Perfect Intrinsic Squeezing at the Superradiant Phase Transition

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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(â† +â)(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 principle ∆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. These coefficients and Ω± are h | θ,ψ,ϕ| i h | θ,ψ,ϕ | i ± 3 determined by an eigenvalue problem [18] derived from co-rotating and counter-rotating terms [35], although we Eq. (5) as simply considered the isotropic Dicke model, Eq. (1), and real g in the present letter. On the other hand, ωa g˜ 0 g˜ w± w± θ (solid curves) depend on g/ω and ω /ω in general, − opt a b a ˜ ˜ 2 ˆ g˜ ω˜b +2D g˜ 2D x± x± while θopt = π/4, i.e., (∆X−π/4,0,π/2) = 0 (â b  − −    = Ω±   . − −h | − − 0g ˜ ωa g˜ y± y± † ˆ 2 − − aˆ + b) 0 /8 always gives the minimum variance in the  g˜ 2D˜ g˜ ω˜ 2D˜ z±  z±  | i  − − b −      normal phase (g<ωa/2) for ωb = ωa.      (12) Next, we try to understand the numerically found per- Two positive eigenvalues correspond to the eigenfrequen- fect and ideal squeezing (∆Xmin =0 at the critical point cies Ω±. We also get two negative eigenvalues Ω±, with ∆Xmin∆Xmax = 1/4) by an analytical expression whose eigenvectors correspond to the creation operators− † of the ground state 0 of the fluctuation Hamiltonian, pˆ . In this letter, we suppose 0 Ω− Ω i.e., Ω− | i ± ≤ ≤ + Eq. (5). Following the discussion by Schwendimann and and Ω+ are the eigenfrequencies of the lower and up- Quattropani [23–25], we consider a unitary operator Uˆ per eigenmodes, respectively. Figs. 1(b,f) show Ω± as that transforms the fluctuation operators aˆ and ˆb into functions of g/ωa. It is known [12, 13] that the lower the eigenmode ones pˆ± as eigenfrequency Ω− vanishes at the SRPT critical point † ˆ ˆ † ˆˆ ˆ † g = √ωaωb/2. In this case, [ˆp−, pˆ ]=1 does not hold, pˆ− UaÛ , pˆ+ UbU . (13a) − ≡ ≡ because Eq. (12) gives two degenerated solutions with For the vacuum 0a,b of the individual fluctuations sat- Ω− = 0 mathematically. In the following, we will show | i isfying aˆ 0 = ˆb 0 = 0, the ground state 0 of the that perfect squeezing is obtained at this critical point. | a,bi | a,bi | i 2 2 coupled system can be expressed as The quadrature variance (∆Xθ,ψ,ϕ) = 0 (Xˆθ,ψ,ϕ) 0 can be evaluated by rewriting the originalh photonic| and| i 0 Uˆ 0a,b , (14) atomic fluctuation operators aˆ, aˆ†, ˆb, and ˆb† with the | i∝ | i † eigenmode operators pˆ± and pˆ± and using Eq. (10). We while there is a freedom of introducing an overall phase numerically searched for the optimal angles θopt, ψopt, factor. This expression certainly satisfies Eq. (10). 2 ˆ and ϕopt that give the minimum variance (∆Xmin) The explicit expression of U for the fluctuation Hamil- 2 ≡ tonian, Eq. (5), derived from the Dicke model has been (∆Xθopt,ψopt,ϕopt ) for given ωa, ωb, and g. In Fig. 1, (c,g) quadrature variances including shown recently by Sharma and Kumar [34] as 2 (∆Xmin) and (d,h) optimal angles θopt, ψopt, and ϕopt Uˆ Uˆ0Uˆ−Uˆ+, (15) are plotted as functions of g/ωa for (c,d) ωb = ωa ≡ and (g,h) ωb = 2ωa. As shown by red solid lines in where the three unitary operators are defined as 2 Figs. 1(c,g), while the minimum variance is (∆Xmin) = ˆ†ˆ† ˆˆ †ˆ ˆ† †ˆ† ˆ Uˆ e−(rb/2)(b b −bb)e−α(â b−b aˆ)e−r(â b −baˆ), (16a) 1/4 (standard quantum limit [26, 27]) in the absence 0 ≡ † † ˆ†ˆ† ˆˆ of the photon–atom coupling (g = 0), it decreases as −(r−/2)(â aˆ −aâˆ) −(r+/2)(b b −bb) Uˆ− e , Uˆ+ e . (16b) g increases and vanishes at the SRPT critical point ≡ ≡ g = √ωaωb/2. After that, in the superradiant phase (g > Uˆ± are one-mode squeezing operators, and Uˆ0 is a prod- 2 √ωaωb/2), (∆Xmin) increases again and approaches 1/4 uct of one-mode squeezing, superposing, and two-mode asymptotically. The variance of its conjugate fluctu- squeezing operators [26, 27]. By a Bogoliubov transfor- 2 2 ation is given by (∆Xmax) (∆X ) , ˆ ˜ ≡ θopt,ψopt,ϕopt−π/2 mation of b for renormalizing the D term in Eq. (5), which diverges at the SRPT critical point (not shown the atomic frequency and coupling strength are modified in the figure). However, as shown by red dashed lines again as in Figs. 1(c,g), we numerically confirmed that the product of these variances satisfy ∆Xmin∆Xmax = 1/4, the ωˇ ω˜ (˜ω +4D˜), gˇ (1 γ)/(1 + γ)˜g, (17) b ≡ b b ≡ − equality in the Heisenberg uncertainty principle, i.e., an q ideal two-mode squeezing is obtained. where γ, giving also rb in Eq.p (16a), is defined as In Figs. 1(c,g), the blue dash-dotted curves represent 2 † 2 ˜ the variance (∆X ) = 0 (â aˆ ) 0 /4 of a pho- 1+4D/ω˜b 1 0,0,π/2 − tonic fluctuation. Such a one-modeh | − variance| i does not γ = tanh(rb). (18) ≡ q ˜ vanish even at the critical point [13, 15, 17] and satis- 1+4D/ω˜b +1 q fies only the inequality ∆X0,0,π/2∆X0,0,0 > 1/4 in the The other factors in Eqs. (16) are defined as Heisenberg uncertainty principle (not shown in the figure). tan(2α)=2ˇg/(ω ωˇ ), (19a) a − b As seen in Figs. 1(d,h), in the present case, the mini- tanh(2r)=2ˇg cos(2α)/(ωa +ˇωb), (19b) mum variance is obtained always for ψopt = 0 (dashed tanh(2r−)=ˇg sin(2α)/ǫ−, (19c) line) and ϕopt = π/2 (dash-dotted line). These two tanh(2r )= gˇ sin(2α)/ǫ , (19d) phases depend on those of the coupling strengths of the + − + 4

