SAP beyond STIRAP: interactions, higher dimensions and shortcuts Albert Benseny Quantum Systems Unit Okinawa Institute of Science and Technology
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
SAP beyond STIRAP: interactions, higher
dimensions and shortcuts
Albert BensenyQuantum Systems Unit
Okinawa Institute of Science and Technology
Outline
• S'up with SAP? Transport in a triple well
• SAP processes for two interacting particles
• Quantum engineering: particle separation
• Speeding up spatial adiabatic passage
Spatial adiabatic
passage in a triple well
• The 3L Hamiltonian has an eigenstate connecting 1 and 3
• This gives us an adiabatic transport process (~STIRAP)
SAP in a triple well
H3L =~
2
0 Ω12 0
Ω12 0 Ω23
0 Ω23 0
|1i |2i |3i
|1i
|2i
|3i
|Di = cos θ|1i − sin θ|3i
tan θ =Ω12
Ω23
|Di : |1i ! |3iθ : 0 ! π/2Ω : Ω23 # Ω12 ! Ω12 # Ω23
counterintuitive
tunneling
sequence
|1i |2i |3iΩ23Ω12
→
SAP in a triple well
−10
−5
0
5
10
−0.2
−0.1
0
0.1
0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
dj
(a)
Energy (b)
t/T
|⟨j|D⟩|2 (c)
Trap movement
Energy spectrum
Dark state
Ω23 ! Ω12 Ω12 ! Ω23
SAP processes of
interacting particles
Phys Rev A 93, 033629 (2016)
drawing: Ayaka Usui
Two atoms in a triple well
• Hamiltonian for two atoms
• Initial and final states: two atoms in a harmonic trap
[Busch et al, Found Phys 28, 549 (1998)]
H = −
1
2
∂2
∂x2
1
+ V (x1, t)−1
2
∂2
∂x2
2
+ V (x2, t) + gδ(x1 − x2)
|Li |Ri
g = −
2√
2Γ(1− Eg/2)
Γ((1− Eg)/2),
(~ = ω = m = 1)
0 ≤ g < ∞
1 ≤ Eg ≤ 2
non-interacting
case
Tonks-Girardeau
regime
Two atoms in three traps: states
• Atoms in different sites
• Atoms in the same site
• Atoms in different sites and levels
• Resonance is no longer guaranteed… Will SAP still work?
E ~ 1
E ~ Eg
E ~ 2
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
• Single particle SAP [Eckert et al, PRA 70, 023606 (2004)]
• Atoms fermionize and are in different states
[Loiko et al., PRA 83, 033629 (2011)]
• Large plateau where F = 1!
Fidelity of 2-particle SAP
Simulations of
TDSE with
exact 1D
Hamiltonian
At weak interactions
• Bose-Hubbard model for the lowest Bloch band
HB =X
j=L,M,R
U
2nj(nj − 1) + 0nj
]
+h
ΩLMb†LbM +ΩMRb
†MbR +Ω
(co)LM b
†2L b
2M +Ω
(co)MR b
†2Mb
2R + h.c.
i
interactions
ground state energy = 1/2
two-particle
co-tunneling
single-particle
tunneling
At weak interactions
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
8 0.2 0.4 0.6 0.8
Eg = 1.25
t/T
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
0
Energy
Energy
|⟨nj|D
⟩|2
The process works if the
interaction is strong enough
to separate the bands
At weak interactions
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Eg = 1.05
Energy
Eg = 1.25
Energy
t/T
|⟨nj|D
⟩|2
t/T
The process works if the
interaction is strong enough
to separate the bands
At strong interactions
• Attractive Fermi-Hubbard Hamiltonian (lowest two bands)
HF =X
j=L,M,R
"
Unj0nj1 +X
i=01
inji
#
+
+X
i=0,1
h
Ω(i)LMa
†LiaMi +Ω
(i)MRa
†MiaRi + h.c.
i
+Ω(co)LM a
†L0a
†L1aM0aM1 +Ω
(co)MRa
†M0a
†M1aR0aR1 + h.c.
interactions
two-particle
co-tunneling
single-particle
tunneling
1/2 or 3/2
At strong interactions
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1.4
1.6
1.8
2
2.2
2.4
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Eg = 1.6
Energy
Eg = 1.85
Energy
t/T
|⟨nji|D
⟩|2
t/T
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
If the interaction is not strong enough,
the bands stay separated.
But... doesn't it stop too early?
