2D Solitons in Dipolar BECs 1I. Tikhonenkov, 2B. Malomed, and 1A. Vardi1Department of Chemistry, Ben-Gurion University2Department of Physical Electronics, School of Electrical Engineering, Tel-Aviv University

Ein Gedi, Feb. 24-29, 2008

Dilute Bose gas at low T

Ein Gedi, Feb. 24-29, 2008

Gross-Pitaevskii descriptionLowest order mean-field theory:Condensate order-parameterGross-Pitaevskii energy functional:minimize EGP under the constraint:Gross-Pitaevskii (nonlinear Schrdinger) equation:

Ein Gedi, Feb. 24-29, 2008

Variational CalculationEvaluation of the EGP in an harmonic trap, using a gaussian solution with varying width b. Kinetic energy per-particle varies as 1/b2 - dispersion.Nonlinear interaction per-particle varies as gn - g/b3 in 3D, g/b in 1D.In 1D with g

SolitonsLocalized solutions of nonlinear differential equations.Result in from the interplay of dispersive terms and nonlinear terms.Propagate long distances without dispersion.Collide without radiating. Not affected by their excitations.

Ein Gedi, Feb. 24-29, 2008

Zero-temperature BEC solitons NLSE in 1D with attractive interactions (g

Zero-temperature BEC solitons

Ein Gedi, Feb. 24-29, 2008

Observation of BEC bright solitons

Ein Gedi, Feb. 24-29, 2008

Observation of BEC solitonsDark solitons by phase imprinting:J. Denschlag et al., Science 287, 5450 (2000).Bright solitonsL. Khaykovich et al. Science 296, 1290 (2002).Bright soliton train:K. E. Strecker et al., Nature 417, 150 (2002).

Ein Gedi, Feb. 24-29, 2008

Instability of 2D solitons without dipolar-interaction

Ein Gedi, Feb. 24-29, 2008

Dipole-dipole interactionvacuum permittivityd - magnetic/electric dipole moment

Ein Gedi, Feb. 24-29, 2008

Units

Ein Gedi, Feb. 24-29, 2008

2D Bright solitons in dipolar BECs P. Pedri and L. Santos, PRL 95, 200404 (2005)

Ein Gedi, Feb. 24-29, 2008

Manipulation of dipole-dipole interactionIn order to stabilize 2D solitary waves in the PS configuration, it is necessary to reverse dipole-dipole behavior, so that side-by-side dipoles attract each other and head-to-tail dipoles repell one another.

Ein Gedi, Feb. 24-29, 2008

Manipulation of dipole-dipole interaction S. Giovanazzi, A. Goerlitz, and T. Pfau, PRL 89, 130401 (2002)The magnetic dipole interaction can be tuned, using rotating fields from +Vd at , to -Vd/2 at The maximum becomes a minimum and 2D bright SWs can be found, provided that the dipole term is sufficiently strong to overcome the kinetic+contact terms, i.e.

Or, for

Ein Gedi, Feb. 24-29, 2008

E for confinement along the dipolar axis z, gaussian ansatz, g=500

Ein Gedi, Feb. 24-29, 2008

Dipolar axis in the 2D planeI. Tikhonenkov, B. A. Malomed, and AV, PRL 100, 090406 (2008)

Ein Gedi, Feb. 24-29, 2008

Dipolar axis in the 2D planeFor gd > 0 stable self trapping along the dipolar axis z:

Ein Gedi, Feb. 24-29, 2008

For gd > 0, what happens along x ?

Ein Gedi, Feb. 24-29, 2008

E for confinement perpendicular to the dipolar axis

Ein Gedi, Feb. 24-29, 2008

3D Propagation and stability

Ein Gedi, Feb. 24-29, 2008

Driven Rotation

Ein Gedi, Feb. 24-29, 2008

Experimental realization52Cr (magnetic dipole moment d=6B)

Dipolar molecules (electric dipole of ~0.1-1D)

For g,gd > 0 :

Ein Gedi, Feb. 24-29, 2008

Conclusions2D bright solitons exist for dipolar alignment in the free-motion plane.

For this configuration, no special tayloring of dipole-dipole interactions is called for.

The resulting solitary waves are unisotropic in the 2D plane, hence interesting soliton collision dynamics.

Ein Gedi, Feb. 24-29, 2008

Incoherent matter-wave Solitons1,2H. Buljan, 1M. Segev, and 3A. Vardi1Department of Physics, The Technion 2Department of Physics, Zagreb Univesity3Department of Chemistry, Ben-Gurion University

Ein Gedi, Feb. 24-29, 2008

What about quantum/thermal fluctuations ?

