A Mathematical Analysis of Dressed Photon in Ground State of Generalized Quantum Rabi Model Using Pair Theory
aa r X i v : . [ qu a n t - ph ] D ec A Mathematical Analysis of Dressed Photon in Ground Stateof Generalized Quantum Rabi Model Using Pair Theory
Masao Hirokawa
Institute of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
Jacob S. Møller
Department of Mathematics, Aarhus University, Aarhus, Denmark
Itaru Sasaki
Department of Mathematical Sciences, Shinshu University, Matsumoto, 390-8621, Japan
Abstract
We consider the generalized quantum Rabi model with the so-called A -term in the lightof the Hepp-Lieb-Preparata quantum phase transition. We investigate the dressed photon inits ground state when the atom-light coupling strength is in the deep-strong coupling regime.We show how the dressed photon appears in the ground state. We dedicate this paper toPavel Exner and Herbert Spohn on the occasion of their 70th birthdays, and Klaus Hepp onthe occasion of his 80th birthday. Quantum electrodynamics (QED) says that the matter coupled with a radiation field emitsa photon when the relaxation (de-excitation) of a quantum state of the matter takes place.In particular, for the relaxation from an excited state to a ground state, the ground stateshould be the vacuum. Preparata, however, claims that there is a coherence domain in whichthe photon cannot be emitted from the matter in certain cases where the matter-radiationcoupling is very strong [22, 8], and the ground state has the photon that should primarilybe emitted outside of the matter. Thus, the photon should be among the dressed photonsin the ground state. Then, the ground state switch from the perturbative ground state tothe non-perturbative ground state. This non-perturbative ground state is called the coherentground state by Preparata [22] and the superradiant ground state by Enz [8]. Preparatafound this phenomena stimulated by and based on the Hepp-Lieb quantum phase transition[10, 11]. We thus call this phenomenon the Hepp-Lieb-Preparata quantum phase transitionin this paper.The Hepp-Lieb-Preparata quantum phase transition is shown by Enz [8] and one ofauthors [12] for a model describing a two-level system coupled with light such as the many-mode-photon version of the Jaynes-Cummings type models as well as Jaynes-Cummingsmodel itself [13, 15, 16]. We note that all the models have the rotating wave approximation.However, in the ultra-strong or deep-strong coupling regime [6] we cannot avoid the effectscoming from the counter rotating terms and the A -term, the quadratic interaction of pho-ton field, for the argument on such a quantum phase transition. It is worthy to note thatsome theoretical observations have lately pointed out the possibility of the quantum phasetransition in a ground state of a model in circuit QED [1, 21, 7].For the model with the A -term and without the rotating wave approximation, Yoshihara et al . recently succeeded in achieving the deep-strong coupling regimes in their circuit QEDexperiment. They observed a quantum phase transition in the ground state in the deep-strongcoupling regime for the generalized quantum Rabi model. Therefore, we are mathematicallyinterested in the generalized quantum Rabi model, and we consider it in the light of the epp-Lieb-Preparata quantum phase transition. There are at least two possibilities: whetherthe dressed photon in the ground state is a bare photon or it is a physical photon. Because,in particular, we pay special attention to the physical photon in the ground state, we copewith the A -term problem following the pair theory for quantum field caused by a neutralsource [9]. For the history of several pair theories, see Ref.[5]. Let H be a separable Hilbert space. We denote by ( , ) H the inner product of the Hilbertspace H . We denote by k k H the norm naturally induced by the inner product ( , ) H .Meanwhile, k k op is the operator norm for a bounded operator acting the Hilbert space.The state space F of the two-level atom system coupled with one-mode light is given by C ⊗ L ( R ), where C is the 2-dimensional unitary space, and L ( R ) the Hilbert spaceconsisting of the square-integrable functions. For any matrix A on C and any operator B acting in L ( R ), we often denote by AB the tensor product of the operators A and B , i.e., AB = A ⊗ B , by omitting the tensor sign ⊗ throughout this paper. For the identity operator I , we write A ⊗ I and I ⊗ B simply as A an B , respectively. We use the notations, | ↑i and |↓i , which stand for the spin-states defined in C by |↑i := (cid:16) (cid:17) and |↓i := (cid:16) (cid:17) . Wedenote by | n i the Fock state in L ( R ) with the photon number n = 0 , , , · · · . That is, | i := ( ω/π ~ ) / exp (cid:2) − ωx / ~ (cid:3) and | n i := √ wγ n H n ( wx ) exp (cid:2) − ( wx ) / (cid:3) ∈ L ( R ), where H n ( x ) is the Hermite polynomial of variable x , γ n = π − / (2 n n !) − / , and w = p mω c / ~ .We often use a compact notation, | s, n i , for the separable