My recent research evolved in several directions, for example rapidly rotating gases and fractional quantum hall physics, novel consequences of strong interactions in lattices, and reduced dimensional quantum gases.

RF spectral density as a function of hopping and chemical potential

The rf spectrum as a function of chemical potential and hopping rate with opacity indicating spectral weight; 20% different initial and final state interactions.

Another fascinating direction has emerged: by studying spectroscopy of ultracold atomic gases, we learn what these probes reveal about many-body systems. Our analysis of experiments in Ketterle's group at MIT has revealed that these spectra directly elucidate the various correlations of superfluid lattice bosons near their instability to the Mott insulating state. The general understanding of this basic but non-trivial system is relevant to other quantum phase transitions.


More details: our "big picture"  goals require improved probes

Cold atoms offer many appealing features for studying many-body systems: (i) one knows exactly the Hamiltonian, which often is a canonical solid state model, such as a Hubbard model; (ii) they are quantum coherent on experimentally relevant timescales --- they couple to no bath;  and (iii) their behavior is readily tunable: for example, one may greatly vary the interaction strength, band structure, and internal spin structure. However, our probes at the moment are rather blunt.  The most common is absorption imaging, which provides real space column-integrated densities.

Localized and delocalized rf excitations

Top: cartoon of superfluid near the Mott instability, for average site fillings slightly greater than 3. Near the Mott state, the superfluid state looks like a Mott insulator with a few delocalized particles or holes dispersed on it. Bottom: After absorbing an rf photon to change an atom's internal state, there are two possible final states, each with different energies, leading to a bimodal rf spectra.

Roughly, each point in the trap corresponds to a different local chemical potential and consequently the density profile gives the equation of state.  Although useful, this fails to directly probe much interesting physics, e.g. the elementary excitations and collective modes of the system.  One may demonstrate how little information density profiles contain by comparing them distinct systems.  One finds that non-interacting fermions, strongly interacting fermions, Bose-Einstein condensates all have qualitatively similar density profiles.  Near the center (at high densities) the density depends on the trap position via a power law, and in the tail the dilute thermal gas yields a Gaussian shape.  Indeed, virtually any physical system displays this generic behavior.


In contrast, spectroscopies cleanly distinguish these scenarios by probing their excitations: each has a characteristic number of excitations, gapless or gapped dispersion, and spectral weight distribution.  Erich and I have concentrated on rf-microwave spectroscopy of bosons in an optical lattice, in which absorbed photons change only the internal state of atoms.  Hazzard and Mueller, PRA 76 063612 (2007) calculated the trap-averaged rf spectra, using a sum rule approach assuming that the local spectrum was a single delta function. As a follow up, we have been investigating the full local structure of the rf spectra.  In work being prepared for publication, we find that the superfluid near the Mott instability displays a multi-peaked spectrum, one peak with "Mott" and one with "delocalized" behavior; the dual nature of these correlations are illustrated to the left. The characteristic spectral densities are to the right and at the top right of the page.

local and trap averaged spectra

Left: spectral density as a function of frequency and distance to trap center (white).  Right: trap averaged spectral density (solid) versus frequency (on vertical axis). Red curves are predictions of the sum rule approach.

I believe rf spectroscopy will continue to play an important role in cold atoms.  For vanishing final state interactions, rf spectra are quasi-hole spectra; for equal final and initial state interactions, it is an analog of Bragg scattering/spectroscopy and transport measurements.  Each are standard, powerful solid state probes.  By varying the final state interaction, we may mimic these techniques and additionally "interpolate" between these two limits.  Consequently, although the present theory and experiment has largely restricted to zero momentum excitations, extending this work to finite momenta would be valuable.  I hope that our work enables an even better understanding of the bosonic Mott insulator and helps pave the way for further application of spectroscopy to cold atomic many body systems.