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arXiv:1001.4358 · Physical Review Letters

Phase-Sensitive Measurements of Order Parameters for Ultracold Atoms through Two-Particle Interferometry

Takuya Kitagawa, Alain Aspect, Markus Greiner, Eugene Demler

Published Mar 16, 2011• 30 citations
arXivHTMLPDF

Abstract

Nontrivial symmetry of order parameters is crucial in some of the most interesting quantum many-body states of ultracold atoms as well as condensed matter systems. Examples in cold atoms include p-wave Feshbach molecules and d-wave paired states of fermions that could be realized in optical lattices in the Hubbard regime. Identifying these states in experiments requires measurements of the relative phase of different components of the entangled pair wave function. We propose and discuss two schemes for such phase-sensitive measurements, based on two-particle interference revealed in atom-atom or atomic density correlations. Our schemes can also be used for relative phase measurements for nontrivial particle-hole order parameters, such as d-density wave order.

DOI
10.1103/physrevlett.106.115302

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Nontrivial symmetry of order parameters is crucial in some of the most interesting quantum many-body states of ultracold atoms as well as condensed matter systems. Examples in cold atoms include p-wave Feshbach molecules and d-wave paired states of fermions that could be realized in optical lattices in the Hubbard regime. Identifying these states in experiments requires measurements of the relative phase of different components of the entangled pair wave function. We propose and discuss two schemes for such phase-sensitive measurements, based on two-particle interference revealed in atom-atom or atomic density correlations. Our schemes can also be used for relative phase measurements for nontrivial particle-hole order parameters, such as d-density wave order.
Physics Interactions between the charge, spin, and orbital degrees of freedom of electrons in condensed-matter systems can give rise to many complex electronic and magnetic phases. The narrow, or often fixed, range of variable materials parameters can be a limitation in probing and understanding the evolution of the order parameters of such complex correlated systems. An array of atoms trapped in an optical lattice has the potential to be extremely flexible in terms of tuning the parameters. Although the atoms tend to be isotropic, leading to somewhat trivial systems, much theoretical work has explored the possibilities of finding ways to imprint and detect more complicated order parameters on the lattice of trapped atoms. It is along such lines that Kitagawa et al. propose a spectroscopic technique based on two-particle interferometry that probes the phase-sensitive correlations between atom-atom interactions across the lattice. They show that the technique should allow the measurement of nontrivial order parameters in entangled ensembles of cold atoms, such as d- or p-wave pairing of electrons found in exotic superconductors and superfluids, and the realization of cold-atom systems that function as complex quantum simulators. Phys. Rev. Lett. 106 , 115302 (2011).
We propose a pump probe experiment for detecting the d-wave superfluid and d-density wave phases of ultracold Fermions on an optical lattice. The pump consists of periodic modulations of the optical lattice intensity which creates quasiparticle pairs in these systems. The changes in the momentum distribution under the drive can be used to measure quasiparticle dispersion and gap anisotropy. Further, we show that the pattern of peaks and dips in the spin selective density-density correlation function provides a phase sensitive probe of the symmetry of the order parameter in these systems.
Order parameters represent a fundamental resource to characterize quantum matter. We show that pair superfluids can be rigorously defined in terms of a nonlocal order parameter, named odd parity, which derivation is experimentally accessible by local density measurements. As a case of study, we first investigate a constrained Bose-Hubbard model at different densities, both in one and two spatial dimensions. Here, our analysis finds pair superfluidity for relatively strong attractive interactions. The odd parity operator acts as the unique order parameter for such phase irrespectively to the density of the system and its dimensionality in regimes of total particle number conservation. In order to enforce our finding, we confirm the generality of our approach also on a two-component Bose-Hubbard Hamiltonian, which experimental realization represents a timely topic in ultracold atomic systems. Our results shed new light on the role of correlated density fluctuations in pair superfluids. In addition, they provide a powerful tool for the experimental detection of such exotic phases and the characterization of their transition to the atomic superfluid phase.
Many-body interference between indistinguishable particles can give rise to strong correlations rooted in quantum statistics. We study such Hanbury Brown-Twiss-type correlations for number states of ultracold massive fermions. Using deterministically prepared ^{6}Li atoms in optical tweezers, we measure momentum correlations using a single-atom sensitive time-of-flight imaging scheme. The experiment combines on-demand state preparation of highly indistinguishable particles with high-fidelity detection, giving access to two- and three-body correlations in fields of fixed fermionic particle number. We find that pairs of atoms interfere with a contrast close to 80%. We show that second-order density correlations arise from contributions from all two-particle pairs and detect intrinsic third-order correlations.
We study the possibility of realizing strong coupled many-body quantum phases in ultracold atomic systems. Motivated by recent experiments, we first analyze the phase diagram of a Bose-Fermi mixture across a Feshbach resonance, and offer an explanation for the collapse of the system observed close to the Feshbach resonance: we find that phase separation leads to a high density phase which causes the collapse. We then focus on the recent attempts to realize the three-dimensional fermionic Hubbard model in an optical lattice. One milestone on the experimentalists' agenda is to access the antiferromagnetic ordered Neél phase, which has so far been hindered by the low ordering temperature. We ask which experimental parameters maximize the antiferromagnetic interactions, which set the scale for the ordering temperature. We find that the maximum is obtained in a regime where the effective Hamiltonian describing the system no longer corresponds to a simple one-band Hubbard model, and we characterize the physics of the system in this regime. The final system we consider is mass imbalanced polarized two-component Fermi gases interacting via a Feshbach resonance. By going to the strongly polarized limit, we use a recently developed method to obtain results which have been shown to be accurate in the mass balanced case, and we find an intriguing set of competing phases in this limit. We discuss what these results imply for the full phase diagram.
We propose a phase-sensitive detection scheme to identify the unconventional ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ symmetry of the condensate wave functions of bosons, which have already been proposed and realized in high bands in optical lattices. Using the impulsive Raman operation combined with time-of-flight imaging, the off-diagonal correlation functions in momentum space give rise to the relative phase information between different components of condensate wave functions. This scheme is robust against the interaction and interband effects, and provides evidence of unconventional Bose-Einstein condensations with nontrivial condensation symmetries.