Photon Berry phases, Instantons, Schrodinger Cats with oscillating parities and crossover from U(1) to Z_{2} limit in cavity QED systems

# Photon Berry phases, Instantons, Schrodinger Cats with oscillating parities and crossover from U(1) to Z2 limit in cavity QED systems

Yu Yi-Xiang, Jinwu Ye , W.M. Liu and CunLin Zhang Department of Physics and Astronomy, Mississippi State University, P. O. Box 5167, Mississippi State, MS, 39762
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Key Laboratory of Terahertz Optoelectronics, Ministry of Education, Department of Physics, Capital Normal University, Beijing 100048, China
August 25, 2019
###### Abstract

The four standard quantum optics models such as Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model were proposed by the old generation of great physicists many decades ago. Despite their relative simple forms and many previous theoretical works, their solutions at a finite , especially inside the superradiant regime, remain unknown. In this work, we address this outstanding problem by using the expansion and exact diagonization to study the Dicke model at a finite . This model includes the four standard quantum optics model as its various special limits. The expansions is complementary to the strong coupling expansion used by the authors in arXiv:1512.08581 to study the same model in its dual representation. We identify 3 regimes of the system’s energy levels: the normal, and quantum tunneling (QT) regime. The system’s energy levels are grouped into doublets which consist of scattering states and Schrodinger Cats with even ( e ) and odd ( o ) parities in the and quantum tunneling (QT) regime respectively. In the QT regime, by the WKB method, we find the emergencies of bound states one by one as the interaction strength increases, then investigate a new class of quantum tunneling processes through the instantons between the two bound states in the compact photon phase. It is the Berry phase interference effects in the instanton tunneling event which leads to Schrodinger Cats oscillating with even and odd parities in both ground and higher energy bound states. We map out the energy level evolution from the to the QT regime and also discuss some duality relations between the energy levels in the two regimes. We also compute the photon correlation functions, squeezing spectrum, number correlation functions in both regimes which can be measured by various experimental techniques. The combinations of the results achieved here by expansion and those in arXiv:1512.08581 by strong coupling method lead to rather complete understandings of the Dicke model at a finite and any anisotropy parameter . Experimental realizations and detections are presented. Connections with past works and future perspectives are also discussed.

## I Introduction

Quantum optics is a subject to describe the atom-photon interactions walls ; scully . The history of quantum optics can be best followed by looking at the evolution of quantum optics models to study such interactions. In the Rabi modelrabi , a single mode photon interacts with a two level atom with equal rotating wave (RW) and counter rotating wave (CRW) strength. To study possible many body effects such as ”optical bombs”, a single two level atoms in the Rabi model was extended to an assembly of two level atoms in the Dicke model dicke . When the coupling strength is well below the transition frequency, the CRW term in the Rabi model is effectively much smaller than that of RW, so it was dropped in the Jaynes-Cummings ( JC ) model jc . Similar to the generalization from the Rabi to the Dicke model, the single two level atom in the JC model was extended to an assembly of two level atoms in the Tavis-Cummings (TC) model tc .

The importance of the 4 standard quantum optics model at a finite in quantum and non-linear optics ranks the same as the bosonic or fermionic Hubbard models and Heisenberg models in strongly correlated electron systems and the Ising models in Statistical mechanics aue ; sachdev . There have been extensive theoretical investigations on the solutions of the four standard quantum optics model. Most of the theoretical works focused on the thermodynamic limit . The TC model was studied at in dicke1 ; popov ; zero ; staircase . A normal to a superradiant phase transition was found and the zero mode due to the broken symmetry identified in the superradiant phase. The Dicke model at was investigated in chaos . A superradiant phase transition with the broken symmetry was found and two gapped modes due to the broken symmetry identified in the superradiant phase. However, there were only very limited works at a finite . The Exact Diagonization (ED) in chaos shows that the level statistics in a give parity sector changes from the Poissonian distribution in the normal phase to the Wigner-Dyson in the superradiant phase at any finite . For the Dicke model, the ground state photon number at the normal to the superradiant quantum critical point QCP was found qcphoton ; china to scale as which is a direct consequence of finite size scaling near a QCP with infinite coordination numbers infinite ; extension . There are also formally ”exact” Bethe Ansatz-like solution for the integrable TC model at a finite betheu1 . Recently, a formal ”exact” solution was found even for the non-integrable Dicke model at ( Rabi model )rabisol . Unfortunately, these ”exact” solutions are essentially useless in extracting any physical phenomena betheu1 ; rabisol .

