Quantum phase transitions and cat states in cavity-coupled quantum dots

We study double quantum dots coupled to a quasistatic cavity mode with high mode-volume compression allowing for strong light-matter coupling. Besides the cavity-mediated interaction, electrons in different double quantum dots interact with each other via dipole-dipole (Coulomb) interaction. For attractive dipolar interaction, a cavity-induced ferroelectric quantum phase transition emerges leading to ordered dipole moments. Surprisingly, we find that the phase transition can be either continuous or discontinuous, depending on the ratio between the strengths of cavity-mediated and Coulomb interactions. We show that, in the strong coupling regime, both the ground and the first excited states of an array of double quantum dots are squeezed Schr\"{o}dinger cat states. Such states are actively discussed as high-fidelity qubits for quantum computing, and thus our proposal provides a platform for semiconductor implementation of such qubits. We also calculate gauge-invariant observables such as the net dipole moment, the optical conductivity, and the absorption spectrum beyond the semiclassical approximation.


I. INTRODUCTION
Placing condensed matter systems in an optical cavity is a promising way of engineering new correlated states of matter via the interaction with quantum fluctuations of the cavity field [1].The main experimental challenge is to achieve the ultrastrong light-matter coupling regime [2,3] that can be reached by external driving [4][5][6][7][8][9][10][11][12], by tuning the cavity in a plasmon or an exciton-polariton resonance [13][14][15][16], or by compressing the mode volume in specially designed resonators [17,18] that can be viewed as LC-circuits [19,20] with a single discrete quasistatic mode, whose frequency ω 0 = 1/ √ LC is not constrained by the resonator dimensions.When the light-matter coupling is strong enough, then even in the ground state the vacuum fluctuations can radically modify electron systems [21][22][23][24].This phenomenon fosters a qualitatively new class of condensed-matter platforms with strongly correlated light-matter excitations.
Superradiance, initially described by R. H. Dicke [25,26], has garnered significant attention to coupled lightmatter systems ever since.There exist various effective models describing cavity-coupled electron systems, known as extended and generalized Dicke models, see, e.g., Refs.[27][28][29][30].An important restriction to such effective models is gauge invariance that must be preserved [31][32][33].Originally, the main signature of the superradiant Dicke phase transition was a photon condensate, the macroscopic occupation of the cavity mode that is not gauge-invariant [34].Nevertheless, the quantum phase transition (QPT) is present and equivalent to the ferroelectric phase transition (FPT), resulting in ordered electric dipole moments, see Refs.[35,36].Important, the FPT is only possible if the Coulomb interaction between the dipoles is included [28,35,[37][38][39].
In this work, we consider a few cavity-coupled double quantum dots (DQDs) with a Coulomb interaction between them, the only non-trivial part of which is the electric dipole-dipole interaction.Choosing a geometry where the dipolar interaction between DQDs is attractive, we find either a first order QPT or a smooth transition depending on the relative strength of the Coulomb and light-matter interactions, leading to ordered phases of the electric dipole moments.The ground and the first excited states are cat states in the smooth transition region.In particular, this is true already for two cavity-coupled DQDs with attractive dipole-dipole (Coulomb) interaction.We suggest such systems as possible semiconductor candidates for a self-correcting cat qubit [40,41] and a realistic platform to study cavityinduced QPTs.Here we calculate the net dipole moment, the optical conductivity, and the absorption spectrum all of which are gauge-invariant.

II. THEORETICAL MODEL
A few identical singly-occupied DQDs are oriented along the line connecting the capacitor plates as shown in Fig. 1.Due to the Coulomb repulsion, DQDs interact with each other directly via the electric dipole-dipole interaction.The double-well shape of the confining potential of each DQD allows us to truncate electron energy levels by the lowest two as long as the higher states are far detuned [42,43].Such an electronic system is described by the following Hamiltonian, where c † i,L/R (c i,L/R ) are the electron creation (annihilation) operators for the two sites (L/R) of the i th DQD, N is the number of DQDs, V b is the bias in each DQD, ∆ is the DQD level hybridization, and is the electric dipole operator.Spin indices are suppressed.The Coulomb interaction is reduced to the dipole-dipole interaction here due to the two-level truncation of each singly-occupied DQD.The dipolar interaction strength between two DQDs is derived in Appendix A where r ij = r ij e z is the distance vector between two DQD centers, ε the dielectric constant, e < 0 the elementary charge, b the DQD length, and eb/2 the dipole matrix element between the lowest two levels of a DQD.If DQDs are assembled along the capacitor axis z, the dipole-dipole interaction is attractive, Screening of U ij due to proximity to the capacitor plates does not affect the sign of U ij but only slightly modifies its absolute value.In what follows, we mostly focus on two (N = 2) DQDs.In this case, only the U 12 ≡ U matrix element of the dipoledipole interaction is important.Throughout the paper, we use cgs-units and also set the Planck and Boltzmann constants to unity, ℏ = k B = 1.
