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### Excitations in Bose-Einstein condensates……a long story

### Excitations in Bose-Einstein condensates……the first 2 years @

### Excitations in Bose-Einstein condensates……the most recent results @

### Excitations in Bose-Einstein condensates……the most recent results @

### Excitations in Bose-Einstein condensates……the most recent results @

### Excitations in Bose-Einstein condensates……the most recent results @

- Collective excitations and hydrodynamic equations
- Collective vs. single-particle
- Excitations in low dimensions
- Collapse, expansion and nonlinear dynamics
- Solitons

- Response of a condensate to a Bragg pulse
- Evaporation of phonons in a free expansion
- Landau damping of collective excitations

- Parametric resonances in optical lattices
- Pattern formation in toroidal condensates
- Stability of solitons in 2D

- Parametric resonances in optical lattices
- Pattern formation in toroidal condensates
- Stability of solitons in 2D

Parametric excitation of a Bose-Einstein condensate in a 1D optical lattice M. Kraemer, C. Tozzo and F. Dalfovo, Phys. Rev. A 71, 061602(R) (2005)

Stability diagram and growth rate of parametric resonances in Bose-Einstein condensates in 1D optical lattices C. Tozzo, M. Kraemer, and F. Dalfovo, Phys. Rev. A 72, 023613 (2005)

Starting point: experiments by Esslinger et al.

T.Stoeferle, et al., PRL 92, 130403 (2004); M.Koehl et al., JLTP 138, 635 (2005); C.Schori et al., PRL 93, 240402 (2004).

Starting point: experiments by Esslinger et al.

T.Stoeferle, et al., PRL 92, 130403 (2004); M.Koehl et al., JLTP 138, 635 (2005); C.Schori et al., PRL 93, 240402 (2004).

ωq= Ω /2

It’s a parametric resonance

Classical example: the vertically driven pendulum.

Stationary solutions: φ = 0 and φ = 180°. In the undriven case, these solutions are always stable and unstable, respectively. But vertical driving can change stability into instability and vice versa.

The dynamics is governed by the Mathieu equation:

(t) = c(t)exp( t), where c(t+1/f) = c(t).

Floquet exponent. If is is real and positive,

then the oscillator is parametrically unstable.

Very general phenomenon (classical oscillators, nonlinear optics,

systems governed by a Non-Linear Schroedinger Equation,

Hamiltonian chaotic systems, etc.)

Previously mentioned in the context of BEC by Castin and Dum, Kagan and

Maksimov, Kevrekidis et al., Garcia-Ripoll et al., Staliunas et al., Salasnich et al., Salmond et al., Haroutyunyan and Nienhuis, Rapti et al.).

Very recent experiments:

Parametric Amplification of Matter Waves in Periodically Translated Optical Lattices N. Gemelke, E. Sarajlic, Y. Bidel, S. Hong, and S. Chu Phys. Rev. Lett. 95, 170404 (2005)

Parametric Amplification of Scattered Atom Pairs Gretchen K. Campbell, Jongchul Mun, Micah Boyd, Erik W. Streed, Wolfgang Ketterle, and David E. Pritchard Phys. Rev. Lett. 96, 020406 (2006)

Important remark: in order to be parametrically amplified, the “resonant” mode must be present at t=0 (seed excitation). The parametric amplification is sensitive to the initial quantum and/or thermal fluctuations.

A deeper theoretical analysis in a simpler case:

no axial trap, infinite condensate, Bloch symmetry

GP equation:

with

Ground state + fluctuations:

j = band index

k = quasimomentum

Dynamics in a periodically modulated lattice

is small.

Assume the order parameter to be still of the form

where

at time t, is the solution of the stationary GP equation for s(t)

is small.

and

Linearized GP gives

This term is a source of excitations in the linear response regime. It is negligible in the range of Ω we are interested in.

Bloch wave expansion:

Floquet analysis:

Look for unstable regions

in the (Ω,k)-plane.

Calculate the growth rate

The lattice modulation enters here

(this equation is the analog of Mathieu

equation of classical oscillators)

Remarks on thermal and quantum seed

In GP simulations the seed is numerical noise or some extra noise added by hand to simulate the actual noise.

In the experimental BECs, the seed can be:

Excitations due to non-adiabatic loading of BEC in the lattice

Imprinted ad-hoc excitations

Thermal fluctuations

Quantum fluctuations

Remarks on thermal and quantum seed

GP theory

Excitations due to non-adiabatic loading of BEC in the lattice yes

Imprinted ad-hoc excitations yes

Thermal fluctuations no

Quantum fluctuations no

Remarks on thermal and quantum seed

Possible approach beyond GP: use the full Bogoliubov expansion with

operators, not c-numbers.

