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
