Outline Background Our method CHRW method Bloch-Siegert shift - - PowerPoint PPT Presentation

Quantum dynamics under strong driving:

Counter-rotating hybridized rotating wave method

Zhiguo Lü (吕智国) Collaborators Prof. Hang Zheng, Prof. H.S. Goan, Dr. YiYing Yan

Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China Conference on Taming Non-Equilibrium Systems: from Quantum Fluctuation to Decoherence（SMR3316） 29 July ~1 Aug, 2019

Outline

 Background  Our method：CHRW method  Bloch-Siegert shift  Driven quantum dynamics  Fluorescence and absorption spectra  Summary

Motivation

The rotating wave approximation of the system-field interactions is a wide-used treatment under serious conditions.

Semiclassical Rabi model

Background

 The physics of the ultrastrong- and deep-strong coupling regimes of light-matter interaction may be realized through state-of-the -art technology such as circuit QED, flux qubit etc.  The analytical method could provide a clear picture to understand the physics of strongly driven systems.

(a)W.D. Oliver, Y. Yu, et. al, Science 310, 1653(2005). (b)C.M.Wilson, G.Johansson, et. al, Phys. Rev. B. 81, 024520(2010). (c) F. Yoshihara,Y.Nakamura, et.al, Phys. Rev. B. 89, 020503(R)(2014). (d) F. Yoshihara, T. Fuse, et. al, Nat. Phys. 13, 44 (2017).

Background

Driven two-level system

NMR Rabi model Tunneling TLS

• M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998)

Background

Rabi rotating wave approximation Floquet theory

• M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998)

Floquet Hamiltonian

1. CHRW approach for semiclassical Rabi models, which is as simple as the usual RWA approach but the results are in good agreement with the numerical calculations. 2. We studied the quantum dynamics of the semiclassical Rabi model without and with static bias, the Bloch-Siegert shift. 3. Because of the simple math structure, the approach my be used for more complex systems.

Our plan

Unitary transformation

Driven two-level system Unitary transformation

Using

Jacobi-Anger relations

Zhiguo Lü, and Hang Zheng, Effects of counterrotating interaction on driven tunneling dynamics: coherent destruction of tunneling and Bloch-Siegert shift, Phy.Rev.A 86, 023831(2012)

The parameter ξ The Hamiltonian Rabi frequency Modified detuning

Diagonalization of the CHRW Hamiltonian Its eigenstates and corresponding eigenenergies are given as follows Floquet mode quasienergies

4 / ~ ) ~ ( ~ 4 / ) ( . 3 ~ . 2 ) 1 ( 2 ~ . 1

2 2 2 2

A A A J A A A

R R

                               

Our renormalized scheme

Checks:quasienergy

Solid line: CHRW Red dot: numerical

𝜕 = 𝜕0 𝜕 = 0.5𝜕0 𝜕 = 10𝜕0

Checks: transition element

Solid line: CHRW Red dashed line: numerical Blue line:RWA

 

4 /

2 2 2 RWA

A      

2 2 2 2 R

) 1 ( ~                           A A J

3 4 2 2 2 R

) ( 32 ) ( 2 ) ( ~                A A

3 4 2 CHRW

• Res

256 4 16         A A

, ~ 2

R

    

Rabi frequency

It is the same as the exact one to fourth order of A

Bloch-Siegert shift

Resonance condition

, ~ 2

R

    

Bloch-Siegert shift

Definition Resonance condition

Bloch & Siegert, PR57, 522 (1940)

The resonance frequency is defined as the frequency at which the transition probability Pup averaged is a maximum.

Bloch-Siegert shift

Bloch-Siegert shift

• J. H. Shirley, Phys. Rev. 138, B979 (1965).

20 Page .

Compared with SC flux qubit:

GHz 400 . 6 2 / 4.154GHz /2 4.869GHz /2

2 2

          

Page . 21

Our results

 

) ( /2

2 2

MHz      

 

2 2 2 2 2 R R

          

2 2 2 2 2 R0

4 1       A

 

3/2 2 2 2 2

16       A

BS 2 2 2 2 2 R

~ ~ ~ A              

2 2 2 2 2 R

~ ~ ~ A              

NCFQ: PRB89, 020503(R)(2014)

2 even ), cos( 3

• dd

), sin(   n t n n t n  

What we calculate: What we dropped:

) ( ) ( ) ( ) ( ) (

) ( ) (

t e e t t t t

t S z t S z z

          



) ( ) ( ) ( t t H t dt d i       ) ( ) ( ) ( t t H t dt d i   

Quantum dynamics

10 20 30 40 50 0.0 0.5 1.0

Pup(t)

t /0=1, A/0=1

)) ( 1 ( 2 1 ) ( t t P

z up

  

