Condensed Matter Theory of Dipolar Quantum Gases
- 1 Introduction.
- 2 The dipole-dipole interaction
- 3 Weakly interacting dipolar Bose gas
4 Weakly interacting dipolar Fermi gas
- 4.1 Effects of dipole-dipole interactions.
- 4.2 Normal (anisotropic) Fermi liquid state.
- 4.3 BCS pairing in a homogeneous single-component dipolar Fermi gas.
- 4.4 BCS pairing in a trapped single-component dipolar Fermi gas.
- 4.5 BCS pairing in a two component dipolar Fermi gas
- 4.6 BCS pairing in a dipolar monolayer
- 4.7 BCS pairing in a bilayer dipolar system
- 4.8 Stability of fermionic dipolar systems
- 5 Dipolar multilayer systems
6 Strongly interacting dipolar gas
- 6.1 Two-dimensional dipoles: phase diagram.
- 6.2 Optical lattices
- 6.3 Advanced Hamiltonian design with polar molecules
7 Dipolar gases in one- and quasi-one dimensional geometries
- 7.1 Dipolar gases in a single tube
- 7.2 Two-species mixtures
- 7.3 Quasi-1D physics: coupling between tubes
- 8 Conclusions and outlook
- 9 Acknowledgements
- 10 Biographies
The realization of Bose Einstein condensates (BEC) and quantum degenerate Fermi gases with cold atoms have been highlights of quantum physics during the last decade. Cold atoms in the tens of nanokelvin range are routinely obtained via combined laser- and evaporative-cooling techniques 1. For high-enough densities ( ), the atomic de Broglie wavelength becomes larger than the typical interparticle distance and thus quantum statistics governs the many-body dynamics of these systems. Characteristic features of the physics of cold atomic gases are the microscopic knowledge of the many-body Hamiltonians which are realized in the experiments and the possibility of controlling and tuning system parameters via external fields. External field control of contact inter-particle interactions can be achieved, for example, by varying the scattering length via Feshbach resonances 2, while trapping of ultracold gases is obtained with magnetic, electric and optical fields 3. In particular, optical lattices, which are artificial crystals made of light obtained via the interference of optical laser beams, can realize perfect arrays of hundreds of thousands of microtraps 4, 5, allowing for the confinement of quantum gases to one-dimensional (1D), 2D and 3D geometries and even the manipulation of individual particles 6, 7. This control over interactions and confinement is the key for the experimental realization of fundamental quantum phases and phase transitions as illustrated by the BEC-BCS crossover in atomic Fermi gases 8, and the Berezinskii-Kosterlitz-Thouless transition 9 for cold bosonic atoms confined to 2D.
Breakthroughs in the experimental realization of BEC and degenerate Fermi gases of atoms with a comparatively large magnetic dipole moment, such as 10, 11, 12, 13, 14, 15, 16, 17 and Dy atoms 18, 19 (dipole moment and 10, respectively, with Bohr’s magneton), and the recent astounding progress in experiments with ultracold polar molecules 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 have now stimulated great interest in the properties of low temperature systems with dominant dipolar interactions (see reviews Refs. 32, 33, 34, 35, 36 for discussions of various aspects of the problem). The latter have a long-range and anisotropic character, and their relative strength compared to, e.g., short-range interactions can be often controlled by tuning external fields, or else by adjusting the strength and geometry of confining trapping potentials. For example, in experiments with polarized atoms, magnetic dipolar interactions can be made to overcome short-range interactions by tuning the effective -wave scattering length to zero using Feshbach resonances 10, 11, 12, 13. This has already led to the observation of fundamental phenomena at the mean-field level, such as, the anisotropic deformation during expansion and the directional stability 37, 18 of dipolar BECs. Heteronuclear polar molecules in a low vibrational and rotational state, on the other hand, can have large permanent dipole moments along the internuclear axis with strength ranging between one tenth and ten Debye (1Debye C m). In the presence of an external electric field (with a typical value of V/cm) mixing rotational excitations, the molecules can be oriented in the laboratory frame and the induced dipole moment can approach its asymptotic value, corresponding to the permanent dipole moment. This effect can be used to tune the strength of the dipole-dipole interaction 35. Additional microwave fields allow for advanced tailoring of the interactions between the molecules, where even the shape of interaction potentials can be tuned with external fields, in addition to the strength. This tunability of interactions forms the basis for the realization of novel quantum phenomena in these systems, in the strongly interacting limit.
