# Study of a degenerate dipolar Fermi gas of Dy atoms

###### Abstract

We study properties of a single-component (spin polarized) degenerate dipolar Fermi gas of Dy atoms using a hydrodynamic description. Under axially-symmetric trapping we suggest reduced one- (1D) and two-dimensional (2D) description of the same for cigar and disk shapes, respectively. In addition to a complete numerical solution of the hydrodynamic model we also consider a variational approximation of the same. For a trapped system under appropriate conditions, the variational approximation as well as the reduced 1D and 2D models are found to yield results for shape, size and chemical potential of the system in agreement with the full numerical solution of the three-dimensional (3D) model. For the uniform system we consider anisotropic sound propagation in 3D. An analytical result for anisotropic sound propagation in uniform dipolar degenerate Fermi gas is found to be in agreement with results of numerical simulation in 3D.

###### pacs:

67.85.Lm,03.75.Ss,05.30.Fk,67.10.Db## 1 Introduction

After the experimental realization of Bose-Einstein condensate (BEC) of Cr [1, 2], Dy [3], and Er [4] atoms with large magnetic dipolar interaction, there have been renewed interest in the study of cold atoms, both theoretically and experimentally. The anisotropic long-range dipolar interaction acting in all partial waves in these atoms is basically different from isotropic short-range S-wave interaction acting in nondipolar atoms. Polar bosonic molecules with much larger permanent electric dipole moment are also candidates for possible BEC experiments [5]. Dipolar BECs have novel distinct properties. Due to the anisotropic dipolar interaction, the stability of a dipolar BEC depends on the scattering length as well as the trap geometry [2, 6, 7]. The shock and sound waves also propagate anisotropically in a dipolar BEC [8, 9]. Anisotropic collapse has been observed and studied in a dipolar BEC of Cr atoms [10]. Anisotropic rotons [11] and anisotropic critical superfluid velocity [12] have been suggested and studied in dipolar BECs. Distinct stable checkerboard, stripe, and star configurations in dipolar BECs have been identified in a two-dimensional (2D) optical lattice as stable Mott insulator [13] as well as superfluid soliton [14] states. Anisotropic solitons in 2D have also been suggested in dipolar BECs [15]. A new possibility of studying universal properties of dipolar BECs at unitarity has been suggested [16].

After the realization of BEC of alkali-metal atoms, degenerate nondipolar gas of fermionic Li [17], K [18], and Sr [19] atoms were observed. Later, superfluid states of paired Li [20] and K [21] Fermi atoms have been investigated. More recently a degenerate dipolar gas of fermionic Dy atoms with large magnetic dipole moment has been created and studied [22]. Realization of quantum degeneracy in Dy atoms should be considered as a doorway for the study of anisotropic superfluidity in dipolar fermions. Fermionic polar molecules, such as KRb, of large permanent electric dipole moment are also considered for this purpose [23]. The KRb molecule in the singlet rovibrational ground state has an electric dipole moment of 0.6 Debye, thus leading to a dipolar interaction larger than in the case of Dy atoms by more than an order of magnitude [1]. One advantage of studying the effect of dipolar interaction in a degenerate dipolar Fermi gas over that in a dipolar BEC is the remarkable stability of the degenerate Fermi gas. A BEC is usually fragile or short-lived for many experiments due to three-body loss through molecule formation. The three-body loss is highly suppressed in a degenerate Fermi gas due to Pauli repulsion among identical fermions. On the other hand a theoretical study of a BEC is much simpler than that of a degenerate Fermi gas [24] due to the existence of a simple order parameter and a simple mean-field Gross-Pitaevskii equation in the former case.

