Universal Relations for Ultracold Reactive Molecules

Universal relations for ultracold reactive molecules

1, 2, 3, 1, 4 1, 5, Mingyuan He, ∗ Chenwei Lv, ∗ Hai-Qing Lin, and Qi Zhou † 1Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA 2Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, 518000, China 3Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 4Beijing Computational Science Research Center, Beijing, 100193, China 5Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, IN, 47907, USA (Dated: May 19, 2020) The realization of ultracold polar molecules in laboratories has pushed both physics and chemistry to new realms [1–8]. In particular, these polar molecules oﬀer scientists unprecedented opportuni- ties to explore chemical reactions in the ultracold regime where quantum eﬀects become profound [9–13]. However, a key question about how two-body losses depend on quantum correlations in an interacting many-body system remains open so far. Here, we present a number of universal relations that directly connect two-body losses to other physical observables, including the momentum distribution and density correlation functions. These relations, which are valid for arbitrary microscopic parameters, such as the particle number, the temperature, and the interaction strength, unfold the critical role of contacts, a fundamental quantity of dilute quantum systems [14] in determining the reaction rate of quantum reactive molecules in a many-body environment. Our work opens the door to an unexplored area intertwining quantum chemistry, atomic, molecular and optical physics, and condensed matter physics.

In a temperature regime down to a few tens of nano- tum systems [14], to establish universal relations between Kelvin, highly controllable polar molecules provide scien- the two-body loss rate and other quantities including the tists with a powerful apparatus to study a vast range of momentum distribution and the density correlation func- new quantum phenomena in condensed matter physics, tion. Previously, two-body losses of zero-range potentials quantum information processing and quantum chemistry, hosting inelastic s-wave scatterings were correlated to the such as exotic quantum phases [15–18], quantum gates s-wave contact [23, 24]. In reality, chemical reactions with fast switching times [19, 20], and quantum chemical happen in a ﬁnite range. Many systems are also char- reactions [9–13]. In all these studies, the two-body loss acterized by high-partial wave scatterings. For instance, is an essential ingredient leading to non-hermitian phe- single-component fermionic KRb molecules interact with nomena. Similar to other chemical reactions, collisions p-wave scatterings [1, 12]. It is thus required to formulate between molecules may yield certain products and re- a theory applicable to generic short-range reactive inter- lease energies, which allow particles to escape the traps. actions. To concretize discussions, we focus on single- For instance, a prototypical reaction, KRb+KRb component fermionic molecules. All our results can be → K2 + Rb2, is the major source causing the loss of KRb straightforwardly generalized to other systems with arbi- molecules. Undetectable complexes may also form, re- trary short-range interactions. sulting in losses in the system of interest [10, 11]. The Hamiltonian of N reactive molecules is written as Whereas chemical reactions are known for their com- ¯h2 plexities, taking into account quantum eﬀects imposes H = [ 2 + V (r )] + U(r r ), (1) −2M ∇i ext i i − j an even bigger challenge to both physicists and chemists. i i>j The exponentially large degrees of freedom and quan- X X

tum correlations built upon interactions make it difficult where M is the molecular mass, Vext(r) is the external to quantitatively analyze the reactions. A standard ap- potential, U(r) is a two-body interaction, as shown in proach is to consider two interacting particles, the re- Fig. 1. The many-body wavefucntion, Ψ(r1, r2, ..., rN ), action rate of which is trackable [21, 22]. Though such satisfies the time-dependent Schrödingerequation, results are applicable in many-body systems when the arXiv:2005.07861v1 [cond-mat.quant-gas] 16 May 2020 temperature is high enough and correlations between dif- i¯h∂tΨ(r1, r2, ..., rN ) = HΨ(r1, r2, ..., rN ). (2) ferent pairs of particles are negligible, with decreasing the temperature, many-body correlations become profound In the absence of electric fields, U(r) is a short-range and this approach fails. A theory fully incorporating interaction with a characteristic length scale, r0. When quantum many-body effects is desired to understand the r > r , U(r) = 0. Chemical reactions happen in an even | | 0 chemical reaction rate. shorter length scale, r∗ < r0. We adopt the one-channel In this work, we show that universality exists in chem- model using a complex U(r) = UR(r)+iUI (r) to describe ical reactions of ultracold reactive molecules. We im- the chemical reaction [22], where U (r) 0. When r > I ≤ | | plement contacts, the central quantity in dilute quan- r∗, UI (r) = 0. Using the Lindblad equation that models 2

