PHYSICAL REVIEW A 87, 062702 (2013) Single-particle momentum distributions of Efimov states in mixed-species systems M. T. Yamashita,1 F. F. Bellotti,2,3,4 T. Frederico,2 D. V. Fedorov,3 A. S. Jensen,3 and N. T. Zinner3 1Instituto de Fı́sica Teórica, UNESP–Univ Estadual Paulista, C.P. 70532-2, CEP 01156-970, São Paulo, SP, Brazil 2Instituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, SP, Brazil 3Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark 4Instituto de Fomento e Coordenação Industrial, 12228-901, São José dos Campos, SP, Brazil (Received 23 March 2013; published 4 June 2013) We solve the three-body bound-state problem in three dimensions for mass imbalanced systems of two identical bosons and a third particle in the universal limit where the interactions are assumed to be of zero range. The system displays the Efimov effect and we use the momentum-space wave equation to derive formulas for the scaling factor of the Efimov spectrum for any mass ratio assuming either that two or three of the two-body subsystems have a bound state at zero energy. We consider the single-particle momentum distribution analytically and numerically and analyze the tail of the momentum distribution to obtain the three-body contact parameter. Our findings demonstrate that the functional form of the three-body contact term depends on the mass ratio, and we obtain an analytic expression for this behavior. To exemplify our results, we consider mixtures of lithium with either two caesium or rubidium atoms which are systems of current experimental interest. DOI: 10.1103/PhysRevA.87.062702 PACS number(s): 34.50.Cx, 03.65.Ge, 21.45.−v, 67.85.−d I. INTRODUCTION Few- and many-body systems in the presence of strong interactions is an increasingly fruitful direction in physics due to several advances in the experimental realization of such systems within cold atomic gases [1]. One aspect of this pursuit concerns the so-called unitary regime in which the two-body scattering amplitude saturates the unitarity bound of basic quantum mechanics. For neutral and nonpolar cold atomic gases, interactions are very short ranged and one can often use the universal limit where interactions are modeled by a zero-range potential which is then subsequently connected to the low-energy two-body scattering dynamics through the scattering length a. In this framework, the unitary regime is simply characterized by a scattering length that is much larger than any other relevant scale in the system under study. The problem of characterizing a strongly interacting system at unitarity in turn lacks a natural scale for the interactions, and one expects universal behavior of structural and dynamical properties that should be applicable irrespective of the details of the short-distance physics, and thus have applicability in many different subfields. An important recent development was the derivation of a number of universal relations that describe the physics of unitary two-component Fermi gases by Tan [2]. It turns out that a parameter dubbed the (two-body) contact C2 emerges in expressions for both structural (energy, adiabatic theorem, pressure, virial theorem [3]) and dynamical observables (inelastic loss [4], radio-frequency spectroscopy [5,6]). This provides a strong connection between the few-body quantities and many-body observables, a connection that has by now been experimentally verified [7,8]. The relations are not particular to the unitary Fermi gas but may also be applied to bosons [9–14] as another recent experimental effort has confirmed [15]. Another direction that has enjoyed access to the unitary regime is the study of universal three-body bound states and the famous Efimov effect [16]. The effect occurs close to the unitary point and implies that a sequence of three-body bound states occurs wherein two successive states always have the same fixed ratio of their binding energies. The effect was first observed in cold atomic gas experiments [17] and has subsequently opened a new research direction dubbed Efimov physics [18]. An interesting recent finding is that the presence of the Efimov effect implies an extension of the universal relations discussed above and the introduction of a three-body contact parameter C3 [11,12]. This parameter vanishes in the case of two-component Fermi gases since the Pauli principle suppresses three-body correlations at short distances [3]. In this paper, we study three-body bound Efimov-type states for systems that contain two identical bosons and a third distinguishable particle. Our goal is to address the contact parameters of such systems when the masses are different and for different strengths of the interaction parameters by computing the single-particle momentum distributions and studying their asymptotic behavior. We will consider only the universal regime, i.e., we approximate all two-body potentials by zero-range interactions. In comparison to the previous stud- ies containing three identical bosons, we find that for different particles there are additional contributions to the asymptotic behavior of the momentum distributions. This suggests that such measurements are a useful probe to discriminate between effects of identical and nonidentical three-body correlations in the many current cold gas experiments that have mixtures of different kinds of atoms in the same trap. The paper is organized as follows. In Sec. II, we introduce the momentum-space formalism that we use and Sec. III contains a discussion of the three-body wave function (or spectator function). In Sec. III, we also present an analytic derivation of the Efimov scaling factors as function of mass ratio in the cases where either two or three of