Long-lived trimers in a quasi-two-dimensional Fermi system
Emma K. Laird, Thomas Kirk, Meera M. Parish, Jesper Levinsen
LLong-lived trimers in a quasi-two-dimensional Fermi system
Emma K. Laird, ∗ Thomas Kirk,
1, 2, ∗ Meera M. Parish, and Jesper Levinsen School of Physics and Astronomy, Monash University, Victoria 3800, Australia London Centre for Nanotechnology, 17-19 Gordon Street, London, WC1H 0AH, United Kingdom (Dated: November 7, 2018)We consider the problem of three distinguishable fermions confined to a quasi-two-dimensional(quasi-2D) geometry, where there is a strong harmonic potential in one direction. We go beyondprevious theoretical work and investigate the three-body bound states (trimers) for the case wherethe two-body short-range interactions between fermions are unequal. Using the scattering param-eters from experiments on ultracold Li atoms, we calculate the trimer spectrum throughout thecrossover from two to three dimensions. We find that the deepest Efimov trimer in the Li system isunaffected by realistic quasi-2D confinements, while the first excited trimer smoothly evolves froma three-dimensional-like Efimov trimer to an extended 2D-like trimer as the attractive interactionsare decreased. We furthermore compute the excited trimer wave function and quantify the stabilityof the trimer against decay into a dimer and an atom by determining the probability that threefermions approach each other at short distances. Our results indicate that the lifetime of the trimercan be enhanced by at least an order of magnitude in the quasi-2D geometry, thus opening the doorto realizing long-lived trimers in three-component Fermi gases.
I. INTRODUCTION
The behavior of three quantum particles interactingwith short-range interactions is a fundamental problemin physics that is relevant to a variety of systems rangingfrom nucleon clusters [1] to quantum magnets [2]. Ourcapability to investigate three-body systems has beengreatly enhanced by recent advances in the manipula-tion and cooling of trapped atoms. Here, one can re-alize a range of cold-atom systems with different quan-tum statistics and different dimensionalities [3]. In allof these scenarios, a key role is played by three-bodybound states, i.e., trimers , whose existence or otherwiseessentially determines the energy spectra and scatteringproperties.Of particular interest has been the so-called Efimov ef-fect [4–6], which corresponds to a series of trimer statesthat become infinitely numerous when the short-rangeinteractions are tuned to be resonant. Such an effectwas first predicted for three identical bosons more than40 years ago [7], and the deepest Efimov trimers havesince been observed in atomic Bose gases [8–11], Bose–Fermi mixtures [12, 13], and three-component Fermigases [14, 15]. However, the Efimov trimers observed inthe cold-atom system are highly unstable towards decayinto deeper bound dimers — indeed, the trimers are typi-cally detected indirectly via three-body loss resonances atlow energy [8]. While stable trimers have been producedby using He atoms [10], this system lacks the tunabilityof metastable ultracold atomic gases. Thus, it remainsan elusive goal to engineer long-lived trimers that canultimately be used as building blocks for quantum simu-lators and correlated phases of matter [16–18].The short lifetime of Efimov trimers in the cold-atomsystem is predominantly due to the large weight of the ∗ Both authors contributed equally to this work. trimer wave function at short distances; i.e., there is ahigh probability that three particles will approach eachother at close range and then decay into a deeply boundmolecule and an unbound atom. Therefore, one can en-hance the trimer stability by engineering a more spa-tially extended three-body wave function. Such a sce-nario can, in principle, be achieved by confining identicalbosons to a two-dimensional (2D) plane [19, 20], a ge-ometry which has been realized in Bose-gas experimentsusing optical lattices [21–23]. In this case, the weak-est bound Efimov states are destroyed by the quasi-2Dconfinement [19, 24, 25] and, in the 2D limit, one hastwo spatially extended “universal” trimer states that arecompletely determined by the low-energy 2D scatteringparameters [26]. However, there are practical difficul-ties in accessing these 2D-like trimers in the Bose systemsince one must start from an attractive quasi-2D Bosegas, which is inherently unstable [27].In this paper, we circumvent this problem by consid-ering the trimers formed from a three-component
Fermi gas. Such a system can be experimentally realized with Li atoms, since the three lowest hyperfine states of Liall have near-resonant s -wave interactions [28], and Efi-mov trimers have already been observed in three dimen-sions (3D): the ground and first-excited trimer stateshave been detected indirectly via three-body loss reso-nances [29–33], while the latter has also been directlyaccessed via radio-frequency association [14, 15]. On theother hand, it is now standard practice to create stabletwo-component Fermi gases confined to a 2D plane [34],and this has already been achieved with Li atoms [35–38]. Thus, it should be feasible to associate quasi-2D s -wave trimers from atoms and dimers in the quasi-2D two-component Li system, similar to what has been done in3D [14, 15]. The advantage of this approach is that weonly require one species of atoms, in contrast to alterna-tive proposals for stable trimers that require two differenttypes of fermion with a large mass ratio [39–41]. a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Using realistic scattering parameters for the three-component Li system, we compute the spectrum oftrimers under different quasi-2D confinements as a func-tion of interaction strength. Since the scattering lengthsfor the three pairwise interactions are different, we mustsolve a more involved set of integral equations for theconfined system, unlike the case of identical bosons [19].While the deepest Efimov trimer is essentially unaf-fected by experimentally realizable confinements, we findthat the first-excited trimer crosses over from a three-dimensional-like (3D-like) Efimov trimer to an extended2D-like trimer as the attractive interactions are de-creased. This behavior is reflected in the real-space wavefunction for the first-excited trimer, and we furthermorequantify the three-body decay rate of the trimer by esti-mating the weight of the wave function at short distances.We find that this weight can be reduced by more thanan order of magnitude compared with the 3D case, thusconfirming our expectation that the quasi-2D geometryenhances the trimer lifetime. We discuss the optimal ex-perimental conditions under which to realize long-livedquasi-2D trimers.The paper is organized as follows: In Sec. II we outlineour model of the Li system and our approach to deter-mining the trimers in both three dimensions and quasitwo dimensions. In Sec. III we discuss the 3D case andthen we present the quasi-2D trimer spectra and wavefunctions, as well as our estimate for the trimer lifetimeas a function of interaction strength. We conclude inSec. IV.