3 2 q cot[η(q)] = 1/vp + q /re, where vp and re are the p- wave scattering− volume and effective range, respectively, −1 both of which are complex for reactive interactions. Con- | r |≪ k U(r) F (0) 2 (1) 4 sequently, ϕ ( r ; ) = ϕm ( r )+q ϕm ( r )+O(q ), m | ij| | ij| | ij| r∗ where r | r | 1 0 r < r k− (0) 0 | ij | F 1 1 rij k −1 ϕm ( rij ) | |, (4) F | | −−−−−−−−→ r 2 − v 3 | ij| p 1 r < r k− 3 (1) 0 | ij | F 1 rij 1 rij 1 ϕm ( rij ) | | + | | + . (5) FIG. 1. A length scale separation in dilute molecules. | | −−−−−−−−→ re 3 vp 30 2 The blue (red) solid spheres represent potassium (rubidium) To simplify expressions, we have considered isotropic p- atoms. Inside the dashed circle are two molecules, the sepa- wave interactions, ϕ( r ) = ϕ ( r ) and G(R ; E ration between which is much smaller than the average inter- | ij| m | ij| ij − 1 ) = Gm(Rij; E ), and suppressed other partial waves particle spacing, r kF− . The enlarged plot of the regime − inside the dashed| circle| is a schematic of the chemical reac- in the expressions, which do not show up in universal rela- tion. The green solid curve represents the real part of the tions relevant for single-component fermionic molecules. interaction, UR(r). The imaginary part of the interaction, Using Eqs. (1, 2, 3), we find that the decay of the total UI (r), is nonzero only in the shaded area, where the reaction particle number is captured by, happens. ¯h 3 ∂ N = κ C , (6) t −8π2M ν ν ν=1 the losses by jump operators, the same universal relations X can also be derived (Supplementary Materials). where the three contacts are written as Universal relations arise from a length scale separation 1 1 2 0 2 in dilute quantum systems, r∗ < r0 kF− , where kF− , C = 3(4π) N(N 1) dR g , (7) 1 − ij| | the inverse of the Fermi momentum, captures the average Z inter-particle separation. When the distance between two 2 0 1 1 C2 = 6(4π) N(N 1) dRijRe(g ∗g ), (8) molecules is much smaller than k− , we obtain, − F Z 2 0 1 1 C = 6(4π) N(N 1) dR Im(g ∗g ), (9) r k− 3 ij | ij | F − Ψ(r1, r2, ..., rN ) ψm(rij; )Gm(Rij; E ), Z −−−−−→ − s m, dRij = d[(ri + rj)/2]drk=i,j and g = X 6 (3) q2sG(R ; E ). As shown later, C deter- R ij −R 1 where rij = ri rj denotes the relative coordinates of mines the leading term in the large momentum tail, − P the ith and the jth molecules, Rij = (ri +rj)/2, rk=i,j similar to systems without losses [25–28]. In contrast, { 6 } is a short-hand notation including coordinates of their C2,3 are new quantities in systems with two-body losses. center of mass and all other particles. ψm(rij; ) is a κν in Eq. (6) are microscopic parameters determined p-wave wavefunction with a magnetic quantum number purely by the two-body physics. In our one-channel m = 0, 1, which is determined solely by the two-body model, their explicit expressions are given by ± 2 2 Hamiltonian, H2 = (¯h /M) + U(rij), as all other 2 − ∇ M ∞ (0) 2 particles are far away from the chosen pair in the regime, κ1 = 2 UI (r) ϕ (r) r dr, (10) 1 − ¯h 0 rij kF− . E is the total energy of the many-body Z | | 2 2 system. =h ¯ q /M, the colliding energy, is no longer M ∞ (0) (1) 2 κ2 = Re UI (r) ϕ ∗(r)ϕ (r)r dr , (11) a good quantum number in a many-body system, and − ¯h2 Z0 a sum shows up in Eq. (3). Since both a continuous M ∞ (0) (1) 2 spectrum and discrete bound states may exist, we use κ3 = 2 Im UI (r) ϕ ∗(r)ϕ (r)r dr , (12) ¯h 0 the notation of sum other than an integral. Z where r = r . If U(r) is modeled by two square well po- Though ψm(rij; ) depends on the details of U(rij) 1 | | when r < r , it is universal when r < r k− , as tentials, one for its real part and the other for its imagi- | ij| 0 0 | ij| F a result of vanishing interaction in this regime. We define nary part, κ1,2,3 can be evaluated explicitly. For simplic- ˜ ˜ ψ (r ; ) = ϕ ( r ; )Y (ˆr ), where Y (ˆr ) is the ity, we set U(r) = UR iUI when r r0 = r∗ and 0 m ij m | ij| 1m ij 1m ij − − | | ≤ p-wave spherical harmonics. Whereas many resonances elsewhere. Changing the ratio r0/r∗ does not change any exist, the phase shift of the scattering between KRb results qualitatively. Figure 2 shows how κ1,2,3 depend molecules is still a smooth function of the energy, due to on U˜I when U˜R is fixed at various values including those the large average line width of these resonances, which far corresponding to small and divergent vp in the absence exceeds the mean level spacing of the bound states [21]. of U˜I . When U˜I = 0, κ1,2,3 = 0. With increasing U˜I , The phase shift, η, then has a well defined expansion, κ1,2,3 change non-monotonically and all approach zero 3 when U˜I is large, indicating a vanishing reaction rate in that such spectroscopy has a universal tail, Γ(ω) 2 1/2 → the extremely large UI limit. [(Ωrf V )/(8π )]C1(¯hω/M)− , where ω is the rf fre- Equation (6) is universal for any particle number and quency, Ωrf is the rf Rabi frequency, and V is the vol- any short-range interactions with arbitrary interaction ume of the system. It is worth mentioning that, for atoms strengths, as well