Three -wave interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients
We present essentially exact solutions of the Schrödinger equation for three fermions in two different spin states with zero-range -wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all -wave scattering lengths.
There has been extensive interest in ultracold atom physics in the last decade review_blume ; review_giorgini ; review_bloch . Ultracold atomic bosonic and fermionic gases are realized experimentally under varying external confinements. In these experiments, the number of particles and the scattering length of the two-body interactions are tunable chin_rmp . Although the complete energy spectrum of the many-body system cannot, in general, be obtained from first principles, the energy spectra of selected few-body systems can, in some cases, be determined within a microscopic quantum mechanical framework busch ; calarco-r ; calarco ; kestner ; blume-greene_prl ; javier-greene_prl . In some cases, the properties of the few-body system have then been used to predict the properties of the corresponding many-body system ho-1 ; ho-2 ; rupak ; drummond_prl ; drummond_pra ; drummond_2d ; salomon-2010 ; zwierlein-2012 ; daily_2012 .
The behavior of atomic and molecular systems depends strongly on the dimensionality of the system olshanii1 ; olshanii2 ; petrov ; esslinger . In three dimensions, e.g., weakly-bound two-body -wave states exist when the -wave scattering length is large and positive but not when it is negative. In strictly one- and two-dimensional geometries, in contrast, -wave bound states exist for all values of the -wave scattering length busch .
In ultracold atomic gases, the de Broglie wavelength of the atoms is much larger than the van der Waals length that characterizes the two-body interactions. This allows one to replace the van der Waals interaction potential in free-space low-energy scattering calculations by a zero-range -wave pseudopotential fermi ; huang ; huang2 . If the particles are placed in an external trap, the validity of the pseudopotential treatment (at least if implemented without accounting for the energy-dependence of the coupling strength) requires that the van der Waals length is much smaller than the characteristic trap length blume_greene_pra ; bolda_pra . In many cases, the use of pseudopotentials greatly simplifies the theoretical treatment. For example, the eigenequation for two particles interacting through a -wave pseudopotential under harmonic confinement has been derived analytically for spherically symmetric, strictly one-dimensional, strictly two-dimensional and anisotropic harmonic potentials busch ; calarco-r ; calarco .
The -wave pseudopotential has also been applied successfully to a wide range of three-body problems, either in free space or under confinement nielsen ; kartavtsev ; rittenhouse-2010 ; mora ; mora-2005 ; werner ; petrov3 . The present paper develops an efficient numerical framework for treating the three-body system under anisotropic harmonic confinement. The developed formalism allows us to study the dependence of the three-body properties on the dimensionality of the system. We focus on fermionic systems consisting of two identical spin-up atoms and one spin-down atom. The dimensional crossover of two-component Fermi gases has attracted a great deal of interest recently sommer-2012 ; kohl-2012 ; thomas-2012 . This paper considers the three-body analog within a microscopic quantum mechanical framework. We note that our framework readily generalizes to bosonic three-body systems. The study of the dimensional crossover of bosonic systems is interesting as it allows one to study how, under experimentally realizable conditions, Efimov trimers braaten that are known to exist in three-dimensional space disappear as the confinement geometry is tuned to an effectively low-dimensional geometry nishidatan .
This paper generalizes the methods developed in Refs. mora ; kestner for three equal-mass fermions in two different pseudospin states under spherically symmetric harmonic confinement to anisotropic harmonic confinement. We develop an efficient and highly accurate algorithm to calculate the eigenenergies and eigenstates of the system up to relatively high energies as functions of the interaction strength and aspect ratio of the trap. Several applications are considered: (i) The BCS-BEC crossover curve is analyzed throughout the dimensional crossover. (ii) For large and small aspect ratios, the energy spectra are analyzed in terms of strictly one-dimensional and strictly two-dimensional effective three-body Hamiltonian. (iii) The second- and third-order virial coefficients are analyzed as functions of the temperature, aspect ratio and scattering length. In particular, we show that the high-temperature limit of the third-order virial coefficient at unitarity is independent of the shape of the trap in agreement with expectations derived through use of the local density approximation. For finite scattering lengths, and for anisotropic harmonic confinement are well approximated by those for isotropic harmonic confinement.
The remainder of this paper is organized as follows. Section II presents a formal solution to the problem of three -wave interacting fermions confined in an axially symmetric harmonic trap. We also consider the extreme cases of strictly one-dimensional and strictly two-dimensional confinement. Sections III and IV apply the formal solution to cigar-shaped and pancake-shaped traps, respectively. We determine a large portion of the eigenspectrum as a function of the scattering length and discuss the transition to strictly one-dimensional and strictly two-dimensional geometries. Section V uses the two- and three-body eigenspectra to calculate the second- and third-order virial coefficients as a function of the temperature and the geometry of the confinement. Finally, Sec. VI concludes.