II. MODEL AND METHODS
We consider three distinguishable fermions with equalmasses m , which we label as 1, 2, and 3. If we takethe case of Li atoms in the three lowest sublevels, theselabels then denote the atoms’ hyperfine states | (cid:105) , | (cid:105) ,and | (cid:105) [28].Our goal is to model the ultracold Li system under theapplication of a strong transverse potential that confinesthe atoms to a quasi-two-dimensional geometry in the x - y plane. We can approximate this potential as harmonicin the z direction, V ( z ) = mω z z , where ω z is the con-finement frequency. The characteristic length scale of thetrap is the confinement length, l z = (cid:112) /mω z (we workin units where (cid:126) = 1), and this always greatly exceedsthe van der Waals range of the background interactions, R vdW . Thus, the underlying short-range interaction po-tential in the gas is unaffected by V ( z ). Before includingthe harmonic trap in the calculation of the trimer ener-gies, we expound the model for the case when ω z = 0and the system is purely 3D. A. 3D System
Throughout this work we model short-range pairwiseinteractions that are close to resonance. In the par-ticular case of Li subjected to an external magneticfield B , there are three nearly overlapping Feshbach res-onances — one between each pair of hyperfine states— in the range B ∼
690 to 840 Gauss [28]. At theresonances the corresponding scattering length diverges,and all three scattering lengths remain much larger than R vdW throughout the range of magnetic fields consideredin this study.To model this system, we consider the Hamiltonianˆ H = (cid:88) q , i (cid:15) q c † q ,i c q ,i + (cid:88) q , q (cid:48) , p i < j g ij e − ( q + q (cid:48) ) / Λ × c † p / q (cid:48) ,i c † p / − q (cid:48) ,j c p / − q ,j c p / q ,i , (1)where we set the volume to unity. Here, c † q ,i ( c q ,i ) is thesecond-quantized operator which creates (annihilates) anatom with 3D momentum vector q and label i = 1 , , (cid:15) q = q / m and q ≡ | q | .The second term of Eq. (1) describes the interactionof two atoms with center-of-mass momentum p , and rel-ative momenta q and q (cid:48) before and after the collision,respectively. To characterize the interactions betweenatoms i and j , we use a separable potential of strength g ij with a Gaussian cutoff at a characteristic momentumΛ [42]. We can relate these two parameters of the modelto the physical parameter of low-energy collisions, the s -wave scattering length a ij , via the process of renormal-ization. This results in [43, 44] a ij = (cid:18) πmg ij + Λ √ π (cid:19) − . (2)Note that, by construction, g ij = g ji and a ij = a ji . Therenormalization condition (2) also allows us to determinethe energy E < T matrix [43]: T − ij ( E ) = 1 g ij − (cid:88) q e − q / Λ E − (cid:15) q = m πa ij − m Λ4 π F (cid:0) mE / Λ (cid:1) , (3)where F ( x ) = e | x | (cid:112) | x | erfc (cid:16)(cid:112) | x | (cid:17) , (4)and erfc( x ) is the complementary error function. Close toresonance where a ij (cid:29) Λ − , the pole condition reducesto the universal two-body energy E = − /ma ij .Apart from the s -wave scattering length, a full descrip-tion of Efimov trimers requires an additional high-energylength scale called the three-body parameter [4], since thetrimer spectrum is unbounded from below in the absenceof a short-distance cutoff. In the model described byEq. (1), the three-body parameter is directly related toΛ, which thus determines the size of the deepest Efimovtrimer.We proceed now to consider the problem of three dis-tinguishable fermions in a 3D system, and we write downa general wave function in the center-of-mass frame, | ψ D (cid:105) = (cid:88) q , q , q β q q q | q , q , q (cid:105) , (5)where the state | q , q , q (cid:105) ≡ c † q , c † q , c † q , | (cid:105) and theamplitude β q q q = δ q + q + q (cid:104) q , q , q | ψ D (cid:105) . Pro-jecting the Schr¨odinger equation, ˆ H | ψ D (cid:105) = E | ψ D (cid:105) ,onto an arbitrary state (cid:104) q , q , q | then yields an expres-sion for the three-body energy E :( E − (cid:15) q − (cid:15) q − (cid:15) q ) β q q q = δ q + q + q (cid:88) { i, j, k } e − | q i − q j | η ( k ) q k . (6)Here, we have defined the three independent functions: η ( i ) q i = g jk (cid:88) q j , q k β q q q e − | q j − q k | , (7)and we have { i, j, k } = { , , } and cyclic permutationsthereof. Rewriting the amplitudes β q q q using Eq. (7),we obtain three coupled expressions from Eq. (6): T − ij (cid:18) E − (cid:15) q (cid:19) η ( k ) q = (cid:88) q (cid:48) e −| q + q (cid:48) / | / Λ e −| q / q (cid:48) | / Λ E − (cid:15) q − (cid:15) q + q (cid:48) − (cid:15) q (cid:48) (cid:104) η ( i ) q (cid:48) + η ( j ) q (cid:48) (cid:105) , (8)where { i, j, k } take the same values as above.In the following, we approximate the Gaussian cutofffunctions appearing in Eq. (8) as e −| q + q (cid:48) / | / Λ e −| q / q (cid:48) | / Λ (cid:39) e − q / Λ e − q (cid:48) / Λ . (9)We have checked the validity of this step by evaluatingthe spectrum with and without the approximation for thecase of three identical bosons. In this case, the relativeerror on the three-body parameter is about 1%, and therelative error on the trimer energy at unitarity is similar.We therefore expect the relative error in the Li scenario to remain very small, as well. In particular, we expectthe error to be further reduced towards the 2D regime(for large B fields), where the trimer has less weight atshort range [19].By relating the T matrix to the scattering length a ij via Eq. (3), we find that E satisfies a matrix equation: a a