state | s i ⊗ | n i for s = ↑ , ↓ and n = 0 , , , · · · .We denote the Pauli matrices by σ x := (cid:16) (cid:17) , σ y := (cid:16) − ii (cid:17) , and σ z := (cid:16) − (cid:17) . Werespectively denote by a and a † the annihilation and creation operators of 1-mode photondefined by a | i := 0, a | n i := √ n | n − i , and a † | n i := √ n + 1 | n + 1 i .We denote by | E i a normalized eigenstate of an operator A with its corresponding eigen-value E , i.e., A | E i = E | E i . For any state ψ , we denote σ x ψ by e ψ , i.e., e ψ = σ x ψ . For anyeigenstate | E i , we denote σ x | E i by | e E i . We note that the operator e A defined by e A = σ x Aσ x is unitarily equivalent to the operator A so that | e E i is its eigenstate with the correspondingeigenvalue E . The flux qubit is demonstrated in a superconducting circuit. The clockwise current andthe counterclockwise current in the superconducting circuit make two states, | (cid:8) i and | (cid:9) i ,respectively. We can regard these two states as a qubit. Then, the Hamiltonian of the qubitis given by H qubit = − ~ ω a σ x + εσ z ) . Here, ~ ω a and ~ ε are the tunneling splitting energy and the bias energy between | (cid:8) i and | (cid:9) i . We respectively represent the states | (cid:8) i and | (cid:9) i with the spin-states |↓i and |↑i fromnow on: |↓i = | (cid:8) i and |↑i = | (cid:9) i . Then, we can regard the Pauli matrix σ x as the spin-chiraltransformation , σ x : |↓i = | (cid:8) i −→ |↑i = | (cid:9) i and σ x : |↑i = | (cid:9) i −→ |↓i = | (cid:8) i . In other words,for the counterclockwise rotation matrix R z ( θ ) = e − iθσ z / and the clockwise rotation matrix R z ( − θ ) = e iθσ z / through an angle θ about the z -axis of the Bloch sphere [3], we have therelation, σ x R z ( θ ) σ x = R z ( − θ ), and therefore, we can say that the states, e ψ and | e E i , are thereflected images on the other side of mirror on the xy -surface for their original states, ψ and | E i , respectively. oshihara et al . consider the Hamiltonian H total = H qubit + ~ ω c (cid:18) a † a + 12 (cid:19) + ~ g σ z (cid:0) a + a † (cid:1) in Ref.[28]. Let us define a unitary operator U xz by U xz := √ (cid:16) − (cid:17) . Then, their Hamil-tonian H total is unitarily transformed to H ( ω a , ε, ω c , g) := U ∗ xz H total U xz = H atom ( ω a , ε ) + H cavity ( ω c ) + H int (g) (2.1)with H atom ( ω a , ε ) = ~ ( ω a σ z − εσ x ), H cavity ( ω c ) = ~ ω c (cid:0) a † a + (cid:1) , and H int (g) = ~ g σ x (cid:0) a + a † (cid:1) .Here the cavity Hamiltonian H cavity ( ω c ) includes the zero-point energy ~ ω c /
2. In the case ε = 0, we call it the quantum Rabi Hamiltonian , and thus, we call H ( ω a , ε, ω c , g) a gener-alized quantum Rabi Hamiltonian . We rewrite the interaction in the (generalized) quantumRabi Hamiltonian using the spin-annihilation operator σ − and the spin-creation operator σ + defined by σ ± := ( σ x ± iσ y ) / ~ g σ x (cid:0) a + a † (cid:1) = ~ g (cid:0) σ − a + σ + a + σ − a † + σ + a † (cid:1) . The opera-tors, σ + a and σ − a † , are the rotating terms, and the operators, σ − a and σ + a † , the counter-rotating terms. The spin-chiral transformation σ x makes the relations, σ x ( σ − a ) σ x = σ + a and σ x ( σ + a † ) σ x = σ − a † . Thus, the counter-rotating terms are the rotating terms on theother side of the mirror.We here give some mathematical notes. By Theorem 4.3 of Ref.[16], the generalizedquantum Rabi Hamiltonian H ( ω a , ε, ω c , g) is self-adjoint on the domain C ⊗ Span( {| n i} ∞ n =0 ),where Span( {| n i} ∞ n =0 ) is a subspace linearly spanned by all the Fock states | n i in L ( R ). We briefly see the Hepp-Lieb-Preparata quantum phase transition in mathematics usingthe Jaynes-Cummings model though the model is not physically valid over the ultra-strongcoupling regime. For simplicity we tune the frequencies, ω a and ω c , to a frequency ω , i.e., ω a = ω c = ω . The Jaynes-Cummings Hamiltonian reads H JC := H atom ( ω,
0) + H cavity ( ω ) + ~ g( σ + a + σ − a † ) . It is easy to solve the eigenvalue problem, H JC ϕ ν = E ν ϕ ν . The eigenstates ϕ ν and thecorresponding eigenvalues E ν are given by (cid:26) ϕ = |↓ , i ,E = 0 , and ϕ ± ( n +1) = ± √ |↑ , n i + 1 √ |↓ , n + 1 i ,E ± ( n +1) = ~ ω ( n + 1) ± ~ g √ n + 1 , n = 0 , , , · · · . Thus, we realize that the separable state ϕ is a unique ground state if 0 ≤ g < ω , andthe entangled state ϕ − is a unique ground state if ω < g < ( √ ω . In the same way,moreover, the entangled state ϕ − ( n +1) is a unique ground state if ( √ n + 1 + √ n ) ω < g < ( √ n + 2 + √ n + 1) ω . In these way, the ground state has some photons provided ω < g,though it has no photon as long as 0 ≤ g < ω . The ground state shifts from a separablestate to an entangled one, and changes how to make the entanglement. Namely, we canconcretely see the Hepp-Lieb-Preparata quantum phase transition for the Jaynes-Cummingsmodel through the facts how many photons the ground state has, and how the entanglementforms the non-perturbative ground state.We are interested in the quantum Rabi model’s case. We now tune the frequencies, ω a and ω c , as in the above. As shown in Ref.[19], the free part of the quantum Rabi Hamilto-nian H ( ω, , ω,