All DQDs are coupled to a single quantized quasistatic LC-cavity mode [44][45][46][47][48][49].The electric field of the cavity mode is almost completely localized in the capacitor and polarized along the DQDs, see Fig. 1.The corresponding vector-potential operator A z is given by where a (a † ) is the annihilation (creation) operator of the cavity mode with frequency ω 0 , E 0 the amplitude of The inset shows the second derivative of the ground state energy with respect to the light-matter coupling g, where the dotted line is obtained by exact diagonalization (ED), the solid line corresponds to the semiclassical (SM) approximation.Panel (b) shows the level crossing corresponding to a first-order ferroelectric QPT, indicated by the vertical dashed red line.The left inset in (b) shows the zoomedin level crossing region (where we introduced a small ratio V b /ω0 = 0.5 • 10 −3 to identify the two otherwise degenerate levels).The right inset in (b) shows the first derivative of the ground state energy ∂EGS(g)/∂g obtained by ED that is discontinuous at the QPT.
the electric field fluctuations, and V eff the effective mode volume.
We describe the coupling of the DQDs to the cavity via the Peierls substitution, where S β = 1/2 N i=1 σ i,β is the orbital pseudospin of the system, σ i,β is the Pauli matrix corresponding to the i th DQD, β ∈ {x, y, z}, g is the dimensionless lightmatter coupling constant, U ij = U .The operators S z and S ± = S x ± iS y satisfy the standard spin algebra: [S ± , S z ] = ∓S ± , [S + , S − ] = 2S z .The pseudospin S describes the collective orbital degree of freedom in the DQD array.For example, the total dipole moment operator maps onto S z : First, we diagonalize the Hamiltonian H, see Eq. ( 5), numerically for N = 2 cavity-coupled DQDs at zero bias V b = 0, truncating the photon Hilbert space at 200 photons to ensure convergence.First six energy levels are shown in Fig. 2. The spectrum demonstrates either continuous coalescence of the energy levels corresponding to a smooth transition to the ferroelectric phase, Fig. 2(a), or a level crossing indicating the first-order ferroelectric QPT, Fig. 2(b).
The zero-temperature phase diagram represented by a 2D plot of the net dipole moment ⟨S z ⟩ as a function of the dipole interaction strength and light-matter coupling constant at ω 0 /∆ = 0.1 [see Fig. 3(a)] shows the firstorder QPT at 5 ∆/ω 0 and the smooth transition otherwise, separated by the critical point (g c ,U c ) marked by the star.It's position remains fixed if plotted in the coordinates (g/q, U/ √ ω 0 ∆), where q = ∆/ω 0 , in agreement with the mean-field analysis shown in Appendix B. At finite temperature, the QPT turns into a smooth transition, see the density plot of ⟨S 2 z ⟩ in Fig. 3(b).Here, we used that ⟨S 2 z ⟩ does not require a small symmetry breaking field V b which is useful for the finite-temperature analysis.We point out that ⟨S 2 z ⟩ is meaningful for N ≥ 2 DQDs.We also stress that there is no QPT at g = 0 implying that this is a cavity-induced phenomenon.