Ein Gedi, Feb. 24-29, 2008

T=0 - Bogoliubov theory (ask Nir) Want to calculate zero temperature fluctuations. Separate: retain quadratic fluctuation terms and add N0 constraint:

Ein Gedi, Feb. 24-29, 2008

T=0 - Bogoliubov theory Bogoliubov transformation:

Ein Gedi, Feb. 24-29, 2008

Bogoliubov spectrum of a bright soliton linearize about a bright soliton solution:

Bogoliubov spectrum of a bright solitonScattering without reflectionTransmittance: Bogoliubov quasiparticles scatter without reflection on the soliton (B. Eiermann et al., PRL 92, 230401 (2004), S. Sinha et al., PRL 96, 030406 (2006)).

Ein Gedi, Feb. 24-29, 2008

Limitations on Bogoliubov theory The condensate number is fixed - no backreaction The GP energy is treated separately from the fluctuations Due to exchange energy in collisions between condensate particles and excitations, it may be possible to gain energyBy exciting pairs of particles from the condensate !direct + exchangepair production

Ein Gedi, Feb. 24-29, 2008

TDHFB approximationretain quadratic terms in the fluctuations, to obtain coupled equations for: separate, like beforeCondensateorder-parameterPair correlation functions - single particle normaland anomalous densities

TDHFB approximation(e.g., Proukakis, Burnett, J. Res. NIST 1996, Holland et al., PRL 86 (2001))

Ein Gedi, Feb. 24-29, 2008

Initial Conditions - static HFB solution in a trapFluctuations do not vanish even at T=0, quantum fluct.

Ein Gedi, Feb. 24-29, 2008

Dynamics - TDHFB equations

Ein Gedi, Feb. 24-29, 2008

Quasi 1D geometryx N = 2.2 104 7Li atoms = 4907 Hz ; a = 1.3 m

x = 439 Hz ; ax = 4.5 m Na3D = -0.68 m Parameters close to experiment: TDHFB can be used only for limited time-scales: Tevolution

TDHFB vs. GP

Incoherent matter-wave solitons

Ein Gedi, Feb. 24-29, 2008

condensate fractionthermal populationNumber and energy conservation

Ein Gedi, Feb. 24-29, 2008

Conclusions Dynamics of a partially condensed Bose gas calculated via a nonlinear TDHFB model

Noncondensed particles (thermal/quantum) affect the dynamics of BEC solitons

Pairing instability - dynamical depletion of a BEC with attractive interactions

Incoherent matter-wave solitons constituting both condensed and noncondensed particles

Analogy with optics: Coherent light in Kerr media zero-temperature BEC Partially (in)coherent light in Kerr media partially condensed BEC

Ein Gedi, Feb. 24-29, 2008

We begin as usual, with the many body Hamiltonian of an interacting Bose gas, consisting of single-particle contributions and a two-body interaction term.

At low energy, when the wavelength of incoming scattering states becomes large compared with the effective range of interaction, the only effect of the potential is to introduce a phase-shift. We thus may replace the actual form of the interaction potential by anything that gives the correct shift. Most simply, one can use a contact interaction potential, linearly dependent on the s-wave scattering length, to ensure that the correct shift is produced. Negative scattering length corresponds to attractive interactions, zero to an ideal gas, and positive to repulsions.

Below the critical temperature for Bose-Einstein condensation, long range order emerges and it becomes sensible to replace the bosonic field operator by a c-number, called the condensate order-parameter or the condensate wavefunction. This is essentially a classical-field approximation, corresponding to a first-quantization assumption that the many-body state is a direct product of identical single-boson solutions.

Consequently, the many-body Hamiltonian in the contact interaction approximation, is replaced by a classical energy functional of the condensate order parameter. In order to obtain stationary states, one needs to minimize this Gross-Pitaevskii functional under the constraint that the total number of particles is conserved.

The result is the Gross-Pitaevskii, or nonlinear Schroedinger equation, where the chemical potential \mu is just a Lagrange multiplier that was added to the energy functional. As you can see, the form is just the usual Schroedinger form plus a cubic nonlinearity.It is quite instructive to do a very simple variational calculation by assuming a Gaussian trial function with varying width b and evaluating the GP energy as a function of this variational parameter.

In 3-D the kinetic energy terms which wants to disperse the wavefunction, grows as 1/b^2 whereas the interaction term per particle grows as 1/b^3. Thus for positive g, the BEC is stable, but for negative g the interactions win at the short range and the condensate is unstable. It is still possible to obtain metastable BECs for sufficiently small particle numbers, but beyond some critical number the minimum becomes an saddle-point and the condensate collapses.