A microscopic description of a degenerate Fermi gas is complicated due to the difficulties with the antisymmetrization of a many-fermion system and finding an appropriate simple order parameter. Drastic approximations are often necessary to achieve this goal for fermions. There have been several different theoretical descriptions for a degenerate nondipolar Fermi gas [25]. There have also been a few studies of the degenerate dipolar Fermi gas employing different types of approximations [26]. However, for a description of some macroscopic observables of a degenerate Fermi gas, a simple hydrodynamic description [27] often seems appropriate which does not require a precise antisymmetrization of the dynamics [24, 28, 29]. The effect of antisymmetrization is included approximately via an appropriate energy or Lagrangian density. A minimization of the energy leads to a well-founded variational approximation [30]. An Euler-Lagrange equation derived from such a Lagrangian provides further improvement over the variational results. Here we present a description of the degenerate spin polarized Fermi gas based on such a hydrodynamic model [27, 28].

The present theoretical formulation for the degenerate dipolar Fermi gas starts in section 2.1 with the standard equations of classical hydrodynamics [27] with the appropriate equation of state including the kinetic energy of fermions filling the Fermi sea and the dipolar interaction among them. The equivalent of the Thomas-Fermi (TF) approximation for the dipolar gas is obtained by setting the velocity field equal to zero in the hydrodynamic equations. An equivalent classical energy density is written using the local-density approximation (LDA) [24]. A quantum pressure term is then introduced in the LDA energy density. With such a quantum pressure term, a nonlinear Schrödinger-type equation is derived for the density of fermions. For a moderate number of fermions (greater than 100 or so), the quantum pressure term is negligibly small and hence has insignificant effect on the result. However, the inclusion of the quantum pressure term allows one to write a dynamical equation for fermions in the form of a nonlinear Schrödinger equation to study the dynamics. The LDA or the TF approximation, on the other hand, allows only the investigation of static properties of the system. For example, these approximations cannot be used to study the sound propagation dynamics in fermions as in section 3.2. In section 2.2, we present a Gaussian variational approximation for the problem described by the LDA energy density. In section 2.3, simplified models are derived in reduced dimensions, appropriate for cigar and disk shapes of the degenerate dipolar gas when there is a strong trap in radial or in axial directions, respectively. In section 2.4, using the hydrodynamic model, we obtain an analytic expression for the anisotropic sound velocity in the degenerate dipolar gas. In section 3.1, we present numerical results for stationary properties shape, size, and chemical potential of a trapped degenerate Fermi gas of Dy atoms, and compare with results from appropriate models in reduced dimensions and variational approximation. In section 3.2, the anisotropic sound propagation in an infinite dipolar degenerate gas is demonstrated numerically and the velocities so obtained are compared with the analytical results of section 2 D. Finally, in section IV we present a summary of our study.

## 2 Analytical Consideration

### 2.1 Hydrodynamic Model

The normal one-component Fermi gas can be in the collisionless regime where collision is rare or in the collisional hydrodynamic regime where frequent collision due to the dipolar interaction allow the system to settle to local equilibrium, where the system can be described by simple hydrodynamic equation rather than a detailed multi-particle description. Here we consider the system in such a configuration. A semi-quantitative estimate for the validity of a hydrodynamic description is given by the condition that relaxation time is small compared to the time scale defined by average trap frequency [8, 9]. For contact interaction this condition can be expressed in terms of the scattering length to measure the strength of atomic interaction [31]. The strength of dipolar interaction is usually measured in terms of the convenient length , where is the magnetic moment of an atom of mass and the permeability of free space. Using this length scale to measure dipolar interaction, the condition for the validity of hydrodynamic description can be expressed as [9, 31]

(1) |

where is the number of atoms, the oscillator length, the temperature, the Fermi temperature and an universal function and can be taken of the order of unity [31] under experimental conditions. Hence for sufficently large and/or sufficiently small the system should enter the hydrodynamic regime. For Dy atoms of magnetic moment with the Bohr magneton, the length where is the Bohr radius and for an oscillator length m and the system is already in the hydrodynamic regime. For polar Fermi molecule KRb of electric dipole moment Debye in the singlet rovibrational ground state [23], the length [1], where is the permitivity of free space, and the system should enter the hydrodynamic regime with larger m (weaker trap) and much smaller of few hundreds.