the two-body subsystems have a two-body bound state at zero energy. We proceed to present our results for the asymptotic momentum distributions in Sec. IV and demonstrate that the subleading correction has a functional form that depends on the mass ratio. The details of the analytic derivations of the asymptotic momentum behavior are given in the Appendix for completeness. Results for relevant experimental mixtures 062702-11050-2947/2013/87(6)/062702(13) ©2013 American Physical Society http://dx.doi.org/10.1103/PhysRevA.87.062702 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) of 6Li-133Cs-133Cs and 6Li-87Rb-87Rb are discussed in Sec. V for different choices of the interaction parameters. Section VI contains conclusions and outlook for future studies. II. FORMALISM We consider a system that has an AAB structure, where the two A particles are identical bosons and the third B particle is of a different kind. When we discuss our results in the following, we will focus on two combinations that are of interest to current experimental efforts in cold atoms: A = 133Cs, B = 6Li and A = 87Rb, B = 6Li. Since we are interested in the universal limit where the range of the two-body potentials can be neglected, we consider purely zero-range interactions in the following. More precisely, if r0 is the range of the two-body potential, we are assuming that the scattering length a is a � r0. For simplicity, we will use units where h̄ = mA = 1 from now on. After partial wave projection, the s-wave coupled subtracted integral equations for the spectator functions χ and the absolute value of the three-body binding energy E3 are given by [19–21] χAA(y) = 2τAA(y; E3) ∫ ∞ 0 dx x y G1(y,x; E3)χAB(x), (1) χAB(y) = τAB(y; E3) ∫ ∞ 0 dx x y [G1(x,y; E3)χAA(x) +AG2(y,x; E3)χAB(x)]; (2) τAA(y; E3) ≡ 1 π [√ E3 + A + 2 4A y2 ∓ √ EAA ]−1 , (3) τAB(y; E3) ≡ 1 π (A + 1 2A )3/2 [√ E3 + A + 2 2(A + 1) y2 ∓ √ EAB ]−1 , (4) G1(y,x; E3) ≡ log 2A(E3 + x2 + xy) + y2(A + 1) 2A(E3 + x2 − xy) + y2(A + 1) − log 2A(μ2 + x2 + xy) + y2(A + 1) 2A(μ2 + x2 − xy) + y2(A + 1) , (5) G2(y,x; E3) ≡ log 2(AE3 + xy) + (y2 + x2)(A + 1) 2(AE3 − xy) + (y2 + x2)(A + 1) − log 2(Aμ2 + xy) + (y2 + x2)(A + 1) 2(Aμ2 − xy) + (y2 + x2)(A + 1) , (6) where x and y denote (dimensionless) momenta. We will use the natural logarithm throughout this paper, i.e. the one with base e. Here, we have introduced the mass number A = mB/mA. The interaction energies of the AA and AB subsystems are parametrized by EAA and EAB , and the plus and minus signs in (3) and (4) refer to virtual and bound two-body subsystems, respectively [22–24]. We map EAA and EAB into the usual scattering lengths aAA and aAB through the relation E ∝ |a|−2. This relation typically holds for broad resonances, and a more detailed mapping needs to be done in the general case [25]. Throughout most of this work we will focus on the region close to unitarity in the AB system, i.e., |aAB | → ∞ or EAB → 0. In light of the fact that experimental information about mixed systems of the AAB type is still sparse, we will consider the two extreme cases (i) EAA = 0 and (ii) a noninteracting AA subsystem. In the numerical work presented later on, we will set μ2 = 1 for the subtraction point (see for instance Ref. [21] for a detailed discussion and references). On the other hand, in the analytical derivations, we will take the limit μ → ∞. We note that this subtraction method is basically equivalent to the procedure employed by Danilov [26] to regularize the original three-body Skorniakov-Ter-Martirosian equation [27]. A very detailed recent discussion of these issues was given by Pricoupenko [28,29]. Defining as �kα (α = i,j,k) the momenta of each particle in the rest frame, we have that the Jacobi momenta from one particle to the center of mass of the other two and the relative momentum of the two are given, respectively, by �qk = mij,k ( �kk mk − �ki + �kj mi + mj ) = �kk and (7) �pk = mij ( �ki mi − �kj mj ) , where {i,j,k} is an even permutation of the particles {A,A′,B} and we have used that �ki + �kj + �kk = 0 in the center-of-mass system. The reduced masses are defined such that mij = mimj mi+mj and mij,k = mk (mi+mj ) mi+mj +mk . In the following we define exactly what we mean by single- particle momentum distributions for particles of types A and B. For a zero-range potential, the three-body wave function for an AAB system, composed by two identical particles A and one different B, can be written in terms of the spectator functions in the basis |�qB �pB〉 as 〈�qB �pB |�〉 = χAA(qi) + χAB(qj ) + χAB(qk) E3 + H0 = χAA(qB) + χAB (∣∣ �pB − �qB 2 ∣∣)+ χAB (∣∣ �pB + �qB 2 ∣∣) E3 + H0 , (8) or in the basis |�qA �pA〉 as 〈�qA �pA|�〉 = χAA (∣∣ �pA − A A+1 �qA ∣∣) + χAB (∣∣ �pA + 1 A+1 �qA ∣∣) + χAB(qA) E3 + H ′ 0 , (9) where H0 = p2 B 2mAA + q2 B 2mAA,B and H ′ 0 = p2 A 2mAB + q2 A 2mAB,A . The re- duced masses are given by mAA = 1 2 , mAA,B = 2A A+2 , mAB = A A+1 , and mAB,A = A+1 A+2 . The momentum distributions for the particles A and B are n(qB) = ∫ d3pB |〈�qB �pB |�〉|2, (10) n(qA) = ∫ d3pA|〈�qA �pA|�〉|2 and they are normalized such that ∫ d3q n(q) = 1. Note that our definition of momentum distributions as well as their normalizations differ from Ref. [12]. In Ref. [12] there is a 062702-2 SINGLE-PARTICLE MOMENTUM DISTRIBUTIONS OF . . . PHYSICAL REVIEW A 87, 062702 (2013) factor of 1/(2π )3 multiplying the definition of n(q), which is normalized to 3, the number of particles. III. ASYMPTOTIC FORMULAS FOR THE SPECTATOR FUNCTIONS We now consider the asymptotic behavior of the spectator function to derive some analytic formulas and compare to cor- responding numerical results. To access the large momentum regime √ E3 q, we take the limit μ → ∞ and E3 = EAA = EAB → 0. The coupled equations for the spectator functions consequently simplify and become χAA(y) = 2 π [ y √ A + 2 4A ]−1 ∫ ∞ 0 dx x y G1a(y,x)χAB (x), (11) χAB(y) = 1 π (A + 1 2A )3/2 [ y √ A + 2 2(A + 1) ]−1 × ∫ ∞ 0 dx x y [G1a(x,y)χAA(x) +AG2a(y,x)χAB (x)], (12) where G1a(y,x) ≡ log 2A(x2 + xy) + y2(A + 1) 2A(x2 − xy) + y2(A + 1) , (13) G2a(y,x) ≡ log (y2 + x2)(A + 1) + 2xy (y2 + x2)(A + 1) − 2xy . (14) We now proceed to solve these equations by using the Ansätze χAA(y) = cAA y−2+is and χAB(y) = cAB y−2+is , (15) where y once again denotes a (dimensionless) momentum. Inserting the functions (15) in the set of coupled equations and performing the scale transformation x = y z, in the integrand of Eqs. (11) and (12), one has the following set of equations: cAA = cAB 2 π √ 4A A + 2 × ∫ ∞ 0 dz z−2+1+is log 2A(z2 + z) + (A + 1) 2A(z2 − z) + (A + 1) , (16) cAB = 1 π (A + 1 2A )3/2√2(A + 1) A + 2 × ∫ ∞ 0 dz z−2+1+is [ cAA log 2A(1 + z) + z2(A + 1) 2A(1 − z) + z2(A + 1) +A cAB log (1 + z2)(A + 1) + 2z (1 + z2)(A + 1) − 2z ] . (17) Inserting Eq. (16) into (17), the set of coupled equations can be written as a single transcendental equation 1 π (A + 1 2A )3/2√2(A + 1) A + 2 × ( AI1(s) + 2 π √ 4A A + 2 I2(s)I3(s) ) = 1, (18) where we have defined I1(s) = ∫ ∞ 0 dz z−1+is log [ (z2 + 1)(A + 1) + 2z (z2 + 1)(A + 1) − 2z ] = 2π s sinh ( θ1s − π 2 s ) cosh ( π 2 s ) , (19) I2(s) = ∫ ∞ 0 dz z−1+is log [ 2A(z2 + z) + A + 1 2A(z2 − z) + A + 1 ] = 2π s sinh ( θ2s − π 2 s ) cosh ( π 2 s ) (A + 1 2A )is/2 , (20) I3(s) = ∫ ∞ 0 dz z−1+is log [ 2A(1 + z) + (A + 1)z2 2A(1 − z) + (A + 1)z2 ] = 2π s sinh ( θ2s − π 2 s ) cosh ( π 2 s ) (A + 1 2A )−is/2 . (21) The angles are given by the equations tan2 θ1 = A(A + 2) and tan2 θ2 = (A + 2)/A with the conditions that π/2 < θ1,θ2 < π . For the special case of equal masses, i.e., A = 1, we have θ1 = θ2, I1 = I2 = I3, and( 1 π √ 4 3 I1(s) ) + 2 ( 1 π √ 4 3 I1(s) )2 − 1 = 0, (22) for which the physically relevant solution is seen to be 1 π √ 4 3 I1(s) = 1 2 . (23) Using Eq. (19), we recover the celebrated Efimov equation for the scaling parameter s of equal mass particles [16,30–32]. Another very interesting and relevant special case is when there is no interaction between the two A particles, in which case we can set cAA = 0 in Eq. (17). The equation for the scale factor [Eq. (18)] now simplifies and we get A π (A + 1 2A )3/2√2(A + 1) A + 2 I1(s) = 1. (24) This equation was first derived in Ref. [30] and later also discussed in Ref. [32]. The derivation of Eqs. (23) and (24) by using the asymptotic forms for the spectator functions reproduces the well-known results for the scaling parameter s. In Fig. 1, we plot the scaling factors exp(π/s) for the case when all three subsystems have resonant interaction, which is the expression in Eq. (18) valid for EAA = EAB = 0 (solid line), and when there is no interaction in the AA subsystem, which is the expression in Eq. (24) valid for EAB = 0 (dashed line). Our results are identical to those shown in Figs. 52 and 53 of Ref. [32]. What is important to notice is that for mA � mB (A 1), the scaling factors are very similar, and both are much smaller than the equal mass case where A = 1. We can therefore see that in the AAB system with heavy A and light B, we should expect many universal three-body bound states (s large or equivalently eπ/s small) irrespective of whether the heavy- heavy subsystem is weakly or strongly interacting. Recent experiments with mixtures of 6Li and 133Cs indicate that there could be a resonance of the 6Li-133Cs subsystem at a point where the scattering length in the 133Cs-133Cs system is close to zero, i.e., weak interaction in the AA subsystem [33,34]. 062702-3 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) FIG. 1. Scaling parameter s as a function of A = mB/mA for EAA = 0 and EAB = 0 (resonant interactions) (solid line) and for the situation where EAB = 0 but with no interaction between AA (dashed line). The arrows show the corresponding mass ratios for 133Cs-133Cs-6Li and 87Rb-87Rb-6Li. Returning to Eqs. (11) and (12), there are two solutions which are complex conjugates of each other, i.e., z±is . Apart from an overall normalization, there is still a relative phase between these two independent solutions. We determine this phase by requiring that the wave function be zero at a certain momentum denoted q∗. This parameter is known as the three-body parameter [31,32]. This is the momentum- space equivalent of the coordinate-space three-body parameter which is now believed to be simply related to the van der Waals two-body interaction of the atoms in question [35–43]. In this case, the asymptotic form of the spectator functions becomes χAA(q) = cAA q−2 sin(s log q/q∗) and χAB(q) = cAB q−2 sin(s log q/q∗). (25) Here, we use q to denote momentum and we see that our boundary condition χ (q∗) = 0 is fulfilled. The asymptotic form of the spectator function should be compared with the solutions of the subtracted equations in the limit of large momentum, constrained by the window κ0 qB μ, where κ0 ≡ √ E3. The spectator functions χAA(q) for Rb-Rb-Li and Cs-Cs-Li compared to the respective asymptotic formula are shown in Fig. 2. In the idealized limit where κ0 = 0 and μ → ∞, the two curves would coincide. We can thus see the effect of finite value of these two quantities on each end of the plots. The window of validity for the use of the asymptotic formulas, i.e., √ E3 q μ, can be clearly seen in these figures. IV. ASYMPTOTIC MOMENTUM DENSITY In this section, we discuss the asymptotic momentum den- sity for n(qB), i.e., the single-particle momentum distribution for the B particle. From Eqs. (8) and (10), we can split the