as any real external potential. It sep- with elastic p-wave interactions, Eq. (14) describes the arates Cν , which fully capture the many-body physics, leading term of the large momentum tail [25–28]. We 4 from two-body parameters, κ , which are independent have not found that the subleading term k − has ν ∼ | | on the particle number and the temperature. Therefore, connections to two-body losses. even when microscopic details of the reactive interaction, Another fundamentally important quantity in con- for instance, the exact expression of U(r), are unknown, densed matter physics is the density correlation function, κ can still be accessed in systems whose C are eas- S(r) = dR n(R + r/2)n(R r/2) , which measures ν ν h − i ily measurable (Supplementary Materials). Equation (6) the probability of having two particles separated by a R also holds for any many-body eigenstates and a thermal distance r. Using Eqs. (3, 4, 5), S(r) can be evalu- 1 average does not change its form. Therefore, Eq. (6) ated explicitly in the regime, r < r k− . To en- 0 | | F does apply for any finite temperatures, provided that the hance the signal-noise-ratio, S(r) can be integrated over reaction rate is slow compared to the time scale of estab- a shell with inner and outer radii, x and x + D, respec- lishing quasi-equilibrium in the many-body system, i.e., tively. Such an integrated density correlation is given by the many-body system has a well defined temperature at x+D P (x, D) = x drS(r), and any time. Under this situation, Cν should be understood as their thermal averages. ∂P (x, D)R 1 1 1 = 2 C1 2 + C2 Interestingly, we have found that κ1 and κ2 can be ∂D D 0 16π x 2 1 → n (15) rewritten as familiar parameters. In fact, κ1 = Im(vp− ) 1 1 1 x 2Re( )C Re( )C + Im( )C . and κ2 = Im[ 1/(2re)] (Supplementary materials). In 1 2 3 − − vp − re re 3 contrast, to our best knowledge, κ3 is a new parameter h i o that has not been addressed in previous works. Similar Again, other partial-waves have been suppressed in the to κ2, κ3 can be expressed as the difference between the expression as their contributions are given by different extrapolation of the two-body wavefunction in the regime spherical harmonics. Fitting ∂P (x, D)/∂D D 0 mea- r > r toward the origin and the realistic wavefunction sured in experiments using the power series in| → Eq. (15) | | 0 at short distance, r < r (Supplementary materials). allows one to obtain all three contacts, C , provided | | 0 1,2,3 Equation (6) can be rewritten as that vp and re are known. If these two parameters are unknown, it is necessary to include higher order terms in ¯h 1 1 1 the expansion (Supplementary Materials). ∂tN = Im(vp− )C1 Im(re− )C2 + κ3C3 . −8π2M − 2 We emphasize that, no matter whether thermody- h (13)i namic quantities and correlation functions can be com- For s-wave inelastic scatterings due to complex zero- puted accurately in theories, equations (6, 13, 14, 15) al- range interactions, the first term on the right hand side low experimentalists to explore how contacts determine of Eq. (13) was previously derived, with v 1 replaced by p− chemical reactions in interacting few-body and many- the complex s-wave scattering length [23]. For a generic body systems. In fact, in the strongly interacting regime short-range interaction, all three contacts and all three where exact theoretical results are not available, these microscopic parameters are required, as shown in Eqs. universal relations become most powerful. (6, 13). These expressions allow us to directly connect It is useful to illuminate our results using some exam- the two-body loss rate to a wide range of physical quan- ples. For a two-body system in free space, the center of tities. mass and the relative motion are decoupled. in Eqs. (7, We first consider the momentum distribution, which 8, 9) becomes a good quantum number, i.e., G(R ; E ) has a universal behavior when k 1/r but much ij 0 becomes a delta function in the energy space.− For larger than all other momentum scales,| | including k , the F scattering states with > 0, we consider the wave- inverses of the scattering length and the thermal wavefunction, Ψ[2](r , r ) = φ (R )ψ(r ), where φ (R ) length. We define the total angular averaged momentum, 1 2 c 12 12 c 12 is a normalized wavefunction of the center of mass n( k ) = m=0, 1 dΩnm(k), where Ω is the solid an- | | and ψ(r12) = 8π/V [i/(cot η i)][cot ηj1(q r12 ) gle, ± − | | − P R n1(q r12 )] Y1m(ˆr12). Figure 3 shows the depen- | | m p C dence of C1 on U˜I whenv ˜p is fixed at various values. n( k ) 1 . (14) P | | → k 2 Results for a bound state are also shown. With increas- | | ing U˜I , C1 approaches a non-zero constant in both cases. 2 Once n( k ) is measured, the first term in Eqs. (6, 13) Here, C2 = 2C1Re(q ). C3 = 0 if we consider a scat- | | 2 is known. If an rf spectroscopy exists for molecules, tering state. In contrast, C3 = 2C1Im(q ) for a bound similar to that for atoms, Eq. (14) also indicates state. Analytical results in the limits, v = 0±, , are p ∞ 4