At sufficiently low temperature the normal Fermi gas enters the degenerate phase. Most macroscopic properties, like shape, density, chemical potential, sound propagation etc., of this system can be described by the Landau hydrodynamical equations [27]. A velocity field is introduced as the gradient of a velocity potential of flow, usually related to phase of the order parameter, by subject to the irrotational condition . The continuity and the flow equations are then given by [27]

(2) | |||

(3) |

where is density at space point and time , is an external trap, usually taken as

(4) |

with the trap frequency and and are anisotropy parameters. The bulk chemical potential is determined by the equation of state of the uniform Fermi gas of dipolar atoms and is given by

(5) |

where the first term on the right-hand side (rhs) is the Fermi energy [24] and the last term is the contribution from dipolar interaction energy [28], with is the dipolar potential.

The hydrodynamic description is valid for a macroscopic observable with its characteristic excitation wave-length much larger than the healing length [24]. A safe condition to satisfy this criterion is to take the wave length to be much larger than de Broglie wavelength at the Fermi surface, i.e., [24, 32]

(6) |

with Fermi momentum defined by . Truly speaking, a degenerate Fermi gas may not be fully irrotational and allows rotational components in the velocity field, not allowed in the present hydrodynamic formulation. This fact should influence only the rotational properties [24] of the degenerate Fermi gas not considered in this paper. We also assume the absence of any velocity dependent frictional force.

An approximate TF profile for density can now be obtained by setting velocity in (3), when [24]

(7) |

where is the chemical potential of the trapped gas. When (7) is solved subject to the appropriate normalization condition, we obtain both the chemical potential and the density .

There is an equivalent description of the trapped degenerate Fermi gas in the LDA, based on the assumption that, locally the dipolar Fermi gas would behave like a uniform gas, so that the energy density can be written as the energy of the uniform system times the local density [24]. For the degenerate dipolar Fermi system, the classical energy density per particle is given by [26, 28]

(8) | |||||

where is the density of fermions normalized as . The first term on the rhs of (8) is the total kinetic energy of the spin polarized fermions filling all levels up to the Fermi sea, the second term is energy in the trap, and the third term describes the dipolar interaction. A minimization of the classical energy (8) leads to the TF condition (7).

As the degenerate Fermi gas is a quantum system, a quantum pressure term when included in (8) yields the following expression for net energy density

(9) |

This gradient correction term [33] to the TF energy density (8) takes into account the additional kinetic energy due to spatial variation of density (near the surface). Such a surface term was first considered by von Weiszäcker [34, 35, 36] in the description a large nuclei. Previous descriptions of a degenerate Fermi gas considered different coefficients in this term [25, 33]. The energy density (9) has successfully used in many problems of Fermi gas [35, 36, 37, 38].

With the Lagrangian density the Euler-Lagrange equation is given by [37]

(10) | |||||

with the chemical potential. The derivarive term in (10) term contributes much less than the dominant “Fermi energy” term and its neglect leads to the TF relation (7). The dipolar interaction in (10) is taken as where is the angle between the and the polarization direction

The condition (1) refers to the validity of a hydrodynamic description of the system () [9, 31]. A reliable stationary description obtained using (10) for , of density and other macroscopic properties, such as sound velocity, as considered in this paper, can be obtained for a smaller number of fermions consistent with condition (6) [24, 32], provided the contribution of the kinetic energy term in (10) is much larger than that of other terms.