III. SEMICLASSICAL DECOUPLING
In order to gain physical insight into our numerical results, we analyze the system in the quasi-thermodynamic limit ω 0 ≪ ∆ (the limit of the classical oscillator), see Refs.[50][51][52][53] for details.Our results remain qualitatively the same even when ∆ ∼ ω 0 , see Appendix C. The photonic semiclassical decoupling is reminiscent of the length-gauge formulation of the problem [54][55][56][57], where H is given by Eq. ( 5), U = exp g δS z (a † − a) , δS z = S z −⟨S z ⟩, ⟨S z ⟩ is the average of orbital pseudospin S z over the ground state of the semiclassical Hamiltonian H sm .The perturbation δV accounts for quantum corrections beyond the semiclassical approximation.In contrast to conventional mean-field treatment where both, the photons and the pseudospin, are treated as classical objects, see e.g.Ref. [50], the orbital pseudospin S in our work remains quantum because we apply our results to a small number of DQDs.
The Hamiltonian H D commutes with S 2 , so we consider states with definite orbital pseudospin S. Single DQD corresponds to N = 1 and S = 1/2, this case is known as the quantum Rabi model [52,58].In case of N = 2 DQDs, see Fig. 1, S can be either 0 or 1.The S = 0 state does not couple to the antenna.If S = 1, the semiclassical Hamiltonian [Eq.( 8)] can be diagonalized analytically, see Appendix B. Here, we show the semiclassical ground-state energy, E sm , of two cavity-coupled DQDs at V b = 0 (symmetric DQDs): where P and Q are defined as follows,  12) and ( 13).The semiclassical approximation is valid if the fidelities are close to one.
Here, we introduced the parameter α = g⟨S z ⟩.If α = 0, the phase is trivial.If α ̸ = 0, the ground state is ferroelectric, i.e., it has a net dipole moment ⟨S z ⟩ ̸ = 0.The ferroelectric QPT is first-order and ⟨S z ⟩ has a finite jump at the transition, see Fig. 3(a), whereas the second-order QPT predicted by the mean field turns into a smooth transition in the ED due the tunneling effect.
As E sm is an even function of α at V b = 0 [Eq.(10)], the ground state of H sm is two-fold degenerate at α ̸ = 0.This degeneracy is best seen from the symmetry P = exp(iπa † a + iπS x ) of the transformed Hamiltonian a) .Note that T (α) is the optical displacement operator that creates the coherent state |α⟩ = T (α)|0⟩, where |0⟩ is the photonic vacuum of H sm , see Eq. (8).While the symmetry breaking in this problem occurs only in the limit ω 0 /∆ → 0, we expect very small lifting of the degeneracy at any finite ω 0 /∆ ≪ 1 such that the parity symmetry P of the Hamiltonian H T is restored.In particular, the ground state, |Ψ G ⟩, and the first excited state, |Ψ E1 ⟩, of the Hamiltonian H T in the semiclassical approximation correspond to P = +1 and P = −1, respectively, where α > 0 corresponds to positive dipole moment ⟨S z ⟩ > 0, χ(±α) are the two lowest-energy eigenstates of H sm , and N is the normalization factor.Indeed, we observe a finite splitting in the ferroelectric phase, see Fig. 2 12) and ( 13).This confirms the semiclassical result that the ground and the first excited states are twocomponent cat states.The parameter α 2 , being an increasing function of g (see the inset in Fig. 4), plays the role of the "cat size".The comparison between the semiclassically calculated phase diagram for the order parameter ⟨ Ŝ2 z ⟩ and the ED is shown in Appendix D. Two lowest energy levels become degenerate in the strong coupling limit g → +∞, see Fig. 2, when the Schrödinger cats become truly classical.In order to use such a system as a cat qubit, a finite energy splitting is required which corresponds to the smooth transition region and restricts the cat size α 2 .On the bright side, the cat states appearing at strong coupling (recently observed in circuit QED [58,60]) are robust to decoherence and can be harnessed to implement quantum gates with high fidelity [61,62].We propose the cavity-coupled DQDs as a new solid-state platform for cat qubits (without driving [63]), as promising candidates for quantum computing [64][65][66].In contrast to atomic systems (e.g., see [67][68][69]), solid-state platforms are scalable and require much less stringent experimental conditions.As shown in Appendix C, the results are resilient to variations of the DQD parameters.Also, when the cavity losses are included within the Lindblad formalism, the phase transition is shown to remain first-order as shown in Appendix E. The behaviour of the cat states is analyzed within the quantum jump (Monte Carlo) method revealing switching between the two cat states which gives rise to a finite coherence time of the cat qubit (see Appendix F).