(a) (b) (c) 8 4 8 0 0.8 3 v~ = !1r3 v~p = !0:2r0 p 0 3 6 v~ = !0:1r3 3 0.6 v~p = !0:001r0 6 p 0 3 v~ = 0r3 v~p = 0:001r0 -0.5 p 0 3 0 4 3 0.4 v~p = 0:1r v~p = 0:1r 0 0 2 3 3 3 v~p = 1000r v~ = !1r v~p = 1000r 0.2 0 -0.02 p 0 1 2 0 2 3 4 3 5 5 5 -1 v~p = !0:1r0 0 -0.04 3 0 1 0 5 v~p = 0r0 0 5 10 10 3 10 10 -0.06 v~p = 0:1r0 2 -1.5 v~ = 1000r3 0 p 0 -0.08 101 102 105 0 -1 -2 0 5 10 15 20 25 0 2 4 6 8 10 12 0 5 10 15 20 25 U~I U~I U~I

FIG. 2. Dependence of the three microscopic parameters on interactions.v ˜p represents the scattering volume when U˜I = 0. ˜ 2 2 3 1 1 UI is in the unit of ¯h /(Mr0). κ1,2,3 are in unit of r0− , r0− and r0− , respectively. Whenv ˜p crosses zero, the location of the maximum of κ1 (κ3) ﬁrst approaches and then leaves the origin, and κ1 (κ3) remains positive (negative). In contrast, κ2 quickly changes from positive to negative at small values of U˜I whenv ˜p crosses zero. In the large U˜I limit, all three parameters vanish, as shown by the insets.

