Exact solvability and unified analytical treatments to qubit-oscillator system

Qing-Hu Chen( 陈 庆 虎 )

Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang Normal University, Jinhua 321004, China 　

& Department of Physics, Zhejiang University, Hangzhou 310027, China 　

《第六届全国冷原子物理和量子信息青年学者学术讨论会》2012 年 8 月 14 日 -18 日 , 浙江师范大学

本人 1966 年出生， 早就不属于青年学者 向青年朋友请教了

arXiv: 1204.3668, Phys. Rev. A, in pressarXiv: 1204.0953arXiv: 1203.2410

Prof. Ke-Lin Wang Department of Modern Physics, University of Science and Technology of China, Hefei 230026

Chen Wang (Ph. D student) Department of Physics, Zhejiang University, Hangzhou 310027

Dr. Yu-Yu Zhang (Former Ph. D student) Center for Modern Physics, Chongqing University ， Congqing 400044

Shu He ( MS student), Prof. Tao Liu School of Science, Southwest University of Science and Technology, Mianyang 621010

Prof. Shi-Yao ZhuBeijing Computational Science Research Center, Beijing 100084

Collaborators

Brief introduction to quantum Rabi model (QRM)

□ The interaction of two-level atom (qubit) with a bosonic mode

( )2 z xH a a g a a

ω is the resonant frequency of the cavity, Δ is the the transition frequency of the qubit, and g is the coupling strength, σx,z is usual Pauli matrix, a(a+) is the boson annihilation (creation) operator. δ=Δ- ω is the detuning.

quantum Rabi model (Cavity QED) qubit-oscillator system (Circuit QED)

In the fully quantum mechanical version

Analytically unsolvable !

Rabi, Phys. Rev. 49, 324 (1936); 51, 652 (1937).

( )2 zH a a g a a

( )g a a

RWA CRTs

□ Jaynes-Cummings (JC) model (1963) under the rotating-wave approximation (RWA) is analytically solvable. The counter rotating terms (CRTs) is omitted

The Rabi model (RM) describes the simplest interaction between light and matter.

Although this model has had an impressive impact on many fields of physics --- many physicists may be surprised to know that the quantum Rabi model has never been solved exactly. In other words, it has not been possible to write a closed-form, analytical solution for it

Outline

1. Exact solution for Qubit-Oscillator Systems (1) Numerically exact (2) Analytically exact (3) Applications No explicitly expression

2. Unified analytical treatments to qubit-oscillator systems explicitly expression but complicated

3. Concise first-order corrections to the RWA explicitly expression but very simple

( ) ( , , , , )nE E n g

|| 0,1, 2.

| 1n

nn

a nn

b n

□ In RWA, the n-th eigenstate is

The ground-state 0

0

2

| | 0,

E

1,

2,

2 2

1,

2,

2 2

1 1( )

2 21 1

( )2 2

, 4 ( 1)

sin ||

cos | 1

cos ||

sin | 1

2 1cos

4 ( 1)

n n

n n

n

nn

n

nn

n

n

n

E n R

E n R

R g n

n

n

n

n

g n

R g n

at resonance, δ=0

1,

2,

1,

2,

1( ) 1

21

( ) 12|1

|| 12

|1|

| 12

n

n

n

n

E n g n

E n g n

n

n

n

n

Part I, Exact solution to the Quantum Rabi model (QRM)

,0

,0

,0 ,0 / 2

,0 ,0 / 2

V e g g e

GS g

V GS e

e g

e g

The atom is excited by the operator

spontaneous emission to GS state

The emission spectrum has two peaks with equal height (the distance of the two peaks, 2g, is the vacuum Rabi splitting).

2g is the energy difference of the 1st and 2nd eigenstates

Vacuum Rabi splitting in the JC model

Wallraff et al., Nature 431, 162(2004).

