Quantum phase transitions and critical behaviors in the two-mode three-level quantum Rabi model
Yan Zhang, Bin-Bin Mao, Dazhi Xu, Yu-Yu Zhang, Wen-Long You, Maoxin Liu, Hong-Gang Luo
QQuantum phase transitions and critical behaviors in the two-mode three-levelquantum Rabi model
Yan Zhang,
1, 2
Bin-Bin Mao,
3, 4
Dazhi Xu, Yu-Yu Zhang, Wen-Long You,
7, 8, ∗ Maoxin Liu,
9, 1, † and Hong-Gang Luo
3, 1, ‡ Beijing Computational Science Research Center, Beijing 100193, China College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China School of Physical Science and Technology & Key Laboratory for Magnetism and MagneticMaterials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China Department of Physics, The University of Hong Kong, Hong Kong, China Center for Quantum Technology Research and School of Physics,Beijing Institute of Technology, Beijing 100081, China Department of Physics, Chongqing University, Chongqing 401330, China College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China School of Physical Science and Technology, Soochow University, Suzhou, Jiangsu 215006, China State Key Laboratory of Information Photonics and Optical Communications & School of Science,Beijing University of Posts and Telecommunications, Beijing 100876, China
We explore an extended quantum Rabi model describing the interaction between a two-modebosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom modelis found when the ratio η of the atom energy scale to the bosonic field frequency approaches infinity.An analytical solution is provided when the two lowest-energy levels are degenerate. According toit, we recognize that the phase diagram of the model consists of three regions: one normal phaseand two superradiant phases. The quantum phase transitions between the normal phase and thetwo superradiant phases are of second order relating to the spontaneous breaking of the discrete Z symmetry. On the other hand, the quantum phase transition between the two different superradiantphases is discontinuous with a phase boundary line relating to the continuous U (1) symmetry. Fora large enough but finite η , the scaling function and critical exponents are derived analytically andverified numerically, from which the universality class of the model is identified. I. INTRODUCTION
The quantum Rabi model describes the interaction be-tween a photon field and a two-level system [1, 2], whichis the simplest model for studying the light-matter inter-action and plays a significant role in quantum optics [3],condensed-matter physics [4], and quantum information[5]. With the rapid experimental progress in accessingthe strong [6, 7], the ultrastrong [8–10], and the deepstrong coupling regimes [11, 12], the quantum Rabi modelhas received much attention since the rotating-wave ap-proximation fails [13–31]. Recently, phase transitions andcritical phenomena have been surprisingly found in thequantum Rabi model although only a single atom is in-volved [32], which requires the infinite frequency ratio ofthe atom to the photon rather than the thermodynamiclimit required by traditional phase transitions. Furtherstudy on the scaling behaviors of the Rabi and the Dickemodels has revealed that these two models belong to thesame universality class [33]. These progresses bring anew insight for the quantum phase transition withoutthe thermodynamic limit.In parallel the interaction between a two-mode cavityfield and a three-level system leads to many important ∗ [email protected] † [email protected] ‡ [email protected] phenomena, such as electromagnetically induced trans-parency [34] and dark state [35], which are profitable inthe precise control of coherent population trapping andtransfer [36]. The three-level system is also importantin quantum information, referred as qutrit. Comparedwith the two-level scheme, the quantum key distributionbased on qutrits is more resistant to attack [37, 