It is convenient to classify the four well known quantum optics models from a simple symmetry point of view: the TC and Dicke model as and Dicke model respectively, while JC and Rabi model are just as the version of the two berryphase ; gold ; comment . In fact, as stressed in gold , there are also two different representations on all of the 4 models: representation with independent two level atoms with a large Hilbert space and spin representation with a smaller Hilbert space . The relations between the two representations were clarified in gold . The dramatic finite size effects such as the Berry phase effects, Goldstone and Higgs modes on Dicke model were thoroughly discussed in both representation in berryphase by expansion and spin- representation gold by representation. Remarkably, we find nearly perfect agreements between the results achieved by and and the ED studies even when gets to its lowest value . The effects of a small CRW term near the Dicke model limit was also studied in gold . In view of the tremendous success of the and expansion in studying many strongly correlated electron systems largen ; strongc , particularly in the Dicke model achieved in berryphase ; gold ; comment , it is natural to apply them to study the Dicke model as originally planned in berryphase .

Due to recent tremendous advances in technologies, the 4 standard quantum optics models were successfully achieved in at least two experimental systems (1) with a BEC of atoms inside an ultrahigh-finesse optical cavity qedbec1 ; qedbec2 ; orbitalt ; orbital ; switch and (2) superconducting qubits inside a microwave circuit cavity qubitweak ; ultra1 ; ultra2 ; qubitstrong or quantum dots inside a semi-conductor microcavity dots . The superradiant phase in the Dicke model was also realized in system (1) with the help of transverse pumping ( Fig.1a ) orbitalt ; orbital ; switch . It could also be realized ”spontaneously ” in the system (2) without external pumping gold . Indeed, by enhancing the inductive coupling of a flux qubit to a transmission line resonator, a remarkable ultra-strong coupling with individual was realized in a circuit QED system qubitstrong . However, in such a ultra-strong coupling regime, the system is described well neither by the TC model nor the Dicke model, but a combination of the two Eqn.1 with unequal RW and CRW strength dubbed as Dicke model in gold . It was also proposed in gprime1 that in the thermal or cold atom experiments, the strengths of and can be tuned separately by using circularly polarized pump beams in a ring cavity. Indeed, based on the scheme, a recent experiment expggprime realized the Dicke model with continuously tunable and . As argued in gold ; berryphase , with only a few qubits embedded in system (2), the finite size effects may become important and experimentally observable. With the recent advances of manipulating a few to a few hundreds of cold atoms inside an optical cavity in system (1) fewboson ; fewfermion , the finite size effects may also become important and experimentally observable in near future experiments on system (1). As advocated in gold , the Hamiltonian Eqn.1 with independent and is the generic Hamiltonian describing various experimental systems under the two atomic levels and a single photon mode approximation. In gold , by the expansion, we focused on the Dicke model Eqn.1 near the limit ( namely, with a small anisotropy parameter and not too far from the critical strength ) at any finite . In a very recent preprint strongED , by the strong coupling expansion and the ED, the authors studied the Dicke model in its dual presentation starting from from the limit . Here, by the expansion and ED, we will study the Dicke model Eqn.1 starting from from the limit which is complementary to the strong coupling expansion in strongED . The combinations of both approaches will lead to rather complete understandings of the Dicke model Eqn.1 in the full range of .

