Invariants of motion as a toolbox for ultracold gases

1. Invariants of motion as a toolbox for ultracold gases Adolfo del Campo Institut für Theoretische Physik Universität Ulm

2. 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

3. Fastexpansionwithoutvibrationalheating - Invariants - Physical Realization

4. 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 Dv in spectroscopy& metrology • Bottleneck step in a “quantum refrigerator cycle” (Rezek et al 2009)

5. Example: the Tonks-Girardeau regime

6. Example: the Tonks-Girardeau regime • Recipy by Olshaniiet al. PRL. 81, 938 (1998), ibid86, 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)‏ • ...

7. 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

8. 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

9. Lewis-Riesenfeldinvariants Lewis & Riesenfeld 1969 conjugate to dI/dt=0 Ermakov equation In general the state is a superposition of “expanding modes” I(t)|n(t)>= |n(t)>

10. “Inverse problem” strategy • Leave w(t) undetermined at first • Impose boundary conditions on b so that • |n(0)> is the eigenstate |un(0)> of H(0) • |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 w(t) from Ermakov eq.

11. Boundaryconditions , , • 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

12. Boundaryconditions , , • 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

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

14. Inverseengineering polynomial 1- Interpolate between 0 and tf with an ansatz, e.g. exponential of polynomial 2- Get w(t) from Ermakov equation The potential may become expulsive (tf<1/(2wf)=25 ms) Energies and frequencies for different tf (polynomial b, ground state) initial final 6 ms intermediate 10 ms 15 ms 25 ms

15. Example Time Evolution:

16. Example Time Evolution:

17. Compare toadiabatictrajectories Adiabaticity condition Linear ramp Better strategy: solve 45 ms for a 1% relative error

18. Comparisonwithbang-bangmethod The “bang-bang” (piecewise constant w) method is optimal for w1, w2>0 Salamon et al. (2009) The “minimal time” (6 ms) can be improved by allowing for imaginary intermediate w polynomial Exp of polynomial bangbang However it is difficult to realize a discontinuous jump

19. Theinverse-invariantmethodworksforallstates n=3 n=2 n=1 Ref: Chen et al. PRL 104, 063002 (2010)

20. Physicalrealization • With highly detuned Gaussian beams the effective potential for the ground state is V(x)=W2 (x,t)/4D • Combining red and a blue detuned lasers with t-dependent intensities we may change w2and make it <0.

21. 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

22. Anharmonicities, 3D and allthat • Ongoing work • Actual traps are 3D and not harmonic (typically Gaussian) • From t-dependent perturbation theory for a good fidelity (.99 with w=150 mm, 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.

23. 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…

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

25. Phys. Rev. A 82, 033430 (2010)

26. Second experiment (G. Labeyrie et al. 2010): Bose-Einstein Condensate arXiv:1009.5868

27. 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

28. II Howto tune theamplitude of thecouplingconstant in a low-dimensional BEC A. del Campo, TBS

29. Tuning interactions • Feschbach Resonances • E. 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. Olshaniiet al., PRL 81, 938 (1998), ibid86, 5413 (‘01) • M. Girardeauet al., Optics Communications 243, 3 (‘04) • Modulating the transverse confinement • K. Staliunas, et al., PRA 70, 011601(R) (‘04) • P. Engelset al., PRL 98, 095301 (‘07)

30. Tuning interactions: invariants of motion 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

31. Tuning interactions: invariants of motion Effective coupling after dimensional reduction Boundary conditions An arbitrary time dependence can be engineered by inverting the Ermakov equation

32. Applications

33. Preserving short-range correlations in TOF • 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

34. 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

35. Nearly sudden quenches Polynomial ansatz Coefficients fixed by boundary conditions Ratio of initial and final couplings

36. Assisted self-similar expansion of a1D BEC • 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

37. 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

38. David Guéry-Odelin People Xi Chen Thanks for your attention! Andreas Ruschhaupt Adolfo del Campo Gonzalo Muga

39. Thanks to • M. V. Berry and J. Eberly • Wheeled animals by Andrée Richmondhttp://www.andreerichmond.com Joe Eberly M. V. Berry

40. And specially to