00 0 a η (3) η (1) η (2) = D M MM D MM M D η (3) η (1) η (2) , (10)where η ( i ) is a column vector with elements η ( i ) q . Above, D is a diagonal matrix in momentum q with elements D q = Λ F (cid:18) mE Λ − q (cid:19) , (11)while the q th element of the matrix multiplication of M onto the vector η ( i ) is (cid:104) M η ( i ) (cid:105) q = (cid:90) ∞ q (cid:48) dq (cid:48) πq e − ( q + q (cid:48) ) / Λ × ln (cid:20) E − ( q − qq (cid:48) + q (cid:48) ) /mE − ( q + qq (cid:48) + q (cid:48) ) /m (cid:21) η ( i ) q (cid:48) . (12)Since we are looking for bound states, we consider the s -wave channel and assume E <
0. Equations (10)–(12)can be solved numerically for E , and the solution for the Li system is discussed in Sec. III.For the case of SU(3)-symmetric interactions where a = a = a ≡ a , the ground state of our systemreduces to that of three identical bosons. Indeed, bydefining ¯ η = η (1) + η (2) + η (3) , Eq. (10) becomes1 a ¯ η = D ¯ η + 2 M ¯ η , (13)which is exactly the equation for three identical bosons. B. Quasi-2D System
We move on to consider the scenario where the atomsare tightly harmonically confined along the z direction.In the absence of interactions, the particles occupy theground state of the harmonic-oscillator potential and thegas is kinematically 2D. On the other hand, interactingfermions can explore all excited levels of the trap [34]. Wethus write down a Hamiltonian where the sums run overnot only each atom’s momentum q (which is now an in-plane vector perpendicular to z ), but also its harmonic-oscillator index n :ˆ H q D = (cid:88) q , n, i (cid:15) q n c † q n,i c q n,i + (cid:88) q , q (cid:48) , p i < j (cid:88) n i , n j , n (cid:48) i , n (cid:48) j N, n ij , n (cid:48) ij g ij e − ( q + q (cid:48) ) / Λ f n ij f n (cid:48) ij (cid:104) n (cid:48) i , n (cid:48) j | N, n (cid:48) ij (cid:105)(cid:104) N, n ij | n i , n j (cid:105)× c † p / q (cid:48) ,n (cid:48) i ,i c † p / − q (cid:48) ,n (cid:48) j ,j c p / − q ,n j ,j c p / q ,n i ,i , (14)where the operator c † q n,i creates atom i with in-planemomentum q and harmonic-oscillator quantum number n ≥ (cid:15) q n ≡ (cid:15) q + (cid:18) n + 12 (cid:19) ω z , (15)where, in a slight abuse of notation, we have now defined (cid:15) q = q / m in terms of the in-plane momentum. Thesecond term of Eq. (14) accounts for the pairwise inter-actions. Since these only depend on the relative motion,we model excitations in the harmonic-oscillator space bytransforming from the individual quantum numbers, n i and n j , to the center-of-mass N and relative n ij quantumnumbers. Hence, the object (cid:104) N, n ij | n i , n j (cid:105) in Eq. (14) isthe two-body Clebsch–Gordan coefficient with the selec-tion rule, n i + n j = N + n ij [45]. We have also defined f n = (cid:88) q z (cid:101) φ n ( q z ) e − q z / Λ , (16)where (cid:101) φ n ( q z ) is the Fourier transform of the harmonic-oscillator eigenfunction. It can be shown that f n = ( − n πl z ) / (cid:112) (2 n )!2 n n ! 1 √ λ (cid:18) − λ λ (cid:19) n , (17)while f n +1 = 0 [19]. Above, λ ≡ (Λ l z ) − is the(squared) ratio between the length scale of the short-distance physics Λ − and the confinement length l z . Thisratio is very small in typical experiments [34].Under a quasi-2D confinement, the threshold energyfor free-atom motion is increased since the zero-point en-ergy of the trap must be taken into account. The dimerenergy, E < ω z /
2, is again given by the pole of therelevant T matrix, T [34]: f T − ij ( E ) = 1 g ij − (cid:88) q , n e − q / Λ f n E − (cid:15) q − ( n + 1 / ω z = m πl z (cid:20) l z a ij − F ( E /ω z − / (cid:21) , (18)where F ( x ) = (cid:90) ∞ du (cid:112) π ( u + 2 λ ) × (cid:34) − √ u + 4 λ e xu (cid:112) (1 + λ ) − (1 − λ ) e − u (cid:35) . (19) Here, because the atoms are moving in the 2D plane be-tween interactions, we give the T matrix for the casewhere incoming and outgoing particles are in the lowestharmonic-oscillator state of the relative motion. Further-more, since we are considering the bare interaction in 3D(see the beginning of Sec. II), we renormalize the T ma-trix by using Eq. (2).For l z /a ij <
0, the two-body system becomes increas-ingly 2D like as | l z /a ij | increases [34]. Assuming thatΛ − (cid:28) l z , we expand in the limit of tight confinement toobtain the expression [18], E (cid:39) − Bω z π e − √ λ e √ π l z /a ij + ω z , (20)with B (cid:39) .