As a result of this progress, in recent years dipolar gases have become the subject of intensive theoretical efforts, and there is now an extensive body of literature predicting novel properties for these systems32, 33, 34, 35, 36. It is the purpose of this review to provide a summary of these recent theoretical studies with a focus on the many-body quantum properties, to demonstrate the connections and differences between dipolar gaseous systems and traditional condensed-matter systems, and to stress the inherent interdisciplinary nature of these studies. This work covers spatially homogeneous as well as trapped systems, and includes the analysis of the properties of dipolar gases in both the mean-field (dipolar Bose-Einstein condensates and superfluid BCS pairing transition) and in the strongly correlated (dipolar gases in optical lattices and low-dimensional geometries) regimes.
We tried our best to include all relevant works of this exciting, ever expanding field. We apologize in advance if some papers (hopefully, not many) are not appearing below.
2 The dipole-dipole interaction
For polarized dipolar particles, interparticle interactions include both a short-range Van der Waals (vdW) part and a long-range dipole-dipole one. The latter is dominant at large interparticle separations and assuming a polarization along the -axis as in Fig. 1(a) the interparticle interaction reads
Here is the electric dipole moment (for magnetic dipoles should be replaced with , with the magnetic dipole moment), is the vector connecting two dipolar particles, and is the angle between and the dipole orientation (the -axis). The potential is both long-range and anisotropic, that is, partially repulsive and partially attractive. As discussed in the sections below, these features have important consequences for the scattering properties in the ultracold gas, for the stability of the system as well as for a variety of its properties.
2.1 Scattering of two dipoles
The long-range character () of the dipole-dipole interaction results in all partial waves contributing to the scattering at low energies, and not only, e.g., the -wave, as is often the case for short-range interactions. In fact, for dipole-dipole interactions the phase shift in a scattering channel with angular momentum behaves as for and small (see, e.g., Refs. 38 and 39).
The effect of the anisotropy of the interaction is instead that the angular momentum is not conserved during scattering: for bosons and fermions the dipole-dipole interaction mixes all even and odd angular momenta scattering channels, respectively. Due to the coupling between the various scattering channels, the potential then generates a short-range contribution to the total effective potential in the -wave channel (). This has the general effect to reduce the strength of the short-range part of the interaction.
Thus, for two bosonic dipolar particles (even angular momenta) the scattering at low energies is determined by both the long-range and the short-range parts of the interaction. This is in contrast to the low energy scattering of two fermionic dipoles (odd angular momenta), which is universal in the sense that it is determined only by the long-range dipolar part of the interaction, and is insensitive to the short-range details.
For a dilute weakly interacting gas the above results allow a parametrization of the realistic interparticle interaction between two particles of mass in terms of the following pseudopotential 40-41 (see also Refs. 42 and 43)
parametrizing the short-range part of the interaction. We note that the long-range part of the
pseudopotential is identical to the long-range part of the original potential
and the scattering length controlling the short-range part depends on the dipole moment.
This dependence is important 44 when one changes the dipole moment,
using external, e.g. electric, fields, as explained below.
The strength of the dipole-dipole interaction can be characterized by the quantity
which has the dimension of length and can be considered as a characteristic range of the dipole-dipole interaction, or dipolar length. This length determines the low energy limit of the scattering amplitudes, and, in this sense, is analogous to the scattering length for the dipole-dipole interaction. For chromium atoms with a comparatively large magnetic moment of (equivalent dipole moment Debye) we have nm. For most polar molecules the electric dipole moment ranges in between and Debye, while ranges from to nm. For example, the dipole moment of fermionic ammonia molecules is Debye with nm, while for it increases to Debye and nm. This latter value of the effective scattering length is an order of magnitude larger than, for example, the one for the intercomponent interaction in the widely discussed case of a two-species fermionic gas of , where nm. Thus, the strength of the dipole-dipole interaction between polar molecules can be not only comparable with but even much larger than the strength of the short-range interatomic interaction.
2.2 Tunability of the dipole-dipole interaction
One spectacular feature of the dipole-dipole interaction is its tunability. In Sect. 2.2.1 we first review methods for tuning the strength and sign of dipolar interactions with an eye to cold atoms, and then in Sect. 2.2.2 we discuss tunability for the specific case of polar molecules, where both the strength as well as the shape of interactions can be engineered.
2.2.1 Tunability of interactions in cold atoms
In Ref. 45 a technique has been developed to tune the strength as well as the sign of dipolar interactions in atomic systems with a finite permanent magnetic dipole moment. This technique uses a combination of a static (e.g., magnetic) field along the -axis and a fast rotating field in the perpendicular -plane such that the resulting time dependent dipole moment is [see Fig. 1(b)]
Here is the rotating frequency of the field and the angle , is determined by the ratio of the amplitudes of the static and rotating fields. The above expression implies that the dipoles follow the time-dependent external field adiabatically. This in turn sets an upper limit on the values of the rotating frequency , which should be (much) smaller than the level splitting in the field. However, if the frequency is much larger that the typical frequencies of the particle motion, over the period the particles feel an average interaction
The latter differs from the interaction for aligned dipoles, Eq. (1), by a factor , which can be changed continuously from to by varying the angle . Thus this method allows to ”reverse” the sign of the dipole-dipole interaction and even cancel it completely for , similar to NMR techniques 46. We note that an analogous technique can be also applied for the electric dipole moments of, e.g., polar molecules. We will review applications of this method below.