In this paper, we study novel quantum phenomena in the Dicke model Eqn.1 in its spin representation at a finite , any interaction strength and anisotropy parameter by the expansion gold and the ED chaos ; gold . As a fixed , as the increases, we identify 3 crossover regimes: the normal, and the quantum tunneling (QT) regime Fig.2. The super-radiant regime at splits into the two regimes at a finite : the and quantum tunneling (QT) regime. In the regime, we perform a (non-)degenerate perturbation to evaluate the energy spectrum. It is the Berry phase which leads to the level crossings between the even and odd parity, therefore the alternating parities on the ground state and excited states. In the QT regime, by the WKB method, we find the emergencies of bound states one by one as the interaction strength increases, then investigate a new class of quantum tunneling processes through the instantons between the two bound states in the compact photon phase. It is the Berry phase interference effects in the instanton tunneling event which leads to Schrodinger Cats oscillating with even and odd parities in both ground and higher energy bound states. We map out the energy level evolution from the to the QT regime. In the regime, the doublets consist of scattering states organized as ( or ) pattern. While in the QT regime, the doublets consist of bound states ( or Schrodinger Cats ) organized as ( or ) pattern. There are also some sort of duality relations between the two regimes in both Hamiltonian and spectrum. We compute the photon correlation functions, squeezing spectrum and number correlation functions in both regimes which can be detected by Fluorescence spectrum, phase sensitive homodyne detection and Hanbury-Brown-Twiss (HBT) type of experiments respectively exciton . When comparing the results with those achieved from the strong coupling expansion in strongED , we find nearly perfect agreements among the expansion, the strong coupling expansion and the ED not only in the QT regime, but also in the regime not too close to the QCP at . The combination of the three methods lead to rather complete physical pictures in the whole crossover regimes from the Dicke to the Dicke model in Fig.2. Experimental realizations, especially the preparations and detections of the Schrodinger Cats in both experimental systems are discussed. Connections with the previous works are speculated and future perspectives are outlined.

## Ii 1/j expansion in the super-radiant phase.

The Dicke modelgold ; comment is described by:

 HU(1)/Z2 = ωaa†a+ωbJz+g√2J(a†J−+aJ+) (1) + g′√2J(a†J++aJ−)

where the are the cavity photon frequency and the energy difference of the two atomic levels respectively, the is the collective photon-atom rotating wave (RW) coupling. The is the counter-rotating wave (CRW) term. We fix their ratio to be . If , Eqn.1 reduces to the Dicke model berryphase ; gold with the symmetry leading to the conserved quantity . The CRW term breaks the to the symmetry with the conserved parity operator . If , it becomes the Dicke model chaos ; qcphoton .

Following gold , inside the super-radiant phase, it is convenient to write both the photon and atom in the polar coordinates . When performing the controlled expansion, we keep the terms to the order of and , but drop orders of or higher. We first minimize the ground state energy at the order and found the saddle point values of and :

 λa=g+g′ωa√j2(1−μ2),    λb=√j(1−μ) (2)

where . In the superradiant phase . In the normal phase , one gets back to the trivial solution .

Observe that (1) in the superradiant phase , , (2) it is convenient to introduce the modes: . (3) Defining the Berry phase in the sector berryphase as where is the closest integer to the , so . Due to the large gap in the , it is justified to drop the Berry phase in the sector. (4) after shifting , we reach the Hamiltonian to the order of :

 H[δρ±,θ±] = D2(δρ+−α)2+D−[δρ−+γ(δρ+−α)]2 (3) + 4ωaλ2a[11+βsin2θ−+β1+βsin2θ+]

where is the phase diffusion constant in the sector, with . The is the coupling between the and sector.

Note that the large expansion condition is . In the limit, it holds for any , but leads to a constraint on at a finite . In the superradiant regime, one can simply set in Eqn.3 which becomes a quadratic theory. It can be easily diagonalized and lead to one low energy gapped pseudo-Goldstone mode and a high energy gapped optical mode. Setting recovers the results for the Dicke model in the superradiant phase at in chaos .

If one neglects the quantum fluctuations of the mode, namely, by setting at its classical value , Eqn.3 is simplified to:

 H+[δρ+,θ+]=D2(δρ+−α)2+2ωaλ2a2β1+βsin2θ+ (4)

where . In the superradiant limit , one can identify the approximate atomic mode:

 ω2−0=4ωaλ2a2β1+βD=4E2Hβ1+β[(g+g′)4−g4c] (5)

which is nothing but the pseudo-Goldstone mode due to the CRW term.

Note that for small , the condition to reach the superradiant regime is more stringent than the large expansion condition .

By neglecting the quantum fluctuations of , the high energy optical mode in the sector can not be seen anymore in Eqn.4. The approximation may not give very precise numbers to physical quantities, but do lead to correct qualitative physical picture, especially the topological effects due to the Berry phase in all the physical quantities.

## Iii U(1) regime and the formation of consecutive bound states in quantum tunneling regime.

As the potential in the sector in Eqn.4 gets deeper and deeper, there are consecutive bound states formations at leading to the ”atomic” energy scale . The QT regime in Fig.2 is signatured by the first appearance of the bound state after which there are consecutive appearances of more bound states at higher energies ( Fig.3a,b ). The regime is the regime in Fig.2.

