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|aa|a>=|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 <ot)> 0.6 10 10 20 z
collapses revivals <σz (t)>
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