905 [46]. When λ → E , satisfies the equation: l z a l z a
00 0 l z a η (3) η (1) η (2) = D M MM D MM M D η (3) η (1) η (2) , (21)where η ( i ) is a tensor in momentum q and harmonic-oscillator quantum number n with elements η ( i ) q,n . Like-wise, D is a diagonal tensor D q,n = F (cid:20) E − (3 q ) / (4 m ) − ( n + 1) ω z ω z (cid:21) , (22)while the matrix multiplication of the tensors M and η gives (cid:104) M η ( i ) (cid:105) q,n = − l z m (cid:90) ∞ q (cid:48) dq (cid:48) × (cid:88) l, l (cid:48) n (cid:48) e − ( q + q (cid:48) ) / Λ f l f l (cid:48) (cid:104) n, l | n (cid:48) , l (cid:48) (cid:105) η ( i ) q (cid:48) ,n (cid:48) (cid:114)(cid:104) E − q + q (cid:48) m − ( n + l + 1) ω z (cid:105) − (cid:16) qq (cid:48) m (cid:17) , (23)where, as in the 3D problem, we have projected onto the s -wave sector. Here, q is the relative momentum in the x - y plane between two atoms’ center of mass and the thirdparticle, while l and n are the harmonic-oscillator indicesthat correspond, respectively, to relative atom-atom andatom-pair motion in the z direction. Due to the raisedthree-body continuum, we now have E < ω z (note thatwe have removed the zero-point motion corresponding tothe center-of-mass motion).In Eq. (23), the scalar quantity (cid:104) n, l | n (cid:48) , l (cid:48) (cid:105) is theatom-pair Clebsch–Gordan coefficient where the quan-tum numbers satisfy n + l = n (cid:48) + l (cid:48) . To evaluate these,we can exploit [19, 47] their relation to Wigner’s d ma-trix [48]: (cid:104) n, l | n (cid:48) , l (cid:48) (cid:105) = d n + l n (cid:48)− l (cid:48) , n − l (4 π/ . (24)Similar to the situation in 3D where there are onlytrimers in the s -wave channel, here the odd and evenatom-pair motions decouple and we find bound states inthe even- n channel only. We solve Eqs. (21)–(23) numer-ically for the Li system and discuss our results in theensuing section.
III. TRIMER STATES IN LI We now investigate the spectrum of trimers formedfrom atoms in the three lowest hyperfine states of Li.Before proceeding, we discuss the scattering parametersappearing in the Hamiltonian (1) for this specific sys-tem. The scattering lengths are obtained by using theformula [28], a ij = a ( ij ) bg [1 + ∆ ( ij ) ( B − B ( ij )0 ) − ][1 + α ( ij ) ( B − B ( ij )0 )] . (25)Here, a ( ij ) bg is the background scattering length, ∆ ( ij ) isthe resonance width, B ( ij )0 is the position of the reso-nance, and α ( ij ) is a correction parameter [49]. The rel-ative error in this expression is expected to be less than1% over the range of magnetic fields between B = 600and 1200 Gauss [28].The second parameter of the model is the three-bodyparameter, i.e., the short-range length scale which en-sures that the ground-state energy is well defined. Thethree-body parameter for Li has been calculated by us-ing numerous models and methods (see pp. 43 and 44of Ref. [6] for a recent summary). For large fields
B >
600 G, the three-body parameter can be calculated byfitting to experimentally measured loss rates [15, 31, 50]and by radio-frequency spectroscopy [14, 15]. The valuesreported vary by roughly 10% for magnetic fields rangingfrom 685 to 895 G [15, 50]. In this work, we apply a Gaus-sian cutoff characterized by the ultraviolet momentumscale, Λ − (cid:39) . R vdW [51] where R vdW (cid:39) a [52]( a is the Bohr radius). Such a model has been appliedin previous studies [15, 51] to fit the loss rate associatedwith the excited trimer crossing into the three-atom con-tinuum, as measured in Ref. [31]. This feature is withinthe range of magnetic fields that we consider. FIG. 1. (a) Spectrum of 3D trimers and illustration ofrescaling: By modifying the momentum cutoff, Λ → Λ /φ ( B ),we map the deeply bound ground-state trimer [purple solidline] in the 3D system to the excited trimer state [greendashed line]. This effectively removes the original groundstate from the problem without changing the low-energyphysics. (b) Probability densities (see Sec. III B) for the ex-cited 3D trimer without rescaling [green dashed line] and theground-state trimer with rescaling [purple solid line], at dif-ferent magnetic fields. The functions plotted are normalizedto 1. In the following, we present results for two confinementstrengths, ω z = 2 π ×
10 kHz and ω z = 2 π ×
50 kHz, corre-sponding to confinement lengths of 7800 a and 3500 a ,respectively. Both of these are within reach of currentexperiments on quasi-2D two-component Li gases [35–38].
A. Trimer Energies
In three dimensions, it has been predicted that thereexist two trimers [31]. Solving Eq. (10) for the trimer en-ergies, we find the spectrum shown in Fig. 1(a), whichagrees with the results of Ref. [33]. Notice how theexcited trimer only exists for magnetic fields (cid:46)
900 G,beyond which it disappears into the three-atom contin-uum. On the other hand, the ground-state trimer is ex-pected to be very deeply bound, with a binding energy
FIG. 2. Energy spectra of trimers comprising atoms in the three lowest hyperfine states of Li, for two different confinementstrengths, ω z = 2 π ×
10 kHz [left] and ω z = 2 π ×
50 kHz [right]. Panels (a) and (b) show the ratio of the confinement length l z to the s -wave scattering length a ij for all three pairs of atoms i and j . In panels (c) and (d) we plot the 3D, quasi-2D, and 2Dtrimer states as the green short-dashed, red solid, and blue long-dashed lines, respectively. Under confinement, the thresholdenergy of the free-atom continuum is increased from E = 0 in the 3D system to E = ω z in the 2D and quasi-2D cases, whichis indicated by the darker rendered area. The lighter rendered area is the atom-dimer continuum of the confined systems, andcorresponds to atoms 1 and 2 forming a dimer. ∼ π ×
30 MHz that remains relatively constant over therange of magnetic fields investigated [53].The large separation of energy scales in the 3D trimerspectrum, Fig. 1(a), presents a significant challenge tocalculating the spectrum in the presence of confinement.In particular, the deepest trimer energy exceeds realisticconfinement strengths by three orders of magnitude, andthus we may expect this state to be essentially unaffectedby the confinement. This, in turn, means that the num-ber of harmonic-oscillator levels taken into account in thenumerics has to greatly exceed 1000 to properly describeall energy scales of the problem — which is in practiceunfeasible. Instead, we take advantage of the fact that weare primarily interested in the excited trimer at low ener-gies. Therefore we can rescale the cutoff, Λ → Λ /φ ( B ), insuch a way that the ground state of the rescaled model co-incides with the excited state of the original model — seeFig. 1(a). This procedure effectively removes the groundstate of the original problem. The rescaling of the short-range parameter is inspired by the system of three identi-cal bosons, where the spectrum at large scattering lengthis characterized by a discrete scaling symmetry, such thatthe low-energy physics is unchanged under a rescaling of the ultraviolet cutoff: Λ → Λ / . φ ( B ) which decreases approximately linearly withincreasing magnetic-field strength, i.e., from φ ( B =840 G) (cid:39) . φ ( B = 900 G) (cid:39) .
