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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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