low dimensions

# Noise correlations in an expanding 1d Bose gas

23/10/11 22:15

Just out in PRA is a cute result concerning the expansion of a 1D gas from a regular array.

The initial state -- a 1D Mott insulator -- acts as an array of incoherent sources. Thus one doesn’t expect to see an interference pattern in the density after the atoms of the gas are released. There

The effect of interactions hasn’t been too important historically, either because one dealt with noninteracting particles (photons, say) and / or because the spatial separation of the sources (different parts of a star in the original experiment!).

That will change in the situation discussed here: cold atoms in one dimension. The reason is that the effect involves the interferences of trajectories that differ by an exchange of a pair of particles -- that’s how statistics enters -- thus the particles must necessarily pass each other and interact.

Now comes the interesting part. It’s known that when bosons have infinite short-range repulsion in one dimension, we can describe them in terms of

The answer turns out to be quite simple to state. In general there are still a series of features in the density-density correlation function, but they are Fano resonances.

A Fano resonance has a lineshape given by

\[\frac{\left[q\Gamma/2+\varepsilon\right]^{2}}{\Gamma^{2}/4+\varepsilon^{2}}\]

where \(\varepsilon\) is the distance from the resonance, \(\Gamma\) is the width and \(q\) is an asymmetry parameter that goes from \(\infty\) (a peak or resonance) to \(0\) (a dip or antiresonance). In this case \(q\) depends on the scattering phase of the pairs of particles contributing to that peak, and so goes from \(0\) at low relative momenta where the interaction appears impenetrable and the fermion picture applies, to \(\infty\) at high relative momenta where the noninteracting bosonic HBT effect is recovered. Simple, eh?

Although the answer is simple, the calculation can only be done because of the integrability of the 1D Bose gas, and in particular the existence of a remarkable formula due to Craig Tracy and Harold Widom for the N-particle propagator.

The initial state -- a 1D Mott insulator -- acts as an array of incoherent sources. Thus one doesn’t expect to see an interference pattern in the density after the atoms of the gas are released. There

*are*fringes, however, in the density-density correlation function. This is the famous Hanbury Brown and Twiss effect. The effect is sensitive to the statistics of the particles: an array of sources gives rise to*peaks*in the correlation function for bosons,*dips*for fermions.The effect of interactions hasn’t been too important historically, either because one dealt with noninteracting particles (photons, say) and / or because the spatial separation of the sources (different parts of a star in the original experiment!).

That will change in the situation discussed here: cold atoms in one dimension. The reason is that the effect involves the interferences of trajectories that differ by an exchange of a pair of particles -- that’s how statistics enters -- thus the particles must necessarily pass each other and interact.

Now comes the interesting part. It’s known that when bosons have infinite short-range repulsion in one dimension, we can describe them in terms of

*fermions*, as least as far as any observable involving density is concerned. Thus with increasing interaction, we must somehow cross over from the bosonic (peaks) to the fermonic (dips) HBT effect. This paper describes how.The answer turns out to be quite simple to state. In general there are still a series of features in the density-density correlation function, but they are Fano resonances.

A Fano resonance has a lineshape given by

\[\frac{\left[q\Gamma/2+\varepsilon\right]^{2}}{\Gamma^{2}/4+\varepsilon^{2}}\]

where \(\varepsilon\) is the distance from the resonance, \(\Gamma\) is the width and \(q\) is an asymmetry parameter that goes from \(\infty\) (a peak or resonance) to \(0\) (a dip or antiresonance). In this case \(q\) depends on the scattering phase of the pairs of particles contributing to that peak, and so goes from \(0\) at low relative momenta where the interaction appears impenetrable and the fermion picture applies, to \(\infty\) at high relative momenta where the noninteracting bosonic HBT effect is recovered. Simple, eh?

Although the answer is simple, the calculation can only be done because of the integrability of the 1D Bose gas, and in particular the existence of a remarkable formula due to Craig Tracy and Harold Widom for the N-particle propagator.