momentum density into nine terms, which can be reduced to four considering the symmetry between the two identical particles A. This simplifies the computation of the momentum density to the form n(qB) = 4∑ i=1 ni(qB), (26) where n1(qB) = |χAA(qB)|2 ∫ d3pB 1( E3 + p2 B + q2 B A+2 4A )2 = π2 |χAA(qB)|2√ E3 + q2 B A+2 4A , (27) FIG. 2. Left: χAA(q) of the sixth excited state for EAA = EAB = 0, E3 = −8.6724 × 10−12E0, solution of the coupled equations (1) and (2) (solid line), compared with the asymptotic formula (15) for Rb-Rb-Li molecule (dotted line). Right: Same as left side for the eighth excited state of Cs-Cs-Li, E3 = −8.9265 × 10−13E0. Here, we have defined E0 = h̄2μ2/mA and we work in units where h̄ = mA = μ = 1 as explained in the text. 062702-4 SINGLE-PARTICLE MOMENTUM DISTRIBUTIONS OF . . . PHYSICAL REVIEW A 87, 062702 (2013) FIG. 3. C/κ0 for mass ratios ranging from A = 6 133 to 25. These results should be multiplied by the factor 3(2π )3 in order to be compared with Ref. [12]. n2(qB) = 2 ∫ d3pB ∣∣χAB (∣∣ �pB − �qB 2 ∣∣)∣∣2 ( E3 + p2 B + q2 B A+2 4A )2 = 2 ∫ d3qA |χAB(qA)|2( E3 + q2 A + �qA · �qB + q2 B A+1 2A )2 , (28) n3(qB) = 2 χ∗ AA(qB) ∫ d3pB χAB (∣∣ �pB − �qB 2 ∣∣)( E3 + p2 B + q2 B A+2 4A )2 + c.c., (29) n4(qB) = ∫ d3pB χ∗ AB (∣∣ �pB − �qB 2 ∣∣)χAB (∣∣ �pB + �qB 2 ∣∣)( E3 + p2 B + q2 B A+2 4A )2 + c.c. (30) The leading-order term C q4 B comes only from n2 and the constant C is simply given by C = 8A2 (A+1)2 ∫ d3qA |χAB(qA)|2. This formula gives C/κ0 = 0.0274 for 133Cs-133Cs-6Li and C/κ0 = 0.0211 for 87Rb-87Rb-6Li. For A = 1, we obtain 3(2π )3C/κ0 = 53.197, to be compared with the “exact” value 53.097 obtained in Ref. [12]. The factor 3(2π )3comes from the difference in choice of normalization. In Fig. 3, we plot the value of C/κ0 for mass ratios ranging from A = 6 133 to 25. The increase is very rapid until A ∼ 5 beyond which an almost constant value is reached. Note that we have used the second excited state to perform these calculations for C. This can explain the small discrepancy between the numerical and “exact” results, which was calculated for an arbitrary high excited state. A. Analysis of subleading terms In Ref. [12] it was shown that the nonoscillatory term of order q−5 B coming from n1 to n4 cancels for A = 1 (equal masses). Here, we will demonstrate that this conclusion does not hold for general A �= 1. Following, we will consider the E3 → 0 limit, i.e., the three-body energy is assumed to be negligible, since we are interested in the imprint of Efimov states on the momentum distribution for excited Efimov states that are very extended and do not feel any short-range effects (as encoded in the three-body parameter q∗ discussed above). (i) n1: Upon inserting (25) in the momentum distributions, we get for n1(qB) the following asymptotic expression: n1(qB) → 2π2 √ A A + 2 |χAA(qB)|2 qB → 2π2 |cAA|2 √ A A + 2 |sin(s log qB/q∗)|2 q5 B . (31) Averaging out the oscillating part yields 1 2 and we have 〈n1(qB)〉 = π2 q5 B |cAA|2 √ A A+2 . (ii) n2: For large qB , this term becomes n2(qB) = 2 ∫ d3qA |χAB(qA)|2( q2 A + �qA · �qB + q2 B A+1 2A )2 = 8A2 q4 B (A + 1)2 ∫ d3qA |χAB(qA)|2 + ∫ d3qA|χAB(qA)|2 [ 2( q2 A + �qA · �qB + q2 B A+1 2A )2 − 8A2 (A + 1)2 1 q4 B ] , (32) where we retain a subleading part since it is of the same order as the leading order of the other terms. This integral can be solved analytically (see the Appendix) and the final expression is 〈n2(qB)〉 = −8π2 |cAB |2 q5 B A3(A + 3) (A + 1)3 √ A(A + 2) . (33) The special case A = 1 yields 〈n2(qB)〉 = −4π2 |cAB |2 /( √ 3q5 B). (iii) n3: The term for n3 is considerably more complicated since it involves an angular integral. The details can be found in the Appendix and the final result is 〈n3(qB)〉 = 4π2cAA cAB q5 B cosh ( sπ 2 ){√ A A + 2 cos ( s log √ A + 1 2A ) × cosh [ s ( π 2 − θ3 )] + sin ( s log √ A + 1 2A ) × sinh [ s ( π 2 − θ3 )]} , (34) where tan θ3 = √ A+2 A for 0 � θ3 � π/2. The special case A = 1 yields θ3 = π/3 and 〈n3(qB)〉 = 4π2|cAA|2 cosh( sπ 6 )/[q5 B √ 3 cosh( sπ 2 )]. (iv) n4: The n4 term is also complicated by angular integrals and again we refer to the Appendix for details. The result is 〈n4(qB)〉 = 8π2|cAB |2A2 s q5 B cosh ( sπ 2 ){ sinh [ s (π 2 − θ4 )] − s A√ A(A + 2)(A + 1) cosh [ s (π 2 − θ4 )] } , (35) 062702-5 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) 〉 〈 〉 〈 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 FIG. 4. (Left) Nonoscillatory contributions for n1 + n2 + n3, and n4 as function of the mass ratio A. Their sum, shown in the inset on the left, cancels exactly for A = 0.2, 1, and 1.57. (Right) The individual contributions n1, n2, n3, and n4 as function of A. where tan θ4 = √ A(A + 2) for 0 � θ4 � π/2. The spe- cial case A = 1 yields θ4 = π/3 and 〈n4(qB)〉 = 8π2|cAA|2[sinh( sπ 6 ) − s/(2 √ 3) cosh( sπ 6 )]/[s q5 B cosh( sπ 2 )]. It is important to note here that the asymptotic forms for the spectator functions have been used in the integrals where the integration is being performed from 0 to ∞. This may a priori cause problems for small momenta. However, a numerical check shows that the different behavior of the spectator functions at low momenta contributes only in an order higher than q−5 B to the integrals. This procedure is the same as that used in Ref. [12]. We now have analytic expressions for all the four terms in Eq. (26). The ratio between the coefficients cAA and cAB is given by Eq. (16), which can be used to eliminate one of these normalization factors. The other one can be determined from the overall normalization of the wave function, which we will not be concerned with here and we will merely set cAB = 1 from now on. In Fig. 4, we plot the contribution −(n1 + n2 + n3) and n4 as a function