In comparison, in 1-D the density grows as 1/b so that at short range the dispersion is sufficiently strong to balance the attraction and stop the collapse. This behavior where the ground state is determined from the interplay of dispersion and nonlinearity, without any contribution from an external trap, is characteristic of a soliton.Solitons are a rather general phenomena in nonlinear wave equations. They are self-localized solutions which come about due to a balance between dispersive terms which want to spread things around and nonlinear terms that glue them together. Since they are self-trapped, the solutions retain their form and may propagate long distances without dispersing.

When two solitary waves collide they go through one another without generating any excitations, making their motion Particle-like. Moving to less devastating solitons, the focus of this talk is the dynamics of solitons made out of matter-waves. The NLSE has the two needed ingredients to generate solitons, namely the dispersive kinetic energy and the nonlinear mean-field interactions.

And indeed for negative g (I.e. attractive interaction) in 1-D, one finds that the ground state of the GP equation is a bright soliton of the usual hyperbolic sec form. The width of the soliton is determined by the healing-length of a uniform bose gas whose density is the peak-density of the soliton and the chemical potential is exactly half of the chemical potential of such a uniform gas.The way to show solitary behavior is to demonstrate self-trapping. A soliton is a self-localized entity, not needing an external potential to remain bound. Thus, if we have generated a true soliton in a trap and then released the trap, it would not disperse over time. In comparison, if we did the same and turned off the interactions along with the trap, by means of a Feshbach resonance, the condensate wavefunction should rapidly disperse. So, this was exactly the experimental procedure in an experiment done by C. Salomone group in ENS. A Q1D condensate with attractive interactions was preapared in an elongated cigar-shaped harmonic trap. Then the trap was turned off and the dynamics was followed. Finally, the experiment was repeated, this time using a Feshbach resonance to tune out the nonlinearity at the same time the trap was turned off. The results clearly show that without interactions the density profile spreads out. In contrast, when the interaction is left on, the localization persists over a long time.In addition to the ENS experiment showing a bright soliton, there is a NIST experiment where dark solitons were produced py imprinting a phase-slip onto a BEC, to produce dark solitons, and an experiment from Rice showing the collapse of a BEC prepared on one side of an elongated trap, into a train of solitons. Again, the procedure for demonstrating solitary behavior was to compare the dynamics with repulsive interaction, giving dispersion, to the dynamics with attractive interactions, giving a train of localized structures.What we set out to do is to look at solitary matter-wave behavior beyond the GP mean-field description. While the condensates are prepared at very low temperatures and the condensate fraction is usually large, there is always a bit of a cloud of thermal atoms surrounding the BEC. One issue was to see how this cloud affects the dynamics when the trap is turned off. The second issue is that even at zero temperature, there are still quantum fluctuations that may grow and lead to the depletion of the condensate during its time evolution. The Bogoliubov treatment remains essentially identical when dealing with quantum-fluctuations. Instead of using an energy functional, we use the quantum Hamiltonian operator and separate the field annihilation operator to a c-number part representing the condensate and an operator correction chich corresponds to quantum-field fluctuations.

Once again we retain only quadratic terms in the fluctuations and add a lagrange multiplier to impose the constraint that the total number of particles is conserved. It is important to note that in the usual Bogoliubov treatement, it is the condensate particle number which is taken to be a constant of motion.Next we write down the Heisenberg equations of motion for the fluctuation operators and use the Bogoliubov transformation to diagonalize the interactions. Since the equations are linearized about the GP solution, the result is the same as for finding the natural linear response frequencies.Plugging the soliton solution into the Bogoliubov equations, one obtains a gapped excitation spectrum where the energy of long-wavelength excitations approaches the condensate chemical potential as k goes to zero. This simply manifests the fact that in order to excite a particle from the soliton, one needs to overcome the its self-binding energy .

The asymptotic form of the excitations is a plane-wave without any reflected wave. The magnitude of the coefficients at plus and minus infty is the same, the only difference being the phase. Thus we obtain the well known result that the only effect of scattering quasiparticles off the soliton is to introduce a phase shift. The excitations of a bright soliton pass through it without reflection. This is why solitons are indifferent to ripples around them. The Bogoliubov theory gives us some flavor of the excitations, but it does not tell the entire story. In particular, due to the fixing of the condensate particle-number, the Bogoliubov treatment misses out the backreaction of excitations on condensed particles.

Suppose we were to treat the condensate number as variable and conserve only the total number of particles. Inspecting the Bogpliubov energy, we see that the mean-field for the excitations contains both a direct scattering term and an exchange term. For scattering inside the BEC however, there is no exchange as all condensed particles are in the same state.

Consequently, if we lifted the constraint on N_0 and replaced it by a constraint on the total number, and if g