2.2.2 Effective Hamiltonians for polar molecules
In the following we will be often interested in manipulating interactions for
polar molecules in the strongly interacting regime. In particular, we
will aim at modifying not only the strength but also the shape of
interaction potentials, as a basis to investigate new condensed matter
phenomena. This usually entails a combination of the following two steps: (i)
manipulating the internal (electronic, vibrational, rotational, …) structure
of the molecules, and thus their mutual interactions, using external static
(DC) electric and microwave (AC) fields, and (ii) confining molecules to a
lower-dimensional geometry, using, e.g., optical potentials, as exemplified in
Fig. 2. Under appropriate conditions, the resulting effective
interactions can be made purely repulsive at large distances (e.g.,
at characteristic distances of 10nm or more), as in the 2D example of
Fig. 3(a). On one hand, this has the effect to suppress
possible inelastic collisions and chemical reactions occurring at short-range
(i.e., at characteristic distances of nm), and on the other
hand it allows to study interesting condensed matter phenomena originating
from the often-unusual form of the two-body (or many-body-) interaction
potentials. In the next few subsections we review techniques for the
engineering of the interaction potentials which will be used in the many-body
context in Sect. 6 below.
Our starting point is the Hamiltonian for a gas of cold heteronuclear molecules prepared in their electronic and vibrational ground-state,
Here the first term in the single particle Hamiltonian corresponds to the kinetic energy of the molecules, while represents a trapping potential, as provided, for example, by an optical lattice, or an electric or magnetic trap. The term describes the internal low energy excitations of the molecule, which for a molecule with a closed electronic shell (e.g. SrO, RbCs or LiCs) correspond to the rotational degree of freedom of the molecular axis. This term is well described by a rigid rotor with the rotational constant (in the few to tens of GHz regime) and the dimensionless angular momentum. The rotational eigenstates for a quantization axis , and with eigenenergies can be coupled by a static (DC) or microwave (AC) field via the electric dipole moment , which is typically of the order of a few Debye.
In the absence of electric fields, the molecules prepared in a ground
rotational state have no net dipole moment, and interact via a
van-der-Waals attraction , reminiscent of
the interactions of cold alkali metal atoms in the electronic ground-states.
Electric fields admix excited rotational states and induce static or
oscillating dipoles, which interact via strong dipole-dipole interactions
with the characteristic dependence given in Eq. (1).
For example, a static DC field couples the spherically symmetric rotational
ground state of the molecule to excited rotational states with different
parity, thus creating a non-zero average dipole moment. The field strength
therefore determines the degree of polarization and the magnitude of the
dipole moment. As a result, the effective dipole-dipole interaction may be
tuned by the competition between an orienting, e.g., DC electric field and the
quantum (or thermal) rotation of the molecule. This method effectively works
for the values of the field up to the saturation limit, at which the molecule
is completely polarized (typically V/cm).
The many body dynamics of cold polar molecules is thus governed by an interplay between dressing and manipulating the rotational states with DC and AC fields, and strong dipole-dipole interactions. In condensed matter physics one is often interested in effective theories for the low-energy dynamics of the many-body system, after the high-energy degrees of freedom have been traced out. The connection between the full molecular -particle Hamiltonian (4) including rotational excitations and dressing fields, and an effective low-energy theory can be made using the following Born-Oppenheimer approximation: The diagonalization of the Hamiltonian for frozen spatial positions of the molecules yields a set of energy eigenvalues , which can be interpreted as the effective interaction potential in the single-channel many-body Hamiltonian
The term represents an effective -body interaction, which can be expanded as a sum of two-body and many-body interactions
where in most cases only two-body interactions are considered. The dependence
of on the
electric fields provides the basis for the engineering of the
many body interactions, as described below.
We note that the attractive part of the interaction potential can induce instabilities in a dipolar gas at the few body level as well as at the many-body level (this latter case will be discussed in Sect. 3 below). For example, for several experimentally relevant mixed alkali-metal diatomic species such as KRb, LiNa, LiK, LiRb, and LiCs 47 there exist chemically reactive channels that are energetically favorable, leading to particle recombination and two-body losses in the gas. The rate of chemical reactions can be strongly enhanced by dipole-dipole interactions which can attract molecules in a head-to-tail configuration [e.g., in Fig. 1(a)] to distances on the order of typical chemical interaction distances, nm 48, 49, 50, 51, 52, 53, 54, 55, 56. One aim of interaction engineering is to control these interactions in order to stabilize the gas against particle losses. This will enable the study of complex condensed matter phenomena in these systems.