of mass ratio A on the left-hand side and the individual contributions of n1, n2, n3, and n4 separately on the right-hand side. What is immediately seen is that for A = 1 we reproduce the result of Ref. [12], i.e., that the q−5 B nonoscillatory term cancels. However, for general A this is not the case and one should expect also a q−5 B term in the asymptotic momentum distribution for systems with two identical and a third particle when three-body bound states are present. This is the main result of our paper and it demonstrates that nonequal masses will generally influence not only the value of the contact parameter attributed to three-body bound states but also the functional form of the asymptotic momentum tail. Curiously, there is an oscillatory behavior around A ∼ 1 of the sum of all contributions. This is shown in the inset of Fig. 4 where we see zero crossings at A = 0.2, 1, and 1.57. It seems quite clear that the oscillatory terms that all depend on the scale factor s are to blame for this interesting behavior, but we have not found an easy analytic explanation for it. What makes this interesting is the fact that if we take ratios of typical isotopes of alkali atoms such as Li, Na, K, Rb, and Cs, one can get rather close to 0.2 or 1.57. For instance, taking one 133Cs and two 85Rb yieldsA = 1.565, while one 7Li atom and two 39K atoms yields A = 0.179. These interesting ratios are thus close to experimentally accessible species. FIG. 5. Square root of the ratio of the (N + 1)th three-body state (measured from threshold εAB ) to the N th state plotted as function of the square root of the ratio of EAB and the energy of the N th three- body bound state. Note the factor sgn(k) which indicates whether the two-body system has a bound (k = +1) or virtual (k = −1) state. This sign is consistent with the convention introduced in Eqs. (3) and (4). The limit cycle, which should be in principle reached for N → ∞, is achieved very fast so that the curve has been constructed using N = 2. EAB is the Cs-Li or Rb-Li two-body energy (Cs-Cs and Rb-Rb two-body energies are zero). The negative and positive parts refer, respectively, to virtual and bound AB states, such that εAB = 0 and εAB ≡ EAB , respectively, on the negative and positive sides. The circles labeled from 1 to 6 mark the points where the momentum distributions have been calculated. 062702-6 SINGLE-PARTICLE MOMENTUM DISTRIBUTIONS OF . . . PHYSICAL REVIEW A 87, 062702 (2013) [ ] FIG. 6. Left: Momentum distribution for the second excited state as a function of the relative momentum of one 133Cs (qA) or 6Li (qB ) to the center of mass of the remaining pair 133Cs-6Li or 133Cs-133Cs. The solid, dashed, and dotted lines were calculated for the two- and three-body energies satisfying the ratios indicated by the points 1 to 3 in Fig. 5. The circles show the set of curves related to qA or qB . Right: Same curves as on the left side multiplied by q4, which shows explicitly the leading decay 1/q4. V. NUMERICAL EXAMPLES We now provide some numerical examples of momentum distributions for the experimentally interesting systems with large mass ratios. We will focus on 133Cs-133Cs-6Li and 87Rb- 87Rb-6Li. Here, we will investigate two extreme possibilities: (i) the heavy-heavy subsystems, i.e., 133Cs-133Cs and 87Rb- 87Rb, have a two-body bound state at zero energy and (ii) the opposite limit where they do not interact. In the first case, the heavy atoms are at a Feshbach resonance with infinite scattering length, while in the second case they are far from resonance and we assume a negligible background scattering length. As was recently demonstrated for the 133Cs- 6Li mixture, there are Feshbach resonances in the Li-Cs subsystem at positions where the Cs-Cs scattering length is nonresonant [33,34]. While this does not automatically imply that the Cs-Cs channel can be neglected, we will make the assumption (ii) here. The formalism can be modified in a straightforward manner to also include interaction in the heavy-heavy subsystem. As before, we denote the system AAB, where A refers to the identical (bosonic) atoms 133Cs or 87Rb, and B to 6Li. By solving Eq. (18), one finds s( 6 133 ) = 2.00588 and s( 6 87 ) = 1.68334 when assuming that all three subsystems have large scattering lengths (solid line in Fig. 1). The situation where the interaction between the two identical particles is turned off is shown by the dashed line in Fig. 1. In this case, s(A) was calculated from Eq. (17) by setting cAA = 0. This yields s( 6 133 ) = 1.98572 and s(6/87) = 1.63454. [ [ FIG. 7. Momentum distribution for the second excited state as a function of the relative momentum of one 87Rb (qA) or 6Li (qB ) to the center of mass of the remaining pair 87Rb-6Li or 87Rb-87Rb. The solid, dashed, and dotted lines were calculated for the two- and three-body energies satisfying the ratios indicated by the points 4 to 6 in Fig. 5. The circles show the set of curves related to qA or qB . Right: same curves of the left side multiplied by q4, which shows explicitly the leading decay 1/q4. 