Measured transmission spectrum showing the vacuum Rabi mode splitting

The collapses and revivals in the evolution of the atomic population inversion

2 / 20| (0) | | | 0 |ag e g

2| | | |n a a

22

| | | |( ) cos(2 1) ( )

!

n

rwan

p t e gt n F tn

Population inversion under RWA can be evaluated analytically

M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997

If initially in Photonic Fock state |e, n>This is the quantum Rabi oscillation.

2 2

2 2

( ) ||

( ) | 1

( ) cos 1

( ) sin 1

a t nt

b t n

a t gt n

b t gt n

If initially in photonic coherent state

collapses revivals

<σz(t)>

Strong coupling Qubit-Oscillator System

Deppe et al., Nature physics 4, 686(2008)

Circuit quantum electrodynamics (QED) system

Experiments: T. Niemczyk et al., Nature Physics 6, 772 (2010)Title: Beyond the Jaynes-Cummings model: circuit QED in the ultrastrong coupling regime

FIG. 1: Quantum circuit and experimental setup.

P. Forn-Diaz and J. E. Mooij et al., PRL 105, 237001 (2010). arXiv: 1005.1559.

Spectrum of the flux qubit coupled to the LC resonator.

Theory of Qubit-Oscillator Systems (Biased QRM) QHC, Tao Liu, and Kelin Wang, arXiv: 1007.1747, Chin. Phys. Lett. 29, 014208 (2012)

/ 2q z xH

ωq is the atomic Larmor frequency, ω is the cavity frequency.

The flux qubit behaves effectively as a two-level system

The model Hamiltonian can be expressed as

2 2 2 2(2 ) sin (2 )q p x q p xI I

(cos sin ) ( )2

qz x zH a a g a a

Δ is the tunnel coupling

g is the qubit-resonator coupling strength, enhanced by Josephson junction inductance

g/ω ~ 10-6

Cavity QED:

10-3 0.01

0.1

Circuit QED

Nature 431, 162 (2004). Nature Physics 4, 686 (2008) Nature Physics 6, 772 (2010).

,A a

B a

Transformation (ω=1) g

2

2

2

2

A A gH

B B g

0

01

tr

tr

N

n An

N n

n Bn

c n

d n

2

2

2

2

m mn n mn

m mn n mn

m g c D d Ec

m g d D c Ed

Ansatz for the wavefunction

Numerically exact solution to QRM for ε≠0 (δΦ≠0) QHC et al, arXiv: 1007.1747

21

2

0 0! !

0 0

n n

A A A

g ga

A a

A an

n n

e

2 2

12

!2 exp 2 (4 )

!

m

mn B A

n m n mB mA

D m n

mm n g g L g

n

Laguerre polynomial

◇ For strong coupling or highly excited states, much better than exact diagonalization in a-space

The optimum fitted parameters of the experimental results

□ The theoretical results are in good agreement with the experimental observations

6 6x

2 2 2 2

515 2 0.82 2 8.13 / 4.25

(2 ) sin (2 )

p r

q p x q p x

I nA g GHz GHz h GHz

I I

0x

( )2 x zH a a g a a

[ , ] 0H )212(

zaaie Parity opertor

The system is of even (+) or odd (-) parity. dn=±cn

The wavefunction is reduced to

Hamiltonian

S-equation

The level transition is only allowed between the even and odd parity

( )E ( )E

2 ( )

0( ) 0trN

m mn nnm g E c D c

0

0( 1)

tr

tr

N

n An

N nn Bn

c n

c n

Numerically exact solution to unbiased QRM: ε=0 QHC et al, PRA 82, 052306(2010)

□ Entanglement : highly nonlocal— shared among pairs of atoms, photons, electrons, etc., they may be remotely located and not interacting with each other.