### 2.2 Variational Approximation

The energy density corresponding to the dimensionless equation (11) can be written as

(12) | |||||

A variational approximation for the problem can be obtained with the following Gaussian ansatz for density [30]

(13) |

where and are the variational widths along radial and axial directions. With this density, the effective energy per particle of the system is

(14) | |||||

where and

(15) |

The variational equations are obtained by minimizing the energy (14) by [2, 30]:

(16) | |||||

(17) |

where

(18) | |||

(19) |

The chemical potential of the system per particle is

(20) | |||||

### 2.3 Approximate density for cigar and disk shapes

In many situations of experimental interest, the Fermi gas could be subject to a strong trap either in the polarization direction or in the transverse radial plane. The system then has a one-dimensional (1D) cigar or 2D disk shape, respectively. In such cases simplified equations in lower dimensions could be very useful [39].

First we consider the reduced 1D equation for a cigar shape. We assume that there is a strong trap in the plane and that the density in this plane is given by the Gaussian ground state [39] of the trap , so that the 3D density satisfies

(21) |

Substituting this density in (11), and multiplying this equation by the corresponding Gaussian and integrating out the dependence we obtain the reduced 1D equation [40, 41]

(22) | |||

(23) |

where and is the chemical potential. An approximate variational solution of (2.3) is possible with the following ansatz for density [41] while the width is determined by solving (17) with and .

Next we consider the reduced 2D equation suitable for a disk shape with a strong axial trap. The dipolar Fermi gas is assumed to be in the ground state [39] , of the axial trap and the 3D density can be approximated as

(24) |

Using this ansatz in (11), and multiplying by the corresponding Gaussian and integrating out the dependence we get the effective 2D equation [41, 42]

(25) |

where An approximate variational solution of (25) is possible with the following ansatz for density [41] while the width is determined by solving (16) with and .

### 2.4 Sound propagation in a uniform Fermi gas

To find the sound velocity in a uniform Fermi gas, we evaluate the Bogoliubov spectrum using the linearized hydrodynamic equations. We consider a stationary Fermi gas in a box with periodic boundary condition with the trap fixing just the allowed plane wave solution. Then (3), after the inclusion of the gradient term of (9), reduces to

(26) |

Now we allow small perturbation in and around their equilibrium values and by and , then (2) leads to

(27) |

In (26), we need to use and , while with

(28) |

where is the stationary value of . Then (26) leads to

(29) |

Assuming the perturbations and in plane-wave forms and has the form with

(30) |

where is the angle between the propagation direction and polarization direction , and the last term in (30) is just the Fourier transform of the dipolar potential. Then (27) and (29) become:

(31) | |||

(32) |

The condition of existence of the solution to this set of equations leads to the Bogoliubov spectrum

(33) |

The sound velocity, defined as for a uniform dipolar Fermi gas can be written as

(34) |

where is the Fermi velocity. For a nondipolar Fermi gas (), (34) leads to the well-known velocity of [24, 29].

The angle-dependent second term under the square root in (34) is responsible for anisotropic sound velocity. Specifically, for degrees, this term is negative corresponding to a decrease in sound velocity. For large dipolar interaction, for greater than a critical value and for large density the sound velocity could be imaginary corresponding to no propagation. However, in this study we shall only consider moderate values of density and dipolar interaction, that allow anisotropic sound propagation in all directions.

## 3 Numerical Results

### 3.1 Stationary properties of trapped dipolar Fermi gas

For a trapped 3D Fermi system we solve (11) numerically after discretization [43]. The divergence of the dipolar term at short distances has been handled by treating this term in momentum () space [6]. For numerical calculation in section 3.1, we consider a degenerate Fermi gas of Dy atoms with and employ the oscillator length m.

The anisotropic dipolar interaction is partially attractive in certain angles and repulsive in others and contributes very little in a spherically symmetric trap. The dipolar interaction contributes attractively in the cigar-shaped configuration along the polarization direction and repulsively in the disk-shaped configuration perpendicular to the polarization direction. Hence we will mostly consider the degenerate dipolar Fermi gas in cigar and disk shapes. The 3D model (11) is very strongly nonlinear with nonlinearity , leading to large nonlinearities of and 3526 for and 10000, respectively.