Ii Formal solution
We consider a two-component Fermi gas consisting of two spin-up atoms and one spin-down atom with interspecies -wave interactions under anisotropic harmonic confinement. We refer to the two spin-up atoms as particles 1 and 2, and to the spin-down atom as particle 3. We introduce the single-particle Hamiltonian for the particle with mass under harmonic confinement,
Here, is measured with respect to the trap center, and in cylindrical coordinates we have . In Eq. (1), and are the angular trapping frequencies in the - and -directions, respectively. The aspect ratio of the trap is defined through . In this paper, we consider cigar-shaped traps with as well as pancake-shaped traps with . Our three-particle Hamiltonian then reads
where accounts for the interspecies -wave two-body interactions,
where and .
Since the trapping potential is quadratic, the relative and center of mass degrees of freedom separate and we rewrite the Hamiltonian in terms of the relative Hamiltonian and the center of mass Hamiltonian , . In the following, we obtain solutions to the relative three-body Schrödinger equation , where
In Eq. (6), is the two-body reduced mass, , and the relative Jacobi coordinates and are defined through and . Depending on the context, we use either and or and to describe the relative degrees of freedom of the three-body system.
To determine the relative three-body wave function , we take advantage of the fact that the solutions to the “unperturbed” relative Hamiltonian are known and consider the Lippmann-Schwinger equation (see, e.g., Ref. kestner )
The Green’s function for the two “pseudoparticles” of mass associated with the Jacobi vectors and is defined in terms of the eigenstates and the eigenenergies of ,
Here, collectively denotes the quantum numbers needed to label the single-particle harmonic osillator states. In cylindrical coordinates, we have with , , and . The single-particle harmonic oscillator eigenenergies and eigenstates read
In the last two equations, and denote Hermite and associated Laguerre polynomials, respectively. Throughout most of Secs. II-IV, we use the oscillator energy and oscillator length [ and ] as our energy and length units.
In Eqs. (7)-(12), we employ cylindrical coordinates since this choice allows us to write the Green’s function compactly. However, the two-body -wave interaction potential is most conveniently expressed in spherical coordinates [see Eq. (4)]. Since the pseudopotential acts only at a single point, namely at , it imposes a boundary condition on the relative three-body wave function (see, e.g., Ref. petrov3 ),
The unknown function can be interpreted as the relative wave function of the center of mass of the interacting pair and the third particle. Similarly, the pseudopotential imposes a boundary condition on the wave function when . Since the wave function must be anti-symmetric under the exchange of the two identical fermions, i.e., , where exchanges particles 1 and 2, the properly anti-symmetrized boundary condition corresponding to reads
Here, we defined
To simplify the right hand side of Eq. (7), we impose the limiting behaviors of for and , and expand in terms of the non-interacting harmonic oscillator functions, . Using Eq. (8) for and orthonormality of the single-particle harmonic oscillator functions, we find
Here, we used that can be written as and introduced the one-body Green’s function for the pseudoparticle of mass that is associated with the relative distance vector ,
The one-body Green’s function with coincides with the solution to the relative Schrödinger equation for two particles under harmonic confinement interacting through the zero-range pseudopotential with -wave scattering length and relative two-body energy . is known for all aspect ratios calarco-r ; calarco (see also Secs. III and IV).
To determine the expansion coefficients , we apply the operation to the left hand side and the right hand side of Eq. (15), i.e., we multiply both sides of Eq. (15) by , then apply the derivative operator and lastly take the limit . Defining
with , we find
The quantity can be interpreted as a non-integer quantum number associated with the interacting pair. If we multiply Eq. (II) by and integrate over , we find an implicit eigenequation for the relative three-body energy or equivalently, the non-integer quantum number ,
Equation (19) can be interpreted as a matrix equation with eigenvalues and eigenvectors kestner ; drummond_prl . In practice, we first calculate the matrix elements in Eq. (19) for a given three-body energy and obtain the corresponding scattering lengths for this energy by diagonalizing the matrix with elements . This step is repeated for several three-body energies. Lastly, we invert to get , i.e., to get the three-body energies as a function of the -wave scattering length.
Equation (19) has a simple physical interpretation. If the interaction between particles 2 and 3 is turned off, the matrix vanishes and the solution reduces to that of an interacting pair (particles 1 and 3) and a non-interacting spectator particle (particle 2). The relative energy of the pair is determined by solving the relative two-body eigenequation . The matrix thus arises from the fact that particle 3 not only interacts with particle 1 but also with particle 2. Correspondingly, the terms in Eq. (19) that contain can be interpreted as exchange terms that arise from exchanging particles 1 and 2 drummond_prl .
For a spherically symmetic system with , the total relative angular momentum quantum number , the corresponding projection quantum number and the parity are good quantum numbers, and the eigenvalue equation can be solved for each and combination separately using spherical coordinates kestner . For a fixed and , and in Eq. (19) are constrained by and . The parity of the three-body system is given by .