38], andthe quantum computation using qutrits shows a fasterspeed and a lower error rate [39, 40]. A qutrit quantumcomputer with trapped ions has been proposed [41]. Inaddition, the three-level system is used to construct aquantum heat engine [42, 43]. To identify the possiblequantum phases and quantum phase transitions involvedin a two-mode three-level model is helpful in further un-derstanding these light-matter interaction models and ex-tending their applications.The two-mode three-level interaction model in thethermodynamic limit has received much attention. Hayn et al. studied quantum phase transitions by a general-ized Holstein-Primakoff transformation and revealed thatit exhibits two superradiant quantum phase transitions,which can be both first and second order [44]. Cordero et al. found that the polychromatic ground-state phasediagram can be divided into monochromatic regions bya variational analysis [45]. Here, we report an analyti-cal calculation of the ground-state phase diagram, scal-ing function, and critical exponents for the two-modethree-level quantum Rabi model by taking a single Λ-typethree-level atom as a prototype. The analytical results a r X i v : . [ qu a n t - ph ] O c t are further verified by an exact numerical diagonaliza-tion.The paper is organized as follows. In Sec. II, an effec-tive model is derived when the ratio between the atomfrequency and the photon frequency approaches infinity.In Sec. III, a ground-state phase diagram is extractedanalytically. In Sec. IV, the mean photon number inthe ground state is analytically derived. In Sec. V, thescaling function and critical exponents are analyticallyderived for finite frequency ratios, and also the numeri-cal diagonalization is used to verify the analytical results.Finally, a brief summary is presented in Sec. VI. II. MODEL HAMILTONIAN
Our two-mode three-level quantum Rabi model de-scribes the interaction between a two-mode quantizedfield and a single Λ-type three-level atom, which is givenby ( (cid:126) = 1)ˆ H = ε | (cid:105)(cid:104) | + ε | (cid:105)(cid:104) | + ε | (cid:105)(cid:104) | + ω ˆ a † ˆ a + ω ˆ a † ˆ a + g ˆ A (ˆ a † + ˆ a ) + g ˆ A (ˆ a † + ˆ a ) , (1)where ˆ A = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | , ˆ A = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | , ε i is theenergy eigenvalue of state | i (cid:105) ( i = 1 , , a † and ˆ a (ˆ a † and ˆ a ) are the creation and annihilation operators ofthe photon mode 1 (mode 2), ω ( ω ) is the frequencyof the photon mode 1 (mode 2), g ( g ) is the couplingstrength between the transition | (cid:105) ↔ | (cid:105) ( | (cid:105) ↔ | (cid:105) ) andthe photon mode 1 (mode 2). Note that the transitionbetween state | (cid:105) and state | (cid:105) in the Λ-type configurationis forbidden.For convenience, we define several variables to makethe Hamiltonian dimensionless: ∆ = ε − ε , δ = ( ε − ε ) /∆ , α = ω /ω , β = g /g , R = 2 g / √ ω ∆ . Also, weset ε = 0 and η = ∆/ω , and then the dimensionlessHamiltonian rescaled by ∆ becomesˆ H (cid:48) = δ | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + 1 η (ˆ a † ˆ a + α ˆ a † ˆ a )+ 12 η / R [ ˆ A (ˆ a † + ˆ a ) + β ˆ A (ˆ a † + ˆ a )] . (2)We rewrite the Hamiltonian in terms of the position andmomentum operators (ˆ x i and ˆ p x i , i = 1 ,
2) via ˆ a i =(ˆ x i + i ˆ p x i ) / √ a † i = (ˆ x i − i ˆ p x i ) / √