□ Entanglement as a resource in new approaches to both computation and communication

Yu and Eberly Science 2009

First application to the entanglement dynamics

CAB in two identical JC atoms without RWA

(a) Bell state 1 for α=π/4(b) Bell state 2 for α=π/12

□ Initiated from Bell state 1, ESD appear in non-RWA; disappear in RWA □ No periodicity of entanglement evolution for large g

Effect of photonic number on ESD

Bell state 1

Bell state 2

□ CAB and Nph

opposite behavior

□ Nph suppress CAB

Possible origin of ESD

xzJC aagaaH )(2

The original JC model

aaiz

aaiJC ee z

)()2/12/(Parity 0],[ JCJCH

To solve it easily, we introduce

11

11

2

1V

zxJC aagaaVVHH )(2

aaixJC eVV

0],[ H

Our analytically exact solution to the Rabi model QHC, Tao Liu, Yu-Yu Zhang, and Ke-Lin Wang, EPL 96, 14003 (2011) arXiv: 1011.3280

Qing-Hu Chen, Kelin Wang, and Shaolong Wan, J. Phys.: Condens. Matter 6, 6599(1994)

One dimensional large Frolich polarons

Ntr=2 Ntr=3

The wavefunction

where

to find the solution for α(q)

Ntr=3

Ntr=2

0

0

( ) exp( ) 0

( ) exp( ) 0

tr

tr

N nnn

N nnn

c a a

c a a

The wavefunction

Eigenstate of the pairty:+1-> even parity- 1-> odd parity

)(EHSchroedinger equation

tr

tr

tr

tr

N

n

ann

N

n

ann

N

n

ann

N

n

ann

eac

eacE

eac

eac

aagaa

aagaa

0

0)(

0

0

0)(

0)(

0)(

0)(

)(2

2)(

mjm

m

j

jm

mmm cEcj

gcmcgcgm )(

011 !

)2()1(

2)1()()(

Analytical solution

Without normalization, we can set c0=1The linear term in a+ in the Fock space can be determined by eigenvalue of the pure coherent state exp(αa+ )|0>, we can set c1=0

Comparing with the coefficients

1 10

1 (2 )( ) ( ) ( 1) 1... 1

( 1) 2 2 !

jmm

m m m m j trj

c m c g c c m Nm g j

2/)( gEenergy

Recurrence equation

( ) exp( ) 0ma a

0

0

( ) exp( ) 0

( ) exp( ) 0

tr

tr

N nnn

N nnn

c a a

c a a

10

trNc

Analytical solution

Then we have (g, ∆,ωare given model parameters)

10

(2 )( ) ( ) ( ) ( 1) 0

2 2 !

tr

tr

tr tr tr

N jN

tr N N N jj

f N c g c cj

A polynomial equation for only one variable α, only real value of αis reasonable.

Zeros of f(α) give α

( )

0

0

/ 2

( ) exp( ) 0

( ) exp( ) 0

tr

tr

N nnn

N nnn

E g

c a a

c a a

Eeigenfunctions

Eigenenergies

( 0,1,2,... )i trc i N can be expressed by α,

First order approximation (Ntr=2)

2

0 2

1 41 1

2 (1 ) 2

gE

Irish PRL07: Generalized RWA,

ED: exact diagonalization in Bosonic Fock space

Ground-state energy

◇ Much better than GRWA method

y=f(α) Even parity, Ntr=59 odd parity, Ntr=60

Full solutions

convergence0

0

( ) exp( ) 0

( ) exp( ) 0

tr

tr

N nnn

N nnn

c a a

c a a

The JC model without the RWA can be mapped to a polynomial equation with a single variable.

Its solutions recover exactly all eigenvalues and eigenfunctions of the model for all coupling strengths and detunings.

In the past 80 years, it is analytically unsolvable.

Remarks on QHC et al, EPL 96, 14003 (2011) arXiv: 1011.3280

4 months later, what happen?

