Many-body physics in the radio frequency spectrum of lattice bosons
We calculate the radio-frequency spectrum of a trapped cloud of cold bosonic atoms in an optical lattice. Using random phase and local density approximations we produce both trap averaged and spatially resolved spectra, identifying simple features in the spectra that reveal information about both superfluidity and correlations. Our approach is exact in the deep Mott limit and in the deep superfluid when the hopping rates for the two internal spin states are equal. It contains final state interactions, obeys the Ward identities (and the associated conservation laws), and satisfies the -sum rule. Motivated by earlier work by Sun, Lannert, and Vishveshwara [Phys. Rev. A 79, 043422 (2009)], we also discuss the features which arise in a spin-dependent optical lattice.
Bosonic atoms in optical lattices, described by the Bose-Hubbard model jaksch:olatt (); fisher:bhubb (), display a non-trivial quantum phase transition between a superfluid and Mott insulator. The latter is an incompressible state with an integer number of atoms per site. In a trap the phase diagram is revealed by the spatial structure of the gas: one has concentric superfluid and insulating shells. This structure has been elegantly explored by radio frequency (RF) spectroscopy campbell:ketterle-clock-shift (), a technique which has also given insight into strongly interacting Fermi gases across the BEC-BCS crossover bloch:many-body-cold-atoms-review (). Here we use a Random Phase Approximation (RPA) that treats fluctuations around the strong coupling Gutzwiller mean field theory to explore the radio-frequency spectrum of lattice bosons.
We find two key results: (1) Our previous sum-rule based analysis hazzard:rf-spectra-sum-rule () of experiments at MIT campbell:ketterle-clock-shift () stands up to more rigorous analysis: in the limit of small spectral shifts, the RPA calculation reduces to that simpler theory. (2) In a gas with more disparate initial and final state interactions (such as Cesium), the spectrum becomes more complex, with a bimodal spectrum appearing even in a homogeneous gas. The bimodality reveals key features of the many-body state. For example, in the limit considered by Sun, Lannert, and Vishveshwara sun:rf-spectra-condensate-probe (), the spectral features are related to the nearest-neighbor phase coherence. In the Gutzwiller approximation, the phase coherence directly maps onto the condensate density. In this paper we provide a physical picture of this result and explain how this bimodality can be observed in a spatially resolved experiment.
i.1 RF Spectroscopy
In RF spectroscopy, a radio wave is used to flip the hyperfine spin of an atom from to . The rate of excitation reveals details about the many-body state because the and atoms have slightly different interactions. Generically the interaction Hamiltonian is , with , where is the number of -state atoms on site . In the simplest mean-field picture, the energy needed to flip an atom on site from state to state is shifted by an energy . Applying this picture to an inhomogeneous gas suggests that the absorption spectrum reveals a histogram of the atomic density. Such a density probe is quite valuable: in addition to the aforementioned examples, it was the primary means of identifying Bose-Einstein condensation in atomic hydrogen fried:h ().
Recently Sun, Lannert, and Vishveshwara sun:rf-spectra-condensate-probe () found a bimodal spectrum in a special limit of this problem, as did Ohashi, Kitaura, and Matsumoto ohashi:rf-spectra-dual-character () in a separate limit, calling into question this simple picture. We give a simple physical interpretation of the bimodality. As illustrated in Fig. 1, the superfluid state near the Mott insulator can be caricatured as a dilute gas of atoms/holes moving in a Mott background. An RF photon can either flip the spin of one of the core atoms, or flip the spin of one of the mobile atoms. The energy of these two excitations will be very different, implying that the RF spectrum should be bimodal. Through our RPA calculation, we verify this feature, calculating the frequencies of the two peaks and their spectral weights. Interestingly, this calculation reveals that the two excitations in our cartoon model are strongly hybridized.
We find that that for parameters relevant to experiments on Rb, that the degree of bimodality is vanishingly small and our previous sum rule arguments hazzard:rf-spectra-sum-rule () accurately describe such experiments. On the other hand, there are opportunities to study other atoms (for example, Na, Cs, Yb) for which the bimodality may be more pronounced. Moreover, if the interactions or tunneling rates can be tuned via a spin-dependent lattice or a Feshbach resonance then this spectral feature will appear in a dramatic fashion.
This bimodal spectrum, with one peak produced by the “Mott” component and another by the “superfluid” component, is reminiscent of the spectrum of a finite temperature Bose gas in the absence of a lattice. As described by Oktel and Levitov oktel:cs-ref (), in that situation one sees one peak from the condensate, and one from the incoherent thermal atoms. We would expect that at finite temperature our “Mott” peak continuously evolves into their “thermal” peak.
