Non-universal bound states of two identical heavy fermions and one light particle
We study the behavior of the bound state energy of a system consisting of two identical heavy fermions of mass and a light particle of mass . The heavy fermions interact with the light particle through a short-range two-body potential with positive -wave scattering length . We impose a short-range boundary condition on the logarithmic derivative of the hyperradial wavefunction and show that, in the regime where Efimov states are absent, a non-universal three-body state “cuts through” the universal three-body states previously described by Kartavtsev and Malykh [O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007)]. The presence of the non-universal state alters the behavior of the universal states in certain regions of the parameter space. We show that the existence of the non-universal state is predicted accurately by a simple quantum defect theory model that utilizes hyperspherical coordinates. An empirical two-state model is employed to quantify the coupling of the non-universal state to the universal states.
The unprecedented control over ultracold atomic Fermi gases in optical lattices has made them prime candidates for studying and engineering novel quantum phases as well as probing fundamental theories such as the BEC-BCS crossover Greinter2003 (); Martin2003 (); Solomon2003 (); Hulet2003 (); Regal2004 (); Martin2004 (). This progress has been facilitated by the tunability of the two-body interactions using Feshbach resonances or by changing the lattice confinement. If the interspecies two-body -wave scattering length is much larger than the range of the underlying two-body potential, the few- and many-body behavior of equal-mass two-component Fermi gases is universal, i.e. completely determined by .
Presently significant efforts are directed at creating ultracold atomic Fermi gas mixtures composed of two chemically distinct species Gupta2011 (); Grimm2010 (); Martin2012 (). This introduces a new parameter, the mass ratio between the two species. This new parameter affects the many-body physics of the system, allowing one to realize novel quantum phases such as the interior gap superfluid Wilczek2003 (). Here we show that at the few-body level this additional degree of freedom leads to new three-body resonances which may destabilize the system, making it harder for experiments to explore novel quantum phases with unequal-mass mixtures. In particular, we study a system of two identical heavy fermions with mass , which interact with a light particle through a short-range potential with positive -wave scattering length .
Previous studies revealed two intriguing properties. First, Kartavtsev and Malykh KM () predicted the existence of a universal trimer state with energy for and the existence of two universal states with energies and for ; and were found to be and , respectively. Second, Endo et al. Endo () investigated how the universal trimer states, which are completely determined by the -wave scattering length and the mass ratio , are connected to Efimov trimers, which have been predicted to exist for . By analyzing the trimer system within the framework of the Skorniakov-Ter-Martirosian equation with a momentum cutoff , Endo et al. predicted the existence of a third class of trimer states, termed crossover trimers, which were shown to continuously connect the universal trimers described by Kartavtsev and Malykh and Efimov trimers.
Our study employs, as in Ref. KM (), the hyperspherical coordinates. However, while Ref. KM () enforced that the hyperradial wavefunction vanishes at hyperradius , we explore the entirety of physically allowed boundary conditions by introducing a short-range three-body or hyperradial phase . We determine the eigenspectrum as a function of the value of the three-body phase , the hyperradius at which the hyperradial boundary condition is imposed and the mass ratio . The universal states of Kartavtsev and Malykh are recovered for and . However, for other boundary conditions we find deviations from universality, which are linked to the existence of a non-universal three-body state. Analogous non-universal three-body states have previously been shown to exist Daily (); DailyPRA () (see also Refs. Petrov (); Nishida (); Werner ()) in the limit. The existence of the non-universal state for positive is described accurately within a quantum defect theory (QDT) framework. Moreover, within a two-state model, deviations from universality are explained as being due to the coupling between the non-universal state and the universal states. Our work provides a simple intuitive Schrödinger equation based description of the energy spectra of heavy-light trimers and an alternative means to understanding the connection between universal trimers and Efimov trimers.
The remainder of this paper is organized as follows: Section II describes the hyperspherical framework. Section III determines the three-body energies as functions of and by solving the hyperradial Schrödinger equation numerically. Section IV develops an analytical description, which accounts for the universal and non-universal states of the energy spectrum. Finally, Sec. V concludes.