See also arXiv: 1103.2461

E. Solano, Physics 4, 68 (2011).

Supplemental Material to Braak PRL paper

use the representation of bosonic creation and anihilation operators in the Bargmannspace of analytical functions in a complex variable z

Zeros of the G-function givesall eigenenergies with parity ±1

transcendental function

R. Koc et al., J. Phys. A: Math. Gen. 35, 9425(2002).

A recent work by Braak has renewed the interest in the old problem of coupling a photon field to a single spin 1/2 state, using the Rabi model. The central statement of this work is that the eigenfunctions in Bargmann representation must be analytic functions in the entire complex plane. Based on this condition, a procedure is derived from the series expansion of the eigenstates which provides a recursive evaluation of the spectrum. …

In the following, it is shown that the use of the extra condition of analyticity of the eigenfunction in Bargmann representation is not necessary.

Travenec: PRA 85, 043805 (2012)

Nevertheless, there are disputes on whether the term exact solvability should be used, if the G functions are given only by some Taylor expansions with coefficients coming from a recurrence scheme. In my opinion, the word integrability should be left rather for models where a sufficient number of integrals of motion are known, which is not the case for Rabi models.

arXiv:1204.3856

Exact solvability of the quantum Rabi models within Bogoliubov operators QHC et al., Phys. Rev. A ( in press, 2012), see also arXiv:1204.3668

A proof of our forthcoming article

Qing-Hu Chen et al., Phys. Rev. A ( in press, 2012)

two Bogoliubov transformations

1. Re-derivation of Braak's solution in a physical way!

If both wavefunctions are the true eigen-function for a non-degenerate eigenstate with eigenvalue E, they should be in principle only different by a complex constant r

where

Final G- function

ε=0

Braak, PRL 2011

□ Within extended coherent states, a recent exact solution to the quantum Rabi model [Daniel Braak, Phys. Rev. Lett. 107, 100401(2011)] can be recovered in an alternative simpler and more physical way, without uses of any extra conditions.

Dear Qinghu,

Many thanks for your interesting paper on the derivation of the G-function for the generalized Rabi model. Your method is certainly simpler than my approach, which was based on symmetry considerations,......

The question is: how do you justify this condition of proportionality which in turn gives the spectrum? In the Bargmann space approach it is the condition of analyticity throughout the whole complex plane which determines whether a stateis an element of the Hilbert space or not - and this leads together with Z_2-symmetry to the G-function. What corresponds to the Bargmann condition in your approach? .

Best regards,

Danielthe extra condition is covered in the vacuum state in the spaceof the Bogoliubov operators. These vacuum states are well definedand known as the coherent states, so the present derivation is morephysical and simpler.

The Juddian solutions originate from the properties of degeneracy

□ Both Hilbert spaces in the two Bogoliubov operators are complete, if truncation is not done, the proportionality is justified naturally for non-degenerate states.

□ Koc et al in J. Phys. A: Math. Gen. 35, 9425(2002) have obtained isolatedexact solutions in the QRM, which are just the Juddian solutionswith doubly degenerate eigenvalues.

The degenerate eigenstates are excluded in principle in the solutions based on the proportionality. It naturally follows that the Juddian solutions are exceptional ones

is not analytic in x but has simple poles at x=0,1,2….

All exceptional eigenvalues have the form En=n-g2

the necessary and sufficient condition for the occurrence of this eigenvalue is fn(x)=0

Comparisons with our previous work I

0

0

tr

tr

N

n An

N

n Bn

c n

d n

□ Qing-Hu Chen, Tao Liu, Yuan Yang, and Kelin Wang, PRA 82, 052306(2010) Qing-Hu Chen, Lei Li, Tao Liu, and Kelin Wang, arXiv: 1007.1747

0

0

!

!

tr

tr

N

n An

N

n An

n e n

n f n

0

0

1 ! '

1 ! '

tr

tr

N n

n Bn

N n

n Bn

n f n

n e n

link coefficients in two ansatz of the wavefunction

◇ No essential differences, except that the avenues to obtain the basically same coefficients are different !

