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FINGER PRINTS OF CLASSICAL CHAOS IN NIST AND AUSTIN COLD ATOMS EXPERIMENTS Shmuel Osovsky & Nimrod Moiseyev Chemistry Department Technion - Israel Institute of Technology The Experiments The experiments were held by using two standing waves which are slightly detuned from each other creating an 1D optical lattice . After laser cooling the atoms and preparing them in a narrow distribution around p=0 the optical lattice was turned on adiabatically so the atoms localize in the potential wells In order to give the atoms the chosen initial momentum the lattice was suddenly spatially shifted using an electro optical modulator. After a short time the atoms return to the centers of the potential wells with kinetic energy the acquired in the process. The classical phase space NIST EXP Hamilton's Equations: ∂H ∂H =q = -p ∂p ∂q The classical phase space AUSTIN EXP Hamilton's Equations: ∂H ∂H =q = -p ∂p ∂q Observation of chaos-assisted NIST experiment, tunneling between islands of Nature 412, p. 52 (July stability”, AUSTIN experiment 2001 Science 293, p. 274, July 2001 The Hamiltonian • The laser frequency was chosen to be detuned from the atomic frequency • We assume a two level system e and g are the excited and ground state, and the only populated state is g 2 H0 = p 2M ( e e + g g ) + ω at e e + 0 g g x (no coupling between C.M and relative coordinate) E ( x, t ) ( E + cos ω + t + E − cos ω − t ) cos k L x 2π c ω+ + ω− kL = ; λ = ; ωL = ; δω = ω + − ω − δ L = ω L − ω at λ ωL 2 Dipole approximation for the field d = g z elec e ˆ interaction with internal motion: The Rabbi frequency for each of the waves is Ω ± = dE ± / Center of mass Hamiltonian: 2 px H = + V0 cos(2 k L x ) + V1 cos(2 k L x ) cos(δω ⋅ t ) 2 V0 = ( Ω 2 + Ω − ) /16δ L + 2 V1 = Ω + Ω − / 8δ L The additional parameter eff is given by [ p, q ] = i eff = ∗ 4k 2 L eff Mδw δw is the frequency difference between the 2 standing waves M is the molecular weight of the cesium atom kL is the mean wave vector of the 2 standing waves Floquet solutions (quasi-momentum = 0) ˆ ψ = i ∂ψ α H α ; Floquet QE states ψ α = e − iEα t / ϕα (q, t ) ∂t ⎛ ∂ ˆ ⎞ϕ = E ϕ ⎛ 2π ⎞ ⎜ −i +H⎟ α ϕα (q, t ) = ϕα ⎜ q, t + ω ⎟ α α ⎝ ∂t ⎠ ⎝ ⎠ The Quantum Phase Space Representation Husimi Distribution Function: ∞ Ψ ( G a u ssia n − ce n tered − a t − p 0 − q 0 ) = ∑ m = −∞ g m e x p ( im q ) ( m − p0 )2 g m = N ex p ( eff − iq 0 m ) 2Ω eff Γα ( p0, q0 ) =|< Ψ (Gaussian) | ϕα (q, 0) >|2 = 2 ( m − p0 ) 2 N 2 ∑C m m ,α exp( eff 2Ω − iq0 m) eff two-state mechanism ( eff = 1.9 α = 10.5 ) here the tunneling wavepacket Ψ is well described by the linear combination of two Floquet states : 1 −iε1t ⎛ Φ + − i ( ε 1 −ε 2 ) t Ψ (t) = e eff ⎜ 1 e eff Φ2 ⎞ ⎟ 2 ⎝ ⎠ Φ1 Φ2 Husimi distribution of the combination of the two states Ψ (0) Ψ (t hopping ) π = eff t hopping ε1 − ε 2 0.8 10 0.7 8 6 0.6 4 0.5 2 0.4 0 0.3 −2 0.2 −4 0.1 −6 0 −8 0 20 40 60 0 1000 2000 3000 4000 α number of time periods As we can see the tunneling oscillations are quiet slow due to the small difference between the two quasienergies With Vitali Averbukh, PRL 89, 253201 (2001) Quasienergies as function of the effective Planck constant: green and blue states repel each other causing the blue ε state