2.2.3 Stabilization of dipolar interactions in 2D
The simplest example of stabilization of dipolar interactions against inelastic collisions is sketched in Fig. 2 and consists of a system of cold polar molecules in the presence of a polarizing DC electric field oriented in the -direction, and of a strong harmonic transverse confinement with frequency and characteristic length . The latter is provided, e.g., by an optical lattice along .
Figure 3(b) shows a countour plot of the interaction potential for two dipoles in this quasi-2D geometry, where
Here represents the distance between the two molecules in cylindrical coordinates, and . The first term is the isotropic vdW potential, assuming the molecules are in their rotational ground state, with a vdW length 58, 59. The second term is the anisotropic dipolar potential, with induced dipole moment and dipolar length .
Figure 3(b) illustrates essential features of reduced dimensional collisions: for finite , the repulsive dipole-dipole interaction overcomes the attractive van-der-Waals potential in the ()-plane at distances , realizing a repulsive in-plane potential barrier (blue dark region). In addition, the harmonic potential confines the particles’s motion in the direction. The combination of the dipole-dipole interaction and of the harmonic confinement thus yields a three-dimensional potential barrier separating the long-distance, where interactions are repulsive, from the short-distance one, where interactions are attractive and inelastic processes can occur. If the collision energy is smaller than the height of the barrier at the saddle point (white circles), the particles’ motion is confined to the long-distance region, where particles scatter elastically. Particle losses are due to tunneling through the potential barrier at a rate . Within a semiclassical (instanton) approximation valid for , the tunneling rate is well approximated by the exponential form
The constant has been recently computed numerically by Julienne, Hanna and
Idziaszek 61 to be , while the ”attempt rate”
62 for the scattering of two isolated dipoles
independent of particles’ statistics. Here is the momentum
for a collision with relative kinetic energy , with the DeBroglie wavelength. For
particles in a crystalline configuration (see Sect. 6.1
below), will be proportional to the frequency of phonon
oscillations around the mean particle positions , with the mean inter-particle distance. The
expression Eq. (7) shows that collisional losses may be
strongly suppressed for any molecular species for a large enough
dipole moment or strength of transverse confinement.
In ultracold collisions one often has the following separation of length scales: , and can be tuned by, e.g, increasing the external DC field. Figure 4(a) and (b) show numerical results for reactive and elastic collision rates of bosonic and fermionic KRb molecules, respectively, and for several strengths of transverse confinement. Here , and are on the order of hundreds of nm, tens of nm, and less than 10 nm, respectively. Because of the moderate D of KRb molecules, here and the semiclassical regime of large of Eq. (7) is not reached. Nevertheless, in stark contrast to collisions in 3D 50, the figure shows that the ratio between elastic and inelastic collision rates increases rapidly with , signaling an increased stability with increasing . For bosons, the exact numerical results (thick lines) approach rapidly the semiclassical instanton limit (thin lines) with increasing . The behavior of the inelastic rates for fermions is explained in detail in Refs. 63, 60.
Recent landmark experimental results from the JILA group with fermionic KRb molecules show a strong suppression of inelastic collisions and increase of elastic ones with , in excellent agreement with the predictions of Fig. 3. This opens the way to the study of strongly correlated phenomena in these systems.
2.2.4 Advanced interaction designing: Blue-shielding
By combining DC and AC fields to dress the manifold of rotational energy levels it is possible to design effective interaction potentials with (essentially) any shape as a function of distance. For example, the addition of a single linearly-polarized AC field to the configuration of Sect. 2.2.3 leads to the realization of the 2D “step-like” potential of Fig. 3(a) (black dashed-dotted line), where the character of the repulsive potential varies considerably in a small region of space. The derivation of this effective 2D interaction is sketched in Fig. 5(a-c) 64, 57. The (weak) DC-field splits the first-excited rotational ()-manifold of each molecule by an amount , while a linearly polarized AC-field with Rabi frequency is blue-detuned from the ()-transition by , see Fig. 5(a). Because of and the choice of polarization, for distances the relevant single-particle states for the two-body interaction reduce to the states and of each molecule. Figure 5(b) shows that the dipole-dipole interaction splits the excited state manifold of the two-body rotational spectrum, making the detuning position-dependent. As a consequence, the combined energies of the bare groundstate of the two-particle spectrum and of a microwave photon become degenerate with the energy of a (symmetric) excited state at a characteristic resonant (Condon) point , which is represented by an arrow in Fig. 5(b). At this Condon point, an avoided crossing occurs in a field-dressed picture, and the new (dressed) groundstate potential inherits the character of the bare ground and excited potentials for distances and , respectively. Fig. 5(d) shows that the dressed groundstate potential (which has the largest energy) is almost flat for and it is strongly repulsive as for , which corresponds to the realization of the step-like potential of Fig. 3(a). We remark that, due to the choice of polarization, this strong repulsion is present only in the plane , while for the groundstate potential can become attractive. The optical confinement along of Sect. 2.2.3 is therefore necessary to ensure the stability of the system.
