Congratulations Yifei!
03/11/11 09:48
In August my student, Yifei Shi, had his first preprint out on the arXiv, concerning the fascinating statistical mechanics of bosons that can form single particle and pair condensates. A few days ago we found out that this work will be appearing in PRL shortly. Congratulations on your first paper, Yifei!
I’ll write more about this work when the paper appears in print…
I’ll write more about this work when the paper appears in print…
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 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.
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...
Magnetism 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.
Magnetism 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.
Strings and things
09/04/11 10:32
Our PRL on the statistical mechanics of 2D polar condensates just came out this week. This is the work of my former postdoc Andrew James (now at Brookhaven), showing that a spin-1 Bose gases has an interesting phase diagram in two dimensions driven by the interplay of two different types of topological defects: vortices and strings.
In general the order parameter in a spin-1 condensate is a complex three component vector. For the polar case, which corresponds to spin-spin interactions of antiferromagnetic sign (the situation prevailing in 23Na), this vector is restricted to be a real vector multiplied by a phase \( \phi=\mathbf{n} e^{i\theta}\). This parametrization has some redundancy: \((\mathbf{n},\theta)\) and \((-\mathbf{n},\theta+\pi)\) describe the same state. An immediate consequence of this is that the elementary vortex in a polar condensate has only a \(\pi\) winding of the phase, and thus half the circulation quantum, of a vortex in a regular superfluid. It must, however, coincide with a disclination in the vector \(\mathbf{n}\).
Below you can see a picture of a pair of such half-vortex / disclination defects, where the blue arrows indicate the phase \(\theta\) and the red arrows the vector \(\mathbf{n}\)
The main point of our paper is that, once you turn on a magnetic field, the quadratic Zeeman effect creates an easy axis anisotropy that causes the \(\mathbf{n}\) to align either parallel or antiparallel to the field. Thus in the picture above the red arrows mostly lie horizontally. However, they still have to reverse going around the center of each of the defects, but now this reversal is confined to a string of a well-defined thickness and energy per length (or tension) set by the field.
The Kosterlitz-Thouless transition mediated by the vortices and the Ising transition mediated by the strings fit together in an interesting way. We’ll have more to say about systems with this kind of phase diagram soon!
In general the order parameter in a spin-1 condensate is a complex three component vector. For the polar case, which corresponds to spin-spin interactions of antiferromagnetic sign (the situation prevailing in 23Na), this vector is restricted to be a real vector multiplied by a phase \( \phi=\mathbf{n} e^{i\theta}\). This parametrization has some redundancy: \((\mathbf{n},\theta)\) and \((-\mathbf{n},\theta+\pi)\) describe the same state. An immediate consequence of this is that the elementary vortex in a polar condensate has only a \(\pi\) winding of the phase, and thus half the circulation quantum, of a vortex in a regular superfluid. It must, however, coincide with a disclination in the vector \(\mathbf{n}\).
Below you can see a picture of a pair of such half-vortex / disclination defects, where the blue arrows indicate the phase \(\theta\) and the red arrows the vector \(\mathbf{n}\)
The main point of our paper is that, once you turn on a magnetic field, the quadratic Zeeman effect creates an easy axis anisotropy that causes the \(\mathbf{n}\) to align either parallel or antiparallel to the field. Thus in the picture above the red arrows mostly lie horizontally. However, they still have to reverse going around the center of each of the defects, but now this reversal is confined to a string of a well-defined thickness and energy per length (or tension) set by the field.
The Kosterlitz-Thouless transition mediated by the vortices and the Ising transition mediated by the strings fit together in an interesting way. We’ll have more to say about systems with this kind of phase diagram soon!
Spin-1 microcondensates paper out
09/03/11 22:29
Just out in PRA today is my paper on spin-1 Bose condensates in what I’ve dubbed the microcondensate regime, meaning that there is no variation in the spin state across the system. In some ways this is analogous to studying a single magnetic domain of a ferromagnet, but considerably richer because of the larger phase space afforded by spin-1.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.