Invariants of motion as a toolbox for ultracold gases

Adolfo del Campo

Institut für Theoretische Physik

Universität Ulm

Contents

Shortcut to adibaticity:

How to perform fast expansions without

vibrational heating

Tuning interactions:

How to tune the amplitude of the coupling

constant in a low-dimensional BEC

Fast expansion without vibrational heating

- Invariants- Physical Realization

Motivation

• Most current experiments with cold atoms involve an adiabatic adjustment (expansions, contractions) as part of the preparation.

-Prepare atoms on a lattice -Reach very low T-Reduce v in spectroscopy&

metrology

• Bottleneck step in a “quantum refrigerator cycle” (Rezek et al 2009)

Example: the Tonks-Girardeau regime

Recipy by Olshanii et al. PRL. 81, 938 (1998), ibid 86, 5413 (2001)

Lieb-Liniger gas with

TG: Predicted by Girardeau J. Math. Phys. 1 516 (1960)

Effective 1D gas of hard-core bosons

strong interactions mimick exclusion principle fermionization

44 years later: the experiment

Paredes et al. Nature 429, 227 (2004)

Kinoshita et al. Science 305, 1125 (2005)...

Example: the Tonks-Girardeau regime

Example: Bose-Fermi duality

Girardeau 1960: The Bose-Fermi map

Dual system:

Symm. operator:

The TG gas:

It's involutive

Any local correlation function is identical for both dual systems

Density profile in the ground state:

Girardeau

Example: Standard Quench

Sudden quench: Minguzzi, Gangardt. PRL 94, 240404 (2005)

Smooth finite-time quench

i=f/10 =0.1 /i 10 /i 100/i

breathing of the cloud

Lewis-Riesenfeld invariants

Lewis & Riesenfeld 1969

conjugate to

Ermakov equation

In general the state is a superposition of “expanding modes”

dI/dt=0

I(t)|n(t)= |n(t)

“Inverse problem” strategy - Leave (t) undetermined at first- Impose boundary conditions on b so that(a) |n(0) is the eigenstate |un(0) of H(0)(b) |n(tf) becomes the nth eigenstate |un(tf) of H(tf)

Formally this requires[H(0),I(0)]=0 smooth driving [H(tf),I(tf)]=0 (no vibrational heating)

- Interpolate b(t)- Get (t) from Ermakov eq.

Boundary conditions• t=0

H(0)=I(0) |n(0)> = |un(0)> Just one expanding mode for initial nth state

At intermediate times H(t) and I(t) do not commute so that |n(t) may have more components in the instantaneous basis

, ,

Boundary conditions• t=0

H(0)=I(0) |n(0)> = |un(0)> Just one expanding mode for initial nth state

At intermediate times H(t) and I(t) do not commute so that |n(t) may have more components in the instantaneous basis

• t=tf

|n(tf)=|un(tf)> , and E is minimized

The state becomes un times the phase factor

, ,

Inverse engineering

1- Interpolate between 0 and tf with an ansatz, e.g. exponential of polynomial

polynomial

Inverse engineering

1- Interpolate between 0 and tf with an ansatz, e.g.

2- Get (t) from Ermakov equation

6 ms

10 ms

15 ms

25 ms

exponential of polynomial

polynomial

The potential may become expulsive (tf<1/(2f)=25 ms)

initial

final

intermediate

Energies and frequencies for different tf (polynomial b, ground state)

Example

Time Evolution:

Hzf 25.2

Hz 22500

mst f 2

2Ψ(t,x)

2V(t,x)

Example

Time Evolution:

Hzf 25.2

Hz 22500

mst f 2

2Ψ(t,x)

2V(t,x)

Compare to adiabatic trajectories

Adiabaticity condition

Linear ramp

Better strategy: solve

45 ms for a 1% relative error

Comparison with bang-bang method The “bang-bang” (piecewise constant ) method is optimal for 1, 2>0 Salamon et al. (2009)

The “minimal time” (6 ms) can be improved by allowing for imaginary intermediate

However it is difficult to realize a discontinuous jump

polynomialExp of polynomial

bangbang

The inverse-invariant method works for all states

n=3

n=2

n=1

Ref: Chen et al. PRL 104, 063002 (2010)

Physical realization

• With highly detuned Gaussian beams the effective potential for the ground state is

V(x)=(x,t)/4

• Combining red and a blue detuned lasers

with t-dependent intensities we may change

and make it <0.