Ii Bose-Hubbard Model
ii.1 Model and RF spectra
In the rf spectra experiments we consider, initially all atoms are in the -internal state and the rf pulse drives them to the -state. Consequently, we consider two-component bosons trapped in the periodic potential formed by interfering laser beams, described by a Bose-Hubbard model jaksch:olatt (),
where and are the annihilation and creation operators for states in the internal state , is the chemical potential, is the external potential with , the vacuum - splitting, absorbed into it, is the state- state on-site interaction strength, and is the hopping matrix element. The interactions are tunable via Feshbach resonances and spin-dependent lattices are also available deutsch (). For this latter setup, the hopping matrix elements may be tuned by the intensity of the lattices, and introducing small displacements of the lattice will reduce the overlap between the Wannier states of and atoms, and therefore may also be an efficient way to control the relative size of and . The interaction will be irrelevant: we will only consider the case where there is a vanishingly small concentration of -state particles. In calculating the response to RF photons we will take . Trap effects will later be included through a local density approximation hazzard:rf-spectra-sum-rule () which is valid for slowly varying traps pollet:mi (); bergkvist:mi (); wessel:mi (); batrouni:mi (); demarco:stability (); dupuis:mi-sf-review (); sengupta:bhubb-rpa (); konabe:out-coupling-single-ptcl-spec (); menotti:trivedi-single-ptcl-spectral-weight (); ohashi:rf-spectra-dual-character ().
Experimentally the RF spectrum is measured by counting the number of atoms transferred from state to when the system is illuminated by a RF pulse. These dynamics are driven by a perturbation
where is proportional to the time-dependent amplitude of the applied RF field multiplied by the dipole matrix element between states and : typically is a sinusoidal pulse with frequency with a slowly varying envelope ensuring a small bandwidth. Due to the small wave-number of RF photons, recoil can be neglected.
For a purely sinusoidal drive, the number of atoms transferred per unit time for short times is
where the sum is over the initial states (occupied with probability ) and the final states, all of which are eigenstates of with energies and . We will restrict ourselves to and the physically relevant case where the initial states contain no -atoms.
ii.2 Sum Rules
We defined to be the vacuum - splitting, the local phase coherence factor is
with and nearest neighbors, the site filling is , and the lattice coordination is . The zero-distance density-density correlation function is
The second term in Eq. (5) may be interpreted as the mean shift in the kinetic energy when the spin of an atom is flipped. In particular, within a strong-coupling mean-field picture is the condensate density, which can therefore be measured with this technique. The second term in Eq. (5) is the shift in the interaction energy.
Our subsequent approximations will satisfy this sum rule. This is non-trivial: for example, even in simultaneous limits of , , and considered in Ref. sun:rf-spectra-condensate-probe (), their results violate this sum rule by a factor of .
Since it plays no role in the remainder of the discussion, we will set to zero the vacuum level splitting: . This amounts to working in a “rotating frame”.
Iii Random phase approximation
iii.1 General setup and solution
To calculate the RF spectrum we employ a time-dependent strong-coupling mean-field theory which includes fluctuations around the static strong-coupling Gutzwiller mean field theory fisher:bhubb (). This mean field theory is exact in the deep Mott limit and in the deep superfluid when , and it yields fairly accurate ground states in the intermediate regime pollet:mi (); bergkvist:mi (); wessel:mi (); batrouni:mi (); demarco:stability (). Refs. menotti:trivedi-single-ptcl-spectral-weight (); ohashi:rf-spectra-dual-character () previously used analogous RPA’s to calculate the Bose-Hubbard model’s quasiparticle spectra and RF spectra with , which reduces to the single particle spectra.
We use the homogeneous time-dependent Gutzwiller variational ansatz
where is the state at site with particles in the state and in the state. The equation of motion for and are derived by minimizing the action , with Lagrangian
where is a Lagrange multiplier which enforces conservation of probablility. At time , where we take , and choose to minimize ,
Solving the subsequent dynamics to quadratic order in , one finds
where the retarded response function is
The Green’s functions satisfy the equations of motion for the ’s in the absence of an RF field, but in the presence of a delta function source, and boundary condition for . The relevant equations are simplest in Fourier space, where obeys
where is a Hermitian matrix. The tridagonal part is
The remaining contribution, , is
Specializing to the case where , the response is given in terms of normalized eigenvectors , with It takes the form of a sum of delta-functions,
The ’s are found at each point in the phase diagram by starting with a trial , solving Eq. (10), then updating via Eq. (11) and iterating. We find that almost all spectral weight typically lies in only one or two peaks. Fig. 2 shows sample spectra. The superfluid near the Mott state displays a multi-modal spectrum, but in the weakly interacting limit only a single peak is seen. An avoided crossing is clearly visible in these plots. Fig. 3 shows the manifold of spectral peaks in the and plane, using height to denote frequency and opacity to denote spectral weight. Taking moments of , we see that Eq. (5) is satisfied.
