Atom-Photon Interactions: Methods and Results

Presented by Gilad Laredo Atom-photons interactions 118137 Spring 2017

2?cos?? Introduction ,E ,E System evolves according to Schr dinger eq. ( ) ( ) t ( ) ( ) t 2 cos b E t a a t a a t d dt = i 2 cos b t E ( ) t 2 transition probability to be in at time ' ' a t ( ) 1 =

But how do we solve this? ( ) ( ) t ( ) ( ) t 2 cos b E t a a t a a t d dt = i Difficult 2 cos b t E ? i t i t i t = + 2cos t e e e By the RWA: ( ) ( ) t ( ) ( ) t i t a a t a a t E be d dt = i Easy i t be E ( ) 1 = RWA = rotating wave approximation

Paper goal Present methods and results obtained while solving the transition probability for: ( ) ( ) t ( ) ( ) t 2 cos b E t a a t a a t d dt = i 2 cos b t E Without the rotating wave approximation time independent Hamiltonian Infinite matrix representation Floquet theorem Time periodic Hamiltonian

time independent Hamiltonian Infinite matrix representation Floquet theorem Time periodic Hamiltonian ( ) ( ) t ( ) ( ) ( ) t ) Generalize the problem: H t T 2 cos b ; E t a a F t t a a + t d dt i = i d dt 2 cos b ( ) ( ) t F t c ( ( ) t = = H H t E Q c c : Constant diagonal matrix ( ) ( ) t e Generalize the problem: Step 1 determine general form of solution (Floquet): = iQt F t d dt ( ) ( ( ) ( ) t F t ( ) ( ) t ( ) t = t T + = H H H ; i F t H : Matrix c c c Step 2 equivalence to (infinite) eigenvalue problem. of periodic functions ) ( ) , ( ) , n k k n + = ( ) n F q F , , kn , k ( ) ( ) ( ) 0 t Step 3 express time-evolution operator: = = 1 2 T ( ; ) U t t F t F 0

's q Eigenvalue equation for Floquet Hamiltonian ( ) H ( ) , ( ) , n k k n + = ( ) n F q F , , kn , k = ( ) t in t ( ) n H H e c , , ( ) n ( ) ( ) = in t iq t ( ) , n exp exp F t F , n Greek letters atomic states Roman letters Fourier components ( ) ( ) t e = iQt F t d dt ( ) ( ) ( ) t F t = Hc i F t

( ) H ( ) , ( ) , - - Floquet Floquet Hamiltonian Hamiltonian H n k k n + = ( ) n F q F F , , kn , k ,E ( ) ( ) t ( ) ( ) t 2 cos b E t a a t a a t An example d dt = i 2 cos b t E ,E 2 0 0 0 0 b 0 b 0 0 0 0 b + 0 0 0 b 0 0 0 0 0 0 b + E b b E 0 0 0 0 0 0 0 0 b 0 0 E b E H F 0 0 0 0 0 0 b 0 0 + E b 0 0 0 E 0 0 0 0 E 0 2 b E

n m = + ( ) H H ; ; n m n , , , F n m = = = = = = = = , , , 2 1 1 2 0 0 0 0 b 0 b 0 0 0 0 b + 0 0 0 b 0 0 0 0 0 0 b + n n n n n E b b E 0 0 0 0 0 0 0 0 b 0 0 E , , , , , 0 0 1 1 2 b E 0 0 0 0 0 0 b 0 0 + E b 0 0 0 a n E 0 0 0 0 n n E 0 2 b E / ; n , 2 , 1 , 1 ,0 ,0 ,1 ,1 , 2 / ; m ( ) 2 cos b E t ( ) , ( ) , n k k n + = ( ) H n F q F , , kn 2 cos b t E , k Floquet state H F

n m = + ( ( ) ) H H ; ; n m n ( ) 0 t , , , F n m Time evolution operator = 1 ( ; ) U t t F t F 0 ( ) ( ) t e = iQt F t Using: the periodic structure of The unitarity of Is a complete set Some calculations U H F ( ) , n { } F / ,n ( ) ( ) n t = i H ; , exp ,0 t t n i t t e 0 0 F n Transition amplitude in Floquet space Time evolution operator of ( ) H Ct