We emphasize that the outlined formalism makes no approximations, i.e., Eq. (19) with given by Eq. (20) describes all eigenstates of [see Eq. (5)] that are affected by the interactions. In particular, all “channel couplings” are accounted for. In practice, the construction of the matrix requires one to choose a maximum for and , or alternatively, a cutoff for the single-particle energy . As has been shown in Ref. drummond_pra , good convergence is achieved for a relatively small number of “basis functions” for . As we show below, good convergence is also obtained for anisotropic confinement geometries.
The formalism outlined can also be applied to strictly one-dimensional and strictly two-dimensional systems. Table 1 defines the one-dimensional and two-dimensional pseudopotentials as well as a number of key properties of the corresponding relative two-body system. Making the appropriate changes in the outlined derivation and using the properties listed in Table 1, we find for strictly one-dimensional systems
where is defined in Table 1,
and . Here, denotes the single-particle energy of the one-dimensional system, , and the one-dimensional even parity single-particle Green’s function,
For , the single-particle Green’s function is given by
where is the Gamma function and the confluent hypergeometric function. The strictly one-dimensional relative three-body wave function is characterized by the parity . For even parity states, i.e., for states with , and in Eq. (21) have to be even. For odd parity states, i.e., for states with , and have to be odd.
Similarly, for strictly two-dimensional systems, expressed in units of and [ and ], we find, in agreement with Ref. drummond_2d ,
where is defined in Table 1,
and . Here, denotes the single-particle energy of the two-dimensional system, . The two-dimensional single-particle Green’s function is defined analogously to the three- and one-dimensional counterparts [see Eqs. (16) and (23)]. For and states affected by the zero-range -wave interactions calarco-r ; calarco , one finds
The strictly two-dimensional relative three-body wave function is characterized by the projection quantum number and the parity , . For a fixed , in Eq. (25) is constrained to the value . The next two sections analyze, utilizing our results for strictly one- and two-dimensional systems, Eq. (19) for cigar- and pancake-shaped traps.
Iii Cigar-shaped trap
To apply the formalism reviewed in Sec. II to axially symmetric traps, we need the explicit forms of the functions and , that is, the relative solutions to the trapped two-body system. For cigar-shaped traps (), it is convenient to write as calarco-r ; calarco
The evaluation of the integrals and is discussed in Appendix A. The superscript “c” indicates that the integrals apply to cigar-shaped systems; for pancake-shaped systems (see Sec. IV), we introduce the integrals and instead.
Although it is possible to calculate numerically for any trap aspect ratio , we restrict ourselves to integer aspect ratios for simplicity. For traps with integer aspect ratio, an exact analytical expression for is known calarco-r ; calarco ,
where is the hypergeometric function abramowitz . Knowing and , Eq. (19) can be diagonalized separately for each combination. We recall from Sec. II that and . The and values are constrained by . Moreover, for and , we have and , respectively.
shows the three-body relative energies for for states with (a) and and (b) and as a function of the inverse scattering length . The non-interacting limit is approached when , and the infinitely strongly-interacting regime for (center of the figure). For each fixed projection quantum number , we include around basis functions. This corresponds to a cutoff of around for the single-particle energy . We find that yields converged values for , Eq. (III). For small ( positive and negative), our eigenenergies agree with those obtained within first-order perturbation theory. Our analysis shows that the energy of the ground state at unitarity has a relative error of the order of . The accuracy decreases with increasing energy. For example, for energies around , the relative accuracy at unitarity is of the order of .
The eigenstates fall into one of two categories: atom-dimer states and atom-atom-atom states. The eigenenergies associated with the former are negative for large positive while those associated with the latter remain positive for large positive . The energy spectra shown in Fig. 1 exhibit sequences of avoided crossings. To resolve these crossings, a fairly fine mesh in the three-body energy is needed. In the limit, the lowest state has negative parity in , i.e., . This is a direct consequence of the fact that the two identical fermions cannot occupy the same single particle state. In the limit, in contrast, the lowest state has positive parity in , i.e., . This is a direct consequence of the fact that the system consists, effectively, of a dimer and an atom.
The main part of Fig. 2
shows the relative energy of the energetically lowest-lying state, the so-called crossover curve, of the three-body system with for various aspect ratios of the trap () as a function of the inverse scattering length . For comparative purposes, we subtract the ground state energy of of the strictly two-dimensional non-interacting system, that is, the energy that the system would have in the -direction if the dynamics in the tight confinement direction were frozen, from the full three-dimensional energy. In Fig. 2, asterisks mark the scattering lengths at which the eigenstate associated with the crossover curve changes from to . With increasing , the parity change occurs at larger (that is, smaller ). The inset of Fig. 2 replots the crossover curves as a function of .
shows the relative two-body energies for a system with , and obtained by solving the eigenequation [see Eq. (III) for ]. Figure 3(b) compares the full three-dimensional energy (solid line) with the energy obtained by solving the strictly one-dimensional eigenequation, Eq. (34), with renormalized one-dimensional scattering length