IV. OPTICAL CONDUCTIVITY AND ABSORPTION SPECTRUM
Two gauge invariant response functions that can be routinely measured are the optical conductivity σ(ω) and the absorption spectrum.The latter is defined [70] as the cavity response to an AC voltage applied to the cavity and is proportional to being the Heaviside step function.The absorption spectrum was thoroughly studied before [19,71,72] as a function of the driving frequency, showing two standard polariton branches.We plot the absorption spectrum in Fig. 5(b), revealing the softening of the lower polariton mode to zero at the smooth transition.
The optical conductivity is calculated by standard means [73] Re where the current operator J along the DQD axis is defined as J = e dz/dt = i [H D , (eb/2) i d i,z ] because the z coordinate operator is replaced by the dipole moment, and in the length gauge [Eq.(7)] is given by Contrary to absorption, the optical conductivity retains a strong frequency comb [74][75][76] deep in the ferroelectric phase, see Fig. 5(a).We stress that such a frequency comb is not present in the semiclassical approximation, see Eq. ( 8).Here, we present an analytic result for Re [σ(ω)], where S is the pseudospin, p n (z) = e −z z n /n! is the Poisson distribution, and E n = nω 0 − U (2S − 1).Equation ( 15) is valid in the ferroelectric phase at arbitrary g and near-full semiclassical polarization |⟨S z ⟩| ≈ S, see Appendix H for details.
Our findings are relevant for state-of-the-art experiments, providing key parameters: ω 0 , ranges from tens of GHz to THz; the splitting in DQDs can vary between ∆ ∼ 0.1 − 10 meV; the Coulomb interaction, |U |, may reach several meV depending on the dot configuration.The light-matter coupling g = W/ω 0 is widely tunable and can significantly exceed unity if the length of each DQD is large and the mode volume is highly compressed [18].

V. CONCLUSION
We analyzed two DQDs coupled to a cavity mode and found a ferroelectric QPT at strong light-matter coupling and attractive dipole-dipole interaction between DQDs due to the Coulomb force.There is a first-order QPT and a smooth transition separated by a critical point.We showed that the ground and the first excited states of two cavity-coupled DQDs in the smooth transition region are cat states protected by a finite energy splitting.We argue that such cavity-coupled DQD systems can be used as cat qubits.The quantum phase transition and the cat states are shown to persist against cavity losses and variation of system parameters.Higher excited states are studied via the optical conductivity which exhibits a frequency comb at strong coupling.
with each other via the Coulomb repulsion, where l = z 3 − z 2 .The dipole-dipole interaction in case of arbitrary N ≥ 2 is derived in a similar fashion.Therefore, the Hamiltonian in Eq. ( A1) is equivalent to H el in the main text (up to a constant energy shift).The distance z 3 − z 2 = l is related to the distance between the DQD centers r 12 from the main text: At l ≫ b we restore the result from the main text U = −(eb) 2 /[εr 3  12 ].The dipole-dipole approximation is exact due to the two-level truncation of DQD energy levels.

Appendix B: Semiclassical analysis
The semiclassical Hamiltonian H sm takes the following form where δS z = S z − ⟨S z ⟩, ⟨S z ⟩ is the average of S z over the ground state of H sm .Within this approximation, photons are decoupled from the orbital pseudospin, so the semiclassical ground state wave function Ψ sm = |0⟩χ sm , where |0⟩ is the photon vacuum, χ sm is the lowestenergy spinor of ⟨0|H sm |0⟩.The semiclassical groundstate energy E sm follows from the characteristic equation det(⟨0|H sm |0⟩−E sm ) = 0.In case S = 1, the characteristic equation is a third-degree polynomial.We introduced the notation α = g⟨S z ⟩, and we chose to measure all energies in √ ω 0 ∆.Then, all three roots of this characteristic equation are real and can be conveniently expressed via the dimensionless parameters (g/q, U/ √ ω 0 ∆) (with FIG. 6.Energy landscape of the semiclassical ground state at ω0/∆ = 0.1.Blue region: the semiclassical lowest energy Esm(α) has its minimum at α = 0. Yellow region: the semiclassical energy Esm(α) has the two minima ±α ̸ = 0, and E ′′ sm (α = 0) < 0. Orange region: the semiclassical energy has the two minima at ±α ̸ = 0, and E ′′ sm (α = 0) > 0. The transition from blue to the yellow region is a second-order quantum phase transition (QPT), whereas from blue to orange it is a 1st-order QPT.The boundary between yellow and orange regions does not correspond to a phase transition as in both regions the global minima are at ±α ̸ = 0 (ferroelectric phase).We note here that the second order QPT predicted by the semiclassical analysis turns into a smooth transition in the exact treatment of the problem.q = ∆/ω 0 ) as follows: where k ∈ {0, ±1}, and P and Q are given by + 12 α q − 36 α q The ground state corresponds to k = 0, i.e.E sm = E k=0 .