A Fast Squeezing/Expansion may spill the water (and the atoms)because of “anharmonicity”Even if the atoms stay, the invariant has been obtained for the harmonic trap

Anharmonicities, 3D and all that• Ongoing work• Actual traps are 3D and

not harmonic (typically Gaussian)

• From t-dependent perturbation theory

for a good fidelity (.99 with w=150 m, 2ms)

• Moreover, there are also invariants for anharmonic potentials

• One can play with t-dependence of intensities (& in principle waists) of 2 or more lasers to minimize

anharmonicity & longitudinal/radial couplings.

Invariants for generalized potentials

• [Lewis&Leach 82]• We may construct invariants for more general

Hamiltonains, • in particular containing x4/b6 and 1/x2 terms

• With enough number of degrees of freedom (e.g. number of lasers) we could in principle make them vanish or control all terms…

First experiment (G. Labeyrie et al. 2010): 87 Rb in Ioffe-Pritchard trap

Phys. Rev. A 82, 033430 (2010)

Second experiment (G. Labeyrie et al. 2010): Bose-Einstein Condensate

arXiv:1009.5868

Can I use it?

Thermal gas

Calogero-Sutherland model

Tonks-Girardeau gas /polarized fermions

Excited Lieb-Liniger gases

1d, 2d, 3d Bose-Einstein condensates

…

Extensions to

Luttinger liquids

Dipolar gases (TF)

Ion chains

II How to tune the amplitude of the coupling constant

in a low-dimensional BEC

A. del Campo, TBS

Feschbach ResonancesE. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, PRA 47, 4114 (‘93); Exp: P. Courteille PRL 81, 69 (‘98), S. Inouye et al., Nature 391, 151 (‘98)

Confinement Induced Resoanances

M. Olshanii et al., PRL 81, 938 (1998), ibid 86, 5413 (‘01)

M. Girardeau et al., Optics Communications 243, 3 (‘04)

Modulating the transverse confinement

K. Staliunas, et al., PRA 70, 011601(R) (‘04)

P. Engels et al., PRL 98, 095301 (‘07)

Tuning interactions

Cigar-shaped cloud

Dimensionality parameter

Dimensional reduction V. M. P´erez-Garc´ıa, H. Michinel and H. Herrero, PRA 57, 3837 (1998).L. D. Carr and Y. Castin, PRA 66, 063602 (2002).W. Bao et al., J. Comp. Phys. 87, 318 (2003).

Transverse self-similar dynamics

Solution of the Ermakov equation: scaling factor

Tuning interactions: invariants of motion

Effective coupling after dimensional reduction

Boundary conditions

An arbitrary time dependence can be engineered by inverting the Ermakov equation

Tuning interactions: invariants of motion

Applications

Probing ultracold gases by TOF

Decay of the interactions under free expansion

cigar-shaped clouds

pancake clouds

negligible after

Lost of correlations in a length scale

Engineering an exponential decay

Preserving short-range correlations in TOF

Example: cigar-shape cloud

Boundary conditions

Required trajectory of the transverse frequency

Not-positive definite

It might require bang-bang like discontinuous jumps in the transverse frequency

Preserving short-range correlations in TOF

Nearly sudden quenches

Polynomial ansatz

Coefficients fixed by boundary conditions

Ratio of initial and final couplings

Self-similar dynamics of BEC

In quasi-1D, outside the Thomas-Fermi regime,

Axial scaling factor obeys

self-similar dynamics requires tuning of the interactions

Which can be induced by a modulation of the transverse confinment

Assisted self-similar expansion of a1D BEC

Outlook

Applications:

Preparation of atomic Fock states by squeezing out of the

trap the excess of atoms

Dipolar gases

Optical lattices

More general potentials, transport, etc.

C.-S. Chuu et al. PRL 95, 260403 (2005)

A. del Campo, J. G. Muga, PRA 78, 023412 (2008)

M. Pons et al. PRA 79, 033629 (2009)

D.Sokolovski, et al.. arXiv:1009.0640

People

Xi Chen

Adolfo del

Campo

Andreas Ruschhaupt

Gonzalo Muga

David Guéry-Odelin

Thanks for your attention!

Thanks to

• M. V. Berry and J. Eberly

• Wheeled animals by Andrée Richmond http://www.andreerichmond.com

M. V. Berry

Joe Eberly

And specially to