−iϕ † where ǫ± and the eigenfrequencies Ω± are expressed as e ˆb ) sin θ])/2 is proportional to the eigenmode operator pˆ−, because pˆ− 0 = 0 and then the quadrature 2 2 2| i (ωa +ˇωb) 2 2 (ωa ωˇb) 2 variance 0 (Xˆθ,ψ,ϕ) 0 becomes zero. Since we can ǫ± gˇ cos (2α) − +ˇg , h | | i ≡ r 4 − ± r 4 freely choose the angles θ, ψ, and ϕ, the perfect squeez- (20) ing can be obtained when the weights of the annihila-

2 2 2 tion and creation operators in the eigenmode operator Ω± = ǫ± gˇ sin (2α). (21) † † − pˆ− = w−aˆ + x−ˆb + y−aˆ + z−ˆb are equal as w− = y− q and x = z . Such equal weights are obtained| | | at| Note that the unitary operator Uˆ can be rewritten as − − critical| points| | accompanied| by a vanishing resonance fre-

Uˆ = Uˆd−Uˆd+Uˆ0, (22) quency in some interacting systems, e.g., weakly interacting Bose gases [36]. In the present case, we can easily ﬁnd ˆ i.e., a product of U0 and two one-mode squeezing opera- that w−/y− = x−/z− = 1 is obtained for Ω− =0 from tors the eigenvalue problem in− Eq. (12). In this way, we can † † † (−r±/2)(dˆ dˆ −dˆ±dˆ±) generally get perfect squeezing in a proper quadrature Uˆ ± Uˆ Uˆ±Uˆ =e ± ± (23) d ≡ 0 0 at critical points in the Dicke model and also in simi- under a new basis transformed from the original one (aˆ lar models with counter-rotating terms and a vanishing and ˆb) by Uˆ0 as resonance frequency.