Black: ours (CHRW) Red: exactly numerical Blue: RWA

Quantum dynamics

10 20 30 40 50 0.0 0.5 1.0

Pup(t)

t 0=1, =1, A=2

Black: ours (CHRW) Red: exactly numerical Blue: RWA

)) ( 1 ( 2 1 ) ( t t P

z up

  

Quantum dynamics

10 20 30 40 50 0.0 0.5 1.0

Pup(t)

t 0=0.5, =1, A=2

Black: ours (CHRW) Red: exact numerical Blue: RWA

)) ( 1 ( 2 1 ) ( t t P

z up

  

Quantum dynamics

10 20 30 40 50 0.00 0.25 0.50

Pup(t)

t 0=1, =0.5, A=0.5

Black: ours (CHRW) Red: exactly numerical Blue: RWA

)) ( 1 ( 2 1 ) ( t t P

z up

  

Quantum dynamics

10 20 30 40 50 0.0 0.5 1.0

Pup(t)

t 0=1, =0.5, A=1

Black: ours Red: exact numerical Blue: RWA

)) ( 1 ( 2 1 ) ( t t P

z up

  

Quantum dynamics

   

     

                               

    t i t i t i t i z x z z x z x

e e A e e A t A t H 2 2 2 2 2 ) cos( 2 2 2 ) ( 2

  2

/

y z

i    



Driven tunneling dynamics

)) ( 1 ( 2 1 ) ( t t P

z up

  

Resonance and near Resonance

Zhiguo Lü and H. Zheng, PHYSICAL REVIEW A 86 86, 023831 (2012)

𝜁 =0

Resonance and near Resonance

𝜁 =0

Far off-resonance and CDT

  5 

𝜁 =0

Far off-resonance and CDT

𝜁 =0

Far off-resonance and CDT

𝜁 =0

10 20 30 40 50 60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

 t Pup

0/ =0 A/ =2

Exact CHRW

Bias-modulated case

 

x z

t A t S       ) sin( 2

• i

) (

 

   

       

  t i t i z

e e A t H 2 ~ ~ ~ 2 1 ) (

2 2 CHRW

2 2 1 2 2

, / ) ( ) ( 2 ~ / ) )]( ( 1 [ ~ / ) )]( ( 1 [

• ~

                            X X Z J A X Z J X Z J

 

) 1 ( ~ ) 1 ( ~ ~ ~ ~ / ) 1 ( ~ ) 1 ( ~

2 2 2 2

              

c c c c

J J A J J A           

   /

2 2 

 A Z

2 2

/ )) ( ) ( 1 ( X Z J Z J Jc   

Zhiguo Lü, Yiying Yan, Hsi-Sheng Goan and Hang Zheng, Bias-modulated dynamics of a strongly driven two-level system, Phys. Rev.A 93, 033803(2016)

36 Page .

Bias modulated dynamics: Resonace

2 2

      

37 Page .

Biased modulated dynamics: near resonace

  1.2 

38 Page .

Biased modulated dynamcis: far off-resonace

39 Page .

Bias-modulated dynamics

Off-resonance

Fluorescence spectrum

artificial atom

• O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A.Pashkin, T. Yamamoto, K. Inomata, Y.

Nakamura, and J. S. Tsai, Science 327 327, 840 (2010).

Eff ffect cts o

• f C

CR i inte terac actio ion o

• n F

Fluo uores escen ence

Our model

• Y. Yan, Zhiguo Lü, and H. Zheng, Effects of counter-rotating-wave terms of the driving field on the

spectrum of resonance fluorescence, Phys. Rev. A88，053821(2013).

Using Born-Markov master equation

Resonance Fluorescence

Definition Two time correlation function In the transformed frame

• B. R. Mollow, Phys. Rev. 188, 1969 (1969).

Results (i) the asymmetry of the sidebands with respect to the central peak,

• D. E. Browne and C. H. Keitel, J. Mod. Opt. 47, 1307 (2000)

Yiying Yan, Zhiguo Lü, and Hang Zheng, PHYSICAL REVIEW A 88 88, 053821 (2013)

(ii) The generation of the higher-order Mollow triplets at

(iii) Shifts of the sidebands

probe-pump spectrum

• B. R. Mollow, Phys. Rev. A 5, 2217 (1972).

Population of the upper state

The time-averaged population as a function of driving frequency

Yinying Yan, Zhiguo Lü*, and H. Zheng, Bloch-Siegert shift of Rabi model, Phys. Rev. A91, 053834(2015).