One can calculate all these by using the Bohr-Sommerfeld quantization condition for a smooth potential where and the is the bound state energy in Fig.3a,b. From Eqn.4, we can see the and the and are the two end points shown in Fig.3a. We find that the bound states emerge at

 ω−0D=(n+1/2)π2ℏ,   n=0,1,2,⋯ (6)

In Eqn.6, setting , one can see when , there is no bound state. This is the regime in Fig.2. Substituting the expression for the phase diffusion constant and the atomic mode in Eqn.5 leads to the condition for the regime:

 4λ2a√β1+β<π4 (7)

Note that for large expansion to apply, one only need to require , So for sufficiently small , there is an appreciable regime before the quantum tunneling (QT) regime in Fig.2.

In this regime, the second term in Eqn.4 breaks the symmetry to symmetry, the Goldstone mode at simply becomes a pseudo-Goldstone mode berryphase ; gold ; comment . But its effects at a finite is much more delicate to analyze. One can treat the second term in Eqn.4 perturbatively either by a non-degenerate at or degenerate perturbation expansion at . The total excitation is not conserved anymore and is replaced by the conserved parity , the energy levels are only grouped into even and odd parities in Fig.4. As shown in gold , at a given sector , at , a first order degenerate perturbation at leads to the maximum splitting at in Fig.4b:

 Δm=±1,U(1)(α=0)=D2−ωaλ2a22β1+β (8)

Using Eqn.5, one can rewrite . For general , one needs order ( with the constraint in a given sector ) degenerate perturbation calculation to find the the maximum splitting at between the and crossing in Fig.4b:

 Δm,U(1)(α=0)=D(m+12)−Rm+1−Rm (9)

where is the gap opening at the crossing in the limit Fig.4a. Setting recovers Eqn.8.

Note that the degenerate pair at the edge at has different parity, so will not be mixed in any order of perturbations gold . It is easy to compute the edge gap at by a non-degenerate perturbation. Obviously, the second term in Eqn.4 connects only , so the first order perturbation vanishes, one need to get to at least second order non-degenerate perturbation gold . There could also be a slight shift in the crossing point between the two opposite parities at

 Δm,U(1)(α=±1/2)=D(m+1)−S(m) (10)

where . It leads to at at the limit shown in Fig.4a. It is easy to see that at a given doublet the maximum gap at is smaller than the edge gap at , . But both are of the same order.

Nonetheless, the important phenomena of Goldstone and Higgs mode at the limit can still be observed in this regime after considering these degenerate and non-degenerate perturbations. Various photon correlation functions in this regime can be evaluated straightforwardly gold .

Now we follow the formation of the bound states just after the regime. When , namely , it just holds the first bound state with , . When , it holds the first bound state ( Fig.3a ) with energy where . It is easy to see .

When , namely , it just holds the second bound state with , . While the first bound state energy is given by . When , it holds two bound states ( Fig.3b ) with the energies:

 E1 = 2ωaλ2a2β1+βsin2θ1a,  ω−0DF(θ1a)=π4ℏ E2 = 2ωaλ2a2β1+βsin2θ2a,  ω−0DF(θ2a)=3π4ℏ (11)

where one can identify the first atomic energy in Fig.3b:

 ϵa=E2−E1=2ωaλ2a2β1+β(sin2θ2a−sin2θ1a) (12)

As expected is different than the in Eqn.5.

As , , while , so the left hand side of Eqn.6 diverges, there are infinite number of bound states shown in Fig.4e.

Note that because the bound state is either localized around or , so the Berry phase in Eqn.4 plays no roles, so can be dropped. However, as to be shown in the following section, it does play very important roles in the quantum tunneling process between the two bound states shown in Fig.3 and 5.