Level crossings in the dark-state
1.235
1.24
0.32 0.33 0.34
Eg = 1.25
(b)
8 0.2 0.4 0.6 0.8
t/T
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Energy
0.2 0.4 0.6 0.8 0
t/T
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1.57
1.58
0.39 0.4 0.41
Eg = 1.6
Energy
(a)
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
Depending on how we
take these crossings,
SAP may fail...
Level crossings
0.2 0.4 0.6 0.8 0
t/T
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1.57
1.58
0.39 0.4 0.41
Eg = 1.6
Energy
(a)
103
106
109
1012
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0
0.2
0.4
0.6
0.8
1
adiabatic
diabatic
T
Eg
pi→j '
∣
∣
∣
R tf
t0hj(t)| d
dt|i(t)ie
iR
t
t0(Ej(τ)−Ei(τ))dτdt
∣
∣
∣
2
∣
∣
∣
R tf
t0hj(t)| d
dt|i(t)idt
∣
∣
∣
2 ,
Transition probability at the crossing
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
dia
batic
ad
iab
atic
Cotunnelling
HB =X
j=L,M,R
U
2nj(nj − 1) + 0nj
]
+h
ΩLMb†LbM +ΩMRb
†MbR +Ω
(co)LM b
†2L b
2M +Ω
(co)MR b
†2Mb
2R + h.c.
i
0.8
1
1.2
1.4
0.2 0.4 0.6 0.80.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
t/T
Energy
(a)
t/T
(b)
t/T
(c)
exact with CT without CT
dark state exists
0
0.2
0.4
0.6
0.8
1
0
|⟨nj|ψ
(t)⟩|2
80.2 0.4 0.6 0.8 0
t/T
(e)
dark state
does not exist
80.2 0.4 0.6 0.8 0
t/T
(e)
0
0.2
0.4
0.6
0.8
1
|⟨nj|ψ
(t)⟩|2
State engineering
• We have used SAP to transport a pair of particles, and any
single particle process can also be used.
• It is very simple to create a NOON state by just doing half the
transport.
• What about separating the particle pair...
Quantum engineering:
particle separation
(work in progress)
1
Quantum particle
dispenser
Irin
a R
eshod
ko
...
We need to allow the bands
to “talk to each other”
Particle separation
Eg = 1.25
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Energy
8 0.2 0.4 0.6 0.8
t/T
Finding the optimal lift
• The lift will depend on the interaction, and its optimal value is
not obvious.
• We numerically find the optimal value of the lift for a few
values of Eg and interpolate over those
Lmax
Eg = 1.4
Particle separation fidelity
Particle separation spectrum
Separating 3 particles
2-particle ground state energy
more bands:
harder...
• Experimentally-realistic systems have already been proposed
for SAP [Morgan et al, PRA 85, 039904 (2012)]
Radio-frequency traps
ω0.2 0.4 0.6 0.8 1.0
000
000
000
×106
/
1
0.8
0.6
0.4
ω(M
Hz)
t/T
Separation fidelity in RF traps
Speeding up SAP
‸+Andreas Ruschhaupt
+Anthony Kiely
(work in progress)
Shortcuts to adiabaticity
• Transitionless quantum driving: Add terms to Hamiltonian to
compensate for non-adiabatic excitations
• For the SAP/STIRAP case...
this is
[Rice+Demirplak J Phys Chem A 107, 9937 (2003)]
[Chen et al, Phys. Rev. Lett. 105, 123003 (2010)]
H = H0 +H1
H0(t) =~
2
0 Ω12(t) 0Ω12(t) 0 Ω23(t)
0 Ω23(t) 0
H1 = i~
X
n
(|∂tλnihλn| − hλn|∂tλni|λnihλn|)
Creating the new coupling
• Break symmetry for additional coupling!
• This additional coupling can be used in SAP to create states
with angular momentum or design an interferometer
[Menchon-Enrich et al, PRA 89, 013626 (2014), PRA 89,
053611 (2014)].
Ω12
1
2 3Ω23
Ω31
Creating the new coupling
• We have the 1-3 coupling, but the shortcut Hamiltonian is
imaginary!
• Couplings are usually real as eigenstates are (usually) real:
• Aharonov-Bohm effect: a charged particle in a magnetic field
acquires a geometric phase when moving between points
32
1
H(t) =~
2
0 Ω12(t) iΩ13(t)Ω12(t) 0 Ω23(t)
−iΩ13(t) Ω23(t) 0
Ω13 h1|H|3i
φij =q
~
Z ~rj
~ri
~A · d~l
trap positions
vector potential
idea! add a geometric phase
Geometric phase: the hard way
• The Aharonov-Bohm phases give us the Hamiltonian
• what we want
• apply a field magnetic such that
but this requires a field with a very specific spatial profile:
very hard!
