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Controlling Quantum Systems

Thomas Busch

Controlling Quantum Systems with Spatial Adiabatic Passage

Motivation: Complex Quantum System Dynamics

Step 1: learn how to control small quantum systems (one or two particles) Step 2: learn how to control large quantum systems (more than two particles)

i~ ∂ ∂t|ψ(x1, x2; t)i = H(t)|ψ(x1, x2; t)i i~ ∂ ∂t|ψ(x1, x2, x3, . . . ; t)i = H(t)|ψ(x1, x2, x3, . . . ; t)i

bottom-up: following experimental progress, clean and controllable systems

Develop Applications of QM Quantum Computing Quantum Simulators Quantum Metrology … Find QM in real life Energy Transport Charge Transfer Quantum Phase Transitions … Understand Fundamental QM Entanglement Non-locality Decoherence …

| i

Guiding Principles Why

quantum systems are very fragile and have a massive Hilbert space identify systems which can be engineered develop techniques for quantum engineering find techniques for scaling quantum systems up do this in close collaboration with experimentalists System of Choice: ultracold atoms

Challenge

superfluid vortices as topological matter creation of non-classical states quantum walks as quantum memories nano-sensors for single atom states

Examples of Projects @ OIST

atom-ion hybrid systems multicomponent superfluids spin-orbit coupled superfluids adiabatic engineering techniques & shortcuts

Spatial Adiabatic Passage

?

How to move an atom? Solution: Tunneling (increase and decrease the distance between traps)

!

Problem: Fragile Process (Rabi Oscillations) success depends on good control of three parameters interaction time approach time minimum distance between traps precise experimental control necessary for high fidelities ( > 99.9999%)

ti ta amin

Spatial Adiabatic Passage

How to move an atom? Solution: Tunneling (increase and decrease the distance between traps) Problem: Fragile Process (Rabi Oscillations) success depends on good control of three parameters interaction time approach time minimum distance between traps precise experimental control necessary for high fidelities ( > 99.9999%)

ti ta amin

Spatial Adiabatic Passage

Sequential

time

same problem, only twice Tunneling

1 2

Quantum Systems Unit: Spatial Adiabatic Passage

Counterintuitive Tunneling

time

100% transfer! ( STIRAP)

1 2

K Eckert, M Lewenstein, R Corbalán, G Birkl, W Ertmer, J Mompart Physical Review A 70, 023606 (2004)

10

Why does this work?

|1i |2i |3i ✏   ✏ Ω12 Ω12 ✏ Ω23 Ω23 ✏  

|1i |1i |2i |2i |3i |3i

Ω23 Ω12

|Ψi = cos θ|1i sin θ|3i

eigenstate connects 1 and 3

tan θ = Ω12 Ω23

sin θ : 0 → 1 cos θ : 1 → 0

TRANSFER:

tan θ : 0 → ∞ θ : 0 → π 2

generalise to many particle systems

All good. Now what?

generalise beyond 1D find shortcuts to avoid adiabatic restrictions identify suitable experimental settings for observation identify non-classical correlations develop into other engineering tools: quantum state preparation, deterministic single atom source …. find shortcuts to avoid adiabatic restrictions

1

generalise to many particle systems

2

12

Shortcut To Adiabaticity

Idea: add terms to Hamiltonian that compensate for diabatic excitations when driving is non-adiabatic

H = H0 + H1 Ω12 Ω23

H0(t) = ~ 2   Ω12(t) Ω12(t) Ω23(t) Ω23(t)  

[ Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010) ]

H1(t) = ~ 2   iΩ13(t) −iΩ13(t)  

Ω13

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Spatial Adiabatic Passage in 2D

now:

Ω12 1 2 3 Ω23

up to

Ω31

symmetry breaking gives additional coupling use this degree of freedom to make new states

Tunneling-induced angular momentum for single cold atoms

• R. Menchon-Enrich, S. McEndoo, J. Mompart, V. Ahufinger and TB,
• Phys. Rev. A 89, 013626 (2014)

create angular momentum

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Shortcut To Adiabaticity

3 2 1

H(t) = ~ 2   Ω12(t) iΩ13(t) Ω12(t) Ω23(t) −iΩ13(t) Ω23(t)  

but: shortcut Hamiltonian is imaginary! cannot get this phase dynamically for transition between eigenstates

Ω13 ⇠ h1|H|3i

use geometric phase!? nice idea, but cannot be implemented for SAP…?!