## Iv Quantum tunneling between the two bound states: Berry phase and instantons.

The instanton solution for a Sine-Gordon model was well known books . From Eqn.4, we can find the classical instanton solution connecting the two minima from or : where is the center of the instanton. Its asymptotic form as is . The corresponding classical instanton action is:

 S0=2ω−0D (13)

The instanton problems in the three well known systems (1) a double well potential (DWP) in a theory (2) periodic potential problem (PPP) (3) a particle on a circle (POC) are well documented in books . The tunneling problem in the present problem is related, but different than all the three systems in the following important ways:. (1) the potential in Eqn.4 is a periodic potential in . In this regard, it is different than the theory, but similar to PPP. (2) The is a compact angle confined in . In this regard, it is different than the PPP, but similar to POC. (3) There are two minima inside the range instead of just one. In this regard, it is different than the POC, but similar to the theory. So the present quantum tunneling problem is a new class one. Furthermore, it is also very important to consider the effects of Berry phase which change the action of instanton to , that of anti-instanton to ( Fig.5 ) where is the Berry phase in Eqn.4.

Taking into account the main differences of the present QT problem from the DWP, PPP and POC studied perviously books , especially the crucial effects of the Berry phase, we can evaluate the transition amplitude from to in Fig.5:

 ⟨π|e−Hτ|0⟩ = (ω−0πDℏ)1/2eω−0τ/2∑n1,n2(JKe−Sintτ)n1n1!(JKe−¯Sintτ)n2n2!δn1−n2,odd (14) = (ω−0πDℏ)1/2eω−0τ/212[e2JKτe−S0cosαπ−e−2JKτe−S0cosαπ]

where ( ) is sum over the instanton ( anti-instanton ), is given by the instanton action in Eqn.13 and is the ratio of two relevant determinants to remove the zero mode of the instantons due to its center listed above Eqn.13. It can be shown that where is extracted from the asymptotic form of the instanton solution in the limit listed above Eqn.13. Similarly, in finding the transition amplitude from back to , one only need to change to in the first line, consequently, the sign to the sign in the second line in Eqn.14.

The two transition amplitudes lead to the Schrodinger ” Cat ” state with even/odd parity:

 |e⟩0,SC = A√2(|θ+=0⟩+|θ+=π⟩), |o⟩0,SC = A√2(|θ+=0⟩−|θ+=π⟩), (15)

where the overlapping coefficient is . They have the energy with the splitting between them given by:

 Δ0(α) = 8ω−0(cosαπ)(ω−0πD)1/2e−2ω−0D (16) ∼ (cosαπ)√Ne−cN

where one can see that it is the Berry phase which leads to the oscillation of the gap and parity in Eqn.15. It vanishes at the two end points and reaches maximin at the middle . Note that the Berry phase is defined berryphase ; gold at a given sector . So Eqn.15 has a background parity . So there is an infinite number of oscillating parities in Eqn.15 as increases.

Because the Berry phase effects remain in the given sector, extending the results in Ref.high , we find the splitting in the -th excited bound states ( in Fig.3b ):

 Δn(α)=1n!(8ω−0D)nΔ0 (17)

where with the corresponding -the Schrodinger ”Cat ” state with even/odd parity and the energy :

 |e⟩n,SC = 1√2(|n⟩L+|n⟩R), |o⟩n,SC = 1√2(|n⟩L−|n⟩R), (18)

where is the th bound state in the left(right) well in Fig.3a,b. Putting and projecting it to the coordinate space at recovers Eqn.15 ( Note that the projection to does not work for is odd ). One can see the higher the bound state in the Fig.3, the larger the splitting is. The main difference than the energy level pattern in the regime is that all the bound states have the ( or ) pattern shown in Fig.4d,e and Fig.6b. This important observation is completely consistent with the results achieved from the strong coupling expansion in strongED after identifying . For example, as explained in strongED , there is an extra oscillating sign in Eqn.5 in strongED achieved from the strong coupling expansion, which is crucial to reconcile the results achieved from the two independent approaches !

## V Photon, squeezing and number correlation functions

Following the procedures for the Dicke model at in gold , treating the second term in Eqn.4 as a small perturbation when is small, using non-degenerate perturbation away from and degenerate perturbation near one can evaluate Photon, squeezing and number correlation functions in the regime outlined in Fig.6a. Here we focus on calculating these correlation functions in the QT regime outlined in Fig.6b.