Braak, PRL 107, 100401(2011). His wavefunctions are unfold by us

10

(2 )( ) ( ) ( ) ( 1) 0

2 2 !

tr

tr

tr tr tr

N jN

tr N N N jj

f N c g c cj

( ) / 2E g

◇ zeros of the both functions defined through different power series can give the exact eigenvalues. Both are analytically exact solutions

Comparisons with our previous work II

□ Qing-Hu Chen, Tao Liu, Yu-Yu Zhang, and Ke-Lin Wang, EPL 96, 14003 (2011) arXiv:1011.3280

0

(2 )( ) 0

!

tr

tr

N j

N jj

f cj

Reduced to

◇ In our practical evaluation, it is not more difficult to locate the zeros for our function than those proposed by Braak, because the poles at x = n emerging in the latter are not present in our earlier solution.

Braak’s G function

Both are not analytical closed-form solutions!

□ Braak’s solution

the eigenvalues are given by the zeros of the above Heun functions, which can not be obtained without truncation in the power series.

□ Enrique Solano called on in viewpoint Physics 4, 68 (2011): An intense dialogue between mathematics and physics will be needed to describe and predict unprecedented physical phenomena, since they might be hidden in the quantum numbers associated with Braak’s integrability criterion and analytical expressions. Otherwise, the present achievement will remain but a mathematical monologue.

◇ the expansion can not be closed naturally like in the JC model under the RWA

0

0

!

!

tr

tr

N

n An

N

n An

n e n

n f n

◇ by our two theories, we can also describe any physical phenomena based on Braak’s solution. Until now, no exceptions!

Wolf, Kollar, and Braak, PRA 85, 053817 (2012), see also arXiv:1203.6039

Braak’s method is absolutely not the unique method to obtain their physical figures 4-7 By our two earlier methods, we can completely reproduce all their figures

It is funny that we have studied the dynamics much earlier, which is however still only on arXivYu-Yu Zhang, Qing-Hu Chen, Shi-Yao Zhu, arXiv: 1106.2191,Wolf, Kollar, and Braak, PRA 85, 053817 (2012), see also arXiv: 1203.6039

Ntr=36

□ Counter rotating effect on collapses and revivals

2 2/20| (0) | | | 0 | 10ag e g n

，

Population inversion i.e. <σz(t)>Evolution at differentg

(Yu-Yu Zhang, QHC, Shi-Yao Zhu, arXiv: 1106.2191 )

P(t) is periodic with 2π ，

g=2 ， periodicity τ0= 4gπ=25.132

the mismatched phase in cos function in the summation at other time , leading to collapses.

(Yu-Yu Zhang, QHC, Shi-Yao Zhu, arXiv: 1106.2191 )

Emission spectrum for g=0.1 and 0.8 at resonance

□ Instead of studying the evolution of < σx> , we studied the vacuum Rabi splitting

Different heights More than 2 peaks

No splitting

V|GS>

□ It is predicted that two quantum phenomena are fundamentally altered.

◇ In the VRS: different heights of peaks more than two peaks the splittings completely vanish in the strong coupling regime.

◇ The collapses and revivals disappear in the intermediate g but reappear periodically in the strong g different from aperiodic collapses and revivals in RWA.

Remarks for, Yu-Yu Zhang, QHC, Shi-Yao Zhu, arXiv: 1106.2191 

□ Braak et. al. discussed the similar dynamics in PRA 85, 053817 (2012) ( arXiv: 1203.6039) . ◇ Their results for the evolution of < σz > are not beyond ours! ◇ Instead of studying the evolution of < σx> as it is related the emission, we studied the vacuum Rabi splitting where new result is given and which is easy to be observed in experiments.

Exact solvability of the two-photon quantum Rabi models within Bogoliubov operators Qing-Hu Chen et al., Phys. Rev. A ( in press, 2012), see also arXiv:1204.3668

Bogoliubov transformation

Similarly, the G-function for two-photon QRM is obtained

◇ find the corresponding Bogoliubov operators to remove the linear terms.

