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arXiv:1408.0206 · Physical Review A

Feshbach modulation spectroscopy of the Fermi-Hubbard model

Andreas Dirks, Karlis Mikelsons, H. R. Krishnamurthy, James K. Freericks

Published Nov 16, 2015• 4 citations
arXivHTMLPDF

Abstract

In the vicinity of a Feshbach resonance, a system of ultracold atoms in an optical lattice undergoes rich physical transformations which involve molecule formation and hopping of molecules on the lattice and thus goes beyond a single-band Hubbard model description. We explore theoretically the response of this system to a harmonic modulation of the magnetic field, and thus of the scattering length, across the Feshbach resonance. In the regime in which the single-band Hubbard model is still valid, we provide results for the doublon production as a function of the various parameters, such as frequency, amplitude, etc., that characterize the field modulation, as well as the lattice depth. The method may uncover a route towards the efficient creation of ultracold molecules and also provide an alternative to conventional lattice-depth-modulation spectroscopy.

DOI
10.1103/physreva.92.053612

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•Oct 5, 2026•0 citations
In the vicinity of a Feshbach resonance, a system of ultracold atoms in an optical lattice undergoes rich physical transformations which involve molecule formation and hopping of molecules on the lattice and thus goes beyond a single-band Hubbard model description. We explore theoretically the response of this system to a harmonic modulation of the magnetic field, and thus of the scattering length, across the Feshbach resonance. In the regime in which the single-band Hubbard model is still valid, we provide results for the doublon production as a function of the various parameters, such as frequency, amplitude, etc., that characterize the field modulation, as well as the lattice depth. The method may uncover a route towards the efficient creation of ultracold molecules and also provide an alternative to conventional lattice-depth-modulation spectroscopy.
Abstract The conversion of ultracold atoms to molecules via a magnetic Feshbach resonance with a sinusoidal modulation of the field is studied. Different practical realizations of this method in Bose atomic gases are analyzed. Our model incorporates many-body effects through an effective reduction of the complete microscopic dynamics. Moreover, we simulate the experimental conditions corresponding to the preparation of the system as a thermal gas and as a condensate. Some of the experimental findings are clarified. The origin of the observed dependence of the production efficiency on the frequency, amplitude, and application time of the magnetic modulation is elucidated. Our results uncover also the role of the atomic density in the dynamics, specifically, in the observed saturation of the atom–molecule conversion process.
We present a theory for ultracold atomic gases in an optical lattice near a Feshbach resonance. In the single-band approximation the theory describes atoms and molecules that can both tunnel through the lattice. Moreover, an avoided crossing between the two-atom and the molecular states occurs at every site. We determine the microscopic parameters of the generalized Hubbard model that describes this physics, using the experimentally known parameters of the Feshbach resonance in the absence of the optical lattice. As an application we also calculate the zero-temperature phase diagram of an atomic Bose gas in an optical lattice.
In this dissertation, we develop simple single-band effective Hamiltonians for a fermionic gas across a Feshbach resonance in both a quasi-low-dimensional optical trap and a three-dimensional optical lattice. The proper theoretical description of these systems is nontrivial due to the multiband nature of the dimers which form in the ground state of the strongly interacting gas. For the quasi-low-dimensional case, we show that the ground state of the gas has a significant population in the excited levels of the transverse trap. Hence, the effective low-dimensional (single-band) Hamiltonian must include the effects of this real population frozen in many transverse modes. We accomplish this through the definition of a "dressed molecule" state which incorporates the excited fraction. For the three-dimensional case, a valid single-band lattice model at first seems impossible for strong interactions. However, we find effective single-band lattice models that remain valid by subtly accounting for the nontrivial structure of the on-site dimer. We perform explicit numerical calculations for two fermions in a double-well to demonstrate the range of validity of the models and their parameter values. In the process of investigation leading to these models we find other interesting results such as a ground state level crossing for three fermions in a single well and a set of Feshbach resonances induced by anharmonicity of the optical potential. The ultracold neutral atom experiments related to the work presented within provide a flexible and well-controlled tool to observe aspects of quantum physics not easily accessible or controllable in other systems. The combination of optical lattice trapping techniques with the ability to manipulate the interactions via a Feshbach resonance allows for the study of novel strongly correlated physics. This work addresses an important and fundamentally interesting issue for these systems and serves as a proper starting point for further theoretical studies in conjunction with experiment.
In systems of ultracold atoms, pairwise interactions can be resonantly enhanced by a new mechanism that does not rely upon a magnetic Feshbach resonance. In this mechanism, interactions are controlled by tuning the frequency of an oscillating parallel component of the magnetic field close to the transition frequency between the scattering atoms and a two-atom bound state. The real part of the resulting s-wave scattering length a is resonantly enhanced when the oscillation frequency is close to the transition frequency. The resonance parameters can be controlled by varying the amplitude of the oscillating field. The amplitude also controls the imaginary part of a, which arises because the oscillating field converts atom pairs into molecules. The real part of a can be made much larger than the background scattering length without introducing catastrophic atom losses from the imaginary part. For the case of a shallow bound state in the scattering channel, the dimensionless resonance parameters are universal functions of the dimensionless oscillation amplitude.