7. However, we can-not compute the scaling parameter for fields B (cid:38)
900 G,since there the excited 3D trimer ceases to exist. Hence,we simply take φ ( B = 900 G) throughout this regime,since the three-body parameter we use is most accurateat ∼
900 G, where the excited trimer disappears and aloss feature is observed [51].In Fig. 1(b) we show how, outside the short-range re-gion, the wave function (see Section III B) of the excitedstate in the original model closely matches that of theground state with the rescaled cutoff. This result corrob-orates the use of our rescaling. While such an approachintroduces effective range corrections to our quasi-2D re-sults [see Eq. (20)], these are expected to be small in theexperimental regime of interest since the rescaled van derWaals range remains much smaller than the confinementlength.In Fig. 2, we present our calculated trimer energiesfor the case of a quasi-2D geometry with confinementstrengths, ω z = 2 π ×
10 kHz and ω z = 2 π ×
50 kHz. Asdiscussed above, we only show the excited trimer. Forboth confinement strengths, we see how the trimer en-ergy is close to that of the 3D trimer for B (cid:46)
900 G. Atlarger magnetic fields, the trimer is stabilized by the con-finement and exists far beyond its regime of existence inthree dimensions. In particular, we see that the bindingenergy of the trimer can be comparable to ω z for a largerange of magnetic fields beyond 900 G. The existence ofthe trimer in this regime may be understood from howthe three-body continuum in quasi-2D is raised by ω z ,which results in an effective long-range attractive well inthe hyperspherical potential [19]. Indeed, this result isanalogous to how the two-body state is stabilized by aconfining potential [46].We may elucidate our results further by consideringthe 2D limit. When all three scattering lengths arenegative and their magnitudes are less than the con-finement length, the few-body states are expected tobe extended in the plane and thus strongly modifiedfrom their three-dimensional counterparts. As shown inFigs. 2(a) and 2(b), for a confinement of ω z = 2 π ×
10 kHzthis condition is satisfied when B (cid:38)
900 G, while for ω z = 2 π ×
50 kHz the 2D condition requires stronger mag-netic fields, B (cid:38) B fields,the trimer energies are expected to approach those pre-dicted from a purely 2D theory. We obtain the 2D limitby taking just one atom-pair harmonic-oscillator statein the three-body equation (21), while still retaining thefull quasi-2D T matrix [i.e., the exact D in Eq. (22)],since this allows us to accommodate any effective rangethat arises from the confinement and acts through thetwo-body physics. Indeed, in Figs. 2(c) and 2(d) we seethat the quasi-2D trimer approaches the 2D limit at largemagnetic fields.We also note how, in the case where the scatteringlengths are equal, it is predicted that two trimers exist inthe 2D limit [26]. For Li, the ratios between the threescattering lengths approach unity for increasingly strongmagnetic fields. Therefore, eventually one would expect asecond quasi-2D trimer to emerge from the continuum inthis regime. However, for the confinement strengths con-sidered here, the second 2D trimer remains very weaklybound on the scale shown in the figure.
B. Wave Functions
We now analyze how the quasi-2D confinement affectsthe trimer wave functions and, in particular, their relativeweight at short distance. Starting with the 3D case, weconsider the real-space atom-pair wave function definedas the following Fourier transform: ψ ( i ) ( R ) = 1 √N i (cid:90) q dqR sin( qR ) η ( i ) q , (26)where the constant N i ensures normalization. Here wetake advantage of the fact that the trimer states satisfy- ing Eq. (10) have s -wave symmetry and thus η ( i ) q does notdepend on the direction of q . This wave function corre-sponds to the scenario where we take two atoms to havezero separation and then consider the motion of this pairwith the remaining atom i . As such, we have three atom-pair wave functions ψ ( i ) ( R ), one for each pair, where R corresponds to the relative atom-pair coordinate. The3D wave functions are illustrated in Fig. 1(b).Likewise, we define the quasi-2D real-space atom-pairwave function, from the solution of the quasi-2D three-body equation (21), as ψ ( i ) ( ρ, z ) = 1 (cid:113) N (q2D) i (cid:90) q dq (cid:88) n f n ( z ) J ( qρ ) η ( i ) q,n , (27)where N (q2D) i is again the normalization and J is theBessel function. Here, z is the atom-pair coordinate inthe transverse direction, while ρ is the separation in theplane. We show these wave functions at three differentmagnetic fields in Fig. 3.To evaluate the weight of the three-body wave func-tions at short distance, we employ the following ap-proximation to convert the atom-pair wave functions,Eqs. (26) and (27), to those describing the full three-particle problem: First, we note that in the case of identi-cal pairwise interactions, the 3D three-atom hyperspher-ical wave function is approximately related to the atom-pair wave function by multiplying ψ ( i ) ( R ) by R / , where R is interpreted as the three-body hyperradius [4]. Sim-ilarly, the 2D three-atom hyperspherical wave functionis obtained by multiplying ψ ( i ) ( ρ,