0) is the Witten Laplacian [26] which describes the N = 2 SUSY quantummechanics, and the renormalized Rabi Hamiltonian H ( ω, , ω, g) + ~ g /ω converges to theHamiltonian describing the spontaneously N = 2 SUSY breaking in the weak operatortopology. Here, the energy − ~ g /ω is the binding energy of the van Hove Hamiltonians H cavity ( ω ) ± ~ g( a + a † ) appearing in neutral static-source theory. Actually, the individualvan Hove Hamiltonian sits in each diagonal element of the total Hamiltonian H total ⌈ ε =0 ; it is .5 1 1.5 2 2.5-4-2246 Figure 1: The energy revel of the quantum Rabi Hamiltonian H ( ω, , ω, g). The unit of thetransverse axis is g /ω , and the unit of the longitudinal axis is ~ ω . referred to as an energy renormalization (see Ch.9 of Ref.[9]). The renormalized Rabi Hamil-tonian recovers the spin-chiral symmetry as the coupling strength grows lager and larger, andthen, the spin-chirality causes spontaneously SUSY breaking. For more details, see Ref.[19].Here, we improve the quantum Rabi Hamiltonians’ convergence; it is actually in the normresolvent sense (see Definition on p.284 of Ref.[23]). That is, as proved below, we can obtainthe following convergence:lim g →∞ k (cid:0) H ( ω a , , ω c , g) + ~ g /ω c − i (cid:1) − − (cid:0) U ∗ g H cavity ( ω c ) U g − i (cid:1) − k op = 0 , (2.2)where U g is the Bogoliubov transformation defined by the displacement operator, U g := e g σ x ( a † − a ) /ω c . We recall the fact that the quantum Rabi Hamiltonian H ( ω a , , ω c , g) hasthe parity symmetry, i.e., [ H ( ω a , , ω c , g) , Π] = 0 for the parity operator Π = ( − a † a σ z ,and the ground state | E ε =00 i of the quantum Rabi Hamiltonian has to be an eigenstate ofthe parity operator, of which eigenvalue is −
1, i.e., Π | E ε =00 i = −| E ε =00 i [17, 18]. Thus, thisconvergence say that for sufficiently large coupling strength, g ≫
1, the ground state | E ε =00 i of the quantum Rabi Hamiltonian H ( ω a , , ω c , g) is well approximated by the ground state U ∗ g | ↓ , i of the Hamiltonian U ∗ g H cavity ( ω c ) U g . Namely, the ground state U xz | E ε =00 i of theoriginal total Hamiltonian H total is well approximated by the coherent states, also called thedisplaced vacuum states [28], as U xz | E ε =00 i ≈ ˆ D ( − g /ω c ) | ↑ , i + ˆ D (g /ω c ) | ↓ , i√ e − g /ω √ ∞ X n =0 (cid:26) ( − g /ω c ) n √ n ! | ↑ , n i + (g /ω c ) n √ n ! | ↓ , n i (cid:27) (2.3)for the displacement operator ˆ D ( ∓ g /ω c ) = e ∓ g( a † − a ) /ω c . This improvement of convergencealso guarantees the convergence of energy spectra as g → ∞ , and our numerical analysis in,for instance, Fig.1. It is worth to noting that Braak has given an analytical expression ofthe eigenvalues of the quantum Rabi Hamiltonian [4] though it had been a long outstandingproblem.Here, we give a proof of the improvement of the convergence. Denote H cavity ( ω c ) by H cav for simplicity. We have U g aU ∗ g = a − g σ x /ω c . Then, we can define a Hamiltonian e H renRabi by e H renRabi := U g (cid:18) H Rabi ( ω a , , ω c , g) + ~ g ω c (cid:19) U ∗ g = H cav + W g , (2.4)where W g = ( ~ ω a / U g σ z U ∗ g . Set the operator R as R := ( e H renRabi − i ) − − ( H cav − i ) − , anduse the 2nd resolvent equation. Then, we have the equation, R = ( H cav + W g − i ) − ( − W g ) ( H cav − i ) − , (2.5) hich implies the equation,( H cav + W g − i ) − = ( H cav − i ) − + ( H cav + W g − i ) − ( − W g ) ( H cav − i ) − . (2.6)Inserting Eq.(2.6) into the term, ( H cavh + W g − i ) − , of the right hand side of Eq.(2.5), wereach the equation, R = − ( H cav − i ) − W g ( H cav − i ) − + ( H cav + W g − i ) − W g ( H cav − i ) − W g ( H cav − i ) − , which implies the inequality, k R k op ≤ k ( H cav − i ) − W g ( H cav − i ) − k op + k ( H cav + W g − i ) − k op k W g k op k ( H cav − i ) − W g ( H cav − i ) − k op . Since k ( H cav + W g − i ) − k op ≤ k W g k op = ~ ω a /
2, the above inequality leads to theinequality k R k op ≤ (cid:18) ~ ω a (cid:19) k ( H cav − i ) − W g ( H cav − i ) − k op . We remember that the operator W g converges to 0 in the weak operator topology [19], and theresolvent ( H cav − i ) − is compact. Our theorem proved in A says that lim g →∞ k (1 + ( ~ ω a / k ( H cav − i ) − W g ( H cav − i ) − k op =0, and therefore,lim g →∞ k (cid:0) H Rabi ( ω a , ω c , g) + ~ g /ω c − i (cid:1) − − (cid:0) U ∗ g H cav U g − i (cid:1) − k op = lim g →∞ (cid:13)(cid:13)(cid:13)(cid:0) U g H Rabi ( ω a , ω c , g) U ∗ g + ~ g /ω c − i (cid:1) − − ( H cav − i ) − (cid:13)(cid:13)(cid:13) op = 0 . Based on this convergence and its proof, we think that the leading term of the quantumRabi Hamiltonian is H cavity ( ω c ) + ~ g( σ + a + σ − a † ) for ultra- or deep-strong coupling regime.Namely, we regard H atom ( ω a ,
0) as a small perturbation.