062702-7 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) FIG. 8. Rescaled momentum distribution for the ground, first, and second excited states as a function of the relative momenta of 133Cs to the center of mass of the pair 6Li-133Cs (qA) and 6Li to the center of mass of the pair 133Cs-133Cs (qB ). The subsystem binding energies are all set to zero. Normalization to unity at zero momentum. We first consider the binding energies. Assuming that the Cs-Cs and Rb-Rb two-body energies are zero, we have, for a system satisfying the universality condition a � r0, that any observable should be a function of the remaining two- and three-body scales, which can be conveniently chosen as E (N) 3 and EAB (the Cs-Li or Rb-Li two-body energy). Here, N denotes the N th consecutive three-body bound state with N = 0 being the lowest one. Thus, the energy of an N + 1 state can be plotted in terms of a scaling function relating only EAB and the previous state. The limit cycle, which should be in principle reached for N → ∞, is achieved rapidly so FIG. 9. Rescaled momentum distribution for the ground, first, second, and third excited states as a function of the relative momenta of 87Rb to the center of mass of the pair 87Rb-6Li (qA) and 6Li to the center of mass of the pair 87Rb-87Rb (qB ). The subsystem binding energies are all set to zero. Normalization to unity at zero momentum. that we can construct the curve shown in Fig. 5 using N = 2 [21,22,44]. The negative and positive parts of the horizontal axis refer, respectively, to virtual and bound two-body AB states. The circles labeled from 1 to 6 mark the points where the momentum distributions have been calculated. The points 1 and 4 represent the Borromean case, the points 2 and 5 are the “Efimov situation,” and in points 3 and 6 AB is bound. Figures 6 and 7 give the momentum distributions of the second excited states for the energy ratio √ EAB/E3 given by the points labeled from 1 to 6 in Fig. 5. According to our previous calculations [45], for fixed three-body energy, the size of the system increases as the number of bound two-body subsystems increase. Thus, it seems reasonable that the momentum distribution for the Borromean case (point 1) decreases slower. This behavior is clearly seen on the left side of Figs. 6 and 7. The distance of one atom to the center of mass of the other two is much larger for 6Li than for 133Cs or 87Rb, due to the large difference of the masses, such that the decrease of the momentum distribution for the heavier atom (qA set) decreases much slower than that for the lighter one (qB set). This also reflects on the momentum from which the leading-order decay 1/q4 starts to be dominant. This difference becomes evident on the right side of Figs. 6 and 7, where we plotted q4n(q). Thus, the q4 term is dominant above (20–40)κ0 for qb and much slower for qA at about (60–100)κ0. Figures 8 and 9 show the rescaled momentum distributions for the ground, first, and second excited states. In these figures, the subsystem energies were chosen to zero, corresponding to the transition point to a Borromean configuration. In this situation, the only low-energy scale is E3 (remember that the high-momentum scale is μ = 1). Therefore, in units in which μ = 1, to achieve a universal regime, in principle, to wash out the effect of the subtraction scale μ, we have to go to a highly excited state (see, for instance, Fig. 2 and the comments inside the text associated to it). However, a universal low- energy regime of n(qB)/n(qB = 0) is seen for momentum of the order of √ E3, even for the ground state which is smaller than excited states. Thus, in practice, the universal behavior of the momentum distribution is approached rapidly. VI. CONCLUSIONS AND OUTLOOK In this work, we have calculated the single-particle mo- mentum distribution of systems consisting of two identical bosonic particles and a third particle of a different kind with short-range interaction in the regime where three-body bound states and the Efimov effect occurs. We analytically calculate the asymptotic momentum distribution as a function of the mass ratio and find that the functional form is sensitive to this ratio. In the case of equal mass, we reproduce the results of Ref. [12], i.e., that the leading term has a q−4 tail while the subleading contribution is q−5 times a log-periodic oscillatory function that is characteristic of the Efimov effect and that depends on the scale factor (and thus on the mass ratio) of the Efimov states. In particular, we find that for general mass ratios, there is a nonoscillatory q−5 contribution which appears to only vanish (and leave the oscillatory contribution behind) when the mass ratio is 0.2, 1, or 1.57. To exemplify our study, we consider 133Cs-133Cs-6Li and 87Rb-87Rb-6Li where we numerically determine the coefficient 062702-8 SINGLE-PARTICLE MOMENTUM DISTRIBUTIONS OF . . . PHYSICAL REVIEW A 87, 062702 (2013) of the q−4 tail which is the two-body contact parameter introduced by Tan [2]. For these examples, we also nu- merically determine the momentum distributions of excited Efimov trimers for both the heavy and light components. Our numerical results demonstrate that the momentum distri- butions of ground, first, and second excited Efimov trimers approach universal behavior very fast at large but also at small momentum, indicating that one does not need to go to highly excited (and numerically challenging) three-body states in order to study the universal behavior of Efimov states in momentum space. Recent experiments have successfully measured the momentum distribution of ultracold atomic gases using time of flight and mapping to momentum space [7] and Bragg spectroscopy [7,8,15]. Observing a constant 1/k5 contribution in a system with nonequal mass three-body states is a considerable challenge since one needs to first subtract the leading-order 1/k4 contribution. The 1/k4 tail can be extracted with good precision as discussed in the experimental papers [7,8,15]. If we assume that this subtraction of leading order can be done without severe increase of uncertainties in the data, then one would need to look at small and intermediate k values for this subleading tail behavior. A natural extension of this work is to consider three-body states in dimensions lower than three. In two dimensions, it is well