The variation of a magnetic field in the vicinity of a zero-energy resonance allows highly vibrationally excited molecules (‘Feshbach molecules’) to be produced from an ultracold atomic gas. In this thesis, we study the dynamics of this process. We begin by studying the dissociation of Feshbach molecules, showing that in the limit of a sudden jump the shape of the spectrum of dissociated atoms can act as a probe of the zero-energy resonance. For some resonances, such jumps are within reach of current experiments. We also study the intermediate region between sudden jumps and asymptotically wide, linear ramps. It is shown from a precise derivation how the latter limit leads to a universal spectrum with a shape independent of the implementation of the two-body physics, provided that the near-resonant scattering properties are correctly modelled. We then turn to the dynamics of Feshbach molecule production from thermal and condensed gases. Our microscopic quantum dynamics approach includes the exact twobody evolution as an input to the many-body calculations. We show that in the long-time limit, and the Markov limit for the interactions, the non-Markovian Boltzmann equation (NMBE) we derive for the one-body density matrix reduces to the normal Boltzmann equation. In the limit of short times and small depletion of the atomic gas, the molecule production efficiency can be calculated by thermally averaging the two-body transition probability density. This thermal averaging technique is applied to studies of the formation of Feshbach molecules using a magnetic field modulation that is near-resonant with the molecular bound state energy. The continuum is shown to have a significant effect on both the dynamics and efficiency of this process. We examine the dependence of the molecule production efficiency on the duration, amplitude and frequency of the modulation, as well as the temperature and density of the gas. This method of producing molecules is effective for a wide range of bound state energies, but requires sufficient variation of the two-body energy levels with magnetic field. Lastly, we implement the NMBE for the case of a fast linear ramp across a Feshbach resonance. The solution of this equation is made feasible by including a large part of the required computation in the kernel, which is calculated in advance. The NMBE allows predictions of the molecule production efficiency which go beyond the thermal averaging technique by accounting for the depletion and rethermalisation of the continuum. In the limit of small depletions, the two approaches give the same results. As the depletion increases, the two approaches differ due to many-body effects limiting the maximum possible molecule production efficiency. We have observed this in our simulations by considering higher-density gases. We have therefore shown the suitability and practicability of this beyond mean-field approach for application to further problems in the production of Feshbach molecules from ultracold gases.
Ultracold atoms in optical lattices provide a versatile platform for studying many-body physics in a defect-free system with tunable parameters. For the analog quantum simulation of the Fermi-Hubbard model, fermionic atoms take on the role of the valence electrons and optical lattices provide the periodic potential in place of the solid-state crystal. The dynamics in the system are described by the tunneling amplitude t between neighboring lattice sites and the interaction energy U of two particles occupying the same lattice site. With the depth of the optical lattices and an external magnetic field, both parameters can be tuned to access a wide range of regimes. Two superimposed lattices with different lattice periods form an optical superlattice, where the most common configuration is a bichromatic superlattice with lattice periods different by a factor of two. Depending on the lattice depths and the relative phase of the lattices, the bichromatic superlattice potential ranges from an array of weakly-coupled double wells to a continuous lattice potential with a staggered energy offset. We developed a calibration technique for measuring the depth of an optical lattice with local resolution. The band structure of the optical lattice is probed by modulating the lattice depth and exciting the atoms to higher bands. However, compared to resolving the band occupations in a time-of-flight measurement, we further excite the atoms to an untrapped band to deplete the optical lattice along equipotential lines where the modulation frequency is resonant with the local band transitions. From the profile of the lattice depth, we obtain the calibration factor for the power regulation, the waist of the underlying lattice beams and the position of the lattice axis. This technique combines several calibration measurements, thereby reducing the measurement time and enabling an automation of the lattice alignment. For the bichromatic superlattice we realized an active stabilization of the superlattice phase that enables long-term measurements with a precise control over the superlattice phase. The instability of the superlattice phase is a result of the different refractive indices for the two lattices in air and the optical elements along the optical path. Using environmental sensors, we monitor the temperature, the pressure, and the relative humidity to apply the expected correction to the superlattice phase. The stabilized superlattice phase enabled the investigation of Floquet-driven double wells by modulating the energy offset through the superlattice phase. We used a near-resonant driving frequency with respect to the interaction energy to enhance the pair-tunneling amplitude in doubly-occupied double wells compared to static double wells and compared to the tunneling amplitude in singly-occupied double wells.