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. (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).  The analytical method could provide a clear picture to understand the physics of strongly driven systems.

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

Our plan 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.

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 Modified Rabi frequency detuning

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

Our renormalized scheme  ~    1 . A A 2 A ( 1 )   A         2 . J ~   0 0 0 0            ~      ~ 2 2 2 2 3 . ( ) A / 4 ( ) A / 4 ~ R 0 R 0

Checks:quasienergy 𝜕 = 𝜕 0 Solid line: CHRW Red dot: numerical 𝜕 = 0.5𝜕 0 𝜕 = 10𝜕 0

Solid line: CHRW Red dashed line: numerical Blue line:RWA Checks: transition element

Rabi frequency         2 2 2 A / 4 RWA 0 2     A              ~ 2 2 2 J A ( 1 )     R 0 0        2 4 A A        ~ 2 2 0 0 ( ) R 0       3 2 ( ) 32 ( ) 0 0 It is the same as the exact one to fourth order of A 2 4 A A        ~ 2  0 , Res - CHRW 0    3   R 16 4 256 0 0 0

Bloch-Siegert shift Resonance condition   ~ 2  0 , R   0

Bloch-Siegert shift Bloch & Siegert, PR57, 522 (1940) Definition The resonance frequency is defined as the frequency at which the transition probability P up averaged is a maximum. Resonance condition

Bloch-Siegert shift

Bloch-Siegert shift J. H. Shirley, Phys. Rev. 138 , B979 (1965).

Page . 20 Compared with SC flux qubit:    /2 4.869GHz    /2 4.154GHz      2 2 / 2 6 . 400 GHz

Page . 21 2   Our results ~      ~    2  2 2  2 A ~   R   2            2 2 2 2 R R 0  2 2 1 A   2 R0    2 2 4         2 2 /2 ( MHz )  2 2 A     BS 3/2    2 2 16 2   ~      ~    2  2 2  2 A ~   R NCFQ: PRB89, 020503(R)(2014)

Quantum dynamics What we dropped:   sin( n t ), odd n 3   cos( n t ), even n 2 What we calculate:             S ( t ) S ( t ) ( t ) ( t ) ( t ) ( t ) e e ( t ) z z z d d          i ( t ) H ( t ) ( t ) i ( t ) H ( t ) ( t ) dt dt

1    ( ) ( 1 ( )) Quantum dynamics P t t up z 2  /  0 =1, A/  0 =1 1.0 Black: ours (CHRW) P up (t) 0.5 Red: exactly numerical Blue: RWA 0.0 0 10 20 30 40 50  t

1    P ( t ) ( 1 ( t )) up z Quantum dynamics 2  0 =1,  =1, A=2 1.0 P up (t) 0.5 Black: ours (CHRW) Red: exactly numerical Blue: RWA 0.0 0 10 20 30 40 50  t

1    P ( t ) ( 1 ( t )) Quantum dynamics up z 2  0 =0.5,  =1, A=2 1.0 P up (t) Black: ours (CHRW) 0.5 Red: exact numerical Blue: RWA 0.0 0 10 20 30 40 50  t

1    P ( t ) ( 1 ( t )) Quantum dynamics up z 2 0.50  0 =1,  =0.5, A=0.5 P up (t) 0.25 Black: ours (CHRW) Red: exactly numerical Blue: RWA 0.00 0 10 20 30 40 50  t

1    P ( t ) ( 1 ( t )) Quantum dynamics up z 2  0 =1,  =0.5, A=1 1.0 P up (t) 0.5 Black: ours Red: exact numerical Blue: RWA 0.0 0 10 20 30 40 50  t

1    P ( t ) ( 1 ( t )) Driven tunneling dynamics up z 2      ( t ) A cos( t )             H x z x z z 2 2 2 2 2   2       A A     i                     i t i t i t i t / e e e e      z y x z 2 2 2 2

𝜁 = 0 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

1 Exact CHRW 0.9  0 /  =0 A/  =2 0.8 0.7 0.6 P up 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60  t

  A      Bias-modulated case S ( t ) -i sin( t ) z x  2   ~ 1 A      ~         2 2 i t i t H ( t ) e e ~  CHRW z 2 2 ~          2 - [ 1 J ( Z )]( ) / X 2   A    0 2 Z / ~           2 [ 1 J ( Z )]( ) / X 0    ~ 2 J c ( 1 J ( Z ) J ( Z )) / X          2 2 A 2 ( ) J ( Z ) / X , X 0 2 1   ~ ~ ~ ~ ~                2 2 2 A ( 1 J ) ( 1 J ) / A 0 c c ~ ~            2 ( 1 J ) ( 1 J ) 0 c c 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)

Page . 36        2 2 Bias modulated dynamics: Resonace

Page . 37 Biased modulated dynamics: near resonace   1.2 

Page . 38 Biased modulated dynamcis: far off-resonace

Page . 39 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 of C CR i inte terac actio ion o on 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. A 88 ， 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).