known that no Efimov effect occurs [31,46–49] and, among other things, this implies that the momentum distribution does not have the subleading oscillatory behavior in two dimensions [50]. For two-dimensional systems with large differences in the masses, it is still possible to have many three-body bound states [51] and this should also be reflected in some way through the asymptotic momentum distribution. Another intriguing question is how the universal tail behavior and the contact relations behave in a crossover between two- and three-dimensional or one- and three-dimensional setups [29,50,52]. While we have studied only short-range interactions in this paper, it would be interesting to consider the momentum tails of few-body bound states in systems with long-range interactions. Recent experiments with heteronuclear molecules have demonstrated that the momentum distribution in dipolar systems can be probed using absorption imaging [53]. Few- body bound states of dipolar particles have been predicted in a large parameter regime for both one- [54,55] and two-dimensional systems [56–59]. As was recently shown, one-dimensional dipolar few-body systems can in some cases be described by using zero-range interaction terms with appropriately chosen effective interactions parameters [60]. This opens up the possibility of using the same formalism with short-range interactions as discussed in this paper but applied in a one-dimensional setup. It should then be possible to derive the contact parameters in the presence of few-body bound states with dipolar particles in one dimension, similarly to what has been done for nondipolar bosons [61] and fermions [62]. ACKNOWLEDGMENTS We thank R. Heck, J. Ulmanis, and R. Peres from the group of M. Weidemüller in Heidelberg for updates on the experimental progress of 6Li-133Cs mixed systems, and F. Werner for discussions on the work presented in Ref. [12]. M.T.Y. and T.F. thank the hospitality of the Department of Physics and Astronomy of the Aarhus University, where part of this work was done, and to the Brazilian agencies CNPq and FAPESP. This work was supported in part by a grant from the Danish Ministry of Science, Innovation, and Higher Education under the International Network program. APPENDIX: DERIVATION OF THE SUBLEADING TERMS 1. n2 term In Eq. (32), we have a subleading term of the form ∫ d3qA |χAB(qA)|2 [ 2( q2 A + �qA · �qB + q2 B A+1 2A )2 − 8A2 (A + 1)2 1 q4 B ] = |cAB |2 ∫ d3qA q4 A [ 1( q2 A + �qA · �qB + q2 B A+1 2A )2 − 4A2 (A + 1)2 1 q4 B ] = 2π |cAB |2 q5 B ∫ dx x2 [ 1 x4 + 1 Ax2 + (A+1 2A )2 − 1(A+1 2A )2 ] , (A1) where in the first equality we have inserted the asymptotic form of |χAB(qA)|2 = |cAB |3q−4 A /2 obtained after averaging over the oscillatory term in Eq. (25). In the second equality, we have performed the angular integral and introduced the variable qA = qBx. We have also used the fact that the integrand is even to extend the integration to the entire real axis. The function under the integral f (x) = 1 x2 [ 1 x4 + 1 Ax2 + (A+1 2A )2 − 1(A+1 2A )2 ] (A2) falls off faster than 1/x for |x| → ∞. We can therefore extend it to the complex domain and consider a contour in the upper- half plane (or lower-half) that includes the real axis and a semicircle of large radius in a counterclockwise orientation. To use the residue theorem, we need to first find the poles of f (x). Since f (x) is regular at x = 0, the only poles are out in the complex plane. The four poles are given by x1 = reiθ1/2, x2 = rei(π−θ1/2), (A3) x3 = rei(π+θ1/2), x4 = re−iθ1/2, 062702-9 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) where r = √ A+1 2A and tan2 θ1 = A(A + 2). If we use the convention that π/2 < θ1 < π as in the main text, then x1 and x2 are the poles in the upper-half plane. The sum of the two residues is Res(f,x1) + Res(f,x2) = − 1 ir3 A(A + 3) (A + 1)2 cos ( θ1 2 ) sin(θ1) . (A4) Using the residue theorem, the subleading term in Eq. (A1) then becomes∫ d3qA|χAB(qA)|2 [ 2( q2 A + �qA · �qB + q2 B A+1 2A )2 − 8A2 (A + 1)2 1 q4 B ] = − 4π2|cAB |2 q5 B2 sin ( θ1 2 ) A(A + 3) (A + 1)2 ( 2A A + 1 )3/2 . (A5) From the definition of θ1 we see that cosθ1 = − 1 A+1 and [2 sin( θ1 2 )]−1 = √ A+1 2(A+2) . The subleading term in n2 is given by 〈n2(qB)〉 = −8π2|cAB |2 q5 B A3(A + 3) (A + 1)3 √ A(A + 2) , (A6) where the special case A = 1 yields θ1 = 2π/3 and 〈n2〉 = −4π2|cAB |2/( √ 3q5 B). 2. n3 term Neglecting the three-body energy and making the variable transformation �qA = �pB − �qB 2 in Eq. (29), we find n3(qB) = 2χ∗ AA(qB) ∫ d3qA χAB(qA)( q2 A + �qA · �qB + q2 B A+1 2A )2 + c.c. (A7) Defining �qA = qB �y, integrating over the solid angle, and replacing the asymptotic form for the spectator functions χAA and χAB , given by Eq. (25), we get n3(qB) = 8π c∗ AA cAB q5 B sin2(s log qB/q∗) × ∫ ∞ 0 cos(s log y) dy y4 + 1 Ay2 + (A+1 2A )2 + 8π c∗ AA cAB q5 B sin(s log qB/q∗) cos(s log qB/q∗) × ∫ ∞ 0 sin(s log y) dy y4 + 1 Ay2 + (A+1 2A )2 + c.c. (A8) Averaging out the oscillatory terms, only the first term of Eq. (A8) gives a nonvanishing result 〈n3(qB)〉 = 4π c∗ AA cAB q5 B ∫ ∞ 0 cos(s log y) dy y4 + 1 Ay2 + (A+1 2A )2 + c.c. (A9) Expressing cosine in the complex exponential form we write that I = ∫ ∞ 0 cos(s log y) dy y4 + 1 Ay2 + (A+1 2A )2 = Re [ ∫ ∞ 0 yis dy y4 + 1 Ay2 + (A+1 2A )2 ] = Re I1, (A10) where Re denotes the real part. The residue theorem can be applied to solve the above integral. We set y = eα in order to extend the interval of integration from −∞ to ∞ and rewrite I1 as I1 = ∫ ∞ −∞ eα(1+is) (eα − eα1 )(eα − eα2 )(eα − eα3 )(eα − eα4 ) dα. (A11) The next steps are about extending the integrand f (α) to the complex plan, finding its poles, and evaluating the residues of the poles. All