Use the Wigner representation of quantum fields.

In this way, the dynamics is still governed by “classical” Bogoliubov-like

equations; the depletion is included through a stochastic distribution of

the coefficients cjk.

Exact results can be obtained by averaging over many different realizations

of the condensate in the same equilibrium conditions. One has

Thermal fluctuations

Quantum fluctuations

Remarks on thermal and quantum seed

Two limiting cases:

Thermal fluctuations.

Possible measurement of T, even when the thermal

cloud is not visible (selective amplification of

thermally excited modes).

Amplification of quantum fluctuations.

Analogous to parametric down-conversion in quantum optics.

Source of entangled counter-propagating quasiparticles.

example:

Dynamic Casimir effect: the environment in which quasiparticles

live is periodically modulated in time and this modulation

transforms virtual quasiparticles into real quasiparticles (as

photons in oscillating cavities).

- Parametric resonances in optical lattices
- Pattern formation in toroidal condensates
- Stability of solitons in 2D

Detecting phonons and persistent currents in toroidal Bose-Einstein condensates by means of pattern formation

M. Modugno, C.Tozzo and F.Dalfovo, to be submitted (today!)

Bose-Einstein condensates have recently been obtained with ultracold

gases in a ring-shaped magnetic waveguide (Stamper-Kurn et al.)

Other groups are proposing different techniques to get toroidal condensates.

Main purpose: create a system in which fundamental properties, like quantized circulation and persistent currents, matter-wave interference, propagation of sound waves and solitons in low dimensions, can be observed in a clean and controllable way. An important issue concerns also the feasibility of high-sensitivity rotation sensors.

Our approach:

Parametric resonances as a tool to measure the excitation spectrum and rotations.

Advantage of toroidal geometry:

Clean response; nonlinear mode-mixing suppressed; periodic pattern formation.

- The condensate is initially prepared in the torus.
- The transverse harmonic potential is periodically modulated in time.
- Both the trap and the modulation are switched off and the condensate expands.

We solve numerically the time dependent GP equation, using the Wigner representation for fluctuations at equilibrium at step (i).

a periodic modulation of the confining potential of a toroidal condensate induces a spontaneous pattern formation through the parametric amplification of counter-rotating Bogoliubov excitations.

- This can be viewed as a quantum version of Faraday's instability for classical fluids in annular resonators.
- The occurrence of this pattern in both density and velocity distributions provides a tool for measuring fundamental properties of the condensate, such as the excitation spectrum, the amount of thermal and/or quantum fluctuations and the presence of quantized circulation and persistent currents.

- Parametric resonances in optical lattices
- Pattern formation in toroidal condensates
- Stability of solitons in 2D

Work in progress

Shunji Tsuchiya, L.Pitaveskii, F. Dalfovo, C.Tozzo

Motion in a Bose condensate: Axisymmetric solitary waves

Jones and Roberts, J. Phys. A 15, 2599 (1982)

Numerical solutions of GP equation.

A continuous family of solitary waves solutions is obtained.

At small velocity: a pair of antiparallel vortices, mutually propelling in obedience to Kelvin’s theorem.

At large velocity: rarefaction pulse of increasing size and decreasing amplitude.

Crow instability of antiparallel vortex pairs

Berloff and Roberts, J. Phys. A 34, 10057 (2001)

1D soliton in 2D:

Instability against transverse modulations (self-focusing)

Kuznetsov and Turitsyn , Zh. Eksp. Teor. Fiz. 94, 119 (1988)

Stability or instability of Jones-Roberts soliton in 2D

Our approach: calculate the real and imaginary (if any) eigenfrequencies of the linearized GP equation (Bogoliubov spectrum)

When U approaches the speed of sound:

Kadomtsev-Petviashvili equation

Linearized Kadomtsev-Petviashvili equation

Work in progress …

(instantaneous) Bogoliubov quasiparticle basis

Multi-mode coupling, induced by s(t)

Coupling parameters:

j-j’bands, same k

Nk constant

j-j’bands, opposite k

Exponential growth of Nk

Assumption: coupling by pairs.

Two-mode approximation

Replace sum over j’ with a single j’ and keep leading terms (small A):

with

and

growth rate

Resonance condition:

Growth rate on resonance:

with

Seed:

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