Considering α as a variational parameter, we analyze the global minima of E sm (α) at all other parameters fixed.We stress that α = g⟨S z ⟩ at extrema of E sm (α).In Fig. 6 we display three different regions: one global minimum (blue), two global minima located at ±α ̸ = 0 with E ′′ sm (α = 0) < 0 (yellow) or with E ′′ sm (α = 0) > 0 (orange).Notice that E sm (α) contains two (three) local minima in the yellow (orange) region.In other words, in the semiclassical analysis the boundary between blue and yellow (blue and orange) regions corresponds to the second-(first-) order ferroelectric quantum phase transition (FPT, we use the terms FPT and QPT interchangeably in this work).The boundary between yellow and orange regions does not correspond to a phase transition, it only shows that the local extremum at α = 0 changes from local maximum to local minimum, while the global minima are located at ±α ̸ = 0.The position of the critical point separating the first-order QPT from the smooth transition remains unchanged if plotted in coordinates (g/q, U/ √ ω 0 ∆) when the quasi-thermodynamic limit q = ∆/ω 0 → ∞ is considered.

Appendix C: Non-equivalent quantum dots
If the DQDs are not equivalent, i.e. have different splittings ∆ i , applied biases V b,i , widths b i (and, hence, couplings to the cavity g i ), the model describing a set of N = 2 DQDs placed in the cavity from the main text takes the following form We note that in the present case the Coulomb term U S 2 z retains its form and only the expression of U via microscopic characteristics of the individual DQDs is altered.
It is clear from Fig. 7 that the first-order phase boundary remains sharp both near the quasi-thermodynamic limit and away from it.In the case of a single DQD, the square of the dipole moment S 2 z is trivial (identity matrix).This is not the case for N ≥ 2 DQDs.In Fig. 8(a),(b) we show the exact (numerical) and the semiclassical color maps of ⟨S 2 z ⟩ for two DQDs at ω 0 /∆ = 0.1 and zero temperature, T = 0.Even though at T = 0 the phase boundaries on the ⟨S z ⟩ and ⟨S 2 z ⟩ color maps are the same, the situation is different at finite temperature T ≫ |V b |.At these temperatures, ⟨S z ⟩ = 0 due to the symmetry restoration effect, while ⟨S 2 z ⟩ is not sensitive to either weak symmetry breaking field V b ≪ ω 0 , or to the symmetry restoration due to the quantum tunneling (instanton) effect.This is why we plot the ⟨S 2 z ⟩ color map at finite temperature in Fig. 3  where D[a](ρ) = aρa † − 1 2 (a † aρ + ρa † a).We use the length-gauge description with the Hamiltonian where H is given in the main text, and U = exp g S z (a † − a) .The only difference between H l.g. and H D from the main text is that here we just performed the gauge transformation from the velocity to the length gauge without subtracting ⟨S z ⟩ in the unitary transformation.In Fig. 9 we see that in the presence of singlephoton losses in the cavity, the open system exhibits a 1st order quantum phase transition in the steady state ρ ss = ρ(t → ∞) that is very similar to what the closedsystem analysis from the main text predicts.Given that the numerical solution of the Lindblad equation requires higher truncation of the photon Hilbert space, we decided to choose ∆/ω 0 = 4, U/ω 0 = −0.4,a choice which also supports the 1st order QPT at a similar value of g as in the main text.Appendix H: Optical conductivity in the ordered phase The optical conductivity σ(ω) is a gauge-invariant observable.Here, we derive Re[σ(ω)] in the leading order in ∆.This result is applicable deep in the ordered phase where ⟨S z ⟩ ≈ S, ⟨S z ⟩ is the ground-state average of the semiclassical Hamiltonian H sm , and S is the value of orbital pseudospin.Here, we assume S ≥ 1, such that the dipole-dipole interaction U S 2 z is non-trivial.Indeed, in this case, effects of the "depolarization" field −∆S x are weak and therefore, they can be treated