Applications：

Multi-chromatic field Bloch-Siegert shift Driven dissipative system Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition, Phys. Rev. B 96, 020501(R) Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime, Phys. Rev. Lett. 105, 237001 Mooij Dynamical effects of counter-rotating couplings on interference between driving and dissipation Phys. Rev. A90, 053850 (2014) superconducting circuits

Summary

 (i) We investigate the dynamics of a driven two-level system (classical Rabi model) using the counter-rotating hybridized rotating-wave method (CHRW), which is a simple method based on a unitary transformation. Since the CHRW approach is mathematically simple as well as tractable and physically clear, it may be extended to some complicated problems where it is difficult to do a numerical study.  (ii)We calculate the Bloch-Siegert (BS) shift over the entire driving-strength range. we demonstrate the signatures caused by the BS shift by monitoring the excited- state population and the probe-pump spectrum under the experiment accessible conditions.  (iii) We investigate the fluorescence spectrum of a two-level system driven by a monochromatic classical field by the Born-Markovian master equation based on a unitary transformation We find the main effects of counter-rotating-wave terms of the driving on spectral features of the fluorescence.

𝜊 = 𝜕 𝛦 + 𝜕 + 𝐵2𝛦𝜕 8 𝛦 + 𝜕 4 + 𝐵4𝛦 8𝛦 − 𝜕 𝜕 192 𝛦 + 𝜕 7 + 𝐵6𝛦𝜕 169𝛦2 − 46𝛦𝜕 + 𝜕2 9216 𝛦 + 𝜕 10 + 𝐵8𝛦𝜕 6799𝛦3 − 2903𝛦2𝜕 + 197𝛦𝜕2 − 𝜕3 737280 𝛦 + 𝜕 13 + 𝐵10𝛦𝜕 443821𝛦4 − 259736𝛦3𝜕 + 32766𝛦2𝜕2 − 776𝛦𝜕3 + 𝜕4 88473600 𝛦 + 𝜕 16 𝛻R

2

= 𝛦 − 𝜕 2 + 𝛦𝐵2 2 𝛦 + 𝜕 − 𝛦𝐵4 32 𝛦 + 𝜕 3 + 𝛦 −13𝛦2 − 3𝛦𝜕 + 𝜕2 𝐵6 1152 𝛦 + 𝜕 7 − 𝛦 381𝛦3 − 77𝛦2𝜕 − 25𝛦𝜕2 + 𝜕3 𝐵8 73728 𝛦 + 𝜕 10 + 𝛦 −19616𝛦4 + 10153𝛦3𝜕 − 45𝛦2𝜕2 − 113𝛦𝜕3 + 𝜕4 𝐵10 7372800 𝛦 + 𝜕 13

The tenth order of A

Enhancement of BS shift

Effects of counter-rotating couplings of the Rabi model with frequency modulation, Phys. Rev. A 96, 033802 (2017).

Enhancement of BS shift

Effects of counter-rotating couplings of the Rabi model with frequency modulation, Phys. Rev. A 96, 033802 (2017).

 Yiying Yan, Zhiguo Lü*, Junyan Luo, Hang Zheng*, Role of generalized parity in the symmetry of the fluorescence spectrum from two-level systems under periodic frequency modulation, Phys. Rev. A100, 013823 (2019)  Yiying Yan, Zhiguo Lü*, Junyan Luo, Hang Zheng*, Multiphoton-resonance-induced fluorescence of a strongly driven two-level system under frequency modulation, Phys. Rev. A97, 033817 (2018)  Zhiguo Lü, Chunjian Zhao and Hang Zheng, Quantum dynamics of two-photon quantum Rabi model, J. Phys. A:

• Math. Theor. 50, 074002(2017).

 Yiying Yan, Zhiguo Lü*, Junyan Luo, Hang Zheng*，Effects of counter-rotating couplings of the Rabi model with frequency modulation,Phys. Rev. A96, 033802 (2017).  Zhiguo Lü, Yiying Yan, Hsi-Sheng Goan and Hang Zheng, Bias-modulated dynamics of a strongly driven two-level system, Phys. Rev.A93, 033803(2016).  Yiying Yan, Zhiguo Lü*, Hang Zheng, Resonance fluorescence of strongly driven two-level system coupled to multiple dissipative reservoirs, Annals of Physics371, 159-182(2016).  Yiying Yan, Zhiguo Lü, Hang Zheng,and Yang Zhao, Exotic fluorescence spectrum of a superconducting qubit driven simultaneously by longitudinal and transversal fields, Phys.Rev. A93, 033812 (2016).  Yinying Yan, Zhiguo Lü*, and H. Zheng, Bloch-Siegert shift of Rabi model, Phys. Rev. A91, 053834(2015).  Yinying Yan, Zhiguo Lü*, and H. Zheng, Dynamical effects of counter-rotating couplings on interference between driving and dissipation, Phys. Rev. A 90, 053850 (2014).  Yinying Yan, Zhiguo Lü*, and H. Zheng, Effects of counter-rotating-wave terms of the driving field on the spectrum of resonance fluorescence, Phys. Rev. A88，053821(2013).  Zhiguo Lü, and Hang Zheng, Effects of counterrotating interaction on driven tunneling dynamics: coherent destruction of tunneling and Bloch-Siegert shift, Phys.Rev.A 86, 023831(2012).

References: