Stability Spectroscopy of Rotons in a Dipolar Bose Gas
We study the stability of a quasi-one-dimensional dipolar Bose-Einstein condensate (dBEC) that is perturbed by a weak lattice potential along its axis. Our numerical simulations demonstrate that systems exhibiting a roton-maxon structure destabilize readily when the lattice wavelength equals either half the roton wavelength or a low roton subharmonic. We apply perturbation theory to the Gross-Pitaevskii and Bogoliubov de Gennes equations to illustrate the mechanisms behind the instability threshhold. The features of our stability diagram may be used as a direct measurement of the roton wavelength for quasi-one-dimensional geometries.
It is widely believed that ultracold, gaseous samples of bosonic atoms or molecules possessing sufficiently large dipole moments will exhibit internal structure reminiscent of the roton in superfluid helium (1). The basic phenomenology of the roton, in analogy with helium, is a local minimum in the quasiparticle dispersion . The existence of such a minimum is predicted to lead to a host of attendant phenomena in these dilute gases, including structured ground-state density profiles (2); (3); (4), reduced and anisotropic critical superfluid velocity (1); (5); (6), enhanced sensitivity to external perturbations (7), abrupt transitions in Faraday patterns (8); (9), short-wavelength immiscibility phases (10), and strongly-oscillatory two-body correlations on the roton length scale (11). Signatures of the roton in Bragg spectroscopy of trapped dipolar Bose-Einstein condensates (dBECs) have been calculated in Ref. (12). While these many exciting predictions are in principle observable in current experiments with highly magnetic atoms (13); (15); (14), or in future experiments with electrically polar molecules (16); (17); (18), rotons remain to be seen directly in dBECs (19).
In a dipolar condensate, the roton represents a mode of finite wavelength that has an anomolously low energy – hence the minimum in . The location of this minimum is given by a momentum approximately equal to , where denotes the harmonic oscillator length of the tightest confinement of the dBEC (usually along the polarization axis). Without this confinement, a homogeneous dBEC would be energetically unstable to collapse due to the attraction between dipoles that are aligned head-to-tail. In the presence of this confinement, the collapse is prevented by the zero-point energy introduced by the confinement, at least up until a critical dipole moment or density. When this critical parameter is exceeded, the condensate collapses in localized “clumps” of size , that is, via a dynamical instability into the roton mode (20). For dBECs that are just barely stable, the energy minimum is achieved at the roton wave vector .
A low-energy roton is therefore a mode that is linked intrinsically to condensate instability. This link suggests that the dBEC may respond nontrivially as an object of conventional spectroscopy, that is, that it would absorb and distribute energy differently from different wavelengths of an applied probe. For example, if one imposes on a dBEC a weak potential of periodicity (via a 1D lattice beam (21), for example), then one expects to trigger an instability most easily when . In this Letter we verify this conjecture via mean-field simulations of a quasi-1D (q1D) dBEC. Strikingly, we find structure in addition to this main peak, analogous to nonlinear spectroscopy of atoms or crystals in strong laser fields. Namely, we see hastened destabilization by probes with wavelengths that are integer multiples of , reminiscent of “multiphoton” scattering. We also observe a similar feature at the shorter wavelength , reminiscent of resonant Raman coupling. Since the stability of a condensate is easy to assess experimentally, the observation of such structures constitutes a direct stability-spectroscopic measurement of the roton wavelength as well as a novel signature of roton physics.
Consider a dBEC that is tightly confined in the and directions by a harmonic trap of frequency , with no trapping potential in the direction. The dipole moments of the constituent atoms or molecules are polarized along . To this initially-stable system is applied a probe in the form of an optical lattice potential
whose periodicity and (dimensionless) lattice depth are tunable parameters. Such a potential can be realized using retroreflected or crossed off-resonant laser beams of peak intensity (21). We take the single-mode approximation, assuming that the order parameter is a Gaussian of width in the directions of tight confinement (22). We suppose that there are atoms spread over the periodic domain , where is the one-dimensional integrated density. Expressed in natural length () and energy () units, the Gross-Pitaevskii equation for this situation reads
The momentum-space form of the q1D interaction potential is
where . Contact interactions contribute to via the scattering length , and dipole-dipole interactions contribute via the dipole length , where is the dipole moment of the bosons. We remark that decreases monotonically as a function of . The q1D chemical potential is related to the three-dimensional chemical potential via . In the absence of the probing potential , the chemical potential and order parameter are easily found to be and , respectively.
To determine the stability of the dBEC for a given lattice periodicity and intensity , we solve Eq. (2), and determine the excitation spectrum within the Bogoliubov approximation. Dynamical instability is indicated by a complex frequency in a given mode, which also identifies the mode that triggers the instability. Numerical results are plotted in Fig. 1, which shows two sample stability spectra for dBECs consisting of Dy atoms. Plotted is the intensity versus periodicity of the probe. The shaded region below each curve represents the stable region, since higher intensitites perturb the condensate more strongly and lead to collapse. In one case (I, yellow region) the scattering length is assumed to be , while in the other (II, brown), it is . These differences correspond to different roton spectra, as shown by the blue lines in Fig. 2. Case II has a “softer” roton than case I, i.e., a lower excitation energy . Along the lines of Ref. (23), we have checked that our quantum and thermal depletions for both case I and case II, at an assumed temperature of nK, remain small compared to the condensate fraction. Similar results can be found for all other species of strongly dipolar condensates, although there is a tradeoff between density and dipole length for a given system to rotonize.
In both cases, the spectra clearly show three features, which are more prominent for softer rotons: i) A weak laser drives the condensate to instability for probe periodicities near the roton wavelength, accounting for the lowest dip in Fig. 1. Indeed, the weakest laser required to de-stabilize the condensate has periodicity . We emphasize that at the threshhold of stability, our simulations show that the lattice probe is not chopping the condensate into smaller, unstable pieces, but rather is weakly perturbing it. ii) A secondary dip in stability appears when the lattice spacing is twice the roton wavelength. In terms of the analogy to optical absorption in a nonlinear medium, this situation is reminiscent of the excitation of the resonant state by absorption of two photons of half the energy required to excite the state directly. iii) Finally, a third dip in stability is observed at half the roton wavelength. We have also observed these structures in full D simulations of horizontal, cigar-shaped dBECs where the roton “wavelength” is less well defined.
The origin of these features becomes clear within perturbation theory. Rigorous perturbation theory for the Gross-Pitaevskii equation was worked out in Refs. (7); (24); (25); (26). Let and represent the order parameter and chemical potential in the absence of the probe . Expanding , , one derives the first-order perturbation equation
Given the condition , the solutions to (4) are easily found to be and
The Hartree-Fock spectrum of a soft-roton dBEC posesses a shallow minimum at , and this minimal energy approaches zero as the roton softens. The effect of this low-lying mode is to strongly perturb the ground state, as suggested by (5). Indeed, a modest perturbation becomes amplified in the density by a factor . The order of perturbation theory additionally introduces density modulations of the general form . That is, the roton can be driven by overtones of the fundamental wave number of the lattice, when , or . The stability minimum corresponding to can be seen in Fig. 1. Even away from one of these resonant conditions, the probe laser introduces density modulations that manifest themselves in a mean-field potential which combines with the probe field itself to make a combined potential
Thus again, whatever influence the periodic probe potential has on propagation of the excited states in the condensate, this effect is amplified for probes in the vicinity of the roton (or a subharmonic thereof, as a higher-order effect).
There remains the task of explaining the feature in the stability spectrum at . This cannot be done by considering the ground state only. Rather, it is necessary to explore how the spectrum of excited states is modified by the combined potential energy (6). The Bogoliubov de Gennes (BdG) equations describing the excitations can be written compactly as
where is the free-particle Hamiltonian, describes direct interactions with the mean-field, and is an integral operator describing exchange interactions. The functions and define the usual Bogoliubov transformation of the quantum fluctuation field operator , and they allow one to write the nontrivial part of the grand canonical Hamiltonian in approximate diagonal form as . In the absence of the perturbation, Eq. (7) is easily solved using complex exponentials parameterized by momentum , yielding the unperturbed energies . The perturbation (1) separates modes of definite parity, so we write the unperturbed modes in a basis of cosine and sine functions: , , , and . The amplitudes and are defined by and , for (28). A system destabilizes when at least one of the excitation energies vanishes.
Assuming an initially stable condensate, the influence of on the excited states can also be approximated in perturbation theory. A perturbation theory for the BdG equations was developed rigorously in Refs. (25); (26); (29) in the phase-density representation. Since we are chiefly concerned with first-order mode destabilization, we develop a tractable perturbation theory that paints a physical picture of mode softening and naturally encompasses the degeneracy of the roton spectrum. After expanding , , and in perturbation series, and then substituting into Eq (7), we derive the first-order perturbation equation for the corrections , , and :
where is the zeroth-order exchange integral operator, is the first-order combined potential given by Eq. (6), and is the first-order exchange integral operator. We can isolate the energy shift on the right-hand side by taking advantage of the fact that the left-hand side vanishes whenever it is acted on by the operator for any . Degenerate perturbation theory is, in situations exhibiting a roton-maxon excitation spectrum, generally necessary for a complete understanding of first-order effects; however, we will see that the stability structure at may be explained without accounting for degeneracy. To simplify our notation, we denote the hermitian matrix on the right-hand side of (8) by .
It turns out that the matrix elements of vanish in most instances, simplifying our analysis. Firstly, the sine modes completely decouple from the cosine modes as anticipated. Moreover, the matrix element of between any modes and vanishes unless the mode-matching condition is satisfied. This follows from reasons of orthogonality, since all matrix elements are evaluated by integrating products of three sine or cosine functions. Satisfying both the mode-matching condition and the degeneracy condition severely limits the number of nonzero matrix elements determining the modes (or degenerate mode mixtures) that shift at first order.
The significance of the mode matching is illustrated schematically in Fig. 3. For any given perturbation probe with wave vector , a standing matter wave of this wavelength is established, defining the combined potential (upper panel). Against this backdrop are shown the cosine and sine modes of wave vector (during some phase of their oscillation) in the lower two panels. We depict these modes because they always satisfy both the mode-matching and degeneracy conditions. The density oscillations of the cosine mode accumulate atoms at the maxima of , causing an increase in the mode energy by an amount . Conversely, the density oscillations of the sine mode collect atoms at the minima of , thereby lowering the mode energy by the amount . Linear combinations of degenerate modes satisfying may also accomplish this spatial appropriation of atoms in an average sense, as the direct terms in the diagonal elements appear as , where and are linear combinations of matched modes. This is the mechanism behind the first-order softening of Bogoliubov modes and degenerate mode mixtures, and it is similar to the discussion of “staggered modes” found in Ref. (30).
Given that the roton mode is a local minimum of the system dispersion, we expect to observe a lower stability boundary for all lattice periodicities in which the roton softens to first order. This occurs when the roton mode-matches to itself () and when it mode-matches to a degenerate phonon (). These two scenarios correspond to the stability dips observed at and , respectively. Of course, the latter structure was already expected due to strongly-enhanced, static density modulation, as discussed previously. The structure at , however, occurs in spite of relatively weak density modulation. In that case, the roton responds directly to the perturbation, rather than to a strongly-amplified mean field potential.
We note that our discussions of density response and mode softening remain applicable despite the increase in quantum and thermal fluctuations for rotonized systems. There exists a minimum roton energy below which either thermal or quantum fluctuations cause a strong reduction in the condensate fraction (23). For unperturbed dBECs with roton energies exceeding , the perturbing lattice will likely drive the system towards an uncondensed or thermal state before reaching dynamic instability. The criterion for the existence of a stable dBEC would then change from to , effectively lowering the boundaries in Fig. 1 without changing the essential structure thereof.
In summary, we have explored the novel stability features of quasi-one-dimensional dipolar condensates in a weak lattice. Rotonized systems destabilize rapidly when the lattice periodicity matches either half of the roton wavelength or a subharmonic of the roton. The observation of these stability dips is a clear signal of rotonization, as well as a direct measurement the roton wavelength.
We thank B. Lev for fruitful discussions. J.P.C. acknowledges support from the US DoD through the NDSEG fellowship program. R.M.W. acknowledges support from an NRC postdoctoral fellowship. J.L.B acknowledges financial support from the NSF.
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