The interactions in the presence of a single AC field are described in detail in Ref. 57, where it is shown that in the absence of external confinement this case is analogous to the (3D) optical blue-shielding developed in the context of ultracold collisions of neutral atoms 65, 66, 67, however with the advantage of the long lifetime of the excited rotational states of the molecules, as opposed to the electronic states of cold atoms. The strong inelastic losses observed in 3D collisions with cold atoms 65, 66, 67 can be avoided via a judicious choice of the field’s polarization, eventually combined with a tight confinement to ensure a 2D geometry (as e.g. in Fig. 2 above). For example, in Ref. 68 it is shown that in the presence of a DC field and of a circularly polarized AC field the attractive time-averaged interaction due to the rotating (AC-induced) dipole moments of the molecules allows for the cancelation of the total dipole-dipole interaction. The residual interactions remaining after this cancelation are purely repulsive 3D interactions with a characteristic van-der-Waals behavior . This 3D repulsion provides for a shielding of the inner part of the interaction potential and thus it will strongly suppress inelastic collisions in experiments.
Recent works 69, 70 have considered the microwave
spectra of alkali-metal dimers including hyperfine interactions. It is an
important open question to determine the effects that the presence of internal
states such as, e.g. hyperfine states, have on the broad class of shielding
techniques described above.
3 Weakly interacting dipolar Bose gas
3.1 BEC in a spatially homogeneous gas.
Let us discuss now the influence of the dipole-dipole interaction on the properties of a homogeneous single-component dipolar Bose gas 111This and the next Sections are substantially revised and updated version of the corresponding part of Ref. 32. This can be most conveniently done in the language of second quantization. For this purpose we introduce particle creation and annihilation field operators and satisfying standard bosonic commutation relation
The corresponding second quantized Hamiltonian of the system then reads
where is the mass of the particles, is the interparticle interaction, and the chemical potential fixes the average density of the gas. We consider the case when the system is away from any ”shape” resonances 38,39,71 and, therefore, replace the original interparticle interaction with the pseudopotential (2). Assuming that the system is dilute, , we can write the Hamiltonian as
where is given by Eq. (1) and [as compared to Eq. (3), we omit the dependence of the scattering length on the dipole moment ]. Note that the scattering length has to be positive, , to avoid an absolute instability due to local collapses 72.
As we will see below, an important parameter that determines the properties of the system described by the Hamiltonian (9) is
It measures the strength of the dipole-dipole interaction relative to the short-range repulsion. In the case , the short-range part of the interparticle interaction is dominant while the dipole-dipole interaction results in only small corrections. For a positive scattering length , the system is stable and exhibits BEC at low temperatures. This case corresponds to earlier experiments 10, 13 with Cr BEC ( 13). It was found that the corrections due to magnetic dipole-dipole interaction between Cr atoms are of the order of .
For the opposite case , the anisotropic dipole-dipole interaction plays the dominant role resulting in instability of a spatially homogeneous system 73,74, 75. This instability can be seen in the dispersion relation between the energy and the momentum of excitations in the Bose-condensed gas, which can be easily obtained within the standard Bogoliubov approach:
Here is the angle between the excitation momentum and the direction of dipoles, and is the Fourier transform of . For , the excitation energies at small and close to become imaginary signalling the instability (collapse). This instability of a spatially homogeneous dipolar Bose gas with dominant dipole-dipole interaction is a result of a partially attractive nature of the dipole-dipole interaction.
3.2 BEC in a trapped gas.
The above consideration shows that the behavior of a spatially homogeneous Bose gas with a strong dipole-dipole interaction is similar to that of a Bose gas with an attractive short-range interaction characterized by a negative scattering length . In the latter case, however, the collapse of the gas can be prevented by confining the gas in a trap provided the number of particles in the gas is smaller than some critical value , (see, e.g., 72). This is due to the finite energy difference between the ground and the first excited states in a confined gas. For a small number of particle this creates an effective energy barrier preventing the collapse and, therefore, results in a metastable condensate. The same arguments are also applicable to a dipolar BEC in a trap, see Refs. 15 and 16, with one very important difference: The sign and the value of the dipole-dipole interaction energy in a trapped dipolar BEC depends on by the trapping geometry and, therefore, the stability diagram contains the trap anisotropy as a crucial parameter.