0) by ρ / [54], whereagain ρ corresponds to the planar hyperradius. In the Li case, we still expect this to be a reasonable approxi-mation since the three interaction strengths are approx-imately equal. Therefore, for the quasi-2D system, wedefine the following weighting function that interpolatesbetween the 2D and 3D limits: ω ( ρ ) = ρ ρ + l z . (28)We then define the relative weight of the trimer at shortdistances as P ρ < ρ = (cid:82) ρ dρ ω ( ρ ) (cid:80) i = 1 | ψ ( i ) ( ρ, z = 0) | (cid:82) ∞ dρ ω ( ρ ) (cid:80) i = 1 | ψ ( i ) ( ρ, z = 0) | . (29)At this stage, several comments are in order: First, inthe following, we take the short-range length scale to be ρ = 10 R vdW ; we have checked that our results are notsensitive to the precise range, by varying the definitionof this length scale up to a factor of 10. Second, thecrossover scale of l z in Eq. (28) is the squared atom-pair confinement length; again, we have checked that ourresults do not depend sensitively on the precise range cho-sen for this interpolation. Third, we evaluate the weightat short range by taking z = 0. This is reasonable when FIG. 3. (upper panel) The relative weight of the trimer wave function at short range ( (cid:46) R vdW ) in a quasi-2D geometry fortwo different confinement strengths, ω z = 2 π ×
10 kHz [red solid line] and ω z = 2 π ×
50 kHz [orange dashed line]. The shadedarea indicates the regime of existence for the excited trimer in 3D. Note that the small kink in the relative weight at B = 900 Gis related to our rescaling and should not be understood as a physical effect. (lower panels) The in-plane quasi-2D probabilitydensities for both confinements at three different magnetic-field strengths. [Refer to the discussion around Eqs. (28) and (29)of the text for our treatment of the quasi-2D wave functions.] The functions plotted are normalized to 1. the wave function is 3D like, since it is then isotropicand we are thus free to choose any direction. Conversely,when the wave function is more 2D like at large distances,then the relevant part of the wave function is exactly the z = 0 component.In Fig. 3, we show our calculated trimer weight inthe short-range regime and the corresponding probabilitydensities at select magnetic fields. Beyond B (cid:39)
900 G,where the excited 3D trimer ceases to exist, the short-range relative weight of the quasi-2D trimer decreases byalmost an order of magnitude for the stronger confine-ment, and four orders of magnitude for the weaker con-finement, over the range of magnetic fields shown. Thus,we expect the lifetime of the trimer to increase accord-ingly. The reduction in the short-range weight is due tothe trimer becoming increasingly spatially extended aswe approach the 2D limit. This is because, unlike in 3D,the trimer now resides in the long-range attractive tailof the hyperspherical potential. This is the same mecha-nism responsible for the longer lifetimes of the trimers ofidentical bosons discussed in Ref. [19]. Note that our ap-proximate expression (29) does not account for how thetwo-body scattering within each pair of atoms changesfrom 3D to 2D. However, if anything, we would expect2D-like two-body scattering to further suppress decay ofthe trimers into atoms and dimers [20, 55, 56].
IV. CONCLUSIONS AND OUTLOOK
In this work, we have considered the problem of threedistinguishable fermions confined to a quasi-2D geome-try. In particular, we have allowed for the possibilitythat the three pairwise interactions are different, as isthe case for the Li system. While trimers comprisingthree dissimilar particles can, in principle, also be manu-factured from bosons, we have exclusively studied thequasi-2D Fermi gas since the corresponding Bose sys-tem has significant instabilities [27]. Furthermore, the Li system has the advantage that the two-componentFermi gas is stable, and the three-component trimers in3D have already been realized in experiment [14]. Thus,by using realistic experimental parameters, we have com-puted the Li trimer spectrum for two different quasi-2Dconfinements. We have focused exclusively on the evolu-tion of the excited trimer from the 3D spectrum, sincethe ground-state trimer is too deeply bound to be sig-nificantly affected by the confinement. We have foundthat the excited trimer evolves into a 2D-like spatiallyextended trimer as the interactions are decreased withincreasing magnetic field. This behavior is also apparentin the approximate three-body wave function we havecalculated for the trimer.Our results indicate that the quasi-2D trimers can belonger lived by at least an order of magnitude comparedwith their 3D counterparts, since these spatially extendedtrimers have a reduced probability that three fermionscan approach each other at short distances and decay intoa deeply bound dimer state. This opens the door to engi-neering long-lived three-body bound states in cold-atomexperiments. In principle, such trimers can be associatedfrom atoms and pairs in a quasi-2D two-component Ligas. To achieve this in experiment, we require all interac-tions l z /a ij < − ω z is around 2 π ×
50 kHz or more, thenthe trimer lifetime will become comparable to that in 3D,while if the confinement is too weak, then the quasi-2Dtrimer will be dissociated by thermal fluctuations. Since the temperature of the confined Li gas is typically of or-der kHz [35–38], we expect the optimal confinement andmagnetic field to be in the ranges 2 π × ACKNOWLEDGMENTS
We are grateful to C. Vale, P. Dyke, and S. Hoinkafor fruitful discussions. J.L. is supported throughthe Australian Research Council Future FellowshipFT160100244. M.M.P. and J.L. also acknowledge finan-cial support from the Australian Research Council viaDiscovery Project No. DP160102739.