First, we compare the results of the reduced 2D density of a disk-shaped degenerate Fermi gas of Dy atoms with as obtained from the 3D and 2D models (11) and (25), respectively, and from the variational approximation to the 3D model. The reduced 2D density is defined by . In figure 1 (a) we plot the reduced 2D density along axis for different . Next, we compare the results of reduced 1D density of a cigar-shaped degenerate Fermi gas of Dy atoms with as obtained from the 3D and 1D models (11) and (2.3), respectively, and from the variational approximation to the 3D model. The reduced 1D density is defined by . In figure 1 (b) we plot the reduced 1D density along axis for different . From figure 1 we find that in both cigar and disk shapes the 1D and 2D models perform fairly well, even for large nonlinearities, when compared with the results of the full 3D model. To see if the condition (6) is satisfied by the shapes in figure 1, we can use the TF estimate for of a trapped degenerate Fermi gas [24]: . For , as in figure 1, the de Broglie wave length at the Fermi surface is . The shapes and sizes in figure 1 are much larger than this value, consistent with the condition (6).

Next we consider a cigar-shaped degenerate Fermi system of Dy atoms with trap parameters . In figure 2 (a) we plot the numerical and variational results for the chemical potential and rms sizes and of this system for . In figure 2 (b) we plot the same for a disk-shaped degenerate Fermi system with trap parameters . Finally, in figure 2 (c) we plot the same quantities for versus the trap anisotropy for . In all cases the variational results are in good agreement with the 3D model. For medium to small number () of trapped Dy atoms as considered here and also of experimental interest, the effect of the dipolar term in Eq. (11) (the last term in this equation) is small compared to the Fermi energy term (the last but one term there). Hence the effect of the dipolar term in figures 2 is small. The plots in these figures only change by less than about two to four percent if we set .

### 3.2 Anisotropic sound propagation in uniform dipolar Fermi gas

The hydrodynamic analytical result for sound velocity (34) in a uniform dipolar Fermi gas shows a clear anisotropy through the angle between the propagation direction and the polarization direction . In this expression for sound velocity there are two competing terms under the square root. The first term involving the Fermi velocity is isotropic and proportional to whereas the second term proportional to dipolar interaction is anisotropic and proportional to density . The anisotropy in sound velocity will manifest strongly for large strength and for large as the anisotropic dipolar term in (34) will increase more rapidly with than the isotropic term. In figure 3 (a) we plot the sound velocity for angles and as a function of density for a degenerate Fermi gas with (Dy atom) and (polar K-Rb molecules in the singlet rovibrational ground state [1]) using the analytical result (34). The sound velocity for nondipolar atoms () is also plotted. From this plot we find that the anisotropy in sound propagation as measured by the difference is sizable only for a relatively large density cm for Dy atoms. However, for polar K-Rb molecules the anisotropy is appreciable for a relatively low density of cm. From figure 3 (a) we see that for K-Rb molecules, the sound velocity is imaginary for for a density of about cm.

To study sound propagation, the present stationary (static) 3D model (11) is generalized to include time variation by replacing by the usual time derivative . Consequently, the infinite dipolar Fermi gas satisfies the following Galilei-invariant equation [37]

(35) |

where density is not normalizable for infinite hydrodynamics. Equation (3.2) is consistent [37] with the time-dependent hydrodynamic equation (3) after the inclusion of the gradient correction term [33].

The anisotropy of the dipolar interaction would be prominent at low to medium density for large dipolar interaction. To illustrate the anisotropic sound propagation we will consider two examples: Dy atoms at a medium density of 10 cm and the polar molecules K-Rb at the low density of 10 cm. First we consider the numerical simulation of sound propagation in the infinite Dy gas () at a background density of m cm. The numerical simulation is initiated with a 3D Gaussian pulse at the center of the uniform 3D background density given by m, m, subject to a weak expulsive Gaussian potential m. With this initial condition (3.2) is solved by real-time propagation [43] to study sound waves. An ellipsoid-like sound wave front is found to emerge outwards upon time propagation. From a numerical study of this wave front the sound velocity in different directions is calculated.