2. Further rescalingthe position and momentum operators by ˆ y i = η − / ˆ x i ,the Hamiltonian becomesˆ H (cid:48) = δ | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + 12 (ˆ y + α ˆ y ) + 12 η (ˆ p y + α ˆ p y )+ √ R ( ˆ A ˆ y + β ˆ A ˆ y ) . (3)Supposing that ˆ p y and ˆ p y are finite, the contributionsfrom the momentum terms disappear in the limit η → ∞ , FIG. 1. Ground-state phase diagram as a function of theparameter ratio α/β and the coupling strength R . The lowerpart is the normal phase. The higher part is the superradiantphase with two regimes of α < β and α > β . The former isonly contributed to by the coupling of mode 2 and transition | (cid:105) ↔ | (cid:105) , labelled as y -type. The latter is only from thecoupling of mode 1 and transition | (cid:105) ↔ | (cid:105) , labelled as y -type. Two distinct regions are separated by a border of α = β . The solid and dashed lines represent the first-order andsecond-order phase transitions, respectively. and then the effective Hamiltonian reduces toˆ H eff = δ | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + 12 ( y + αy )+ √ R ( ˆ A y + β ˆ A y ) . (4)Because of the absence of momentum operators, ˆ y andˆ y can be replaced with the eigenvalues y and y inEq. (4). In consequence, the ground-state phase diagramand the mean photon number are obtained. III. PHASE DIAGRAM
The analytical diagonalization of the effective Hamil-tonian (4) is equivalent to solving a third-order algebraicequation. An explicit form of the ground-state energycan be readily obtained for a special case that the state | (cid:105) and the state | (cid:105) are degenerate (i.e., δ = 0), E ( y , y ) = 12 ( y + αy ) + 12 [1 − (cid:113) R ( y + β y )] . (5)Through the first-order and the second-order deriva-tives ( ∂E∂y , ∂E∂y ; ∂ E∂y , ∂ E∂y ), the ground-state energy E is further determined. The detailed derivation of theground-state energy is given in Appendix A. The ob-tained ground-state phase diagram is depicted in Fig. 1.One can see that the phase diagram shows several dis-tinct regimes.In the regime of α < β , the critical point of phasetransition is R <,c = (cid:114) αβ . (6)When R is less than or equal to R <,c , y = 0 and y = 0, and the ground state stays in a normal phasewith E = 0. When R is larger than R <,c , the normalphase becomes unstable and bifurcates into two degener-ate stable solutions y = 0 , y , ± = ± (cid:115) β R − α α β R , (7)with the ground-state energy E = −
14 ( αβ R + β R α ) + 12 . (8)It implies that the ground state enters the so-called su-perradiant phase. While E and ∂E ∂R are continuous, ∂ E ∂R is discontinuous at R = R c as shown in Fig. 2(a),which reveals that the phase transition from the normalphase to the superradiant phase is of second order.In the regime of α > β , the critical point of phasetransition is R >,c = 1 . (9)When R is less than or equal to R >,c , the ground statecorresponds to E = 0 with y = 0 and y = 0, whichis the normal phase. When R is larger than R >,c , theground state bifurcates into two degenerate stable solu-tions y , ± = ± (cid:114) R − R , y = 0 , (10)with the ground-state energy E = −
14 ( 1 R + R ) + 12 . (11)The ground-state energy is a constant in this case, inde-pendent of α and β . Likewise, the normal-superradiantquantum phase transition is also a second-order phasetransition.When α = β , both y and y are nonzero in theground state when R exceeds the critical point R = ,c = 1,which satisfy the relation: y + αy = R − R . (12)The ground-state energy is accordingly given by E = −
14 ( 1 R + R ) + 12 . (13)This quantum phase transition still has the second-ordernature. However, across the boundary line ( α = β ) be-tween the two superradiant phases, the model undergoes (a) ∂ E ∂R (b) ∂E ∂γ FIG. 2. (a) ∂ E ∂R , the second-order derivative of the ground-state energy with respect to the coupling strength R . Itreflects a second-order normal-superradiant phase transitionat the critical points. (b) ∂E ∂γ , the first-order derivative ofthe ground-state energy with respect to the parameter ratio γ = α/β , reflecting a first-order transition between the twosuperradiant phases of α < β and α > β . a first-order quantum phase transition because the first-order partial derivative ∂E ∂γ ( γ = α/β ) is discontinuous,as shown in Fig. 2(b). Here, the two-mode three-levelquantum Rabi model presents two kinds of typical spon-taneous symmetry breaking, which are separately charac-terized by the