Perfect Intrinsic Squeezing at the Superradiant Phase Transition Critical Point Kenji Hayashida,1, 2 Takuma Makihara,3 Nicolas Marquez Peraca,3 Diego Fallas Padilla,3 Han Pu,3 Junichiro Kono,1, 3, 4 and Motoaki Bamba5,6, ∗ 1Department of Electrical and Computer Engineering, Rice University, Houston 77005, USA 2Division of Applied Physics, Graduate School and Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan 3Department of Physics and Astronomy, Rice University, Houston 77005, USA 4Department of Material Science and NanoEngineering, Rice University, Houston 77005, USA 5Department of Physics, Kyoto University, Kyoto 606-8502, Japan 6PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan (Dated: September 8, 2020) The ground state of the photon–matter coupled system described by the Dicke model is found to be perfectly squeezed at the quantum critical point of the superradiant phase transition (SRPT). In the presence of the counter-rotating photon–atom coupling, the ground state is analytically expressed as a two-mode squeezed vacuum in the basis of photons and atomic collective excitations. The variance of a quantum fluctuation in the two-mode basis vanishes at the SRPT critical point, with its conjugate fluctuation diverging, ideally satisfying the Heisenberg uncertainty principle. When photons strongly couple with an ensemble of system would behave at the critical point is not well un- atoms, there is a threshold coupling strength above which derstood. a static photonic field (i.e., a transverse electromagnetic In the present letter, we show that perfect squeezing is field) and a static atomic field (i.e., an electric polariza- obtained in a photon–atom two-mode basis at the SRPT tion) are expected to appear spontaneously. This phe- intrinsically, i.e., in the ground state of the coupled sys- nomenon, known as the superradiant phase transition tem. It means that, in contrast to the usual squeezing (SRPT) [1, 2], can occur not only at finite temperatures generation in dynamical and nonequilibrium situations but also at zero temperature. Since its first proposal in [26, 27], the SRPT can provide strong squeezing stably 1973, the SRPT has long attracted considerable attention in equilibrium situations and has a potential to develop from both experimental and theoretical researchers. In decoherence-robust quantum technologies. addition to the experimental demonstration of nonequi- We consider the isotropic Dicke model [28, 29] whose librium SRPT in atoms confined in optical cavities [3, 4], Hamiltonian is given by the possibilities of realizing the original SRPT in ther- † N 2g † mal equilibrium [5–7] using various physical platforms ˆDicke/~ = ω aˆ aˆ+ω Sˆ + + (ˆa +ˆa)Sˆ . (1) H a b z 2 √ x including a superconducting circuit [8] and a magnonic N system [9], have been demonstrated theoretically in re- Here, aˆ is the annihilation operator of a photon with a cent years. ˆ N resonance frequency ωa. Sx,y,z are the spin- 2 opera- Although the finite-temperature SRPT is a classical tors representing an ensemble of N two-level atoms with phase transition in the sense that it is driven by thermal a transition frequency ωb. g is the coupling strength fluctuations [10, 11], quantum aspects of the SRPT have and assumed to be real and positive for simplicity [31]. been investigated in terms of quantum chaos [12, 13], In terms of the lowering and raising operators Sˆ± † ≡ entanglement entropy [14], and individual photonic and Sˆx iSˆy = Sˆ∓ , the last term (photon–atom coupling arXiv:2009.02630v1 [quant-ph] 6 Sep 2020 ± { } † atomic squeezings [13, 15–17], i.e., quantum fluctuation term) in Eq. (1) can be rewritten as 2g(ˆa +ˆa)Sˆx/√N = † † of the photonic (atomic) field is suppressed in one quadra- g(ˆa +ˆa)(Sˆ+ + Sˆ−)/√N. Among these four terms, aˆ Sˆ− ture whereas its conjugate fluctuation is enlarged while and Sˆ+aˆ are called the co-rotating terms and responsi- † satisfying the Heisenberg uncertainty principle. ble for the vacuum Rabi splitting; while aˆ Sˆ+ and aˆSˆ− Due to the ultrastrong photon–atom coupling [18–20], are called the counter-rotating terms and responsible for which means that the coupling strength (vacuum Rabi the vacuum Bloch-Siegert shift [32, 33]. As we will show splitting; anti-crossing frequency) is a considerable frac- later, it is these counter-rotating terms that are respon- tion of photonic and atomic resonance frequencies and sible for the two-mode