(a) (b) 5 �! �" # �$ # �% # 3 6 3 v~p = !100r0 10 210 v~p = 1r0 v~ = !1r3 v~ = 1:5r3 + $ (' 4 p 0 p 0 ' $ ($ 360 2� �! �# �� + � v~ = 0r3 3 v~ = 3r3 72 2� � � �� $ ) p 0 10 p 0 � → 0 ! # � 4 4� Re −�) Im( )�* 3 200 3 ! & $ $ ) v~p = 1r0 v~p = 1000r0 +2 4� Re −� � 4 4� Re −�) Re( )�* �! 3 3 ) * � v~p = 1000r0 ! 1 10-4 10-2 101 1 C C 190 � → 0 ' $ ($ + $ (' 2 ! ( 72 2� �! �# �� 360 2� �! �# �� 0 $ $ $ % $ 180 �! → ∞ 24 4� �) �#�� 12 4� �) �� 0 1 170 0 10-1 100 102 100 102 104 TABLE II. Analytical expressions for thermal averaged con- U~ U~ I I tacts Cν T in diﬀerent limits. Line 1 and 2 show the results h i in the weakly interacting regime. When vp is positive, bound FIG. 3. Contacts of a two-body system. (a) C1 (in unit of states exist and their contributions are included in Line 1. 4 ˜ r0/V ) of a scattering state as a function of UI (in unit of Line 3 includes the results at resonance. ND is the number 2 2 2 1/2 ¯h /(Mr0)) whenv ˜p is ﬁxed at various values. q = 0.01/r0. of dimers. λT = [(2π¯h )/(MkB T )] is the thermal wave (b) C1 (in unit of r0) of a bound state as a function of U˜I (in length. 2 2 unit of ¯h /(Mr0)).

0 n and n are the eigenenergies of the relative mo- � � � tion with and without interactions, respectively. � → 0± 12 4� � � /� 2�Re(� ) 0 Based on results of the two-body problem, thermal averaged contacts are derived using Cν T = � → ∞ 12 4� � � /� 2�Re(� ) 0 h i Z 1e2µ/(kB T )( e Ec/(kB T ))( C ( )e n/(kB T )). − Ec − n ν n − bound 2 4� Re(−�) 2�Re(� ) 2�Im(� ) Using N = kBT ∂µ ln Z, we eliminate µ and obtain Cν T as a function ofPN and T . AnalyticalP expressionsh in thei TABLE I. Analytical expressions for contacts Cν of two par- limits, vp = 0±, , are shown in Table II. ticles in diﬀerent limits. Line 1 and line 2 show the results Table II may∞ shed light on some recent experiments in the weakly interacting regime and those at resonance, re- conducted in the weakly interacting regime [9, 12]. spectively. vp 0 ( ) means vp 0 + 0i ( + 0i) on the ± ± complex plane.→ Line∞ 3 includes the→ results for∞ bound states, Though vp > 0, it is likely that bound states are not oc- in which a single angular momentum m is considered. cupied, i.e., the system is prepared at the upper branch. Therefore, C3 = 0. In a homogenous system, we obtain,

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[30] S. M. Yoshida and M. Ueda, Phys. Rev. A 94, 033611 (2016). Supplementary Material for “Universal relations for ultracold reactive molecules”

Mingyuan He,1, 2, 3 Chenwei Lv,1 Hai-Qing Lin,4 and Qi Zhou1, 5 1Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA 2Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, 518000, China 3Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 4Beijing Computational Science Research Center, Beijing, 100193, China 5Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, IN, 47907, USA (Dated: May 19, 2020)

The Lindblad equation We consider a Lindblad master equation, dρ ~ = i[H, ρ] + [ρ], (S1) dt − D where H is the Hamiltonian that describes the unitary part of the time evolution and the dissipator describes the loss due to inelastic collisions, D

3 3 1 [ρ] = d x d x Γ( x x ) 2Ψ(x )Ψ(x )ρΨ†(x )Ψ†(x ) Ψ†(x )Ψ†(x )Ψ(x )Ψ(x ), ρ , (S2) D − 1 2 2 | 1 − 2| 2 1 1 2 − { 1 2 2 1 } Z (3) Ψ(x) is the fermionic ﬁeld operator satisfying Ψ(x), Ψ†(x0) = δ (x x0). (1/2)Γ( x1 x2 ) describes a ﬁnite range { } 3 − | − | dissipation. The loss rate of the total particle number, dN/dt = d x(d/dt)Tr(n(x)ρ), n(x) = Ψ†(x)Ψ(x), is written as R dN 1 3 3 3 1 = Tr d xΨ†(x)Ψ(x) d x d x Γ( x x ) 2Ψ(x )Ψ(x )ρΨ†(x )Ψ†(x ) dt − 1 2 2 | 1 − 2| 2 1 1 2 ~ Z Z Ψ†(x )Ψ†(x )Ψ(x )Ψ(x ), ρ −{ 1 2 2 1 } (S3) 1 3 3 3 = d x d x d xΓ( x x )Tr ρ[Ψ†(x )Ψ†(x ), Ψ†(x)Ψ(x)]Ψ(x )Ψ(x ) − 1 2 | 1 − 2| 1 2 2 1 ~ Z 2 3 3 = d xd x0Γ( x0 x ) Ψ†(x)Ψ†(x0)Ψ(x0)Ψ(x) . | − | ~ Z

(3) This equation is valid for any ﬁnite range dissipator. In the approximation of zero-range dissipators, Γ = gδ (x x0), it reduces to [1–3] −

dN 2 3 = g d x Ψ†(x)Ψ†(x)Ψ(x)Ψ(x) . (S4) dt ~ Z

Reference [1] considered two-component fermions and obtained d N1 /dt = d N2 /dt = [~/(2πm)]Im(1/a)C, where h i h i − N1 (N2) is the number of spin-up (spin-down) fermions, a is the s-wave scattering length, and C is the s-wave contact. We emphasize that Eq. (S3) is equivalent to results derived from the Hamiltonian with a complex interaction, as shown in Eq. (S7) in the next section, provided that we identify UI and Γ, i.e., UI ( x0 x ) = Γ( x0 x ). Thus, universal relations derived from the Lindblad equation are the same as those shown| in− the| main text,| − since| arXiv:2005.07861v1 [cond-mat.quant-gas] 16 May 2020 the probability of having more than two particles within a distance smaller than r0 is negligible in dilute systems 1 satisfying r k− . 0 F Decay rate In the presence of complex short-range interactions, a many-body wavefunction satisﬁes

N 2 ~ 2 i~∂tΨ(r1, r2, ..., rN ) = [ + Vext(ri)] + U(rij) Ψ(r1, r2, ..., rN ). (S5) −2M ∇i i=1 i>j h X X i 2