† † dˆ− Uˆ0aÛˆ , dˆ+ Uˆ0ˆbUˆ . (24) In summary, we found that perfect and ideal squeez- ≡ 0 ≡ 0 ing is an intrinsic property associated with the zero- In the case of ωa = ωb and in the normal phase temperature SRPT in the Dicke model. Phenomenolog- (g < ω ω /2, a¯ = ¯b = r = γ = 0, ωˇ = ω , √ a b b b b ically, owing to a possible divergence of quantum fluctu- and gˇ = g), we can easily find that the ground state ation at a critical point, its conjugate fluctuation can be 0 Uˆ 0 is an ideal two-mode squeezed vacuum. | i ∝ | a,bi perfectly squeezed under satisfying the Heisenberg uncer- From Eqs. (19), (20), and (21), in the limit of ωb + → tainty principle. Such an ideal quantum behavior should ω +0 , we get Ω± = ω (ω 2g), α = π/4, r =0, a a a ± − be obtained only in limited systems with a vanishing reso- tanh(2r−) = g/(ωa g), and tanh(2r+) = g/(ωa + g). − −p ˆ nance frequency and counter-rotating terms, and we con- Since the unitary operator U0 is simply a superposing op- firmed that the Dicke model is one of such systems. (π/4)(â†ˆb−ˆb†aˆ) erator as Uˆ0 = e , the new basis dˆ± defined in Eq. (24) are just the equal-weight superpositions of In contrast to the standard squeezing generation the original fluctuation operators as dˆ± = (â ˆb)/√2. ± processes in dynamical and nonequilibrium situations Then, the ground state is simply expressed as 0 [26, 27], the phenomenon of intrinsic squeezing we de- ˆ ˆ ˆ | i ∝ U 0a,b = Ud−Ud+ 0a,b , i.e., squeezed by r± in the scribed here does not diminish in time and is stably ob- | i | i ˆ two-mode (superposed) basis d±, and the variances of tained in equilibrium situations. While perfect intrinsic ˆ quadratures defined by d− =c ˆ−π/4,0 are obtained as spin squeezing has been reported in some spin models 2 2 2r− 2 (∆Xmin) = (∆X−π/4,0,π/2) =e /4 and (∆Xmax) = such as the Lipkin–Meshkov–Glick model [37], the XY 2 −2r− (∆X−π/4,0,0) =e /4. Then, ∆Xmin∆Xmax =1/4 is model [38], and the transverse-field Ising model [39], this satisfied for any g. When the coupling strength reaches work presented the first photon–matter coupled model in − the critical point as g ωa/2+0 , the lower eigen- which perfect intrinsic squeezing arises. Intrinsic squeez- → + frequency becomes Ω− 0 , and the perfect squeezing ing has a potential for improving continuous-variable → is obtained as r− in the dˆ− basis. Therefore, quantum information technologies [40, 41], which have → −∞ 2 the quadrature variance (∆Xmin) vanishes at the SRPT been developed mostly in photonic systems, by making critical point, as we demonstrated in Fig. 1. them more resilient to decoherence. For practical ap- In the general case with ωa = ωb case (and in the su- plications, including quantum metrology [39], intrinsic 6 perradiant phase), we can mathematically confirm that squeezing at finite temperatures, for finite number (N) perfect squeezing can be obtained from the expression of atoms, and in the presence of coupling with a bath 0 Uˆ 0a,b of the ground state described by the uni- should be investigated in the future. | i ∝ | i tary operator Uˆ in Eq. (22), while the basis dˆ± is not simple superpositions of the original fluctuation opera- M.B. acknowledges support from the JST PRESTO tors aˆ and ˆb but includes also their creation operators program (grant JPMJPR1767). J.K. acknowledges sup- aˆ† and ˆb†. Instead of such a straightforward but compli- port from the U.S. National Science Foundation (Cooper- cated analysis, we can understand the perfect squeezing ative Agreement DMR-1720595), and the U.S. Army Re- at the SRPT critical point g = √ωaωb/2 in the following search Office (grant W911NF-17-1-0259). H.P. acknowl- manner. edges support from the U.S. NSF and the Welch Foun- The perfect squeezing can be obtained generally when dation (Grant No. C-1669). We thank Tomohiro Shitara iϕ −iϕ † iψ iϕ the quadrature Xˆθ,ψ,ϕ = [(e aˆ+e aˆ )cos θ+e (e ˆb+ for a fruitful discussion. 5

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