via the perturbation theory.Note that the light-matter coupling constant g can be of arbitrary value and the perturbation expansion is performed only in the small parameter ∆/|U S|.It is more convenient to present the derivation within the velocity gauge, see Eq. ( 5) in the main text.The current operator J along the DQD axis z follows from the fact that the coordinate operator is given by z = bS z in the Peierls gauge, where b is the separation between left and right minima within each DQD: First, we calculate the current-current correlator Π(g, t), where J H (t) = e iHt Je −iHt is the Heisenberg representation of the current operator and θ(t) the Heaviside step function.Within leading order in ∆, Π(g, t) is given by the following average: where J(t) = e iH0t Je −iH0t is the interaction representation of the current operator, H 0 is given by Eq. ( 4) in the main text.First we note that S z (t) = S z as [H 0 , S z ] = 0.The interaction representations of S + and a are the following: where The statistical average of the exponential operators then follows directly from the Campbell-Baker-Hausdorff formula, F (g, t) ≡ e g(a(t)−a † (t)) e −g(a−a † ) (H6) = e −g 2 (2N ph +1) exp 2g 2 N ph cos(ω 0 t) + g 2 e −iω0t , where N ph = [e βω0 − 1] −1 is the average photon number at finite temperature T = 1/β.As ⟨S + (t)S + ⟩ = ⟨S − (t)S − ⟩ = 0, we find Π 0 (g, t) = F (g, t)Π 0 (g = 0, t), (H7) where Π 0 (g = 0, t) is the current-current correlator of the electron system decoupled from photons.We emphasize that the factorization in Eq. (H7) holds in the limit ∆ ≪ ⟨K(S z )⟩, i.e. when the hopping ∆ can be treated as a small perturbation.In the limit ∆ = 0, H = H 0 , see Eq. ( 4) in the main text, and the ground state at U < 0 is the state with the maximal pseudospin projection Interestingly, the Poissonian structure of the frequency comb in Re [σ(ω)] is similar to the down-conversion in circuit QED [74,75], and to the replica bands recently discussed in the context of light-matter interaction [76].In Fig. 12 we show the comparison between the optical conductivity Re[σ(ω)] calculated via exact numerical diagonalization and the analytical result (Eq.( 15) from the main text) in the ordered phase.
FIG. 1.(a) Sketch of the system: Two DQDs embedded in a split-ring resonator.The cavity mode is polarized along the arrow, E0 = (0, 0, E0).(b) Two double-well potentials for each DQD.The two lowest energy levels are shown.The two minima of each DQD are marked with L (Left) and R (Right), respectively.The DQD axes are aligned along the z axis.

FIG. 2 .
FIG. 2.Numerically exact six lowest energy levels of H, Eq. (5), are shown for ω0/∆ = 0.1 and attractive dipolar interaction (a) U/ω0 = −5, (b) U/ω0 = −1.5 as a function of light-matter coupling g.Panel (a): energy levels merge pairwise, indicating the smooth transition to the ordered state.The inset shows the second derivative of the ground state energy with respect to the light-matter coupling g, where the dotted line is obtained by exact diagonalization (ED), the solid line corresponds to the semiclassical (SM) approximation.Panel (b) shows the level crossing corresponding to a first-order ferroelectric QPT, indicated by the vertical dashed red line.The left inset in (b) shows the zoomedin level crossing region (where we introduced a small ratio V b /ω0 = 0.5 • 10 −3 to identify the two otherwise degenerate levels).The right inset in (b) shows the first derivative of the ground state energy ∂EGS(g)/∂g obtained by ED that is discontinuous at the QPT.
FIG. 3. (a) Phase diagram of the numerically exact net dipole moment ⟨Sz⟩ for N = 2 DQDs at ω0/∆ = 1/q 2 = 0.1 and zero temperature as a function of the dipole interaction strength and light-matter coupling constant.An infinitesimal symmetry-breaking field −V b Sz with V b / √ ω0∆ = 10 −3 is introduced.(b) Phase diagram of ⟨S 2 z ⟩ at finite temperature T /ω0 = 0.1.