In this section we will consider the dressed photon in a ground state of the total Hamiltonianwith the A -term, the quadratic coupling, defined by H A := H ( ω a , ε, ω c , g) + ~ C g g (cid:0) a + a † (cid:1) , (3.1)where C g is a function of the coupling strength g. Judging from reference to Eq.(15) ofRef.[28], the coupling strength g and the function C g are given byg = M ( I p ) I p I zpf and C g = I p I zpf I − I , where I p and I zpf are respectively the persistent-current and the zero-point-fluctuation cur-rent with I zpf ≪ I p , I cM is the critical current of a single effective Josephson junction, and M ( I p ) the mutual inductance between the flux qubit and the LC oscillator: M ( I p ) = Φ π q I − I . We can rewrite C g as C g = I p I zpf I − I = 2 π Φ q I − I g or C g = I p I zpf I − I = (cid:18) π Φ (cid:19) I p I zpf g . .5 1 1.5 2 2.5-4-2246 Figure 2: The energy revel of the total Rabi Hamiltonian H A . The left is for the case C g =0 . × g, and the right for the case C g = 0 . × g . The unit of the transverse axis is g /ω , and theunit of the longitudinal axis is ~ ω . Thus, we may approximate it at C g = C g ℓ , ℓ = 1 ,
2, with a constant C , where we have tochange the dimension of C to meet each case. In the latter case, C ≪ ≫ I p I zpf to make g large enough. The numerical computation of the energylevel of the total Hamiltonian H A is in Fig.2.We denote by the eigenstates | E ν i , ν = 0 , , , · · · , of the total Hamiltonian H A , and by E ν the corresponding eigenenergies with E ≤ E ≤ E ≤ · · · . Return the representationused by Yoshihara et al . [28], U xz H A U ∗ xz = H total + ~ C g g( a + a † ) . (3.2)Then, we have the matrix representation, U xz H A U ∗ xz = (cid:18) H + A − ~ ε/ − ~ ω a / − ~ ω a / H − A + ~ ε/ (cid:19) , where H ± A = H cavity ( ω c ) ± ~ g (cid:0) a + a † (cid:1) + ~ C g g (cid:0) a + a † (cid:1) . In other words, following the representation in Ref.[6], we define the annihilation operator α and creation operator α † by α ♯ := σ x a ♯ . Then, we have the matrix-valued CCR, [ α, α † ] = 1.We denote by Π ± the orthogonal projection on the eigenspace of the parity operator Πwith the corresponding eigenvalue ±
1. Then, we have the relations, Π = − Π − + Π +1 and1 = Π − + Π +1 . Using these relations, we have the well-known representation, H A = H − Π − ⊕ H + Π +1 − ~ ε σ x , (3.3)where H − and H + are the photon Hamiltonians given by H ± = ~ ω c (cid:18) α † α + 12 (cid:19) + ~ g( α + α † ) + ~ C g g( α + α † ) ± ~ ω a − α † α . (3.4)We regard the term, ~ ω c (cid:0) α † α + 1 / (cid:1) + ~ g( α + α † ) + ~ C g g( α + α † ) , with the self-interactionas the leading term, and the terms, ± ( ~ ω a / − α † α and − ( ~ ε/ σ x , as its perturbation asg ≫ N bare0 = h E | a † a | E i = h E | α † α | E i s for the bare photons. Thus, from the point of view of the photon field with the uncountablymany modes, the ground-state expectation N bare0 includes the number of virtual photons aswell as that of non-virtual photons. Here ‘virtual photon’ is a technical term in the theoryof elementary particle, and ‘non-virtual’ is its antonym of the virtual though some peopleuse ‘real’ instead of non-virtual. The virtual photon causes the Coulomb force in quantumelectrodynamics, though it cannot directly be observed in any experiment. Thus, we shouldinvestigate the effect coming from the fluctuation caused by bare photons.Let ∆Φ be the fluctuation of the bare-photon field Φ = ( a + a † ) / √ ω c at the groundstate | E i , i.e., ∆Φ = p h E | (Φ − h E | Φ | E i ) | E i . We can show the relation,(∆Φ) ≤ N bare0 + 1 ω c . (3.5)In addition, according to the way in Eq.(12.24) of Ref.[9], we can obtain the following lowerbound in the case C g = C g ℓ , ℓ = 1 ,
2, with a constant
C > N bare0 ≥ √ ω c p ω c + 4 C g ℓ +1 + p ω c + 4 C g ℓ +1 √ ω c − ! − ǫ (g) , (3.6)where 0 < ǫ (g) → (cid:26) / (8 Cω c ) ℓ = 1 , ℓ = 2 , as g → ∞ . We will give the proofs of Eqs.(3.5) and (3.6) in B. Thus, we have to avoid the increase inthe number of bare photons which tends to divergence. Also, we have to mind and cope withthe effect coming from the quadratic coupling. Therefore, we will follow the pair theory [9]to consider the physical state. The pair theory is primarily to expose non-virtual photons tous by removing virtual photons from the ground state. H A In this part, we mathematically study the Hopfield-Bogoliubov transformation to cope withthe quadratic coupling.Following Eqs.(12.17) and (12.19) of Ref.[9], the annihilation and creation operators, b and b † , for physical photons are introduced so that the relations, a = M b + M b † and a † = M b + M b † , (3.7)are satisfied, where M = 12 (cid:26)r ω c ω g + r ω g ω c (cid:27) and M = 12 (cid:26)r ω c ω g − r ω g ω c (cid:27) . Then, we can prove that for arbitrary ω c , g , C g there is a unitary operator U such that U ∗ H A U = H ( ω a , ε, ω g , e g) , (3.8)where ω g = q ω + 4 C g g ω c and e g = g r ω c ω g . This unitary operator U is the Hopfield-Bogoliubov transformation [20, 21] used in Ref. [28].We here note that this Hopfield-Bogoliubov transformation becomes unitary without anyrestriction. The cavity frequency and the coupling strength are respectively renormalized as ω g and e g so that they include the effect coming from the A -term. In particular, the term,2 p C g g ω c , in the renormalized cavity frequency ω g plays a role of a mass of photon (cf. § H ( ω a , ε, ω g , e g) the renormalized Hamiltonian with A -term effect. sing this unitarily-equivalent representation, in the same way as getting Eq.