The above physical pictures in the QT regime inspire us to decompose photon and atomic operators as:

 a=aLτz,     b=−bLτz (19)

where are confined to the left well in Fig.3 and the stand for the Left/Right quantum wells in Fig.3. It is the component which contains the important Berry phase effects and the quantum tunneling process between the Left and Right quantum well with the tunneling Hamiltonian . Note that the two states here is the two Schrodinger Cat states of the strongly interacting atom-photon system instead of the two levels of an atom in the original Dicke Hamiltonian Eqn.1. The two energies and should appear in the single photon correlation function ( which contain the magnitude , the phase and the Ising correlation functions as shown in Fig.2 ) with the corresponding spectral weights respectively. By using this decomposition, we will perform the expansion to evaluate all the relevant photon and atomic correlation functions.

Because the symmetry is broken in either left or right well in Fig.3, one can ignore the periodicity in and expand the atom and photon operators as:

 aL = λa+iλaθa+δρa2λa−λaθ2a2−(δρa)28λ3a+⋯ bL = λb+iλbθb+δρb2λb−λbθ2b2−(δρb)28λ3b+⋯ (20)

Using the Holstein-Primakoff (HP) representation of the angular momentum operator , one can evaluate the atomic spin correlation functions. Here, we focus on evaluating the photon correlation functions.

Using Eqn.4, 5, we can calculate the phase-phase, density-density and density-phase correlation functions in the imaginary time :

 ⟨θa(τ)θa(0)⟩ = D2ω−0e−ω−0τ, ⟨θ2a⟩=D2ω−0 ⟨δρa(τ)δρa(0)⟩ = ωaλ2a2ω−02β1+βe−ω−0τ, ⟨(δρa)2⟩=ωaλ2a2ω−02β1+β ⟨δρa(τ)θa(0)⟩ = −i4e−ω−0τ=−⟨δθa(τ)ρa(0)⟩ (21)

Because the is a phase confined on , we can define . For , we can set standing for the left/right well in Fig.3 and find the correlation function in the state:

 ⟨e|τz(τ)τz(0)|e⟩=e−Δlτ (22)

where is the splitting at state in Fig.3 and 6.

From the decomposition Eqn.19, 20, 21 and Eqn.22, one can evaluate the photon correlation functions:

 ⟨a(τ)a†(0)⟩=[λ2a−ω−08ωa(1+β2β)−ωa8ω−0(2β1+β)]e−Δ0τ +18[ω−0ωa(1+β2β)+ωaω−0(2β1+β)−2]e−(ω−0+(Δ1+Δ0)/2)τ (23)

where the first term containing the ground state splitting has the corresponding spectral weight , while the second term containing the atomic energy plus the average of the splittings at and in Fig.6 has the spectral weight .

Very similarly, one can evaluate the anomalous photon correlation functions:

 ⟨a(τ)a(0)⟩=[λ2a−ω−08ωa(1+β2β)−ωa8ω−0(2β1+β)]e−Δ0τ +18[ωaω−0(2β1+β)−ω−0ωa(1+β2β)]e−(ω−0+(Δ1+Δ0)/2)τ (24)

where the first term containing the ground state splitting has the same spectral weight as its counterpart in Eqn.23, while the second term containing the atomic energy plus the average of the splittings at and in Fig.6 has a different spectral weight , maybe even different sign than its counterpart in Eqn.23.

One can also compute the photon number correlation functions:

 ⟨n(τ)n(0)⟩−⟨n⟩2=ωaλ2a2ω−02β1+βe−(ω−0−(Δ1−Δ0)/2)τ (25)

where is the photon number at the ground state. It contains the atomic energy minus the difference of the splittings between and in Fig.6 and has a spectral weight .

So all the parameters of the cavity systems such as the doublet splittings and the atomic energy are encoded in the photon normal Eqn.23 and anomalous Green function Eqn.24 and photon number correlation function Eqn.25. They can be measured by photoluminescence, phase sensitive homodyne and Hanbury-Brown-Twiss ( HBT ) type of experiments exciton respectively.

## Vi Comparison with the results from the Exact Diagonization and the strong coupling expansion.

When in Eqn.1, it is not convenient to perform the ED in the coherent basis anymore used in china , so we did the ED in the orginal ( Fock ) basis. In the Fock space, the complete basis is where the is the number of photons and the is the Dicke states. In performing the ED, following chaos , one has to use a truncated basis in the photon sector where the is the maximum photon number in the artificially truncated Hilbert space. As long as the low energy levels in Fig.7 and Fig.8 are well below , then the energy levels should be very close to the exact results without the truncation ( namely, sending