Ii System Hamiltonian
We model the interactions between the heavy and light particles by a zero-range two-body pseudopotential with -wave scattering length . If we denote the heavy fermions of mass as and and the light particle of mass as , the Hamiltonian is given by
where is the position vector of the th particle, is the Laplacian of the th particle and with
Here is the reduced mass of the heavy-light pair, , and . The pseudopotential imposes the Bethe-Peierls boundary condition on the three-body wave function in the limit .
To solve the Schrödinger equation for the Hamiltonian , we separate off the center of mass motion and write the relative wave function in terms of the hyperradius and five hyperangles, collectively denoted by KM (); Rittenhouse2010 (). The hyperrradius , which provides a measure of the overall size of the system, is defined by , where is the three-body reduced mass associated with the hyperradius, , and denotes the center of mass vector. We expand the relative wavefunction in terms of a set of weight functions and adiabatic channel functions , which depend parametrically on the hyperradius Macek (),
The adiabatic channel functions satisfy the hyperangular Schrödinger equation at fixed ,
In Eq. (4), denotes the grand angular momentum operator, which accounts for the kinetic energy associated with the hyperangles Rittenhouse2010 (). Inserting Eq. (3) into the relative Schrödinger equation yields a set of coupled equations for the weight functions .
In the following, we employ the adiabatic approximation, which neglects the coupling between the different channels Rittenhouse2010 (); Greene1980 (); Lin1995 (). This approximation has been shown to provide a qualitatively correct description for a number of three-body systems KM (); Esry1996 (); Esry1996He (); Jonsell2002 (). In this approximation, the hyperradial Schrödinger equation for the lowest adiabatic channel reads
where is the diagonal correction to the adiabatic energies,
We determine the three-body energies using a two step process. First we find the hyperangular eigenvalues and coupling elements . Then we solve the radial Schrödinger equation, Eq. (5). For states with symmetry and zero-range interactions, the scaled hyperangular eigenvalues can be obtained semi-analytically by solving the transcendental equation KM (); Rittenhouse2010 ()
The hyperangular eigenvalues are completely determined by , and . In the limit , goes to a constant. For the purpose of this study we are only concerned with positive values of in the small limit, i.e. we only consider mass ratios . Figure 1 shows the
scaled eigenvalue as a function of for . As we discuss in more detail below, the value of determines the allowed short-range boundary condition of . In the large limit, vanishes and the quantity approaches , where denotes the dimer binding energy, .
Equation (5) has two linearly independent solutions, which scale as and in the small limit and are referred to as the regular and irregular solutions, respectively. For and , the irregular solution is not normalizable and thus does not contribute, implying that goes to zero as Petrov (); Nishida (); Werner (). On the other hand, in the regime both solutions are well-behaved and must be included when constructing the general solution Petrov (); Nishida (); Werner (). We parameterize the short-range boundary condition of the wavefunction at using the logarithmic derivative ,
where . Using this parameterization, we cover all possible short-range phases. In the special case of and , this boundary condition and the resulting three-body energies agree with those of Ref. KM (). We refer to this boundary condition as “universal boundary condition” since the three-body states are completely determined by the regular solution. We refer to the corresponding states as “universal states” with energies for , and and for . In the next section we discuss deviations from universality that occur if is not equal to . These deviations increase with increasing (for a fixed ) and increasing (for a fixed ).
Iii Numerical Treatment
In this section we examine the behavior of the three-body bound states with symmetry, obtained numerically using the shooting algorithm, as a function of the logarithmic derivative boundary condition for selected mass ratios. The three-body energies for and are shown in Figs. 2(a) and (b), respectively.