(1.18) ofRef.[12], we can estimate the ground state energy E ( H A ) of the Hamiltonian H A as − ~ p ω + ε + ~ ω g − ~ e g ω g ≤ E ( H A ) ≤ − ~ e − e g /ω g2 p ω + ε + ~ ω g − ~ e g ω g . (3.9)We can show how the Hopfield-Bogoliubov transformation is obtained as a unitary opera-tor. We are now considering just one-mode photon, not uncountably many modes light. Thus,we only have to consider the Schr¨odinger equation, in particular, the Schr¨odinger operatorsin our case instead of considering the field equation in pair theory: Return the representation U xz H A U ∗ xz in Eq.(3.2) again. We recall that the unitary operator U xz is only for the spin,and our Hopfield-Bogoliubov transformation U only for photon. We can represent the anni-hilation operator a and the creation operator a † of bare photons using the position operator x and the momentum operator p as a = r mω c ~ x + i r m ~ ω c p and a † = r mω c ~ x − i r m ~ ω c p. (3.10)Use these representations, and rewrite the Hamiltonians H ± A as H ± A = 12 m p + mω x ± ~ g r mω c ~ x + 2 mC g g ω c x = 12 m p + mω x ± ~ e g r mω g ~ x. We denote by ϕ n the normalized eigenstates of (1 / m ) p + ( mω / x , and by ψ n thenormalized eigenstates of (1 / m ) p + ( mω / x for n = 0 , , , · · · : ϕ n = √ w c γ n H n ( w c x ) e − ( w c x ) / and ψ n = √ w g γ n H n ( w g x ) e − ( w g x ) / , where H n ( x ) are the Hermite polynomials of variable x , γ n = π − / (2 n n !) − / , and w ♯ = p mω ♯ / ~ , ♯ = c , g. Since the sets, { ϕ n } ∞ n =0 and { ψ n } ∞ n =0 , are respectively a complete or-thonormal basis of the function space L ( R ), we can make a unitary operator U by thecorrespondence, U ϕ n = ψ n . We here note the relations, aϕ = 0 and aϕ n = √ nϕ n − for n = 1 , , · · · .Introducing the annihilation and creation operators, b and b † , for physical photons by b = r mω g ~ x + i s m ~ ω g p and b † = r mω g ~ x − i s m ~ ω g p, (3.11)we have the relations, bψ = 0 and bψ n = √ nψ n − for n = 1 , , · · · . Meanwhile, the equations, U aϕ = 0 and U aϕ n = √ nU ϕ n − , lead the equations, ( U aU ∗ ) ψ = 0 and ( U aU ∗ ) ψ n = √ nψ n − , which implies the relation U aU ∗ = b . Taking its conjugate, we have the relation, U a † U ∗ = b † . Then, using Eqs.(3.10) and (3.11), we easily obtain the relations, U aU ∗ = b = 12 ( c + c ) a + 12 ( c − c ) a † ,U a † U ∗ = b † = 12 ( c − c ) a + 12 ( c + c ) a † , (3.12)where c = p ω g /ω c and c = p ω c /ω g . It is easy to show Eqs. (3.7) and (3.8) with the helpof Eqs.(3.12). We investigate how the ground state has some physical photons that should primarily beemitted to the outside of the matter. hanks to the unitarity of the Hopfield-Bogoliubov transformation Eq.(3.8), we can definethe normalized eigenstates of the renormalized total Hamiltonian H ( ω a , ε, ω g , e g) with the A -term effect by | E ren ν i := U ∗ | E ν i , and then, the eigenenergy E ren ν of each eigenstate | E ren ν i is,of course, E ν . The result in Ref.[18] says that E is always less than E , i.e., E < E . Weconsider the renormalized ground-state expectation of physical photon, N ren0 = h E ren0 | a † a | E ren0 i = h E | b † b | E i , and how the ground state of the renormalized total Hamiltonian with the A -term effect hasphysical photons.We first find its upper bound as N ren0 ≤ e g ω = ω c ω g = ω − / (cid:18) ω c g / + 4 C g g / (cid:19) − / . (3.13)Eq.(3.13), which is proved bellow, tells us that how many physical photons the ground statecan have at most, and that it depends on the function C g . Thus, we have N ren0 → → ∞ in the case C g = C g ℓ , ℓ = 1 ,
2, while N bare0 → ∞ .Meanwhile, we can give a lower bound of the renormalized ground-state expectation. Letus define a function L ren (g) of the coupling strength g by L ren (g) := e g ω g − vuut p ω + ε (cid:16) − e − e g /ω (cid:17) ω g . Then, as proved at the tail end of this subsection, for the parameters ω a , ε , ω c , and gsatisfying L ren (g) ≥
0, a lower bound is given by N ren0 ≥ L ren (g) . (3.14)Now we have the upper bound e g /ω g and the lower bound L (g) . Fig.3 shows theirnumerical analyses for C g = C g or C g , ω a = 0 . ω c = 0 .
75. According to these numericalanalyses, we realize that the ground-state expectation N ren0 has a gently-sloping peak whenthe constant C is sufficiently small, though the peak disappears as the constant is large.That is, the hight of the peak is tall for a small C , though it is low for a large C . The tallpeak tells us that there is a chance so that the ground state can have a physical photon.This situation reminds us of that of the Hepp-Lieb quantum phase transition [21] though wehave not yet understood whether this is caused by a quantum phase transition.Next our problem is to investigate where the photons that the ground state has comefrom, and how the ground state has them. We will hereafter prove the following expressionof the renormalized ground-state expectation of physical photon. For an arbitrary couplingconstant g, we have N ren0 = |h e E ren0 | E ren0 i| e g ω + ~ e g ∞ X ν =1 |h e E ren ν | E ren0 i| ( E ν − E + ~ ω g ) (3.15)by using the pull-through formula. We note that we do not use the perturbation theory toprove this expression. For the operator-theoretical pull-through formula, see Ref.[14]. In thesame way, for the ground-state expectation of bare photon, we have the expression, N bare0 = |h e E | E i| g ω + ~ g ∞ X ν =1 |h e E ν | E i| ( E ν − E + ~ ω c ) . Eq.