3.2.1 Ground state.
The Hamiltonian for a trapped dipolar Bose gas reads
is the trapping potential and we again use the pseudopotential (2) for the short-range part of the interparticle interaction assuming that the system is away from ”shape” resonances. For the trapping potential we consider the experimentally most common case of an axially symmetric harmonic trap characterized by the axial and radial trap frequencies. The aspect ratio of the trap is defined through the ratio of the frequencies: , where and are the axial and radial sizes of the ground state wave function in the harmonic oscillator potential (13), respectively. For one has a pancake-form (oblate) trap, while the opposite case corresponds to a cigar-form (prolate) trap. Taking into account the anisotropy of the dipole-dipole interaction, one can easily see that the aspect ratio should play a very important role in the behavior of the system.
The standard mean-field approximation corresponds to taking the many-body wave function in the form of a product of single-particle wave functions:
The condensate is then described by the condensate wave function normalized to the total number of particles, , and governed by the time-dependent Gross-Pitaevskii (GP) equation
The validity of this approach was tested in Refs. 44 and 41 by using many-body diffusion Monte-Carlo calculations with the conclusion that a GP equation with the pseudopotential (2) provides a correct description of the gas in the dilute limit . Note that, being the product of single-particle wave functions, the many-body wave function (14) does not take into account interparticle correlations at short distances due to their interaction, which takes place at interparticle distances . This change of the wave function is taken into account in Eq. (15) by the contact part of the pseudopotential (2) [the fourth term in the right-hand-side in Eq. (15)] but ignored in the last term of Eq. (15) because the main contribution to the integral comes from large interparticle distances (of order the spatial size of the condensate).
Let us first consider stationary solutions of Eq. (15), for which and obeys the stationary GP equation
where . Numerical analysis of Eqs. (15) and (16) was performed in Refs. 40,73, 74,76-77 on the basis of numerical solutions of the non-linear Schrödinger equation (16) together with variational considerations with the Gaussian ansatz for the condensate wave function. In Refs. 44 and 41 the problem was treated using diffusive Monte-Carlo calculations, while the authors of Ref. 78 apply the Thomas-Fermi approximation that neglects the kinetic energy and allows to obtain analytical results.
We begin the discussion of the results with the case of a dominant dipole-dipole interaction, , such that the third term in the left-hand-side of Eq. (16) can be neglected. This case demonstrates already all important features of the behavior of dipolar condensates. The general case will be briefly discussed at the end of this section.
Let us introduce the mean-field dipole-dipole interaction energy per particle
which together with the trap frequencies and are important energy scales of the problem. One can easily see that the value of the chemical potential and the behavior of the dipolar condensate are determined by the aspect ratio of the trap , the quantity , and the parameter . Notice also that the anisotropy of the dipole-dipole interaction results in squeezing the cloud in the radial direction and stretches it in the axial one (along the direction of dipoles) in order to low the interaction energy. For this reason the aspect ratio of the cloud is always larger than the aspect ratio of the trap. Here and are the axial and the radial sizes of the cloud, respectively.
We now summarize the results of the stability analysis of the dipolar condensate with (Eq. (16) with ) 74,76,79 (see also Ref. 77 for the stability analysis in a general harmonic trap). The mean-field dipole-dipole interaction is always attractive, , for a cigar shaped trap causing instability (collapse) of the gas if the particle number exceeds a critical value . This critical value depends only on the trap aspect ratio . It was found that the shape of the cloud with close to is approximately Gaussian with the aspect ratio for a spherical trap (), and for an elongated trap with .
For a pancake shaped trap with , the situation is more subtle. In this case there exists a critical trap aspect ratio , which splits the pancake shaped traps into soft pancake traps () and hard pancake traps (). For soft pancake traps one has again a critical number of particles such that the condensates with are unstable. For close to and , the aspect ratio of the cloud approaches the aspect ratio of the trap, . Note that in this case the collapse occurs even in a pancake shaped cloud with positive mean dipole-dipole interaction due to the behavior of the lowest quadrupole and monopole excitations (see Section 3.2.2).