# Some research themes

29/09/11 11:24

A couple of the broad themes in my work over the last few years...

The electron has the quantum mechanical property of spin, causing it to behave as a tiny bar magnet. The phenomenon of magnetism – so familiar to us from childhood – hinges upon the cooperative behavior of myriad spinning electrons.

Atoms also have spin, typically more than the electron. Can a gas of atoms display magnetism? This is one of the principal questions I have been addressing theoretically. The short answer is yes, but our idea of magnetism has to be broadened substantially beyond the simple ferromagnetism of (say) iron due to the high spin of the particles involved. Additionally, because spin is usually conserved in a gas during the lifetime of an experiment, many of the manifestations of magnetism that I have discovered are dynamical rather than static.

In a broader context, the interplay of magnetism and superfluidity lies at the heart of a diverse array of problems in modern condensed matter physics, perhaps most notably in the continuing mystery of high-temperature superconductivity. Eventually this syzygy may have technological applications, notably in magnetometry.

Cold atoms have proven to be a remarkably pliant experimental material, cheerfully accommodating almost every whim of the experimentalist. Atoms have been confined by laser light to lie flat in a plane, or single file in tubes, or in periodic arrays.

The influence of the (effective) spatial dimension on the properties of matter is one of the principal themes of condensed matter physics, and it appears several times in the work of my group. As an illustrative example, consider the motion of an impurity particle in a fluid. In three dimensions the impurity has some added mass from the fluid it drags with it. In one dimension, however, the particle will accelerate when subject to a force, but then come to a halt and reverse its motion, undergoing oscillations akin to Bloch oscillations in a periodic potential. Hopefully this counterintuitive prediction will be tested in upcoming experiments.

Ultracold systems are also promising arenas for the study of nonequilibrium phenomena. Bearing in mind that equilibrium systems are all alike, while every non-equilibrium system is out of equilibrium in its own way, I have tried to frame the most significant questions in a forthcoming book chapter.

Magnetism in atomic gases – new quantum phasesMagnetism in atomic gases – new quantum phases

The electron has the quantum mechanical property of spin, causing it to behave as a tiny bar magnet. The phenomenon of magnetism – so familiar to us from childhood – hinges upon the cooperative behavior of myriad spinning electrons.

Atoms also have spin, typically more than the electron. Can a gas of atoms display magnetism? This is one of the principal questions I have been addressing theoretically. The short answer is yes, but our idea of magnetism has to be broadened substantially beyond the simple ferromagnetism of (say) iron due to the high spin of the particles involved. Additionally, because spin is usually conserved in a gas during the lifetime of an experiment, many of the manifestations of magnetism that I have discovered are dynamical rather than static.

In a broader context, the interplay of magnetism and superfluidity lies at the heart of a diverse array of problems in modern condensed matter physics, perhaps most notably in the continuing mystery of high-temperature superconductivity. Eventually this syzygy may have technological applications, notably in magnetometry.

__Extreme conditions: low dimensions; non-equilibrium__Cold atoms have proven to be a remarkably pliant experimental material, cheerfully accommodating almost every whim of the experimentalist. Atoms have been confined by laser light to lie flat in a plane, or single file in tubes, or in periodic arrays.

The influence of the (effective) spatial dimension on the properties of matter is one of the principal themes of condensed matter physics, and it appears several times in the work of my group. As an illustrative example, consider the motion of an impurity particle in a fluid. In three dimensions the impurity has some added mass from the fluid it drags with it. In one dimension, however, the particle will accelerate when subject to a force, but then come to a halt and reverse its motion, undergoing oscillations akin to Bloch oscillations in a periodic potential. Hopefully this counterintuitive prediction will be tested in upcoming experiments.

Ultracold systems are also promising arenas for the study of nonequilibrium phenomena. Bearing in mind that equilibrium systems are all alike, while every non-equilibrium system is out of equilibrium in its own way, I have tried to frame the most significant questions in a forthcoming book chapter.