(3.15) implies the inequality, N ren0 ≤ |h e E ren0 | E ren0 i| e g ω + ~ e g ∞ X ν =1 ~ ω ( E ν − E + ~ ω g ) ( E ν − E ) . (3.16) upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.05 × g, ω a =0.1, ω c =0.75 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.05 × g , ω a =0.1, ω c =0.75 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.1 × g, ω a =0.1, ω c =0.75 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.1 × g , ω a =0.1, ω c =0.75 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.5 × g, ω a =0.1, ω c =0.75 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 upp e r (r e d ) a nd l o w e r ( b l u e ) bound s strength g of interactionC g =0.5 × g , ω a =0.1, ω c =0.75 Figure 3: The red solid line is the upper bound e g /ω , and the blue line is the lower bound L (g) . Here we set C g = C g or C g , ω a = 0 . ε = 0, ω c = 0 . ere, we note the followings before we prove Eqs.(3.15) and (3.16). The set of {| e E ren ν i} ∞ ν =0 is a complete orthonormal system of the Hilbert space σ x F = F by Theorem 4.3 of Ref.[16].Thus, there is an ν ∗ such that h e E ν ∗ | E i 6 = 0 since | E i 6 = 0. This fact gives a lower bound ofthe ground-state expectation of physical photon: ~ e g |h e E ren ν ∗ | E ren0 i| ( E ν ∗ − E + ~ ω g ) ≤ N ren0 . (3.17)In the case ε = 0, we have ν ∗ = 0 since we can show h e E ren0 | E ren0 i = 0 using the paritysymmetry, [ H ( ω a , , ω g , e g) , Π] = 0, for the parity operator Π. Eqs.(3.15) and (3.17) tellus that the ground state has at least a physical photon with the transition probability, |h e E ren ν ∗ | E ren0 i| , form the states | e E ren ν ∗ i on the other side of the mirror.We now prove Eqs.(3.15) and (3.16): We first give the so-called pull-through formula forthe quantum Rabi Hamiltonian. We note the equation,[ H ( ω a , ε, ω g , e g) − E , a ] | E ren0 i = ( H ( ω a , ε, ω g , e g) − E ) a | E ren0 i . Meanwhile, we compute the commutator in the left hand side of the above as[ H ( ω a , ε, ω g , e g) − E , a ] = − ~ ω g a − ~ e g σ x . Combining these equations, we obtain the pull-through formula: a | E ren0 i = − ~ e g ( H ( ω a , ε, ω g , e g) − E + ~ ω g ) − σ x | E ren0 i . (3.18)Using Eq.(3.18), we have the expression of the ground-state expectation of photon as N ren0 = ~ e g k ( H ( ω a , ε, ω g , e g) − E + ~ ω g ) − σ x | E ren0 ik F , (3.19)which implies the upper bound Eq.(3.13). Using the completeness of the set {| E ren ν i} ∞ ν =0 , wehave the expansion, σ x | E ren0 i = ∞ X ν =0 h E ren ν | σ x | E ren0 i| E ren ν i . (3.20)Inserting Eq.(3.20) into Eq.(3.19), we reach Eq.(3.15).To obtain Eq.(3.16), we have only to estimate the transition probability, |h e E ren ν | E ren0 i| .Define the spin-chiral transformed Hamiltonian e H ( ω a , ε, ω g , e g) by e H ( ω a , ε, ω g , e g) := σ x H ( ω a , ε, ω g , e g) σ x . Then, we have the equation, e H ( ω a , ε, ω g , e g) = H ( − ω a , ε, ω g , e g) . (3.21)Since | e E ren ν i is an eigenstate of the transformed Hamiltonian e H ( ω a , ε, ω g , e g) with the eigenen-ergy E ν , Eq.(3.21) leads to the following: E ν h e E ren ν | E ren0 i = h e E ren ν | H ( − ω a , ε, ω g , e g) | E ren0 i = h e E ren ν | H ( ω a , ε, ω g , e g) − ~ ω a σ z | E ren0 i = E h e E ren ν | E ren0 i − ~ ω a h e E ren ν | σ z | E ren0 i , which implies the equation, ( E ν − E ) h e E ren ν | E ren0 i = − ~ ω a h e E ren ν | σ z | E ren0 i , and thus,( E ν − E ) |h e E ren ν | E ren0 i| = ~ ω a |h e E ren ν | σ z | E ren0 i| . Therefore, we can obtain our desired estimate: |h e E ren ν | E ren0 i| ≤ ~ ω ( E ν − E ) . e now prove Eq.(3.14). By the variational principle, we have the inequality, E = h E ren0 | H ( ω a , ε, ω g , e g) | E ren0 i = h E ren0 | H atom ( ω a , ε ) | E ren0 i + h E ren0 | H cavity ( ω g ) | E ren0 i + h E ren0 | H int (g) | E ren0 i≥ − ~ p ω + ε + ~ ω g N ren0 + ~ ω g ~ e g h E ren0 | σ x ( a + a † ) | E ren0 i≥ − ~ p ω + ε + ~ ω g N ren0 + ~ ω g − ~ e g |h E ren0 | σ x ( a + a † ) | E ren0 i| . Using the Schwarz inequality, the fact that the creation operator a † is the adjoint operatorof the annihilation operator a , and Eq.(3.21) of Ref.[16], we have the inequality, |h E ren0 | σ x ( a + a † ) | E ren0 i| ≤ k a | E ren0 ik F = 2 p N ren0 . Combining above two inequalities, together with Eq.(3.9), leads to the inequality,2 ~ e g p N ren0 − ~ ω g N ren0 ≥ ~ F (g) , where F ren (g) = 12 p ω + ε (cid:16) e − e g /ω − (cid:17) + e g ω g . Set X and γ as X := p ω g N ren0 and γ := pe g /ω g . Then, dividing the both sides by ~ ,the above inequality is rewritten as 2 γX − X ≥ F ren (g). Thus, X has to satisfy γ − p γ − F ren (g) ≤ X ≤ γ + p γ − F ren (g), which implies s e g ω g − s e g ω g − F ren (g) ≤ p ω g N . Thus, we obtain our desired statement.
We consider the possibility that the ground state becomes an entangled state.Eq.(3.17) says that once the coupling strength is turned on (i.e., g > N ren0 > | s ∗ i and the Fock state | n ∗ i with a photon number n ∗ > h n ∗ , s ∗ | E ren0 i 6 = 0 . (3.22)More precisely, let us express | E ren0 i as | E ren0 i = P ∞ n =0 c ↑ n | ↑ , n i + P ∞ n =0 c ↓ n | ↓ , n i . We havethe expression of the renormalized ground-state expectation as N ren0 ≡ h E ren0 | a † a | E ren0 i = ∞ X n =1 n | c ↑ n | + ∞ X n =1 n | c ↓ n | . Since we know N ren0 >
