Emergence of Adiabatic Diffr.

Condensate Dynamics

Tdelay [ms]0 0.2

0

0.5

0.5 0.9 1.7

N

r

(Nb+

Nr)

τpump

=2π/5 µs τprobe

=2π/3 µs

In-situ atom interferometry

• measure oscillation period of 67ms (2-photon recoil)• decay consistent with wave-packet separation in the trap

Ωr

b

r

pΩp

Tdelay

Thold

τprobeτ

pumpMic

row

ave

P

ow

erp [ħ

k]

-404

0 010 10

-404 b

r

0 2

0 5000

0

1.0 Relative N

umber

τ [2π/Ω]10

0.5

0.5

±6±4±2

0

0 10

Connecting to Kapitza-Dirac DiffractionFor strong coupling Ω>V

0, internal and external

dynamics decouple.Regain evelopes for standard Raman-Nath diffraction:

=

2)( 02

2tVJtP nkn

Ω/V0= 0.01 0.05 15

τ [2π/Ω]

Time dynamics from nonadiabatic to adiabatic

Driven Dynamics beyond the Born-Oppenheimer Approximation:

Nonadiabatic Diffraction of Matter Waves

Phys. Rev. A 92, 023628 (2015)

Jeremy Reeves, Ludwig Krinner, Michael Stewart, Arturo Pazmino, and Dominik SchnebleDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA

δ/2π

[kH

z]

Nr/Nr+b

020406080

100120140

0 0.5 1.0

0 2 4

0

2

4

1

3

b

r6.8 GHz

V0

Band Spectroscopy

Ω/2π=8.1 kHzτ=60 µs V0=30Er

ΩΩ−

+=δ

δ)(sin/222

ˆ 20

2 kzVm

pHI

The Hamiltonian in the presence of Rabi coupling

0 0.4 0.8 1.2 1.60

0.5

1.0

τ [2π/Ω]

Nr/N

r+b

Ω/2π=8.8 kHz

Ω/2π=17 kHz

0

0.2

0.4

0.6

0.8

Frac

tion

γ0Ωτ=π

τ [2π/Ω]

p [ħ

k]

-4-2024

0 1 2 3

Nr/Nr+b Nb,±2/2Nb

Ω Ω=0.72Ω ψbϕn=0,r

Weak coupling Ω<E

r(2)=14.8 kHz

Reduction of oscillation amplitude is inconsistent with Rabi dynamics.

Coherent oscillations between 0ħk and ±2ħkmomentum states.The internal and external dynamics oscillate in phase, locked together.

Strong coupling Ω>E

r(2)

Rabi oscillations with Ω reduced by the Franck-Condon overlap.

Osc. contrast ~100%, up to small detuning effects.

Experimental Schemeand Orbital Transfer

A BEC is coupled to thedeep wells of a state-selective optical lattice via microwave drivenhyperfine ground statetransitions.

Resonant Driving

Selectively transfer atoms into bands of the lattice.Band occupation isprobed with a “band map”(right). Ωτ=π in the V

0=0 case, line shapes

are given by the Fouriertransform of the pulseand Franck-Condon overlap of the band and free particle states.

45

-30

θ=π/2 θ=0

30

-5

22 )(/)(2cos

sincoscossin

zz

br

δδθ

θθθθ

χχ

+Ω−=

−=

−

+

E [E

r]

z [d/2]

• resonance condition satisfied twice per lattice site

Bare state basis

Ω

Ω

−

+

),,(00),,(

zVzV

δδ

=H

• weak variation of the mixing angle; neglect kinetic energy• coupling (larger than all other energy scales) splits the adiabatic potentials

• diagonalization of potential transforms momentum operator:

Adiabatic dressed states

E [E

r]

• Born-Oppenheimer approximation fails • mixing angle varies rapidly and its gradient cannot be neglected

−

−⇒

−−−+

+−++

ˆˆˆˆˆˆ

pppppp

χχχχχχχχ

p

z [d/2]0-1 1

Ω = 0

Ω =V0

E [E

r]

30

-50-1 1

Ω = 0.1 V0

z [d/2]0-1 1

Nonadiabatic case: weak coupling

0 1 2 3

-1

0

1

2

ħΩ [Er(2)]

e0

e1

e+

e-

E [E

r(2) ]

• weak coupling gives light shifts ±γn=0Ω/2 as expected for a two level system • upper states see off-resonant light shift ±γn=0Ω

2/4δ• For Ω>E

r(2) higher momentum states get mixed into the dressed states, and

the resulting “free” eigenstates of the dressed system take on periodicity of the lattice

Dressed State Picture

e0

e1