FIG. 5 .
FIG. 5. Density plot of (a) the optical conductivity Re [σ(ω)] /σ0 and (b) the absorption spectrum (photon spectral function A(ω)) for N = 2 DQDs at U/ω0 = −5 and ω0/∆ = 0.1 corresponding to the smooth transition region.The optical conductivity shows a frequency comb at strong coupling.The normalization parameter σ0 = (eb) 2 and each delta peak is replaced by a Lorentzian with the broadening Γ/ω0 = 0.1.The absorption spectrum shows the lower polariton softening to zero in the smooth transition region alongside a reduction in its spectral weight.In the case of the first-order QPT, the photon spectral function shows a jump instead of softening to zero (see Appendix G).At g = 0, the polariton frequencies are ω0 and the eigenvalues of −∆Sx + U S 2 z .
A1) where the 2N sites comprising N DQDs are located at the positions z k (k = 1, . . ., 2N ) and numbered continuously, i.e. c 1 = c 1,L , c 2 = c 1,R , c 3 = c 2,L , c 4 = c 2,R , . . ., where c i,L/R are the electron annihilation operators introduced in the main text, ∆/2 is the DQD level hybridization (the hopping amplitude), V b the bias in each DQD, b the DQD length, i.e. z 2 − z 1 = z 4 − z 3 = ... = b, n k,0 = ⟨GS 0 |n k |GS 0 ⟩ is the average occupation of the k th site, |GS 0 ⟩ is the ground state of the non-interacting Hamiltonian.The sum in the kinetic energy is restricted to odd numbers only as there is no hopping between the DQDs.The Coulomb interaction is described by W kk ′ = e 2 /(ε|z k − z k ′ |), ε is the dielectric constant.As we assume that each DQD is singly-occupied, there are no other interaction terms.If there are only N = 2 singly-occupied DQDs, then n 1 + n 2 = n 3 + n 4 = 1 and the inter-DQD Coulomb interaction can be represented in terms of the product n 2 n 3 only.On the other hand, the dipole-dipole interaction 2U [(n 1 − n 2 )/2][(n 3 − n 4 )/2] from the main text can be simplified to (U/2)(1 − 2n 2 )(2n 3 − 1) = −2U n 2 n 3 +U (n 2 +n 3 )−U/2.Comparing the coefficients of the bilinear term n 2 n 3 in the Coulomb term and in the dipole-dipole interaction term, we find the dipole-dipole interaction strength,
(b) in the main text.
FIG. 8. (a) Colour map of ⟨S 2 z ⟩ obtained by the exact diagonalization.(b) Colour map of ⟨S 2z ⟩ obtained from the semiclassical solution, see the main text.In both figures ω0/∆ = 0.1.The phase boundary predicted by the semiclassical approximation approaches the exact one in the quasithermodynamic limit ω0/∆ → 0.

FIG. 9 .FIG. 11 .
FIG. 9. (a,b): steady state solutions (photon number and the average value of the square of the net dipole operator, the later is a gauge invariant quantity) of the Lindblad equation with single-photon losses.(c,d): exact diagonalization of the closed system exhibiting a first-order quantum phase transition.The parameters are U/ω0 = −0.4,∆/ω0 = 4, and V b /ω0 ≈ 10 −2 .
Fidelities |⟨GS|ΨG⟩| and |⟨E1|ΨE1⟩| in the smooth transition region as a function of light-matter coupling g, where |GS⟩ and |E1⟩ are the exact ground and the first excited states (in the length gauge), |ΨG⟩ and |ΨE1⟩ are the semiclassical cat states, see Eqs. ( Fig. 4 as function of g justify the semiclassical treatment in the ferroelectric phase, where |GS⟩ and |E1⟩ are exact (numerical) ground and first excited states, |Ψ G ⟩ and |Ψ E1 ⟩ are corresponding semiclassical cat states, see Eqs. ( [59]Restoration of the parity symmetry P is due to the tunneling (instantons) between two semiclassical ground states[59].The fidelities |⟨GS|Ψ G ⟩| and |⟨E1|Ψ E1 ⟩| plotted in