order parameters y and y . One can seethat the normal phase possesses a discrete Z symme-try, but the ground-state energy functionals either E ( y )or E ( y ) shows a double-well structure when the cou-pling strength exceeds the critical points, which impliesa breaking of the Z symmetry. In the case of α = β , theground-state energy functional E ( y , y ) presents a con-tinuous U (1) symmetry. When the coupling strength isabove the critical point, both y and y become nonzero,which reflects a breaking of the continuous symmetry.The U (1) symmetry was also found in the two-level sys-tem [46], in which the light-atom interaction, however,requires the consideration of both the electric and themagnetic components for the electromagnetic field. Suchcontinuous symmetry breaking is related to the Nambu-Goldstone mode [47]. IV. MEAN PHOTON NUMBER
The photon number is a common observable in exper-iments and usually used to characterize the states of thelight-matter interaction systems. In the Dicke quantumphase transition experiments, an abrupt increase of themean intracavity photon number marks the onset of thenormal-superradiant phase transition [48, 49]. To calcu-late the mean photon number of the ground state, werewrite the Hamiltonian in Eq. (4) as (with δ = 0),ˆ H eff = 12 ( y + αy ) + 12 ˆ M , (14)where ˆ M = √ Ry √ βRy √ Ry √ βRy . (15)When α < β , the lowest eigenvalue of ˆ M is 1 − β R α for y = 0 and y , ± = ± (cid:113) β R − α α β R , and the correspondingeigenstate (normalized) is a , ± b , ± c , ± = ± − (cid:113) β R + α β R (cid:113) β R − α β R . (16)Thus the ground-state wavefunction of ˆ H eff can be ex-pressed as ψ = 1 √ a , + b , + c , + + a , − b , − c , − . (17)Based on the wavefunction ψ , the mean photon numbersof the two modes above the critical point are (cid:104) ˆ a † ˆ a (cid:105) = 12 η (cid:104) ψ | ˆ y | ψ (cid:105) = 0 (18)and (cid:104) ˆ a † ˆ a (cid:105) = 12 η (cid:104) ψ | ˆ y | ψ (cid:105) = 12 η ( y , + + y , − ) = η β R − α α β R . (19)When α > β , the lowest eigenvalue of ˆ M is 1 − R for y , ± = ± (cid:113) R − R and y = 0, and the correspondingeigenstate (normalized) is a , ± b , ± c , ± = ± − (cid:113) R +12 R (cid:113) R − R . (20)Hence, the ground-state wavefunction of ˆ H eff can be ex-pressed as ψ = 1 √ a , + b , + c , + + a , − b , − c , − . (21)Then, the mean photon numbers of the two modes abovethe critical point are (cid:104) ˆ a † ˆ a (cid:105) = 12 η (cid:104) ψ | ˆ y | ψ (cid:105) = 12 η ( y , + + y , − ) = η R − R (22)and (cid:104) ˆ a † ˆ a (cid:105) = 12 η (cid:104) ψ | ˆ y | ψ (cid:105) = 0 . (23) (a) mode 1 (b) mode 2 FIG. 3. Mean photon number as a function of α/β and R . Inthe normal phase below the critical points, the mean photonnumbers of both modes are zero. Above the critical points,as for α/β <
1, the superradiant phase of
R > (cid:112) α/β ischaracterized by the mode-2 mean photon number of (cid:104) ˆ a † ˆ a (cid:105) /η (namely, y -type); As for α/β >
1, the superradiant phaseof
R > (cid:104) ˆ a † ˆ a (cid:105) /η ( y -type). ( β = 1 . We choose (cid:104) ˆ a † ˆ a (cid:105) /η and (cid:104) ˆ a † ˆ a (cid:105) /η as the order param-eters for α < β and α > β , respectively. Figure 3presents the mean photon numbers of the two photonmodes. One can see that the photon field is a vacuum inthe normal phase and simultaneously the atom is at | (cid:105) for α < β and | (cid:105) for α > β inferring from Eqs. (16)and (20). Above the critical points, both the photon fieldand the atom are excited and the so-called superradiantphase transition takes place. In each regime of α < β and α > β , there is only one coupling involved. Moreprecisely, only the coupling between the photon mode 2and the transition | (cid:105) ↔ | (cid:105) contributes to the groundstate when α < β , whereas only the coupling betweenthe photon mode 1 and the transition | (cid:105) ↔ | (cid:105) con-tributes to the ground state