Relation to Quantized Field Theory ( ) ( ) n t = i H ; , exp ,0 U t t n i t t e 0 0 F n Transition amplitude in Floquet space Time evolution operator of ( ) H Ct The transition amplitudes in Floquet space is the Fourier series coefficients of the time evolution operator(!) the resolvant operator is the Fourier transform of ( ) ( ) 0 0 ; ( G z U t t z H = = + 1 ( ) ; U t t 0 ( ) 1 = + + + ) V G G VG G VG VG 0 0 0 0 0 0

( ) ( ) H , exp ,0 G z n i t t 0 F But is in Floquet space (not Hilbert-Fock quantum space ) F H Q= + + H H H H atom field interaction ( ) ( ) 1 2 1 q m p a e p a e + j t j t + * k k N 0 , , k k , , k k 2 , k , k 0 H H Two differences between and : Q F H 1. start form n=0 while at Q H H F 2. off-diagonal elements depend on n . Q

= = = = = = = = , , , 2 1 1 2 0 0 0 0 b 0 b 0 0 0 0 b + 0 0 0 b 0 0 0 0 0 0 b + n n n n n E b b E 0 0 0 0 0 0 0 0 b 0 0 E , , , , , 0 0 1 1 2 b E 0 0 0 0 0 0 b 0 0 + E b 0 0 0 a n E 0 0 0 0 n n E 0 2 b E / ; n , 2 , 1 , 1 ,0 ,0 ,1 ,1 , 2 / ; m

H H Main differences between and : H and are similar but NOT the same! F H Q F Q H 1. off-diagonal elements depend on n . Q H 2. start form n=0 while at Q H But, our writer didn t give up. F Jon H. Shirley solution : consider in the vicinity of some very large photon number N H Q why? ( ) In the matrix region, the off-diagonal terms change slowly. if is an eigenvalue of then + F H 1 a a N , is also eigenvalue. n

1 N Therefore: = + + + N H H H H H 1 atom field int Floquet Q Authors claim : The quantum state is approximately isomorphic to the Floquet state for . ;m N m + ;N m Floquet states can be interpret as quantum states containing a definite though very large number of photons.

= + + + N H H H H H 1 atom field int Floquet Q = = = = = = = = , , , 2 1 1 2 0 0 0 0 b 0 b 0 0 0 0 b + 0 0 0 b 0 0 0 0 0 0 b + n n n n n E b b E 0 0 0 0 0 0 0 0 b 0 0 E , , , , , 0 0 1 1 2 b E 0 0 0 0 0 0 b 0 0 + E b 0 0 0 a n E 0 0 0 0 n n E 0 2 b E / ; n , 2 , 1 , 1 ,0 ,0 ,1 ,1 , 2 / ; m

+ N H H ( 1 Could we reconstruct from the semiclassical amplitude ? Floquet ) 0 t t Q ; U 2 0/4 e in t n e n Using coherent states. Assuming large number of photons. peaked at Using the periodicity properties of to get rid of N. ( ) 0 ; , n Quantum-Floquet theory: Equivalent to semiclassical theory. Admits interpretation in terms of quantized field. 0 n 2 ! n n A n 2 0/ 2 n A N and extremely small elsewhere H F ( ) in t = H exp ,0 U t t n i t t e 0 F

H More AMAZING things with F Rabi formula + correction: Valid for large times No averaging over a continuum (Fermi s golden rule) Bloch-Siegert shift: easier derivation Simple derivation of higher orders Transition probability for multiple quantum transitions (non-directly connected states) P , t b = 0.25 b = 0.5 ;0 ;1 ;2 ;3 Applied to more than 2 level quantum state.

All of this because - i t i t i t = + 2cos t e e e Thank you.