Quantum Scattering
in aCollider for Ultracold Atoms

Niels Kjærgaard

Department of Physics
QSO – Centre for Quantum ScienceDodd-Walls Centre for Photonic and Quantum Technologies

Quantum Scattering
in aCollider for Ultracold Atoms

Niels Kjærgaard

Department of Physics
QSO – Centre for Quantum ScienceDodd-Walls Centre for Photonic and Quantum Technologies

Colliding indistinguishable bosons (87Rb)

Quantum Scattering
in aCollider for Ultracold Atoms

Niels Kjærgaard

Department of Physics
QSO – Centre for Quantum ScienceDodd-Walls Centre for Photonic and Quantum Technologies

Anatomy of an Atom (87Rb): Internal structure

Atoms in magnetic field: Zeeman
87Rb Ground state splitting in magnetic field

Certain states minimize their energy by seeking a low magnetic field!

Magnetic atom trapConfinement

We can detect our atoms via absorption of resonant laser lights+d partial wave interferenceA steerable opticaltweezer platform(can use B-field asexternal tuning
parameter forinteractions)Dipole trap / Optical tweezer

Otago ultracold atom collider

Crossed laser trap for atoms

Laser beam can move atoms

Split operation: Moving two traps

colliding 87Rb87Rb (boson) angular scattering

40K (fermion) angular scattering

40K (fermion) angular scattering

No sideways scattering

Quantum scattering
Incoming (reduced) particle
potential

Quantum scattering

Quantum scattering
40K(�9 , 9+9 , 9�)

2 2 2 2

87Rb(|2,2⟩+|2,2⟩)

Quantum scattering

2+++

Example: s+p wave interference
constructive

87Rb(|2,2⟩+|1, − 1)

destructive

Round up the usual suspects

Round up the usual suspects

Note:We state energies in units of kB(1 µK ~ 100 peV)

Turning point

Wave functions above threshold
ShaperesonanceCorresponds to an S-matrix pole

Free space radial solution (to Schrödinger eq.)V(r)0.5 1 1.5 23r

Add wellV(r)0.5 1 1.5 23r

Phase shift with respect to free spaceV(r)0.5 1 1.5 23r

Bound state solutions – pure exp. decay
V(r)Scattering states

Bound state

Bound state

0.5 1 1.5 23r

Antibound stateV(r)Scattering states

Bound state

Bound state

0.5 1 1.5 23r

Antibound state – pure exp. growthV(r)Scattering states

Bound state
Antibound state
Bound state

0.5 1 1.5 23r

Scattering Matrix
V(r)0.5 1 1.5 23r

Scattering Matrix
V(r)0.5 1 1.5 23r

Analytic continuation S-matrix
V(r)

Im

Re

0.5 1 1.5 23r

Analytic continuation S-matrix
V(r)

Im

Re

0.5 1 1.5 23r

Bound and antibound states ⟺ poles on
imaginary axis of analytically continued S-matrix
V(r)

Im

Re

0.5 1 1.5 23r

Bound and antibound states ⟺ poles on
imaginary axis of analytically continued S-matrix
V(r)

Im

Re

0.5 1 1.5 23r

Weakening potential:
effect in complex k and E planes
ImReWeakening potential:
effect in complex k and E planes
ImImReReWeakening potential:
effect in complex k and E planes
ImImReReWeakening potential:
effect in E planeImImReReResonance stateAtomic Pole DanceS-matrix Pole (d-wave shape resonance)

Turning pointclosedlocallyopen

locallyopen

locallyopen

locallyopen

TunableB“q” (shape)

Fano profiles

E-B map

Round up the usual suspects

K-Rb: broad elastic s-wave Feshbach resonanceThomas et al., Nature Communications (2018)19 uK19 uK

19 uK19 uK19 uK52 uK

19 uK

52 uK

19 uK

52 uK

19 uK

52 uK

19 uK

E/kB=52 µK
Scattered fraction

Beutler-Fano
Line shape

Constructive

Destructive

B(G)Record resonant B-field vs energy
Antibound state

Fano-profile pos

Record resonant B-field vs energy

Record resonant B-field vs energy

Caption: “Effect of virtual state on Feshbach resonance…”

Record resonant B-field vs energy
Antibound state(virtual state)

Prediction

Bound state

Experiment

Prediction

Monday Talk

Revisitin MQDT framework

Open channel(s)Closed channel(s)Revisitin MQDT framework• Things only getcomplicated at short range• Complicated coupled multichannel short-range
interaction can be captured
by a single energyindependent quantity
unlocks QDT’s use of only the long-range vdW potential

Quantum defect theory frameworkSine at long range (phase shifted)

Increasing energy: faster sine at long range

Increasing energy: faster sine at long range

What about short range where well is VERY deep?

Short range where well is VERY deep:
Wave function looks the same!

Short range where well VERY deep:
Wave function looks the same!

Quantum defect theory framework

Solution linear combination

FreephaseSolution linear combination

Solution linear combination

Two pairs of reference functions

Solution linear combination

Choice advantageous for numerics

Analytic rotationof QDT parameters? Why pick
Because this choice reproduces
as the backgroundphase shift inClosed channel QDT parameter

The above can be shown to lead to aparametrization of the resonance position

Prediction for widths of Fano profiles

Microscopy of a Feshbach resonance

Email fram Ravi Rau:
“I read you very nice posting on shape and Feshbachresonances in a QDT analysis.
It is a very rich paper and Fig.7 impressive.”

TeamAmita
Deb

James Croft

Ryan
ThomasBiancaSawyerCraig
Chisholm

MilenaHorvathMatthew
ChilcottContributions• Pristine system forelucidating paradigms ofquantum scattering

Contributions• Pristine system forelucidating paradigms ofquantum scattering

Interplay between S-matrix poles:• Feshbach resonance + antibound state• Feshbach resonance + shape resonanceContributions• Pristine system forelucidating paradigms ofquantum scattering

Interplay between S-matrix poles:• Feshbach resonance + antibound state• Feshbach resonance + shape resonance• MQDT analysis

Contributions• Pristine system forelucidating paradigms ofquantum scattering

Interplay between S-matrix poles:• Feshbach resonance + antibound state• Feshbach resonance + shape resonance• MQDT analysis

Non-Hermitian effective Hamiltonian

Look at above table in light ofwhat we found for:

Non-Hermitian effective Hamiltonianstrong coupling

Exceptional point