APPENDIX: Three-Body Problem in a Quasi-2D System
Here, we derive Eqs. (21)–(23) of the main text which determine the bound states of three distinguishable fermionsinteracting in a quasi-2D geometry.We write down a general wave function at zero center-of-mass momentum, | ψ q D (cid:105) = (cid:88) q , q , q n , n , n β q q q n n n | q n , q n , q n (cid:105) , (30)where the state | q n , q n , q n (cid:105) ≡ c † q n , c † q n , c † q n , | (cid:105) and the amplitude β q q q n n n = δ q + q + q (cid:104) q n , q n , q n | ψ q D (cid:105) . For three particles, we transform from the individual harmonic-oscillator indices { n , n , n } to thenew indices { n ij , n ijk , N } . These correspond, respectively, to the relative motion of two atoms in the z direction, z ij = z i − z j , the relative motion between their center of mass and the third atom, z ijk = ( z i + z j ) / − z k , and thecenter-of-mass motion of all three atoms, Z = ( z i + z j + z k ) / H q D | ψ q D (cid:105) = E | ψ q D (cid:105) onto an arbitrary state, we obtain the following expression for the three-body energy E :( E − (cid:15) q n − (cid:15) q n − (cid:15) q n ) β q q q n n n = (cid:88) q (cid:48) , q (cid:48) , q (cid:48) { i, j, k } (cid:88) n (cid:48) , n (cid:48) , n (cid:48) n ijk , n ij , n (cid:48) ij g ij e − | q i − q j | e − | q (cid:48) i − q (cid:48) j | δ q k , q (cid:48) k δ q + q + q f n ij f n (cid:48) ij × (cid:104) n , n , n | N = 0 , n ijk , n ij (cid:105)(cid:104) N = 0 , n ijk , n (cid:48) ij | n (cid:48) , n (cid:48) , n (cid:48) (cid:105) β q (cid:48) q (cid:48) q (cid:48) n (cid:48) n (cid:48) n (cid:48) , (31)where (cid:15) q n is defined in Eq. (15) and we have { i, j, k } = { , , } and cyclic permutations. Note that since we areworking in the center-of-mass frame, we make the simplification N = 0.We can remove two harmonic-oscillator indices from the problem by defining three independent functions [47], η ( k ) q (cid:48) k ,n ijk = g ij (cid:88) q , q , q n ij , n , n , n e − | q i − q j | δ q k , q (cid:48) k f n ij (cid:104) , n ijk , n ij | n , n , n (cid:105) β q , q , q n ,n ,n , (32)which allow us to rewrite Eq. (31) as( E − (cid:15) q n − (cid:15) q n − (cid:15) q n ) β q q q n n n = (cid:88) n ijk , n ij { i, j, k } e − | q i − q j | δ q + q + q f n ij (cid:104) n , n , n | , n ijk , n ij (cid:105) η ( k ) q k ,n ijk . (33)To proceed, we divide by ( E − (cid:15) q n − (cid:15) q n − (cid:15) q n ), and then act with the operator g ij (cid:88) q , q , q n (cid:48) ij , n , n , n e − | q i − q j | δ q k , q (cid:48) k f n (cid:48) ij (cid:104) , n ij (cid:48) k , n (cid:48) ij | n , n , n (cid:105) ( · ) (34)0on the left three separate times, where { i, j, k } take the same values as in Eqs. (31)–(33). This yields a separateequation for each of the three η ( k ) functions, and we give one of these below: η (3) q (cid:48) ,n (cid:48) = g (cid:88) q , q , q n ijk , n ij , n (cid:48) { i, j, k } e −| q i − q j | / (4Λ ) e −| q − q | / (4Λ ) E − (cid:15) q − (cid:15) q − (cid:15) q − ( n (cid:48) + n (cid:48) + 1) ω z δ q , q (cid:48) δ q + q + q f n ij f n (cid:48) × (cid:104) , n (cid:48) , n (cid:48) | , n ijk , n ij (cid:105) η ( k ) q k ,n ijk . (35)To arrive at Eq. (35), we make use of the fact that (cid:88) n , n , n | n , n , n (cid:105) E − (cid:15) q n − (cid:15) q n − (cid:15) q n (cid:104) n , n , n | = 1 E − (cid:15) q − (cid:15) q − (cid:15) q − ˆ H z , (36)in which ˆ H z is the non-interacting Hamiltonian for the one-dimensional harmonic oscillator.Evaluating the δ -functions, we then obtain three coupled expressions of the form f T − ij (cid:20) E − (cid:15) q − (cid:18) n + 12 (cid:19) ω z (cid:21) η ( k ) q ,n = (cid:88) q (cid:48) , n (cid:48) (cid:104) η ( i ) q (cid:48) ,n (cid:48) + η ( j ) q (cid:48) ,n (cid:48) (cid:105) (cid:88) l, l (cid:48) e −| q + q (cid:48) / | / Λ e −| q / q (cid:48) | / Λ E − (cid:15) q − (cid:15) q + q (cid:48) − (cid:15) q (cid:48) − ( n + l + 1) ω z f l f l (cid:48) (cid:104) n, l | n (cid:48) , l (cid:48) (cid:105) , (37)with the same values for { i, j, k } . Above, the left-hand side contains the T matrix appearing in Eq. (18) and theharmonic-oscillator wave function f in Eq. (17). The harmonic-oscillator quantum numbers, l and n , correspondrespectively to relative atom-atom and atom-pair motion in the z direction, while q is the relative atom-pair momentumin the x - y plane (and similarly for the primed variables).The Gaussian cutoff functions can be approximated the same way as in the 3D problem — see Eq. (9). Afterprojecting to the s -wave, Eq. (37) then leads to the system of equations (21)-(23) in Sec. II B. [1] P. Bedaque, H.-W. Hammer, and U. van Kolck, Nucl.Phys. A , 357 (2000).[2] Y. Nishida, Y. Kato, and C. D. Batista, Nat. Phys. ,93 (2013).[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[4] E. Braaten and H.-W. Hammer, Phys. Rep. , 259(2006).[5] F. Ferlaino and R. Grimm, Physics , 9 (2010).[6] P. Naidon and S. Endo, Rep. Prog. Phys. , 056001(2017).[7] V. Efimov, Phys. Lett. B , 563 (1970).[8] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl,C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola,H.-C. N¨agerl, and R. Grimm, Nature (London) , 315(2006).