to intersect with the red. heff The different effective Planck constant for which the different mechanisms are observed are: eff eff < 1.9 two - state mechanism eff = 1.921 locking eff < 1.9 three - state mechanism Conclusion The cold atom setup lets us observe three mode of the dynamical tunneling: • The suppression of tunneling coming from the degeneracy of two Floquet states • Slow tunneling by the two –state mechanism • The enhancement of tunneling by interaction with an intermediate state. The intermediate state in the 3-state dynamics doesn’t have to be “chaotic” in order to enhance the tunneling. However chaos is essential to the problem since chaos creates the two regular islands to beginwith Finger prints of chaos in NIST/AUSTIN experiments QUANTUM PHASE SPACE ENTROPY (Korsch-Mirbach PRL 1995) QE yields NO INFORMATION for each phase space point ∑ Γα α ,k ( p0 , q0 ) = 1 Quantum phase-space density kernel for the k-quasi momentum Bloch state: ρ ( p0 , q 0 ; p , q ) = ∑ Γ α , k ( p0 , q0 ) Γ α , k ( p , q ) (k ) α Quantum phase-space entropy: S ( k ) ( p, q ) = − ∫ ρ ( k ) ( p0 , q0 ; p, q) ln ρ ( k ) ( p0 , q0 ; p, q )d p0 dq0 What is the Reason for the decay of the oscillations of the mean momentum observed in the experiments? Why don’t we see the same effect in our calculations? Bloch-Floquet The Floquet-Bloch eigenstates are given by: Ψ (q,t) = e e B-F ikq k -iEα t / eff Φ (q,t) (k) α where Φαk ) (q, t ) = ∑ Cαk,m (t )eimq is an eigenfunction of the ( ( ) m Bloch-Floquet operator ˆ H ∂ (q, t ) = −i eff + 2 (∂ / ∂q − ik )2 eff B-F ∂t 2M −α (1 + cos(2π t / T ))cos(q) The initial state was chosen to be ⎛ 1 ⎞ 1 4 0.5 − k2 ( m − p0 ) 2 ψ (t=0)= ⎜ ⎟ ⎝ πσ ⎠ ∫ dke 2σ ∑ exp( eff 2Ω − iq 0 m ) e imq − 0.5 m eff The time propagated WP at t = n 1 4 0.5 k2 ⎛ 1 ⎞ − ψ (t = nT ) = ⎜ ∫ 2σ e Ψ k (t = nT ) ikq ⎟ dke ⎝ πσ ⎠ −0.5 where iε ε ,k nT QE − Ψ k (t = nT ) = ∑ e eff Dαk ) (t = nT )Φ αk ) ( q , t = 0) ( ( α ( m − p0 ) 2 Dα (t)=∑ (C (k ) (k) α ,m (t )) * exp( eff − iq0 m) m 2Ω eff The scaled momentum is given by: P (t = nT ) = eff − i ( εα ,k + ε QE ) nT QE ' 2 α ,k 0.5 k − ∫ dke σ ∑ ( Dαk' )* (nT )Dαk ) (nT )e ( eff ∑ (m + k )Cαk' ,)* Cαk,m ( m ( ) −0.5 α ,α ' m when looking at the momentum with different values of σ we see that the rate of dephasing strongly depend on it Raizen 10 0.0001 8 0.001 0.01 6 4 2 0 −2 −4 −6 0 100 200 300 400 number of time steps 2 Philips 0.0001 0.001 1.5 0.01 1 0.5 0 −0.5 −1 −1.5 −2 0 5 10 15 20 25 30 35 40 NIST EXPERIMENT vs OUR THEORETICAL RESULTS 2.5 theory experiment 2 1.5 1 <P> 0.5 0 −0.5 −1 −1.5 0 5 10 15 20 25 30 number of time periods As the WP oscillates through the chaotic sea it accumulates a random phase which causes the decay of the amplitude of the oscillations The projection of the α k-th Bloch state on ψ k ( n ) is iεα ,k n QE − iγ α ,k (n) Wα ,k (t = n) = Wα ,k (n) e =e eff D (k ) α t=0 t = 10 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 K 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 10 20 30 40 50 10 20 30 40 50 NIST EXPERIMENT −4 t=0 t=5T x 10 6 1 4 0.5 2 0 0 −0.5 −2 −4 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 t=10T t=15T 1 1 0.5 0.5 image(population) 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 real(population) Finger prints of