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