冷原子物理和量子信息学术讨论（PPT讲稿）Exact solvability and unified analytical treatments to qubit-oscillator system

《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 Exact solvability and unified analvtical treatments to qu bit-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 arXiv: 1204.3668, Phys. Rev. A, in press arXiy:1204.0953 arxiv;1203.2410

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 press arXiv: 1204.0953 arXiv: 1203.2410

Collaborators Prof Ke-Lin wans g Department of modern physics, University of Science and technology of china, Hefe 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 Zhu Beijing Computational Science Research Center, Beijing 100084

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 Zhu Beijing Computational Science Research Center, Beijing 100084 Collaborators

Brief introduction to quantum rabi model (Qrm) Rabi, Phys.Rev.49,324(1936)51,652(1937) u The interaction of two-level atom(qubit)with a bosonic mode H==.+oa a+gla +ao o is the resonant frequency of the cavity, A is thethe transition frequency of the qubit and g is the coupling strength, Oxz is usual Pauli matrix, a(a) is the boson annihilation(creation )operator. d=A-0 is the detuning quantum Rabi model( Cavity QED) qubit-oscillator system (Circuit QED) In the fully quantum mechanical version Analytically unsolvable a Jaynes-Cummings (JC)model(1963)under the rotating-wave approximation (RWA) is analytically solvable. The counter rotating terms(CrTs) is omitted H=0+aa+g(ao+.+g(ao +ao_) RWA CRTS

Brief introduction to quantum Rabi model (QRM) □ The interaction of two-level atom (qubit) with a bosonic mode ( ) 2 H a a g a a    z x  + + = + + + 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 H a a g a a     z + + − +  = + + + 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

PhysIcS Physics4,68(2011 Viewpoint The dialogue between quantum light and matter Enrique Solano Departamento de Quimica Fisica, Universidad del Pais Vasco-Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao Dain 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. I other words, it has not been possible to write a closed-form, analytical solution for it

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 B)Applications No explicitly expression Em=E(n, E, 4,,g) 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

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 ( ) ( , , , , ) n E E n g =   

Part l, Exact solution to the quantum Rabi model (Qrm) D In RWA, the n-th eigenstate isp -/a,In> ,n+1> En=O(n+ Rn at resonance, 0=0 Ern=a(n+o+=R E,n=O(m+)-g√hn+1 6=△-O,Rn=√62+482(n+1) E2n=0(n+3)+gVn+1 -sin e, n) 1(-|n) cos n+1 n+1 cose,n) In 2. sin n+ 1) 2n√2(n+ gvn+ △ cOS (R-6)2+4(m+1) The ground-state 0.个

| | 0,1,2. | 1 n n n a n n 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 2 1 1 ( ) 2 2 , 4 ( 1) sin | | cos | 1 cos | | sin | 1 2 1 cos 4 ( 1) n n n n n n n n n n 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 2 1 ( ) 1 2 1 | | 2 | 1 1 | | 2 | 1 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)

Vacuum Rabi splitting in the JC model The atom is excited by the operator Measured transmission spectrum showing the vacuum Rabi mode splitting S S GS 1 |2) 0 VIGS e o 92/4 (e,O)+|g,0))/2 0.1 O))/2 spontaneous emission to Gs state 0.04 o 0.02 data The emission spectrum has two peaks with 6026.036.046.056.066.07 equal height(the distance of the two peaks Frequency, VRF(GHz) 2g is the vacuum Rabi splitting) Walraff et al. nature 431 2g is the energy difference of the lst and 2nd 162(2004 eigenstates

( ) ( ) , 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 a()|n) If initially in Photonic Fock state e,n> b()|n+1) This is the quantum rabi oscillation. a(o)=cos (gtV/n+I b(O)=sin(g√n+ If initially in photonic coherent state (0)>ag>e aa+-a2/2 10>g> n==|a|2 population inversion under rWa can be evaluated analytically C 12n plt) -|a2 ∑cos(g、m+1)=F(On) M.O. Scully and M. S. Zubairy, Quantum optic Cambridge University Press, Cambridge, 1997

The collapses and revivals in the evolution of the atomic population inversion 2 / 2 0 | (0) | | | 0 | a g e g     + − − =  =   2 n a a    | | | | + = = 2 2 | | | | ( ) cos(2 1) ( ) ! n rwa n p t e gt n F t n    − = + =  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 n t b t n a t gt n b t gt n     =   +    = + = + If initially in photonic coherent state

1.01 Collapses evivals 06 02 w(C) 02h 0.6 10 10 20 z

collapses revivals

Strong coupling Qubit-Oscillator System d. c. SQUID measurement lines Resonator 2C C 米 50 mK 000 Deppe et al, Nature physics 4, 686(2008) Circuit quantum electrodynamics(QED) system

Strong coupling Qubit-Oscillator System Deppe et al., Nature physics 4, 686(2008) Circuit quantum electrodynamics (QED) system