when α > β . Actually, thetwo-mode three-level quantum Rabi model degeneratesinto the quantum Rabi model in these two parameterregimes. In each parameter regime, there exists a darkstate, which is state | (cid:105) for α < β and | (cid:105) for α > β ,as seen in Eqs. (16) and (20). The dark state was alsostudied in a semi-classical two-mode three-level quantumRabi model [50].The critical exponent can be obtained by the defor-mation of the mean photon number with the reducedcoupling strength r = ( R − R c ) /R c , (cid:104) a † a (cid:105) /η = R − R ∝ r, (24) (cid:104) a † a (cid:105) /η = β R − α α β R ∝ r. (25)They indicate that the critical exponent of the mean pho-ton numbers is 1, which is the same as the quantum Rabimodel [33, 51]. It suggests that the present model be-longs to the same universality class with the quantumRabi model. V. SCALING BEHAVIOR
The finite-size scaling behavior of continuous phasetransitions is important near the critical point (the sizehere is characterized by the frequency ratio η ). We de-rive the scaling function and critical exponents of thismodel. For finite η , after diagonalizing the atomic part,the Hamiltonian in Eq. (3) becomes ( δ = 0)ˆ H (cid:48) (ˆ y , ˆ y ) = 12 η (ˆ p y + α ˆ p y ) + 12 (ˆ y + α ˆ y )+ 12 [1 − (cid:113) R (ˆ y + β ˆ y )] . (26)In the case of α < β , the wavefunction near the crit-ical point R <,c is very localized around y = 0 when η is very large. Thus, the Hamiltonian might be approx-imated by a second-order expansion in the vicinity of y = 0, ˆ H (cid:48) (ˆ y ) ≈ α η ˆ p y − αr ˆ y + α y . (27)Defining ˆ y = η − / ˆ z and r = η − / r (cid:48) , the Hamiltoniancan be rewritten asˆ H (cid:48) (ˆ z ) = η − / ( α p z − αr (cid:48) ˆ z + α z ) . (28)In this respect there exists a rescaled ground-state wave-function φ (ˆ z , r (cid:48) ), which is independent of η . Based onthis wavefunction, the mean photon number is (cid:104) ˆ a † ˆ a (cid:105) /η = η − / (cid:104) φ | ˆ z | φ (cid:105) . (29)In the case of α > β , the wavefunction near R >,c is also very localized around y = 0 for a very large η .Likewise, we expand the Hamiltonian to the second orderin the vicinity of y = 0,ˆ H (cid:48) (ˆ y ) ≈ η ˆ p y − r ˆ y + 14 ˆ y . (30)Analogously defining ˆ y = η − / ˆ z and r = η − / r (cid:48) , wehave ˆ H (cid:48) (ˆ z ) = η − / ( 12 ˆ p z − r (cid:48) ˆ z + 14 ˆ z ) . (31)In this case, the mean photon number is (cid:104) ˆ a † ˆ a (cid:105) /η = η − / (cid:104) φ | ˆ z | φ (cid:105) . (32)The detailed derivation of the Hamiltonian from Eq. (26)to Eqs. (27) and (30) is given in Appendix B.Based on the above analytical derivation, the scalingfunction for the mean photon number of the two-modethree-level quantum Rabi model is obtained as follows, N ( η, r ) = η − / f ( η / r ) . (33) where N ( η, r ) is the photon numbers of (cid:104) ˆ a † ˆ a (cid:105) /η and (cid:104) ˆ a † ˆ a (cid:105) /η .According to the standard finite-size scaling law [52],the divergent correlation length at critical points enablesthe scaling behavior of a physical quantity P at differentfinite η : P ( η, r ) = η − κ/ν f ( η /ν r ) , (34)where κ is the critical exponent of P ( P ∝ r κ ), ν isthe critical exponent of the correlation length ( ξ ∝ r − ν ).Comparing Eq. (33) with Eq. (34), we infer that κ and ν of the mean photon number for the two-mode three-level quantum Rabi model are 1 and 3/2, respectively.The finite-size critical scaling exponent κ of the meanphoton numbers is the same as that obtained from thelimit η → ∞ , as is concluded from Eqs. (24) and (25).To check our analytical prediction, we numerically di-agonalize the two-mode three-level quantum Rabi model.We calculate the mean photon numbers of the two pho-ton modes in the case of δ = 0. To determine criticalpoints and critical exponents numerically, we take thelogarithm of Eq. (34) asln P ( η, r ) = ( − κ/ν ) ln η + ln f ( η /ν r ) . (35)At the critical point r = 0, ln P ( η, r ) and ln η are linearlyrelated as ln P ( η,