Dotted, dashed and dash-dotted lines show as a function of the three-body phase for , and , respectively. In Fig. 2 the three-body energies have been scaled by the zero-range two-body binding energy . The three-body state becomes unbound with respect to the break-up into a dimer and an atom for . For and , the system supports one three-body bound state. For and , is nearly constant for . At , referred to as the critical angle , the energy rapidly goes to a large negative value and a second bound state, whose energy is approximately equal to , is supported for . We refer to the feature in the vicinity of as “three-body resonance”. As increases [see dashed and dash-dotted lines in Fig. 2(a)] the width of the three-body resonance increases. Note, however, that the deviations from are small for all considered, except for three-body phases very close to . As increases [see Fig. 2(b)], the overall behavior of the energy spectrum is unchanged. The key trends with increasing are that, at a fixed , the energy away from becomes more negative and both and the “width” of the three-body resonance increase (see also symbols in Figs. 5 and 7).
For , the three-body system supports a second bound state. As an example, Fig. 3 shows the three-body bound state energy for and as a function of .
Away from , there exist two bound states whose energies depend weakly on . For , the corresponding hyperradial functions possess nodes and node, respectively. For , the corresponding hyperradial functions possess node and nodes, respectively. This reflects the fact that a new bound state is being pulled in at .
In general, we find that depends relatively weakly on as long as . Moreover, the three-body energy depends relatively strongly on in the vicinity of but comparatively weakly on away from . This suggests that the states near and away from can be classified as non-universal and universal, respectively. This interpretation is corroborated by our analysis of the hyperradial wavefunction for , and . The main part of Fig. 4 shows that the wavefunction has an appreciable amplitude in the small region, signaling non-universal behavior.
Away from , in contrast, has a vanishingly small amplitude in the small region (see inset of Fig. 4). We find that depends fairly weakly on when is away from , lending further support to our assertion that the three-body system behaves universally in this region. For larger , we observe deviations from universality for a larger range of three-body phases. Similar non-universal behavior has been previously reported for the three-body system at unitarity Daily (); DailyPRA () and for the four-body system with positive -wave scattering length Blume4b ().
Iv Analytical Treatment
Section IVA applies a QDT framework to predict the short-range phase at which the non-universal state first becomes bound. Section IVB develops a two-state model to describe the behavior of the universal and non-universal states as a function of .
iv.1 QDT Treatment
To predict the short-range phase at which the non-universal state first becomes bound with respect to the break-up into a dimer and an atom we apply QDT Daily (); DailyPRA (); Fano (); Seaton (). In the short-range limit, the wavefunction can be approximated by Daily (); DailyPRA ()
where and denote the Bessel functions of the first and second kind, respectively. The quantum defect controls the relative contribution of the regular solution and the irregular solution . A new three-body state is expected to be pulled in when the hyperradial solution is dominated by the irregular solution, i.e. for . The critical angle is then given by
where denotes the real part and is evaluated for and .
The solid, dotted and dashed lines in Fig. 5 show determined using the QDT framework for , and , respectively.
It is interesting to note that is very weakly dependent on for small . The symbols in Fig. 5 show the critical angle obtained by analyzing the numerical solutions of the hyperradial Schrödinger equation, Eq. (5), for . The numerical results are in excellent agreement with the QDT prediction. For (see inset of Fig. 5), the dependence of on becomes more pronounced. Figure 5 shows that the three-body system supports a non-universal bound state not only for , but also for [for one has ]. Thus, it may be surprising at first sight that the system supports a non-universal state for since the irregular solution cannot be normalized if for . However, since we impose the boundary condition at a finite and not at , the resulting wavefunction can, even though it contains an admixture of the irregular solution, be normalized. Correspondingly, non-universal states can exist for . In fact, we find that the system supports a non-universal bound state even for , i.e. for mass ratios where universal states are not supported.
iv.2 Two-state model
In this section we develop a two-state model that describes the behavior of the universal and non-universal states as a function of . In our model the three-body resonance is an avoided crossing at between the universal state with energy and a non-universal state with energy . In Fig. 6 we plot the quantity as a function of .
The thick solid, dotted and dashed lines correspond to and , respectively, with and . The circles correspond to , and .