0, we eventually obtain that there are s ∗ ∈ {↓ , ↑} and a naturalnumber n ∗ > |h n ∗ , s ∗ | E ren0 i| = | c s ∗ n ∗ | = 0. That is, we reach Eq.(3.22).We immediately realize that the quantum state | s ∗ , n ∗ i is not a ground state, and there-fore, there is another quantum state | s, n i so that |h n, s | E ren0 i| = 0. We try to seek it now.We define the orthogonal projection operators, P | ♯ i and P | i , for ♯ = ↑ , ↓ by P | ♯ i := h ♯ | i| ♯ i and P | i := h | i| i . We have the inequality, P | i ≥ − a † a , as shown in the proof of Lemma4.6 of Ref.[2] since a † a = P ∞ n =1 nP | n i and 1 = P ∞ n =0 P | n i for the orthogonal projectiondefined by P | n ii := h n | i| n i , we have |h , ↑ | E ren0 i| + |h , ↓ | E ren0 i| = h E ren0 | P |↑i ⊗ P | i | E ren0 i + h E ren0 | P |↓i ⊗ P | i | E ren0 i = h E ren0 | I ⊗ P | i | E ren0 i≥ h E ren0 | − a † a | E ren0 i = 1 − N ren0 . y this inequality, we can say that there is a spin-state | s i so that h , s | E ren0 i 6 = 0 if N ren0 < . (3.23)Eqs.(3.22) and (3.23) tell us that the expansion of the ground state | E ren0 i by the states | ♯, n i must include a superposition of the state | s , i and | s ∗ , n ∗ i when the ground-stateexpectation of photon satisfies 0 < N ren0 <
1. Therefore, at least a quantum states, | s , i and | s ∗ , n ∗ i , make an entanglement in the expansion of the ground state | E ren0 i then. As forthe possibility of the entangled ground state, see the following section. For the generalized quantum Rabi model with the A -term, we gave an upper bound anda lower bound of the renormalized ground-state expectation of physical photon. We showedhow the ground state has at least one physical photon, and the possibility of entanglement inthe ground state. As shown in Eq.(2.3) for the quantum Rabi model, its ground state is wellapproximated by the coherent state for sufficiently large coupling strength. Therefore, weconjecture that both the bare and physical ground states makes a highly entangled groundstate. We will show this conjecture and its leading term. Our results say that we may stablystore a physical photon in the ground state for a properly large coupling strength at least.Quoted from Ciuti’s lecture [7]: The ground state is the lowest energy state; the photons inthe ground state cannot escape the cavity. A Theorem for Improvement of Convergence
In this appendix, we state and prove the theorem that we used in § Theorem : Let A and B be compact operators on a Hilbert space H . If the operatorsequence W n is bounded (i.e., sup n k W n k op < ∞ ), and converges to as n → ∞ in the weakoperator topology, then the operator AW n B converges to in the uniform operator topology,i.e., in the operator norm. Proof : Let { ϕ j } ∞ j =1 be a complete orthonormal system of the Hilbert space H . ByTheorem VI.13 of Ref.[23], the finite rank operator B N := P Nj =1 ( ϕ j , · ) H Bϕ j converges tothe compact operator B as N → ∞ in the uniform operator topology. By the triangleinequality, we have k AW n B k op ≤ k AW n B N k op + k AW n ( B − B N ) k op ≤ k AW n B N k op + k A k op (sup n k W n k op ) k B − B N k op (A.1)for arbitrary natural number N . We can estimate the first term of the above as k AW n B N k op = sup k ψ k H =1 k AW n B N ψ k H = sup k ψ k H =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 ( ϕ j , ψ ) H AW n Bϕ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ≤ N X j =1 k AW n Bϕ j k H , where we used the Schwarz inequality, | ( ϕ j , ψ ) H | ≤ k ϕ j k H k ψ k H = 1. Since the operators W n converges to 0 in the weak operator topology, the vectors { W n Bϕ j } n are weakly conver-gent sequence. Since the operator A is compact, the vectors { AW n Bϕ j } n are a convergentsequence in the Hilbert space H (i.e., the sequence { AW n Bϕ j } n converges in the norm ofthe Hilbert space H ) by Theorem VI.11 of Ref.[23]. Thus, we havelim n →∞ k AW n B N k op = 0 for arbitrary natural number N . (A.2) ombining Eqs.(A.1) and (A.2) leads to the inequality,lim n →∞ k AW n B k op ≤ k A k op (sup n k W n k op ) k B − B N k op for arbitrary natural number N . Taking the limit N → ∞ in the above, we conclude ourtheorem. (cid:3) B Estimate of N bare0We easily have (∆Φ) ≤ ω c h E | ( a + a † ) | E i (B.1)since (∆Φ) = h E | Φ | E i − h E | Φ | E i , and h E | ( a + a † ) | E i = 2 N bare0 + 1 + h E | a + ( a † ) | E i . (B.2)Meanwhile, by Eqs.(3.21) and (3.22) of Ref.[16], we have k a | E ik F = p N bare0 and k a † | E ik F = p N bare0 + 1. With the help of the Schwarz inequality, these equations tell us |h E | ( a ♯ ) | E i| ≤ k a | E ik F k a † | E ik F ≤ q N bare0 q N bare0 + 1 ≤ N bare0 + 12 . (B.3)Combining Eqs.(B.1)–(B.3) leads to Eq.(3.5).Since b = U aU ∗ and b † = U a † U ∗ , we have the relations U ∗ aU = M a + M a † and U ∗ a † U = M a + M a † by Eq.(3.7). Using these relations, we can rewrite the ground-state expectation of bare photonas N bare0 = h E ren0 | ( U ∗ a † U )( U ∗ aU ) | E ren0 i = ( M + M ) N ren0 + M + M M h E ren0 | a + ( a † ) | E ren0 i . (B.4)In the same way as showing Eq.(B.3), we have the inequality, |h E ren0 | a + ( a † ) | E ren0 i| ≤ p N ren0 p N ren0 + 1. Since M > M <
0, we obtain the estimate from below byEqs.(3.13), (B.4), and (B.5): N bare0 ≥ ( M + M ) N ren0 + M + 2 M M p N ren0 p N ren0 + 1 ≥ M + 2 M M ( N ren0 + 1) ≥ M − ǫ (g) , (B.5)where ǫ (g) = 12 ( ω − ω ω c ω g ! (cid:18) e g ω g + 1 (cid:19)) = 12 ((cid:18) ( ω g / g ( ℓ +1) / ) − ( ω c / g ( ℓ +1) / ) ω c ( ω g / g ( ℓ +1) / ) (cid:19) (cid:18) ω c ( ω g / g ( ℓ +1) / ) g − ℓ + 1g ( ℓ +1) / (cid:19)) . Therefore, we finally obtain our desired lower estimate from Eq.(B.5).