0) = ( − κ/ν ) ln η + ln f (0) . (36)The scaling parameter κ/ν can be determined by a linearfitting. The remaining parameter ν can be obtained bya collapse of the data points with different η values ontoa single scaled curve.Figures 4(a) and 5(a) show the logarithm of the meanphoton number as a function of the logarithm of η forthe two regimes of α < β and α > β , respectively.The red-square line represents the linear behavior andthe determined critical points are R <,c = 0 . R >,c = 1 . R <,c = (cid:112) α/β and R >,c = 1, respectively.The fitted values of κ/ν are 0.624 and 0.616, and ν iscorrespondingly obtained to be 1.582 and 1.592, whichreproduce the analytical values κ/ν = 2 / ν = 3 / η collapse onto a well-definedsingle curve. It indicates that the analytical solution re-veals the correct ground-state phase diagram of the two-mode three-level quantum Rabi model and captures itsscaling invariance near the critical points. VI. CONCLUSIONS
Based on the analytical solution and the numericaldiagonalization, we attain the ground-state phase dia-gram, scaling function, and critical exponents of the two-mode three-level quantum Rabi model. The phase di-agram is divided into the three regions of one normal
11 12 13 14−10.5−9.5−8.5−7.5 ln η l n [ h a † a i / η ] R = 0 . R = 0 . R = 0 . (a) linear fitting −2 −1 0 1 200.511.52 η . r η . [ h a † a i / η ] η = 2 η = 2 η = 2 η = 2 (b) scaling invariance FIG. 4. α < β (a) ln[ (cid:104) ˆ a † ˆ a (cid:105) /η ] as a function of ln η . Thered-square line exhibits the linear relation, which indicatesthe critical point of 0.7454. Two green dashed straight linesassist us to observe. (b) Scaling of (cid:104) ˆ a † ˆ a (cid:105) /η makes all thedata points of different η values collapse onto a single curve.( α = 0 . β = 1 .
11 12 13 14−10.5−9.5−8.5−7.5 ln η l n [ h a † a i / η ] R = 1 . R = 1 . R = 0 . (a) linear fitting −2 −1 0 1 200.511.5 η . r η . [ h a † a i / η ] η = 2 η = 2 η = 2 η = 2 (b) scaling invariance FIG. 5. α > β (a) ln[ (cid:104) ˆ a † ˆ a (cid:105) /η ] as a function of ln η . Thered-square line shows the linear relation at the critical pointof 1.0000. (b) Scaling of (cid:104) ˆ a † ˆ a (cid:105) /η makes all the data points ofdifferent η values collapse onto a single curve. ( α = 1 . β = 0 . phase and two superradiant phases. The phase transi-tions could take place by adjusting the frequency ratio ofthe two photon modes and the relative strength of thetwo photon-atom couplings. The normal-superradiantquantum phase transitions are found to be second or-der and related to the spontaneous breaking of Z sym-metry. In addition, the model undergoes a first-orderphase transition across the boundary line between thetwo superradiant phases, where a spontaneous contin-uous U (1) symmetry breaking is discovered. Differentfrom the traditional phase transitions in the thermody-namic limit, here the quantum phase transition is realizedalternatively when the frequency ratio η of the atomictransition and the photon field approaches infinity. Thefinite- η scaling function is derived and the obtained crit-ical exponent of the mean photon number is the same asthat determined in the limit η → ∞ . Based on this, theuniversality class of this model is identified. This workis helpful in further understanding the quantum phasetransitions and exploring some potential applications forsuch single-atom systems. The present results are ob-tained from the degenerate case for two lowest states, thegeneral case of the two-mode three-level quantum Rabimodel needs further exploration. ACKNOWLEDGMENTS
This work was supported by NSFC (Grants No.11604009, No. 11705008, No. 11474211, No. 11674139,No. 11834005, and No. 11504298) and NSAF (Grant No.U1530401). D.X. was also supported by the Beijing In-stitute of Technology Research Fund Program for YoungScholars.