Motivated by Fig. 6, we write the energy of the non-universal state as , where is a dimensionless scaling constant; in our analysis we use . The term proportional to determines the slope at which the non-universal state crosses the universal state. In our parameterization of , should be interpreted as the three-body phase at which the universal and non-universal states cross. Our numerical results show that this crossing point is nearly identical to the QDT three-body phase at which the non-universal state first becomes bound. Hence we use the QDT value in our two-state model. The dimensionless two-state Hamiltonian is then
where is the coupling between the universal and non-universal states. Within this model, is real and the scaled three-body energies are given by the eigenvalues of the two-state Hamiltonian. We apply the two-state model separately to the universal states with energies and , yielding separate values for the two universal states. The values of are obtained by fitting the energy spectrum predicted by the two-state Hamiltonian to the numerically determined energies. Thin dashed lines in Fig. 6 show the results for the two-state model. It can be seen that the two-state model provides a quantitatively correct description of the three-body spectra. Moreover, it provides an intuitive physical picture in which deviations from universality arise due to the coupling of the universal states to the non-universal state.
The circles, diamonds and squares in Fig. 7 show for and and , respectively, as a function of .
The value of increases with increasing for fixed . Moreover, the dependence of on the value of increases with increasing . Since determines the coupling between the universal state and the non-universal state, it can be used to quantify the deviations from universality. For mass ratios larger than those considered in this work, the heavy-light trimer system supports three-body bound states with Efimov character Efimov1970 (); Efimov1973 (); Braaten2006 (). Within the zero-range framework employed here, the exact number and energy of the Efimov trimers supported depends on the short-range hyperradial boundary condition. If the hyperradial boundary condition is fixed and is varied, the energy spectrum changes smoothly from (i) deviating from the universal spectrum only in a small region around to (ii) deviating from the universal spectrum for a fairly large range of to (iii) supporting three-body states with Efimov character for . Thus, the deviations from universality discussed in this paper can be interpreted as connecting the universal states predicted by Kartavtsev and Malykh and the Efimov trimers.
Our findings are in qualitative agreement with the results reported in Ref. Endo (). In that work, the authors imposed a momentum cutoff in the Skorniakov-Ter-Martirosian equation and used the quantity ( or ) to determine the “boundary” between the universal trimers predicted by Kartavtsev and Malykh and the crossover trimers in the parameter space. The momentum cutoff was introduced in two ways, using a sharp and a Gaussian cutoff. We speculate that in our formulation a change in the three-body phase and/or hyperradius corresponds to a change in . A crossover trimer, in turn, corresponds to a trimer whose energy deviates appreciably from . While the deviations from are, in our formulation, linked to the coupling of the universal state to a non-universal state, the treatment by Endo et al. does not seem to yield an analogous physical picture. Additionally we speculate that while Ref. Endo () employs two different models for the momentum cutoff , it does not explore the entirety of the parameter space. In the future it will be interesting to investigate the precise connection between the formulations in the coordinate and momentum spaces by, e.g., comparing the wavefunctions. Such a comparison is needed to check if the above correspondencies are correct.
In this paper we studied a system of two identical heavy fermions of mass and a light particle of mass with zero-range two-body interspecies interactions. In particular, we looked at deviations of the three-body bound state energies from the universal energies for and and for as a function of the hyperradial short-range boundary condition. We imposed a short-range phase using a logarithmic derivative boundary condition at various hyperradii . This parameterization allowed us to explore the full range of possible short-range boundary conditions.
We found that (i) for the universal states with energies and , predicted by Kartavtsev and Malykh KM (), are recovered; (ii) the three-body states deviate from universality in the vicinity of a three-body resonance located at the short-range phase , at which the non-universal state is first bound; (iii) the deviations from universality increase with increasing mass ratio (at fixed ) and with increasing (at fixed ); (iv) QDT accurately predicts the values of ; (v) a two-state model quantitatively describes the behavior of the universal and non-universal states as a function of ; and, finally, (vi) the non-universal bound state exists for even though universal bound states are not supported in this regime.
This work was supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory. DB acknowledges support by the NSF through grant PHY-1205443.
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