Tutorial for Bogoliubov operators approach

It can be encouraged to extended to various spin-boson systems with multi-level, even multi-mode.

◇ Eliminating the ratio constant of these wavefunctions will give the transcendental functions, which would be defined through power series in model parameter dependent quantities with coefficients related recursively.

◇ Finally, zeros of these transcendental functions would give the eigenvalues exactly, where numerical solutions to the one-variable (or finite variables in other multi-level systems for example) nonlinear equation must be required.

◇ expand the wavefunctions in terms of each Bogoliubov operator respectively.

Part II, Unified analytical treatments to qubit-oscillator Shu He, Yu-Yu Zhang, and QHC et al, arXiv: 1204.0953

( )2 2z x zH a a g a a

□ Hamiltonian for the whole system reads

□ Recent most cited work

E. K. Irish, PRL 99, 173601 (2007)Generalized rotating-wave approximation (GRWA) ε=0

E. Solano’s group: PRL105, 263603(2010), an expansion in deep strong coupling regime (DSC) ε=0

M. Grifoni’s group, Phys. Rev. A 82, 062320 (2010). Van Vleck perturbation theory (VVP) ε≠0…..

( ) ( , , , , )nE E n g ◇ Goal: without any numerical effort

,A a

B a

After transformation g

2

2

2

2

A A gH

B B g

0

01

tr

tr

N

n An

N n

n Bn

c n

d n

2

2

(2 )2

(2 )2

m mn n mn

m mn n mn

m g c D d Ec

m g d D c Ed

Ansatz for the wavefunction

Recall: Our previous numerically exact solution: Qing-Hu Chen, Tao Liu, and Kelin Wang, arXiv: 1007.1747, Chin. Phys. Lett. 29, 014208 (2012)

□ We alternatively present some analytical results in the framework of the aboveFormalism, and recover many previous analytical results!

21

2

0 0! !

0 0

n n

A A A

g ga

A a

A an

n n

e

Variational study for ε=0

g to relaxed to a variational parameter α, the trial state is chosen as the most simple state Ntr=0, i.e. the vacuum state

0

0A

B

2 21 1

2 20 0 0 0g a g a

A a B ae e

◇ The energy expectation

◇ Minimizing the energy gives

◇ In the weak coupling limit, we can obtain α and the ground state energy

Which are exactly the same as Eqs. (7) and (8) obtained inYuanwei Zhang et al., PRA 83, 065802 (2011)

◇ In the strong coupling limit, we can obtain α and the ground state energy2

0,g E g

Recovery of E. Solano’s group: PRL 105, 263603(2010), ε=0 Perturbation theory based on the exact solution in the strong coupling limit

neglecting the qubit tunneling term

the exact eigenstates

the eigenvalues for the m state.

the second-order perturbative study. E. g. for even parity

Finally we have

Which are exactly the same as that in Eq. (5) in E. Solano’s group: PRL105, 263603(2010),

0 ( ) zH a a g a a 0

( ) 0

0 ( )z

a a g a aH

a a g a a

Recovery of M. Grifoni’s group, PRA 82, 062320 (2010). ε≠0 Van Vleck perturbation theory (VVP)

For any value of the qubit bias ε≠0 , the Hamiltonian with a vanishing tunneling element Δ = 0 can be diagonalized in terms of two eigenstates

the corresponding eigenvalues are

For finite Δ, the perturbative matrix elements becomes

the full Hamiltonian can be diagonalized perturbatively to second-order in Δ by using VVP theory

l is εdependent, ambiguity

Recovery of E. K. Irish, PRL 99, 173601 (2007), GRWA ε=0

2 ( )

0( ) 0trN

m mn nnm g E c D c

200 01 02 03 04

210 11 12 13 14

220 21 22 23 24

230 31 32 33 34

240 41 42 43 44

1

2

3

4

g D E D D D D

D g D E D D D

D D g D E D D

D D D g D E D

D D D D g D E

Eg. Ntr=4

◇ First-order approximation, the determinant takes the 2-by-2 block form

◇ Moreover, further analytical treatment is easily implemented in this scheme!e.g. Second-order approximation, the determinant takes the 3-by-3 block form Beyond the RWA, which can not be implemented within any renormalized RWA