Appendix A: Derivation of the ground-state energy
To determine the ground-state energy of Eq. (5), wecalculate its first derivative with respect to y and y ,and let both them equal zero (cid:40) ∂E ( y ,y ) ∂y = 0 ∂E ( y ,y ) ∂y = 0 . (A1)It yields y (1 − R √ R ( y + β y ) ) = 0 y ( α − R β √ R ( y + β y ) ) = 0 . (A2)The solutions have four cases:Case 1: (cid:40) y = 0 y = 0; (A3)the corresponding ground-state energy E = 0.Case 2: (cid:40) y = 0 α − R β √ R ( y + β y ) = 0; (A4)we thus have y , ± = ± (cid:115) β R − α α β R . (A5)This solution exists only when R (cid:62) (cid:113) αβ , and the corre-sponding energy E = − ( αβ R + β R α ) + (cid:54) (cid:40) − R √ R ( y + β y ) = 0 y = 0; (A6)hence, y , ± = ± (cid:114) R − R . (A7)It exists only when R (cid:62)
1, and the corresponding energy E = − ( R + R ) + (cid:54) − R √ R ( y + β y ) = 0 α − R β √ R ( y + β y ) = 0 (A8)requires α = β . On substituting this relation intoEq. (5), we have E ( y , y ) = 12 ( y + β y ) + 12 [1 − (cid:113) R ( y + β y )] . (A9)Defining y + β y = S , the ground-state energy expres- sion becomes E ( S ) = 12 S + 12 [1 − (cid:112) R S ] . (A10)We let its first-order derivative d E ( S )d S = 0, then S = R − R (A11)It indicates a continuous set of y and y with the degen-erate energy of E = − ( R + R ) + (cid:54)
0, which existsalso when R (cid:62) α < β R (cid:54) (cid:113) αβ { y = 0 , y = 0 } E = 0 R > (cid:113) αβ { y = 0 , y , ± = ± (cid:113) β R − α α β R } E = − ( αβ R + β R α ) + α > β R (cid:54) { y = 0 , y = 0 } E = 0 R > { y , ± = ± (cid:113) R − R , y = 0 } E = − ( R + R ) + α = β R (cid:54) { y = 0 , y = 0 } E = 0 R > { y + β y = R − R } E = − ( R + R ) + They are verified by the positive second-order derivatives of ∂ E ( y ,y ) ∂y > ∂ E ( y ,y ) ∂y > y and y . Appendix B: Derivation of the finite- η Hamiltonian
For a large enough but finite η , we derive the Hamilto-nian of this model near the critical point from Eq. (26).When α < β , the superradiant phase is y -type inview of the ground-state phase diagram (see Fig. 1). Al-though the contributions of the ˆ y terms are not strictlyzero for finite η , ignoring the ˆ y terms in Eq. (26) shouldbe a good approximation when η is large enough. TheHamiltonian is given byˆ H (cid:48) (ˆ y ) = α η ˆ p y + α y + 12 [1 − (cid:113) β R ˆ y ] . (B1)The wavefunction of this Hamiltonian is very localizedbecause the contributions of the momentum term be-comes very small when η is very large. In particular,the wavefunction is localized around y = 0 near thecritical point. Thus, we express the Hamiltonian by asecond-order expansion in the vicinity of y = 0,ˆ H (cid:48) (ˆ y ) ≈ α η ˆ p y + 12 ( α − β R )ˆ y + 14 β R ˆ y . (B2)Considering that the coupling strength R is close to thecritical point, namely, R ≈ (cid:112) α/β , the Hamiltonian canbe further processed asˆ H (cid:48) (ˆ y ) ≈ α η ˆ p y + √ α ( √ α − βR )ˆ y + α y . (B3) Recalling the reduced coupling strength r = ( R − R c ) /R c ,we have ˆ H (cid:48) (ˆ y ) ≈ α η ˆ p y − αr ˆ y + α y . (B4)When α > β , the superradiant phase is y -type. Sim-ilarly, when η is large enough, the Hamiltonian becomesˆ H (cid:48) (ˆ y ) = 12 η ˆ p y + 12 ˆ y + 12 [1 − (cid:113) R ˆ y ] . (B5)Its wavefunction near the critical point is very localizedaround y = 0 when η is very large, and we expand theHamiltonian to the second order in the vicinity of y = 0,ˆ H (cid:48) (ˆ y ) ≈ η ˆ p y + 12 (1 − R )ˆ y + 14 R ˆ y . (B6)Using R ≈
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