15

Geometric Phase

assume charged particle moving in a magnetic field that is constant everywhere (or localised) Brief Reminder: Aharanov Bohm Effect phase is added when particle moves from to

~ ri

~ rj

ij = q ~ Z ~

rj ~ ri

~ A · d~ l

are the positions of the wells and is the magnetic vector potential

~ ri

~ A total phase in a closed loop:

magnetic flux through the closed path around the triangle

Φ = 12 + 23 + 31 = q ~ I ~ A · d~ l = q ~ΦB

3 2 1

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Geometric Phase

HAB = −~ 2   Ω12e−iφ12 Ω13eiφ31 Ω12eiφ12 Ω23e−iφ23 Ω13e−iφ31 Ω23eiφ23  

what we want:

H = −~ 2   Ω12 −iΩ13 Ω12 Ω23 iΩ13 Ω23  

engineer field such that φ12 = φ23 = 0

φ31 = −π 2

and requires field with specific spatial profile

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Geometric Phase

change basis using only local phases

U =   e

i 2 (φ12+φ23)

e

i 2 (−φ12+φ23)

e− i

2 (φ12+φ23)

 

so that we get H0

AB = UHABU 1 = −~

2   Ω12 Ω31eiΦ Ω12 Ω23 Ω31eiΦ Ω23   and therefore only need

Φ = −π 2 ΦB = I ~ A · d~ l = −~⇡ 2q

can be achieved with homogeneous field distribution

• nly relevant value is the total phase (or total flux)

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Spatial Adiabatic Passage in 2D

3 2 1

with shortcut pulse phase time no shortcut pulse time phase

full transfer (adiabatic) low transfer (fast)

0.1 0.3 0.5 0.7 0.9

can also be inverted to measure magnetic fields!

generalise to many particle systems

All good. Next…

generalise beyond 1D find shortcuts to avoid adiabatic restrictions identify suitable experimental settings for observation identify non-classical correlations develop into other engineering tools: deterministic single atom source generalise beyond 1D find shortcuts to avoid adiabatic restrictions

1

generalise to many particle systems

2

20

Interactions

resonance not guaranteed dark state not guaranteed

g = ∞

g = 0

non-interacting bosons strongly interacting bosons (non-interacting fermions)

21

Weak Interactions

Three-well Bose-Hubbard model: diagonalise and find two energy bands

E = 1

particles are in different wells

E = 1 + U

particles are in same well

2 2 2 1 1 1 1 1 1

22

Weak Interactions

interaction leads to band separation co-tunneling restores the dark state

E = 1.05 E = 1.25

level crossings make following the dark state effectively impossible

23

Strong Interactions

Three-well Fermi-Hubbard model: particles are in different wells particles are in same well

E = 2

E = 2 − |U|

diagonalise and find two energy bands

24

Strong Interactions

|U| = 0.3 |U| = 0.15

level crossings make following the dark state effectively impossible interaction leads to band separation co-tunneling restores the dark state

25

Exact Diagonalisation

interaction band is isolated, but crossings still exist! adiabatic and diabatic dynamics can lead to full transfer

26

Summary

Spatial Adiabatic Passage possess an experimentally implementable Shortcut to Adiabaticity Interactions can lead to band-separation that allow to use single particle ideas for many-particle systems Adiabatic techniques are not necessarily slow or limited to single particles.

Collaborations

Albert Benseny Irina Reshodko Tara Hennessy Lee O’Riordan Yongping Zhang Angela White Rashi Sachdeva Thomas Fogarty James Schloss Andreas Ruschhaupt Anthony Kiely Jeremie Gillet TB