Typical anisotropic sound propagation in a Dy degenerate gas of density 10 cm is shown in figure 4. The sound propagation is illustrated via contour plots of 3D density in the plane and of density in the plane. In figures 4 (a), (b), (c), and (d), to illustrate the anisotropic sound propagation in the plane, we show the contour plot of at times and , respectively, with s the time scale and m the length scale used in the numerical solution of (3.2). However, the propagation in the radial plane is isotropic. This isotropic sound propagation is shown in figures 4 (e), (f), (g), and (h) via the contour plot of at times and , respectively. A clean wave front of high density is identified in these contour plots encompassing a region of low density a typical panorama in sound propagation.

The sound propagation along direction (polar angle ) has a velocity () larger than that for a nondipolar system () of same density, whereas that in the plane (polar angle ) has a velocity smaller than that for a nondipolar system as can be seen from (34). The analytical sound velocity for nondipolar atoms of density cm and atomic mass 161 is 8.88 mm/s compared to the numerically obtained velocity of 8.5 mm/s. For Dy atoms of density cm with , the analytical radial sound velocity is mm/s (numerical 7.0 mm/s), and the analytical axial sound velocity is mm/s (numerical 10.0 mm/s). In figure 3 (b), we present the variation of axial () and radial () sound velocities versus as obtained from numerical simulation and analytical consideration (34). In figure 3 (c), the analytical result for velocity versus the polar angle is compared with the numerical result. The agreement between the analytical result (34) for sound velocity and the result of numerical simulation is satisfactory considering the very large nonlinearities present in the system due to large density () and large dipolar interaction.

To have a larger anisotropy in sound propagation than in a gas of Dy atoms at a lower density a stronger dipolar interaction, as in polar molecules, is needed. A convenient polar fermionic molecule molecule K-Rb has a permanent electric dipole moment 0.052 Debye () for the triplet rovibrational ground state and 0.566 Debye () for the singlet rovibrational ground state [1]. For illustration, we consider a uniform gas of K-Rb molecules of density cm with . The numerical simulation is initiated with a 3D Gaussian pulse at the center of the uniform 3D background density given by m, m, subject to a weak expulsive Gaussian potential m. With this initial condition (3.2) is solved by real-time propagation [43] to study sound waves. Typical anisotropic sound propagation in a K-Rb degenerate gas of density 10 cm is shown in figure 5. Because of the stronger dipolar interaction, a larger anisotropy has appeared at a lower density in figure 5 compared to figure 4. For K-Rb molecules of density cm with , the analytical radial sound velocity is mm/s (numerical 2.4 mm/s), and the analytical axial sound velocity is mm/s (numerical 3.4 mm/s).

## 4 Summary

We developed a 3D theoretical formulation for describing certain macroscopic observables of a degenerate dipolar Fermi gas appropriate for studying stationary properties, such as, shape, size, energy, chemical potential etc. of a trapped system. The effect of dipolar interaction is negligible in the spherically-symmetric configuration and the dipolar interaction manifests strongly in the asymmetric cigar and disk shapes. Simple reduced equations in 1D and 2D suitable for studying the trapped degenerate dipolar Fermi gas in cigar and disk shapes, respectively, are derived. Also, a Gaussian variational approximation for studying these macroscopic properties is derived. We apply the present formalism to study the stationary properties of a trapped degenerate Fermi gas of Dy atoms. The stationary properties of the 3D model under appropriate conditions are found to be in satisfactory agreement with those from the reduced 1D and 2D models as well as with variational approximation.

Using the present 3D model we also obtained analytical results for anisotropic sound velocity as a consequence of the anisotropic dipolar interaction in a dipolar Fermi gas in agreement with numerical simulation of the 3D (3.2). The sound velocity is larger along the polarization direction than in the transverse plane.

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