For hard pancake traps, it was argued in Refs. 74 and 76 that the dipolar condensate is stable for any because the dipole-dipole interaction energy is always positive. On the other hand, by using more advanced numerical analysis and larger set of possible trial condensate wave functions, the authors of Ref. 79 found that the dipolar condensate in a hard pancake trap is also unstable for sufficiently large number of particles. Similar conclusions were drawn in Ref. 77. It was found that the critical values of the parameter for the instability to occur are orders of magnitude larger than in soft pancake and cigar shaped traps. In addition, the regions in parameter space were discovered where the maximum density of the condensate is not in the center of the cloud such that the condensate has a biconcave shape. (Analogous behavior of the condensate in a general three-dimensional harmonic trap were found in Ref. 77, see also 80 and 81.) These regions exist also in the presence of a small contact interaction with , but their exact position and size depend on . It is important to mention that condensates with normal and biconcave shapes behave differently when the instability boundary is crossed. The condensate with a normal shape develops a modulation of the condensate density in the radial direction, so-called ”radial roton” instability similar to the roton instability for the infinite-pancake trap () 82, see Section 3.2.3. On the other hand, it is the density modulations in the angular coordinate that lead to the collapse of biconcave condensates - a kind of “angular roton” instability in the trap. In the latter case one has spontaneously broken cylindrical symmetry.
The behavior of the trapped dipolar condensate can be simply captured by means of a Gaussian variational ansatz for the condensate wave function :
where the equilibrium radial size and the cloud aspect ratio can be found by minimizing the energy. Note that in order to describe biconcave shaped condensates, one has to consider (see Ref. 79) a linear combination of two wave functions: the first one is a Gaussian (18) and the second one is the same Gaussian multiplied by , where is the Hermite polynomial of the second order.
For large values of the parameters , where or , are large (but still ), one can use the Thomas-Fermi approximation to find the chemical potential and the shape of the cloud 78. This case corresponds to the small the kinetic energy, as compared to other energies, and, therefore, we can neglect the corresponding term with derivatives in Eq. (16). The GP equation then becomes
The solution of this equation reads
with the chemical potential
where is the density of the condensate in the center of the trap and
The energy of the condensate is
and the radii of the condensate in the radial and axial directions and are
and the corresponding aspect ratio of the cloud can be found from the equation
Note, that the above equation coincides with the equation on the aspect ratio for the Gaussian variational ansatz (18) when the kinetic energy contribution is neglected, as shown in Ref. 73. It was also found that the Thomas-Fermi approximation agrees well with numerical results when used to analyze the stability of the condensate. However, the critical number of particles cannot be found in the Thomas-Fermi approximation because both terms in the expression (21) for the energy have the same dependence on the number of particles after taking into account the expressions (22) and (23) for and .
Let us now briefly discuss the stability of a dipolar condensate in the general case with . It is obvious that for an attractive short-range interaction with the condensate can only be (meta)stable for a small number of particles. For a repulsive short-range interaction with and weak dipole-dipole interaction , the condensate is always stable. For the dipolar condensate can be only metastable for number of particles smaller than a critical value, , which depends on and the trap aspect ratio . This means that the (metastable)condensate solution provides only a local minimum of the energy, while the global minimum presumably corresponds to a collapsed state with or, for , a kind of density modulated state.
3.2.2 Collective excitations and instability.
We have already mentioned that collective excitations play an important role in the stability analysis of a dipolar condensate. They also determine the dynamics of the gas and, therefore, are of experimental interest.
For a trapped dipolar condensate, the analysis of excitations is usually performed on the basis of the Bogoliubov-de Gennes equations which can be obtained by linearizing the time-dependent GP equation (15) around the stationary solution . This can be achieved by writing a solution of Eq. (15) in the form
where the second term describes small () oscillations of the condensate around with (complex) amplitudes and . To the first order in the linearization of Eq. (15) gives the Bogoliubov-de Gennes equations
where is given by Eq. (2). The solution of these linear equations provides the eigenfunctions (,) with the amplitudes and obeying the normalization condition
and the corresponding eigenfrequencies of the collective modes. The Bogoliubov-de Gennes equations (25) and (26) can also be obtained by diagonalizing the Hamiltonian (3.2.1) in the Bogoliubov approximation, which corresponds to splitting the field operator into its mean-field value and the fluctuating quantum part expressed in terms of annihilation and creation operators and of bosonic quasiparticles (quanta of excitations):
The normalization condition for the amplitudes and ensures the bosonic nature of the excitations: The operators and obey the canonical Bose commutation relations.
Nonlocality of the dipole-dipole interaction results in an integrodifferential character of the Bogoliubov-de Gennes equations (25) and (26), making it hard to analyze them both analytically and numerically. A simpler way is to study the spectrum of small perturbations around the ground state solution of the time-dependent GP equation (15) (see Ref. 83 for this approach to atomic condensates). Using this approach in combination with the Gaussian variational ansatz 73, 84 or the Thomas-Fermi approximation 85, it is possible to obtain analytic results for several low energy excitation modes.