squeezing [18, 21–25]. is required for realizing the SRPT, it is known that the Since the Dicke model can be treated effectively as an ground state of the photon–matter coupled systems be- infinite-dimensional system [10] in the thermodynamic comes a two-mode squeezed vacuum [18, 21–25] even in limit (N ), the SRPT can be analyzed under the the normal phase (zero expectation values of the pho- mean-field→ framework ∞ [2, 12, 13, 34]. In the present let- tonic and atomic fields). One might expect that some ter, we follow the Holstein–Primakoff-transformation ap- critical quantum behavior should exhibit at the onset of proach [12, 13, 34], which is suited for zero-temperature the SRPT. However, how the squeezing property of the analyses. The spin operators are rewritten by a bosonic 2 ˆ ωb = ωa ωb = 2ωa annihilation operator b of the atomic collective excita- 1.5 tions as (a) (e) a † † 1/2 1 Sˆ ˆb ˆb N/2, Sˆ− (N ˆb ˆb) ˆb. (2) a z → − → − 0.5 The appearance of the superradiant phase, where non- b b zero aˆ = √Na¯ and ˆb = √N¯b (a,¯ ¯b R) appear 0 h i h i − ∈ spontaneously, at the zero temperature can be easily con- 4 (b) (f) firmed through the classical energy ¯/(~N) = ω a¯2 + Ω / ω H a + a ¯2 ¯ ¯2 ωbb 4ga¯b 1 b obtained from Eq. (1). The zero- 2 Ω+ / ωa temperature− classical− state satisfies ∂ /∂a¯ = ∂ /∂¯b = p H H Ω / ω Ω / ω 0, from which we have 0 - a - a (c) (g) ΔX ΔX ΔX ΔX 2g 0, g √ωaωb/2 min max min max ¯ ¯2 ¯2 ≤ 0.25 a¯ = b 1 b , b = 1 ωaωb ωa − 1 2 , g> √ωaωb/2 ( 2 − 4g p 2 2 (ΔX ) 2 (ΔXmin ) (3) min (ΔX ) 2 0,0,π/2 (ΔX0,0,π/2 ) 0 These are plotted as a function of g/ωa in Fig. 1(a) and (d) (h) φ / 7 φopt / 7 (e) with ωb = ωa and ωb = 2ωa, respectively. The zero- 0.5 opt temperature SRPT occurs at -θ / 7 -θ / 7 0.25 opt opt g = √ωaωb/2, (4) ψ / 7 ψ / 7 0 opt opt i.e., in the ultrastrong coupling regime [18–20]. 0 0.5 1 0 0.5 1 1.5 The quantum fluctuations around the zero- Coupling strength g / ω Coupling strength g / ω temperature classical state are described by replacing a a aˆ and ˆb with √Na¯ +a ˆ and √N¯b + ˆb, respectively FIG. 1. For (a-d) ω = ωa and (e-h) ω = 2ωa, we plot, as a [12, 13, 34]. After this replacement,− aˆ and ˆb are now b b function of g/ωa, (a,e) order parameters a¯ and ¯b, (b,f) eigen- considered as the fluctuation operators. The Dicke frequencies Ω±, (c,g) quadrature variance, and (d,h) optimal Hamiltonian, Eq. (1), is then expanded as angles θopt, ψopt, and ϕopt that give the minimum variance 2 (∆Xmin) [red solid curves in (c,g)]. The minimum variance ˆ/~ ω aˆ†aˆ +˜ω ˆb†ˆb +˜g(ˆa† +ˆa)(ˆb† + ˆb)+ D˜(ˆb† + ˆb)2 H ≡ a b vanishes at the SRPT critical point (g = √ωaωb/2) under + O(N −1/2) + const., (5) satisfying the equality in the Heisenberg uncertainty princi- ple ∆Xmin∆Xmax = 1/4 [red dashed line in (c,g)] with the 2 2 where the coefficients are modified by the order parame- variance (∆Xmax) conjugate to (∆Xmin) . ters a¯ and ¯b as g(1 2¯b2) ga¯¯b respect to the ground state 0 of the fluctuation Hamil- g˜ − , D˜ , ω˜b ωb +2D.˜ (6) | i ≡ 1 ¯b2 ≡ 1 ¯b2 ≡ tonian, Eq. (5). Here, we consider annihilation operators − − pˆ± of eigenmodes (i.e., polariton modes) that diagonalize In the following,p we considerp the thermodynamic limit Eq. (5) as (N ) and focus only on the leading terms in Eq. (5), → ∞ ˆ ~ † † −1/2 which gives rise to a quadratic Hamiltonian in terms of / = Ω−pˆ−pˆ− + Ω+pˆ+pˆ+ + O(N ) + const., (9) aˆ and ˆb. H where Ω± are the eigenfrequencies. The ground state 0 By describing the photonic and atomic fluctuations us- | i ing Eq. (5), we first demonstrate the perfect intrinsic two- is determined by requiring mode squeezing numerically. We consider a general su- pˆ± 0 =0. (10) perposition of the two fluctuation operators defined with | i two angles θ and ψ as Due to the presence of the counter-rotating terms aˆˆb, iψ aˆ†ˆb†, ˆbˆb, and ˆb†ˆb†, originating from those in the Dicke cˆθ,ψ aˆ cos θ +e ˆb sin θ. (7) ≡ model in Eq. (1), the eigenmode operators are obtained We define a quadrature [26, 27] by this bosonic operator by a Bogoliubov transformation [18, 21–25, 34] as with a phase ϕ as † † pˆ± = w±aˆ + x±ˆb + y±aˆ + z±ˆb . (11) ˆ iϕ † −iϕ Xθ,ψ,ϕ = (ˆcθ,ψe +ˆcθ,ψe )/2. (8) For positive eigenfrequencies Ω± > 0, the coefficients 2 2 2 2 2 2 We evaluate the variance (∆Xθ,ψ,ϕ) 0 (Xˆθ,ψ,ϕ) 0 must satisfy w± + x± y± z± = 1 in order 2 2 ≡ h | | i− | † | | | − | | − | | 0 Xˆ 0 = 0 (Xˆ ) 0 of this quadrature with to yield [ˆp±, pˆ ]=1.