iii.2 Limiting Cases
Although finding the spectrum in Eq. (19) is a trivial numerical task, one can gain further insight by considering limiting cases. First, when and the system possesses an symmetry. In this limit we find that is constant for . Thus our approximation gives a spectrum which is proportional to . This result coincides with the exact behavior of the system: the operator is a ladder operator, , and can only generate excitations with energy (set equal to zero in our calculation). The fact that our approximations correctly capture this behavior is nontrivial: in a field theoretic language one would say that our equation of motion approach includes the vertex corrections necessary for satisfying the relevant “Ward identities” pethick:pseudopot-breakdown (); baym:self-consistent-approx (); zinn-justin:qft ().
The current Rb experiments are slightly perturbed from this limit, with and . We find that the -function is shifted by a frequency proportional to , but that the total spectral weight remains concentrated on that one frequency: the sum of the spectral weights at all other frequencies scale as . Consequently it is an excellent approximation to treat the spectrum as a delta-function, and our RPA calculation reduces to the results in hazzard:rf-spectra-sum-rule (). We emphasize however that other atoms, such as Cesium, can be in a regime where is large.
We gain further insight by considering the superfluid near the Mott phase with . Here one can truncate the basis to two states with total particle number and on each site. Then the ’s and ’s can be found analytically: one only needs to solve linear algebra problems. In the , limit, this is similar to Ref. sun:rf-spectra-condensate-probe ()’s approach, but includes the hopping self consistently, allowing us to satisfy the sum rule Eq. (5). This truncation is exact in the small limit, and yields
if (here, only the peak has non-zero spectral weight). We omit the cumbersome analytic expressions for the spectral weights . The spectrum consist of two peaks – hybridized versions of the excitations caricatured in Fig. 1. One can identify and as the energies of those caricature processes, recognizing that the hybridization term, , grows with . The avoided crossing between these modes is evident in Fig. 2.
iii.3 Inhomogeneous spectrum
We model the trapped spectrum through a local density approximation. We assume that a given point in the trap has the properties of a homogeneous gas with chemical potential . In Fig. 4 we show the density profile and the spectrum corresponding to each point in space. Also shown is the trap averaged spectrum. The bimodality of the homogeneous spectrum is quite effectively washed out by the inhomogeneous broadening of the trap. On the other hand, if one images the atoms flipped into the state as in Ref. campbell:ketterle-clock-shift (), there is a clear qualitative signature of the bimodality. If one excites the system with an RF pulse whose frequency lies between the resonant frequencies of two Mott plateaus, one will excite two “shells” of atoms. These shells should be clearly visible, even in column integrated data.
Iv Conclusions and discussion
In this paper we have shown that the RF spectra of a homogeneous Bose gas in an optical lattice will have two (or more) peaks in the superfluid state when the parameters are tuned close to the superfluid-Mott insulator phase transition. Physically, this bimodality is a result of the strong correlations in the system. These correlations result in two distinct forms of excitations (which are strongly hybridized): those involving “core” atoms, and those involving delocalized atoms. When is small, such as in the experiments on Rb, this bimodality is absent.
Our approach, based upon applying linear response to a time dependent Gutzwiller mean field theory, is both simple and quite general. It allows arbitrary interactions between both spin states, and it allows arbitrary spin-dependent hopping rates. The major weakness of the theory is that it fails to fully account for short range-correlations: the atoms are in a quantum superposition of being completely delocalized, and being confined to a single site. The physical significance of this approximation is most clearly seen when one considers the case where the final-state atoms have no interactions, , and see no trap or lattice. Imaging the -atoms after a time-of-flight is analogous to momentum resolved photoemission jin:photoemission (), and would reveal the dispersion relationship of the single-particle excitations. The fact that the spectrum consists of two sharp peaks means that all of the non-condensed atoms are approximated to have the same energy. One will also see that their momentum is uniformly distributed throughout the first Brillioun zone. In the strong lattice limit, where the bandwidth is small, this approximation is not severe.
We thank Sourish Basu, Stefan Baur, Stefan Natu, Kuei Sun, Smitha Vishveshwara, Henk Stoof, Ian Spielman, and Mukund Vengalattore for useful discussions. This material is based upon work supported by the National Science Foundation through grant No. PHY-0758104, and partially performed at the Aspen Center for Physics.
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