limits, with the crossover scale determined by the condition c₂x = 1.

Phase diagram of the XY spin model

The ferromagnetic XY model realizes a ferromagnetic phase with a net magnetization along $\widehat{x}$ ($\widehat{y}$) for Δ > 0 (Δ < 0), which spontaneously breaks the ${{\mathbb{Z}}}_{2}$ symmetry of ${\widehat{H}}_{{{\rm{spin}}}}$, present at any Δ ≠ 0 ⁷⁰. The QPT between the two phases is continuous with the QCP at Δ = 0, realizing an enhanced SO(2) symmetry. Unitarily equivalent Ising models are recovered in the limits Δ → ± 1.

Thermodynamic limit and multi-mode generalization

The quantum phase transitions discussed in this work implicitly assume a thermodynamic limit in which both the quantum magnet and the cavity participates⁶⁴, as discussed in Supplementary Note 4. If strictly implemented, this will result in a quasi-continuum of photonic modes, which is expected to limit the regime of validity of the single-mode approximation, especially if the matter sector is in a charge-itinerant phase⁷¹. While we formally focus on the strong coupling limit of the cavity coupled Hubbard model^72,73,74, where charge fluctuations are localized, and the matter sector is a quantum magnet (see Supplementary Note 4), multi-mode aspects still require careful considerations. In Supplementary Note 4, we discuss the limits of the single-mode approximation and demonstrate that the multi-mode version of our model, as applicable to cavity magnonics settings⁵⁴, continues to exhibit qualitatively similar physics. We summarize the results below.

The simplest multi-mode generalization of the cavity coupled to a critical degree of freedom is governed by the Hamiltonian

$$\widehat{H}_{{{\rm{multi}}}}={\sum }_{k=1}^{M}{\omega }_{k}\widehat{a}_{k}^{{\dagger} }\widehat{a}_{k}-h\widehat{S}_{T}^{z}+\frac{1}{\sqrt{N}}{\sum }_{k=1}^{M}{g}_{k}(\widehat{a}_{k}+\widehat{a}_{k}^{{\dagger} })\widehat{S}_{T}^{x}+\widehat{H}_{{{\rm{spin}}}},$$

(11)

where k labels the M modes present within the cavity, ω_k specifies the mode profile, g_k is the distribution of collective light-matter coupling for the kth mode, and $\widehat{S}_{T}^{\mu }={\sum }_{j}\widehat{S}_{j}^{\mu }$ is the μ-th component of the collective spin. For our minimal requirement of the existence of a quantum phase transition in the spin sector, the spin–spin interaction part of the Hamiltonian is assumed to have a collective form, $\widehat{H}_{{{\rm{spin}}}}=-(J/N){(\widehat{S}_{T}^{x})}^{2}$. As detailed in Section VI of the SI, the condition for SRPT in this multi-mode models is ${\sum }_{k=1}^{M}{g}_{k}^{2}/{\omega }_{k}=(h-J)$, which implies that with growing proximity to the magnetic QCP, it becomes easier to obtain a superradiant state. This is in qualitative agreement with our single-mode model.