the roots in the denominator of f (α) are in the complex plane, out of the real axis, and are given by α1 = log r + iθ3, α2 = log r − i(π − θ3), (A12) α3 = log r − iθ3, α4 = log r + i(π − θ3), with r = √ A+1 2A and tan θ3 = √ A+2 A for 0 � θ3 � π/2. We extend f (α) to the complex plane and choose the closed path as a rectangle of vertices −R, +R, +R + iπ , and −R + iπ (for R → ∞), which encompasses the poles α1 and α4 in the upper-half plane. We are left with four integrals, namely, J1 which extends along the real axis from −R to +R, J2 from +R to +R + iπ , J3 from +R + iπ to −R + iπ , and J4 from −R + iπ to −R. In the limit R → ∞, we find that J1 = I1, J3 = e−sπ I1, and J3 and J4 → 0. In this way, we find that I1 = 2πi 1 + e−πs [Res(f,α1) + Res(f,α4)] = πA 1 + e−πs √ 2 (A + 2)(A + 1) × ( eis(log r+iθ3)−ıθ3 + eis[log r+i(π−θ3)]+iθ3 ) , (A13) where r and θ3 are defined after Eq. (A12). It is necessary to split the real Re and imaginary Im parts to achieve our goal. Manipulating the trigonometric and hyperbolic functions we get Re I1 = π 2 cosh ( sπ 2 ){√ A A + 2 cos(s log r) cosh [ s ( π 2 − θ3 )] + sin(s log r) sinh [ s ( π 2 − θ3 )]} , (A14) Im I1 = −π 2 cosh ( sπ 2 ){√ A A + 2 sin(s log r) cosh [ s ( π 2 − θ3 )] + cos(s log r) sinh [ s ( π 2 − θ3 )]} . (A15) 062702-10 SINGLE-PARTICLE MOMENTUM DISTRIBUTIONS OF . . . PHYSICAL REVIEW A 87, 062702 (2013) Finally, from Eqs. (A9), (A10), and (A14), the nonoscillating part of n3(qB) is given by 〈n3(qB)〉 = 4π2cAA cAB q5 B cosh ( sπ 2 ){√ A A + 2 cos ( s log √ A + 1 2A ) cosh [ s ( π 2 − θ3 )] + sin ( s log √ A + 1 2A ) sinh [ s ( π 2 − θ3 )]} , (A16) where tan θ3 = √ A+2 A for 0 � θ3 � π/2. The special case A = 1 yields θ3 = π/3 and 〈n3(qB)〉 = 4π2|cAA|2 cosh( sπ 6 )/ [q5 B √ 3 cosh( sπ 2 )]. 3. n4 term Although Eqs. (29) and (30) are similar, it is not possible to extend the results in the preceding section to obtain the nonoscillating term of n4(qB). Defining �pB = �qB 2 �y and dropping the three-body energy, Eq. (30) becomes n4(qB) = 4π qB ∫ ∞ 0 y2dy( y2 + A+2 A )2 ∫ +1 −1 dx χ∗ AB(qBx−)χAB(qBx+) + c.c., (A17) where x± = 1 2 √ 1 + y2 ± 2yx. Replacing the spectator function by its asymptotic form (25) in the integral above, we are left with three terms, which read as n4(qB) = 8π |cAB |2 sin2 ( s log qB q∗ ) q5 B ∫ ∞ 0 y2dy( y2 + A+2 A )2 ∫ +1 −1 dx x2+x2− cos(s log x+) cos(s log x−) + 8π |cAB |2 cos2 ( s log qB q∗ ) q5 B ∫ ∞ 0 y2dy( y2 + A+2 A )2 ∫ +1 −1 dx x2+x2− sin(s log x+) sin(s log x−) + 4π |cAB |2 sin ( s log qB q∗ ) cos ( s log qB q∗ ) q5 B ∫ ∞ 0 y2dy( y2 + A+2 A )2 ∫ +1 −1 dx x2+x2− sin[s log(x+x−)]. (A18) As it was done for n3(qB), averaging out the oscillatory term, only the two first terms on the right-hand side of Eq. (A18) give a nonvanishing contribution. The angular integration is performed using that∫ dx ( β + x β − x )±is/2 (β2 − x2)−1 = ± ( β + x β − x )±is/2 (iβs)−1 (A19) and the nonoscillating part of n4(qB) is given by 〈n4(qB)〉 = 32π |cAB |2 is q5 B ∫ ∞ 0 y dy( y2 + A+2 A )2 (1 + y2) [( y + 1 |y − 1| )is − ( y + 1 |y − 1| )−is] . (A20) As was pointed out in [12], the absolute value complicates the calculation of this integral. Circumventing this problem, we follow the same trick as in [12], where the integral is split in two pieces: y ∈ [0,1] and y ∈ [1,∞[ and a new variable is introduced in each piece. We set y = x−1 x+1 in the first piece and y = x+1 x−1 in the second piece. Notice that in both cases x ∈ [1,∞[. Now, we are able to apply the residue theorem to calculate the nonoscillating part of n4(qB). First of all, we introduce a new variable α, such that x = eα and α ∈ [0,∞[. As the resulting integrand is even, it allows us to extend the domain of integration to the entire real axis, i.e., α ∈ ] − ∞,∞[. The integral in Eq. (A20) reads as I = i A2 (A + 1)4 Im [ ∫ ∞ −∞ eα(1+is)(e2α − 1)[(A + 1)2(e6α + 1)/2 + (3A2 − 2A − 1)(e2α + e4α)/2] (eα − eα5 )(eα − eα6 )[(eα − eα1 )(eα − eα2 )(eα − eα3 )(eα − eα4 )]2 dα ] (A21) = i A2 (A + 1)4 Im I1, (A22) where Im denotes the imaginary value. Extending the integrand f (α) to the complex plane, we find that all the roots in its denominator are on the imaginary axis and given by α1 = iθ4, α2 = i(π − θ4), α3 = −iθ4, α4 = −i(π − θ4), α5 = i π 2 , α6 = −i π 2 , (A23) 062702-11 M. T. YAMASHITA et al. PHYSICAL REVIEW A 87, 062702 (2013) where tan θ4 = √ A(A + 2) for 0 � θ4 � π 2 . Notice that α5 and α6 are simple poles while α1, α2, α3, and α4 are poles of second order [see Eq. (A22)]. To evaluate the contour integral, we choose the closed path as in the calculation of n3(qB), namely, a rectangle of vertices −R, +R, +R + iπ , and −R + iπ (for R → ∞), which encompasses the poles α1, α2, and α5 in the upper-half plane. Once more, we are left with four integrals, i.e., J1 which extends along the real axis from −R to +R, J2 from +R to +R + iπ , J3 from +R + iπ to −R + iπ , and J4 from −R + iπ to −R. In the limit R → ∞ we find that J1 = I1, J3 = e−sπ I1, and J3 and J4 → 0. In this way, we find that I1 = 2πi 1 + e−πs [Res(f,α1) + Res(f,α2) + Res(f,α5)]. (A24) Calculating the residues is tedious, except for the case of α5 where Res(f,α5) = 0. After some algebraic work, the real and imaginary parts of I1 are given by Re I1 = π (A + 1)3A 4 √ A(A + 2) cosh ( sπ 2 ) cosh [ s ( π 2 − θ4 )] , (A25) Im I1 = π (A + 1)4 4 √ A(A + 2) cosh ( sπ 2 ) × {√ A(A + 2) sinh [ s (π 2 − θ4 )] − s A A + 1 cosh [ s (π 2 − θ4 )] } . (A26) Combining Eqs. (A20), (A22), and (A26), the nonoscillat- ing part of n4(qB) finally reads as 〈n4(qB)〉 = 8π2|cAB |2A2 s q5 B cosh ( sπ 2 ){ sinh [ s (π 2 − θ4 )] − s A√ A(A + 2)(A + 1) cosh [ s (π 2 − θ4 )] } , (A27) where tan θ4 = √ A(A + 2) for 0 � θ4 � π/2. 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