HAB = −~
2
0 Ω12e−iφ12 Ω13e
iφ31
Ω12eiφ12 0 Ω23e
−iφ23
Ω13e−iφ31 Ω23e
iφ23 0
φ12 = φ23 = 0 φ31 = −π/2
H = H0 +H1 = −~
2
0 Ω12 −iΩ13
Ω12 0 Ω23
iΩ13 Ω23 0
|1i
|2i |3i
|1i
|2i |3i
Geometric phase: the easy way
• Change basis using only local phases
we get
• We only need a total phase in the closed loop
this can be achieved with an
homogeneous magnetic field:
H0
AB = UHABU−1
= −
~
2
0 Ω12 Ω31eiΦ
Ω12 0 Ω23
Ω31e−iΦ Ω23 0
U =
ei
2(φ12+φ23) 0 0
0 ei
2(−φ12+φ23) 0
0 0 e−
i
2(φ12+φ23)
ΦB =
I~A · d~l =
ZZ~B · d~S = −
~
2q
Φ = φ12 + φ23 + φ31 = −π/2
|1i
|2i |3i
q
• Shortcut pulse is a π pulse:
Why the i ?
|1i
|2i |3i
q
SAP
shortcut
for constructive
interference....
Ω12
Ω23
Ω31
t/T
Shortcut fidelity
• With the right phase, we have 100% fidelity for all T.
• We can use this also to measure the magnetic field!
with shortcut pulse
phase
tota
l tim
e
0.1
0.3
0.5
0.7
0.9
no shortcut pulse
tota
l tim
e
full transfer (adiabatic)
low transfer (too fast) 0.1
0.3
0.5
0.7
0.9
appropriate
phase/magnetic field
phase
Lewis-Riesenfeld invariants
• For the Hamiltonian is
• Because this is a closed algebra, we can use Lewis-
Riesenfeld invariants to drive the dynamics.
[Lewis+Riesenfeld, J Math Phys 10, 1458 (1969)]
[Chen et al, Phys Rev Lett 104, 063002 (2010)]
Φ =π
2K1 =
0 1 0
1 0 0
0 0 0
K2 =
0 0 0
0 0 1
0 1 0
K3 =
0 0 −i
0 0 0
i 0 0
Lewis-Riesenfeld invariants
• A Lewis-Riesenfeld invariant (LRI) fulfills
and has time-independent eigenvalues.
• Any solution to the TDSE can be written as
• Inverse engineering:
Follow an eigenstate of a LRI instead of of H.
Ensure that
We’re free to chose the dynamics in between
∂I
∂t+
i
~[H, I] = 0.
|ψ(t)i =X
k
ck |ψk(t)i
eigenstates of the LRI
[I(0), H(0)] = [I(T ), H(T )] = 0
• In our case, the LRI take the form
[Chen et al, Phys Rev Lett 104, 063002 (2010)]
• with eigenstates
For we recover the
SAP dark state, and…
Lewis-Riesenfeld invariants
I =~
2(− sinϕ sin θK1 − sinϕ cos θK2 + cosϕK3) .
θ =1
2(−Ω23 sec θ cotϕ− Ω13) +
1
2tan θ cotϕ (Ω23 sin θ − Ω12 cos θ)
ϕ =1
2(Ω12 cos θ − Ω23 sin θ)
|φ0i =
cos θ sinϕ
i cosϕ
− sin θ sinϕ
|φ±i =1p2
sin θ i cosϕ cos θ
sinϕ
cos θ ± i cosϕ sin θ
Ω13 = −2θ
ϕ = π/2
• Initial/final state:
• Boundary conditions:
Creating an arbitrary superposition
P1
P2
P3
F
t/T
|Ψ(t = 0)i = |1i
|Ψ(T )i = (|1i+ i|2i − |3i)/p3
θ(0) = −
π
2
θ(T ) = − arctan√
2
α(0) = 0
α(T ) =π
2.
Ω12
Ω23
Ω31
t/T
To summarise…
Summary
• SAP is not necessarily limited to single particles or long times.
• Interactions can create a band separation that allows to use
single particle ideas in many-particle systems[Phys Rev A 93, 033629 (2016)]
• We have extended SAP to
separate particles
• Spatial adiabatic passage possesses an
experimentally implementable shortcut
to adiabaticity
0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
T = 4000
T = 12000
Eg
F
ThomasIrina
Jérémie
Yongping
‸Anthony Kiely
Andreas Ruschhaupt
Related Documents