C Numerical-Analysis Method for Eigenvalues of RabiHamiltonian
We recall that in the case ε = 0, the quantum Rabi Hamiltonian H Rabi ( ω a , ω c , g) := H ( ω a , , ω c , g) has the parity symmetry: [ H Rabi ( ω a , ω c , g) , Π] = 0 for the parity operator = σ z ( − a † a . Let us denote by F ± the eigenspace of the parity operator with the eigen-value, ±
1. We can decompose the state space F as F = F − ⊕ F +1 , and then, we canalso decompose the quantum Rabi Hamiltonian into the direct sum of the two self-adjointoperators H ± acting in the individual state space F ± in the same way as in Eq.(3.3), H Rabi ( ω a , ω c , g) = H − ⊕ H + , (C.1)where H ± is obtained by subtracting the A -term from H ± in Eq.(3.4). Following this de-composition, in the numerical analysis for the eigenvalue problem for the Rabi Hamilto-nian H Rabi ( ω a , ω c , g), we have only to seek the eigenvalues of H ± . We can give the matrix-representation H ± matrix of the operator H ± in the following. Using the complete orthonormalbasis, |↑ , i , |↓ , i , |↑ , i , |↓ , i , · · · , for H + , and using the complete orthonormal basis, |↓ , i , |↑ , i , |↓ , i , |↑ , i , · · · , for H − , they are H ± matrix := d ± ∆ √ g √ g d ∓ ∆ √ g √ g d ± ∆ √ g √ g d ∓ ∆ . . .. . . . . . , (C.2)where ∆ = ~ ω a / g = ~ g, and d n = ~ ω c ( n + 2 − ). With these matrix representations we canseek the eigenvalues of the operators H ± . To achieve this, we make good use of the eigenvalueproblem for the symmetric Jacobi matrix: For given complex sequences, a = { a n } ∞ n =0 and b = { b n } ∞ n =0 , the matrix J ( a, b ) := a b ¯ b a b ¯ b a b ¯ b a b ¯ b a . . .. . . . . . is called the symmetric Jacobi matrix. We assume the following two conditions:(A.1) a n → ∞ as n → ∞ .(A.2) ( | b n | + | b n − | )( | a n | + 1) − → n → ∞ .Thanks to these assumption, the operator consisting of the off-diagonal entries is the infinites-imally small with respect to the operator consisting of the diagonal entries, the Kato-Rellichtheorem [24] guarantees the operator J ( a, b ) to be self-adjoint acting in the Hilbert space ofthe square-summable sequences. For any natural number N we define a matrix J N ( a, b ) by J N ( a, b ) := J ( a, b ) (cid:12)(cid:12)(cid:12) b N =0 = a b ¯ b . . . . . .. . . . . . b N − ¯ b N − a N a N +1 b N +1 ¯ b N +1 . . . . . .. . . . . . . efining matrices K N ( a, b ) and R N ( a, b ) by K N ( a, b ) = a b ¯ b . . . . . .. . . a b N − ¯ b N − a N and R N ( a, b ) = a N +1 b N +1 ¯ b N +1 a N +2 b N +2 ¯ b N +1 a N +3 . . .. . . . . . , we can decompose the matrix J N ( a, b ) into the direct sum of these matrices: J N ( a, b ) = K N ( a, b ) ⊕ R N ( a, b ) . (C.3)Under the conditions (A.1) and (A.2), we will prove that the matrix J N ( a, b ) converges tothe symmetric Jacob matrix J ( a, b ) in the norm resolvent sense:lim N →∞ k ( J N ( a, b ) + i ) − − ( J ( a, b ) + i ) − k = 0 , (C.4)and that the n -th eigenvalue µ n ( K N ( a, b )) of the matrix K N ( a, b ) converges to the n -theigenvalue µ n ( J ( a, b )) of the symmetric Jacob matrix J ( a, b ): µ n ( J ( a, b )) = lim N →∞ µ n ( K N ( a, b )) . (C.5)We prove Eqs.(C.4) and (C.5) from now on. We set J , J N , T , K N , and R N as J := J ( a, b ), J N := J N ( a, b ), T := J ( a, K N := K N ( a, b ), and R N := R N ( a, b ) for simplicity. By the2nd resolvent equation we have k ( J N + i ) − − ( J + i ) − k ≤ k ( J − J N )( J + i ) − k≤ k ( J − J N )( T + i ) − k k ( T + i )( J + i ) − k . (C.6)We note that the assumptions (A.1) and (A.2) guarantees k ( T + i )( J + i ) − k < ∞ in the righthand side of the inequality Eq.(C.6). Meanwhile, since J − J N = b N ( | N + 1 ih N | + | N ih N + 1 | )by the definition, we reach the following inequality and the limit with the help of the condition(A.2): k ( J − J N )( T + i ) − k ≤ | b N | (cid:16) k| N + 1 ih N | ( T + i ) − k + k| N ih N + 1 | ( T + i ) − k (cid:17) = | b N | ( | a N + i | − + | a N +1 + i | − ) −→ N → ∞ . (C.7)Eqs.(C.6) and (C.7) lead to Eq.(C.4).Thanks to Theorem VIII.23 of Ref.[23], we have the convergence,lim N →∞ µ n ( J N ) = µ n ( J ) . (C.8)Denote by D N the diagonal matrix of which diagonal entries are a N +1 − | b N +1 | , a N +2 −| b N +1 | − | b N +2 | , a N +3 − | b N +2 | − | b N +3 | , a N +4 − | b N +3 | − | b N +4 | , · · · . Then, by the condition(A.2) we have the inequality and the limit, R N ≥ D N ≥ min n ≥ N +1 ( a n − | b n − | − | b n | ) −→ ∞ as N → ∞ . Thus, with this limit the mini-max principle (see Theorem XIII.1 of Ref.[25]) says thatlim N →∞ µ n ( R N ) = ∞ . (C.9)Remember that an eigenvalue of the matrix J N is an eigenvalue of the matrix K N or aneigenvalue of the matrix R N due to Eq.(C.3). Thus, Eq.(C.9) says that for sufficiently large N we have µ n ( J N ) = µ n ( K N ) . (C.10) qs.(C.8) and (C.10) lead Eq.(C.5). Acknowledgments
M. H. thanks Kouichi Semba, Tomoko Fuse, and Fumiki Yoshihara for the valuable dis-cussions on their experimental result, and Daniel Braak, I˜niago Egusquiza, Elliott Lieb,and Kae Nemoto for their useful comments. He also acknowledge J. S. M.’s hospitality atAarhus University, and Franco Nori’s at RIKEN. He also acknowledges the support fromJSPS Grant-in-Aid for Scientific Researches (B) 26310210 and (C) 26400117. M. H and J. S.M. acknowledges the support from the International Network Program of the Danish Agencyfor Science, Technology and Innovation.
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