As an illustration, let us consider a Gaussian variational wave function
The variational parameters here are the complex amplitude , the widths , the coordinates of the center of the cloud , and the quantities and related to the slope and the curvature, respectively. The normalization of the wave function to the total number of particles provides the constraint
To find the equations governing the variational parameters, we notice that the time-dependent GP equation (15) are equivalent to the Euler-Lagrange equations for the action
with the Lagrangian
where is the phase of [the modulus of was excluded by using Eq. (28)] and we set and for simplicity (this corresponds to ignoring the so-called sloshing motion of the condensate). The standard Euler-Lagrange variational procedure
with (, ) provides equations of motion for the parameters , and :
The above equation describes the motion of a particle with a unit mass in the potential
Therefore, the frequencies of small amplitude oscillations around the stationary solution can be read from the second derivatives of the potential at its minimum. In this way one can obtain the frequencies for the first three compressional excitation modes. In Refs. 73 and 84, these frequencies and the corresponding shapes of the cloud oscillations were found for a cylindrical symmetric trap, , , see Fig. 6.
In the considered cylindrical geometry with dipoles are oriented along the -axis, the projection of the angular momentum on the -axis is a good quantum number that can characterize the mode: One has for modes and and for mode . The modes and as often called breathing and quadrupole modes, respectively, and we will follow this convention here. (In the Thomas-Fermi approximation, one can find analytical expressions for these modes, see Ref. 85.) Important is that with increasing the strength of the dipole-dipole interaction, the quadrupole mode 3 demonstrates the tendency towards instability, and becomes unstable when reaches some critical value. This character of instability via softening of the mode 2 is similar to that in a Bose gas with a short-range attractive interaction ().
The situation for a dipolar gas with dominant dipole interactions is more complicated 79,84, 86. It was found (see Ref. 84) that the instability of collective modes of a dipolar BEC reminds that of a gas with an attractive short-range interaction only if the trap aspect ratio is larger than the critical one, (numerically was found ): The lowest frequency ”breathing” mode 2 becomes unstable when the parameter . The variational approach discussed above provides the scaling behavior of its frequency near the critical point (see Ref. 84): , with , which is very close to the experimental value for Chromium BEC 15.
For intermediate values of above (), the mode which drives the instability (the lowest frequency mode) is a superposition of breathing and quadrupole modes with the exponent still close to . The mode has the breathing symmetry (mode 2) for far below , while it changes and becomes quadrupole-like (mode 3) as approaches the critical value .
For close to () the lowest frequency is the quadrupole mode 3. The frequency of this mode tends to zero as approaches the critical value, , with the exponent if is not too close to . When approaches , one has , and . Finally, when , the frequency of the lowest frequency quadrupole mode can be zero only for 86. (Note that this result cannot be reproduced within the Gaussian variational ansatz, which in general does not provide reliable results close to the instability, see Ref. 84.)
Collective modes for the case were analyzed in Ref. 79 on the basis of the Bogoliubov-de Gennes equations (25) and (26). The two possible type of solutions for the stable condensate were already mentioned above: A pancake (normal) shaped condensate (the maximum condensate density is in the center of the trap), and a biconcave shaped condensate (the maximum condensate density is at some distance from the center of the trap). It was found that in the case of a pancake condensate, the mode which drives the instability has zero projection of angular momentum on the -axis, , and consists of a radial nodal pattern. The number of the nodal surfaces increases with decreasing (flattening of the condensate). This “radial roton” mode in a confined gas can be viewed as an analog of the roton mode in an infinite-pancake trap from Ref. 82, see below. In a biconcave condensate near the instability, the lowest frequency mode has non-zero projection of the angular momentum on the -axis, . This mode is an “angular roton” in the trap: For a biconcave-shaped condensate, the maximum density is along the ring, and an angular roton corresponds to density modulation along this ring. The instability in this case corresponds to the collapse of the condensate due to buckling of the density in the angular coordinate, and, therefore, breaks the cylindrical symmetry spontaneously (see Ref. 79 for more details).
3.2.3 Roton instability of a quasi 2D dipolar condensate.
Let us now discuss the effects of the long-range and anisotropic character of dipole-dipole forces in the physically simpler case of an infinite pancake shaped trap, with the dipoles perpendicular to the trap plane 82. It was found that a condensate with a large density can be dynamically stable only when a sufficiently strong short-range repulsive interaction is present. Otherwise, excitations with the certain in-plane momenta become unstable when the condensate density exceeds the critical value . Interestingly, the excitation spectrum of a stable condensate with the density has a roton-maxon form similar to that in the superfluid helium (see also Ref. 87 for the quasi-2D version of this problem).
The time-dependent GP equation for the condensate wave function of dipolar particles harmonically confined in the direction of the dipoles (-axis) reads
where is the confining frequency. Let us assume the ground state to be uniform in the in-plane directions such that the ground state wave function is independent of the in-plane coordinate . We can then integrate over in the dipole-dipole term of Eq. (33) with the result