Driven Dynamics beyond the Born-Oppenheimer Approximation...
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Emergence of Adiabatic Diffr. Condensate Dynamics T delay [ms] 0 0.2 0 0.5 0.5 0.9 1.7 N r (N b +N r ) τ pump =2π/5 µs τ probe =2π/3 µs In-situ atom interferometry • measure oscillation period of 67ms (2-photon recoil) • decay consistent with wave-packet separation in the trap Ω r b r p Ω p T delay T hold τ probe τ pump Microwave Power p [ħk] -4 0 4 0 0 10 10 -4 0 4 b r 0 2 0 5 0 0 0 0 1.0 Relative Number τ [2π/Ω] 10 0.5 0.5 ±6 ±4 ±2 0 0 10 Connecting to Kapitza-Dirac Diffraction For strong coupling Ω>V 0 , internal and external dynamics decouple. Regain evelopes for standard Raman-Nath diffraction: = 2 ) ( 0 2 2 t V J t P n k n Ω/V 0 = 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 Schneble Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA δ/2π [kHz] N r /N r+b 0 20 40 60 80 100 120 140 0 0.5 1.0 0 2 4 0 2 4 1 3 b r 6.8 GHz V 0 Band Spectroscopy Ω/2π=8.1 kHz τ=60 µs V 0 =30E r Ω Ω − + = δ δ ) ( sin / 2 2 2 ˆ 2 0 2 kz V m p H I The Hamiltonian in the presence of Rabi coupling 0 0.4 0.8 1.2 1.6 0 0.5 1.0 τ [2π/Ω] N r /N r+b Ω/2π=8.8 kHz Ω/2π=17 kHz 0 0.2 0.4 0.6 0.8 Fraction γ 0 Ωτ=π τ [2π/Ω] p [ħk] -4 -2 0 2 4 0 1 2 3 N r /N r+b N b,±2 /2N b Ω Ω=0.72Ω ψ b ϕ n=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 ±2ħk momentum 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 Scheme and Orbital Transfer A BEC is coupled to the deep wells of a state- selective optical lattice via microwave driven hyperfine ground state transitions. Resonant Driving Selectively transfer atoms into bands of the lattice. Band occupation is probed with a “band map” (right). Ωτ=π in the V 0 =0 case, line shapes are given by the Fourier transform of the pulse and Franck-Condon overlap of the band and free particle states. 45 -30 θ=π/2 θ=0 30 -5 2 2 ) ( / ) ( 2 cos sin cos cos sin z z b r δ δ θ θ θ θ θ χ χ + Ω − = − = − + E [E r ] z [d/2] • resonance condition satisfied twice per lattice site Bare state basis Ω Ω − + ) , , ( 0 0 ) , , ( z V z V δ δ = 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 − − ⇒ − − − + + − + + ˆ ˆ ˆ ˆ ˆ ˆ p p p p p p χ χ χ χ χ χ χ χ ˆ p z [d/2] 0 -1 1 Ω = 0 Ω =V 0 E [E r ] 30 -5 0 -1 1 Ω = 0.1 V 0 z [d/2] 0 -1 1 Nonadiabatic case: weak coupling 0 1 2 3 -1 0 1 2 ħΩ [E r (2) ] e 0 e 1 e + e - E [E r (2) ] • weak coupling gives light shifts ±γ n=0 Ω/2 as expected for a two level system • upper states see off-resonant light shift ±γ n=0 Ω 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 e 0 e 1
Transcript of Driven Dynamics beyond the Born-Oppenheimer Approximation...
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