[9] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hut-son, Phys. Rev. Lett. , 190401 (2014).[10] M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin,L. P. H. Schmidt, M. Sch¨offler, A. Czasch, W. Sch¨ollkopf,R. E. Grisenti, T. Jahnke, D. Blume, and R. D¨orner, Sci-ence , 551 (2015).[11] C. E. Klauss, X. Xie, C. Lopez-Abadia, J. P. D’Incao,Z. Hadzibabic, D. S. Jin, and E. A. Cornell, Phys. Rev.Lett. , 143401 (2017).[12] R. Pires, J. Ulmanis, S. H¨afner, M. Repp, A. Arias, E. D.Kuhnle, and M. Weidem¨uller, Phys. Rev. Lett. , 250404 (2014).[13] S.-K. Tung, K. Jim´enez-Garc´ıa, J. Johansen, C. V.Parker, and C. Chin, Phys. Rev. Lett. , 240402(2014).[14] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz,G. Z¨urn, and S. Jochim, Science , 940 (2010).[15] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon,and M. Ueda, Phys. Rev. Lett. , 143201 (2011).[16] A. Rapp, G. Zar´and, C. Honerkamp, and W. Hofstetter,Phys. Rev. Lett. , 160405 (2007).[17] Y. Nishida, Phys. Rev. Lett. , 240401 (2012).[18] T. Kirk and M. M. Parish, Phys. Rev. A , 053614(2017).[19] J. Levinsen, P. Massignan, and M. M. Parish, Phys. Rev.X , 031020 (2014).[20] J. P. D’Incao, F. Anis, and B. D. Esry, Phys. Rev. A ,062710 (2015).[21] Z. Hadzibabic, P. Kr¨uger, M. Cheneau, B. Battelier, andJ. Dalibard, Nature (London) , 1118 (2006).[22] P. Clad´e, C. Ryu, A. Ramanathan, K. Helmerson, andW. D. Phillips, Phys. Rev. Lett. , 170401 (2009).[23] C.-L. Hung, X. Zhang, N. Gemelke, and C. Chin, Nature(London) , 236 (2011).[24] M. T. Yamashita, F. F. Bellotti, T. Frederico, D. V. Fe-dorov, A. S. Jensen, and N. T. Zinner, J. Phys. B: At.,Mol. Opt. Phys. , 025302 (2015).[25] J. H. Sandoval, F. F. Bellotti, M. T. Yamashita, T. Fred- erico, D. V. Fedorov, A. S. Jensen, and N. T. Zinner, J.Phys. B: At., Mol. Opt. Phys. , 065004 (2018).[26] L. W. Bruch and J. A. Tjon, Phys. Rev. A , 425 (1979).[27] B. Huang, A. Zenesini, R. Grimm, V. Ngampruetikorn,M. M. Parish, and J. Levinsen, in preparation (2018).[28] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen,S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Si-moni, E. Tiesinga, C. J. Williams, and P. S. Julienne,Phys. Rev. Lett. , 103201 (2005).[29] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz,and S. Jochim, Phys. Rev. Lett. , 203202 (2008).[30] J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites,and K. M. O’Hara, Phys. Rev. Lett. , 165302 (2009).[31] J. R. Williams, E. L. Hazlett, J. H. Huckans, R. W. Stites,and K. M. O’Hara, Phys. Rev. Lett. , 130404 (2009).[32] T. Lompe, T. B. Ottenstein, F. Serwane, K. Viering,A. N. Wenz, G. Z¨urn, and S. Jochim, Phys. Rev. Lett. , 103201 (2010).[33] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon,and M. Ueda, Phys. Rev. Lett. , 023201 (2010).[34] J. Levinsen and M. M. Parish, Annu. Rev. Cold At. Mol. , 1 (2015).[35] P. Dyke, E. D. Kuhnle, S. Whitlock, H. Hu, M. Mark,S. Hoinka, M. Lingham, P. Hannaford, and C. J. Vale,Phys. Rev. Lett. , 105304 (2011).[36] A. T. Sommer, L. W. Cheuk, M. J. H. Ku, W. S. Bakr,and M. W. Zwierlein, Phys. Rev. Lett. , 045302(2012).[37] I. Boettcher, L. Bayha, D. Kedar, P. A. Murthy, M. Nei-dig, M. G. Ries, A. N. Wenz, G. Z¨urn, S. Jochim, andT. Enss, Phys. Rev. Lett. , 045303 (2016).[38] D. Mitra, P. T. Brown, P. Schauß, S. S. Kondov, andW. S. Bakr, Phys. Rev. Lett. , 093601 (2016).[39] O. I. Kartavtsev and A. V. Malykh, J. Phys. B: At., Mol.Opt. Phys. , 1429 (2007).[40] J. Levinsen, T. G. Tiecke, J. T. M. Walraven, and D. S.Petrov, Phys. Rev. Lett. , 153202 (2009).[41] S. Endo, P. Naidon, and M. Ueda, Phys. Rev. A ,062703 (2012). [42] We take the cutoff Λ to be the same for all three pairs[51]. This is reasonable when the van der Waals rangesof the interactions are all similar, as in the case of Li.[43] F. Werner, L. Tarruell, and Y. Castin, Eur. Phys. J. B , 401 (2009).[44] C. Mora, Y. Castin, and L. Pricoupenko, C. R. Phys. , 71 (2011).[45] Y. Smirnov, Nucl. Phys. , 346 (1962).[46] D. S. Petrov and G. V. Shlyapnikov, Phys. Rev. A ,012706 (2001).[47] E. K. Laird, Z.-Y. Shi, M. M. Parish, and J. Levinsen,Phys. Rev. A , 032701 (2017).[48] E. P. Wigner, Group Theory and Its Application to theQuantum Mechanics of Atomic Spectra , Academic Press,New York (1959).[49] For completeness, the respective values of a bg , ∆, B ,and α are − a , 300 G, 834 .
149 G, and 0 . − for channel (1 , − a , 122 . .
43 G, and0 . − for channel (1 , − a , 222 . .
22 G, and 0 . − for channel (2 ,
3) [28].[50] B. Huang, K. M. O’Hara, R. Grimm, J. M. Hutson, andD. S. Petrov, Phys. Rev. A , 043636 (2014).[51] P. Naidon and M. Ueda, C. R. Phys. , 13 (2011).[52] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[53] The ground-state trimer can furthermore be strongly af-fected by non-universal effects beyond our theory [33].However, its binding energy is still ∼ π ×
10 MHzwhich greatly exceeds experimentally realistic confine-ment strengths.[54] E. Nielsen, D. V. Fedorov, and A. S. Jensen, Few-BodySyst. , 15 (1999).[55] K. Helfrich and H.-W. Hammer, Phys. Rev. A , 052703(2011).[56] V. Ngampruetikorn, M. M. Parish, and J. Levinsen, Eu-rophys. Lett. , 13001 (2013).[57] C. J. Bradly, B. C. Mulkerin, A. M. Martin, and H. M.Quiney, Phys. Rev. A90