2, 1

21, 1, 1

01

mm m m

m m m m

m g D E D

D m g D E

give the GRWA results exactly

Shu He, Yu-Yu Zhang, and QHC et al, arXiv: 1204.0953 Comparsions for energy levels for zero bias

The energy levels as a function of coupling constant for different qubit bias ε

The energy levels as a function of coupling constant for different qubit bias ε

The energy levels as a function of coupling constant for different qubit bias ε

Dynamics for qubit population difference for ε=0

Dynamics for qubit population difference for ε≠0

◇ Many previous analytical treatments ( 2 PRL, 2 PRA) can be recovered in the present scheme unifiedly.

Summary for, Shu He, Yu-Yu Zhang, and QHC et al, arXiv: 1204.0953 

◇ We extend the GRWA to the finite-bias case. The results is much better than VVP in the weak and intermediate coupling regime, which is more experimentally interesting.

◇ At the experimentally accessible coupling regime, the dynamics of the qubit by GRWA can always work well, but VVP theory is not valid in the present-day experimentally interesting coupling regime, except for very large static biases.

◇ For ε=0, the GRWA is further improved to BRWA, which is more close to the exact ones at large detuning while the GRWA deviates strongly.

Part. III, Concise first-order corrections to the RWA for ε=0 Shu He & Qing-Hu Chen et al, arXiv: 1203.2410

Many approaches are on the basis of various polaron-like transformations or shifted operators, which are basically photonic coherent states approaches .

1 1m m A

m A

c m

c m

The original RWA results [from only two bare states] are missed or lost in these approaches, so the effects of counter rotating-wave terms are not clear!

Eg. The wavefunction in GRWA

21

2

0 0! !

0 0

m m

A A A

g ga

A a

A am

m m

e

Although vevry accurate solutions can be obtained, the infinite ‘bare’ states should be involved!

||

| 1n

n

c n

d n

Only two ‘bare’ states under RWA

We are not interested in the more accurate solutions with infinite ‘bare’ states now What emerges first beyond the RWA?

The wavefunction in the photonic Fock states S. Swain, J. Phys. A 6, 1919 (1973).

( )2 x zH a a g a a

Transformed Hamiltonian

0

0( 1)

M

nn

M nnn

c n

c n

Wavefunction with even (odd) parity

The parity is not considered by Swain, so he developed a continued fraction approach very complicated!

First approximation

They are just the RWA energy levels

Transforming back to the original frame gives

which are just the eigenstates under the RWA

The second order approximation the resonant case δ=0

◇ m = 2k with even parity and m = 2k + 1 with odd parity

◇ m = 2k+1 with even parity and m = 2k with odd parity

◇ Unified expressions RWA results

Applications I: vacuum Rabi splitting

◇ Our analytical results for the main peak height agree excellently with the exact ones in a wide coupling regime (0 < g < 0.2).

Peak distance

Applications I: Berry phase

The GS Berry phase: Plastinaet al, EPL 76, 182 (2006).

◇ the analytical Berry phase in the second-order approximation agrees quite well with the exact ones for a wide coupling regime(0 < g < 0.5).

□ by the bosonic Fock space and parity symmetrythe analytical results are presented at different stages:

◇ The first approximation in the present formalism reproduces exactly the RWA results. ◇ The effect of the CRTs emerges clearly just in the second order approximation. they play dominant role g <= 0.2, CQED: gmax = 0.12

□ Effects of CRTs on various phenomena in the ultra-strong coupling regime. Analytical results to the vacuum Rabi splitting and the Berry phase agree well with the exact ones in a wide coupling regime.

Summary for Shu He & Qing-Hu Chen et al, arXiv: 1203.2410  

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