chaos in NIST/AUSTIN experiments: CONCLUSIONS • Although the systems are far from its classical limit the QE Floquet-Bloch states can be assigned as REGULAR INNER, REGULAR OUTER and CHAOTIC STATES. • In both experiments the quantum and the classical phase space entropies are alike, although the classical phase space is a mixed regular-chaotic space • The initial WP accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum • THIS IS ANOTHER TYPE OF FINGERPRINTS OF CHAOS IN QUANTUM DYNAMICS THAT PRESUMBALY WAS MESURED IN THE NIST AND AUSTIN EXPERIMENTS FOR THE FIRST TIME Derivation of the optical lattice potential: Floquet theory (with Dr. Milan Sindelka) N-electron atom interacting with a linearly polarized standing laser wave, beyond the dipole approximation: P2 1 ∑{ } N N −1 H (t ) = + 2 M 2me j =1 p 2 x, j +p 2 y, j +e 2 ∑r j< j ' j − rj ' − Ze 2 ∑r j =1 −1 j 2 { [ ] } N + ∑ pz , j − ( e / c ) A0z ( X ) cos ωt − ( e / c ) ∂A0z ( X ) / ∂X ( x j − X ) cos ωt j =1 ( R, P ) = position vector and momentum of the nucleus ( rj , p j ) = position vector and momentum of an electron How to get an effective potential governing the translational motion of an atom ? Dressed electronic states from the Floquet theory: [H el ( X , t ) − i (∂ / ∂t ) ]ϕ α ( X , t ) = Vα ( X ) ϕα ( X , t ) ; ϕα ( X , t + 2π / ω ) = ϕα ( X , t ) . Term H el ( X , t ) results from H (t ) by neglecting the kinetic part P 2 / 2 M . Time is treated here as an additional dynamical coordinate. Adiabatic approximation: An effective Hamiltonian for the nuclear motion, α H eff = P 2 / 2 M + Vα ( X ) . Second-order time-independent perturbation theory: An explicit formula for Vα ( X ) . Optical lattice potential [ A0z ( X ) = A0 cos(k L X ) , two level approximation] a) Dominant contribution of the dipole transition term 2 Ω dip eA0 N V (X ) = 2 cos ( k L X ) ; Ω dip = e ∑ pz , j g . 4∆ L cme j =1 b) Dominant contribution of the “quadrupole” transition term 2 N Ω qd eA0 V (X ) = sin 2 ( k L X ) ; Ω qd = k L e ∑ x j pz , j g . 4∆ L cme j =1 Quadrupole optical lattice: A proposal for an experiment Ultracold calcium atoms, laser frequency detuned from the 4s-3d transition. 4.5e-13 Quadrupole optical lattice Floquet Proposed experimental parameters: λ ≈ 657 nm PT2 4e-13 ∆ L ≈ 100 kHz (cf. the natural linewidth ) 3.5e-13 3e-13 V(x) [a.u.] 2.5e-13 2e-13 For the laser field amplitude 1.5e-13 A0 ≈ 102 A0 (NIST/AUSTIN) 1e-13 5e-14 the quadrupole optical lattice possesses 0 0 2000 4000 6000 x [a.u.] 8000 10000 12000 14000 a similar strength as the usual (dipole) lattices. Φ 3 ε 3 = −9.4775 Φ2 ε 2 = −8.1155 Φ1 ε 1 = −8.8729 Ψ (0) Ψ ( t hopping ) π t hopping = eff ∆ 0.9 10 0.8 8 0.7 6 0.6 4 0.5 2 0.4 0 0.3 −2 0.2 −4 0.1 −6 0 −8 0 10 20 30 40 50 0 200 400 600 800 α number of time periods As we can see the interaction with the intermediate state has subspecialty reduced the tunneling period. t=0 t= 10 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 K 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 10 20 30 40 50 10 20 30 40 50 AUSTIN EXPERIMENT t=0 t = 10 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 K 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 10 20 30 40 50 10 20 30 40 50 NIST EXPERIMENT

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