For any ﬁnite size system, net current vanishes at the boundary. We obtain

4 2 ∂tN = dRijdrijUI (rij) Ψ(Rij, rij) , (S6) ~ i>j | | X Z which is equivalent to a second quantization form using fermionic operators,

2 3 3 ∂ N = d xd x0U ( x0 x ) Ψ†(x)Ψ†(x0)Ψ(x0)Ψ(x) . (S7) t I | − | ~ Z

2 2 Using Eq. (3) in the main text, and ψm(rij; ) = [ (~ /M) + U(rij)]ψm(rij; ), we obtain − ∇rij

r0 2 2i dRij drij Ψ(Rij, rij) UI (rij) j>i | | Z Z0 X r0 dRij drijΨ∗ (Rij, rij) Gm (Rij; E ) ψm(rij; ) − j>i m, − Z Z0 (S8) X r0 X + dRij drijΨ(Rij, rij) ∗Gm∗ (Rij; E ) ψm∗ (rij; ) j>i m, − Z Z0 2 X r0 X ~ 2 2 = dRij drij Ψ∗ (Rij, rij) rij Ψ(Rij, rij) Ψ(Rij, rij) rij Ψ∗ (Rij, rij) . M j>i 0 ∇ − ∇ X Z Z h i Note that, for the system with isotropic interactions U(r) = U( r ) and ψ (r ; ) = ϕ( r ; )Y (ˆr ), one has [4] | | m ij | ij| 1m ij 3 r0 r0ς 2 vp = − , (S9) 3 r0ς + 1 1 1 r2 1 r5 1 r0 = 0 + 0 [ϕ(0)(r)]2r2dr, (S10) r −r − 3 v 45 (v )2 − e 0 p p Z0 where ς = ∂ ln[rϕ(r; )]/∂r . Based on Eqs. (4, 5) in the main text and Eq. (S10), we deﬁne { }|=0

ss0 2 s s0 s 2s 00 C1m = (4π) N (N 1) dRijgm∗gm, gm = q Gm(Rij; E ),C1m = C1m, (S11) − − Z X and obtain

1 1 1 01 10 01 10 2 Im vp− C1m + Im re− C1m + C1m iκ3 C1m C1m + O (E E∗) m − 2 − − − X 2 (S12) (4π) 2M r0 2 = 2 dRij drij Gm (Rij; E ) ψm(rij; ) UI (rij), ~ i>j 0 m, − Z Z X X which leads to Eqs. (10, 11, 12) in the main text,

2 1 M ∞ 2 (0) κ = Im v− = drr ϕ (r) U (r), (S13) 1 p − 2 I ~ Z0 1 M ∞ 2 (0) (1) κ = Im r− /2 = Re drr ϕ ∗(r)ϕ (r)U (r) , (S14) 2 − e − 2 I ~ Z0 r0 2 2 (0) (0) 2 M ∞ 2 (0) (1) κ3 = Imϕ (r) Imϕ ˜ (r) r dr = 2 Im drr ϕ ∗(r)ϕ (r)UI (r) , (S15) − 0 − ~ 0 Z h i h i Z whereϕ ˜(0) (r) is a wave function obtained from extending the actual wave function ϕ(0) (r) outside the potential (r > r0) into the regime r < r0. We obtain Eq. (13) in the main text,

~ 1 1 1 ∂ N = Im(v− )C Im(r− )C + κ C , (S16) t −8π2M p 1 − 2 e 2 3 3 h i 01 01 where C1 = m C1m, C2 = 2 m Re(C1m) and C3 = 2 m Im(C1m). The above equation leads to Eqs. (7, 8, 9) in the main text by considering Gm(Rij; E ) = G(Rij; E ). P P − P− 3

Momentum distribution N ik ri 2 Similar to systems with real interactions [5], using n (k) = i=1 j=i drj driΨ(r1, r2, ..., rN ) e− · , We obtain 6 | | R R 1 P Q k k r− F 0 C n(k) | | 1 Y (kˆ) 2, (S17) −−−−−−−−→ 3 k 2 | 1m | m | | X which leads to C n( k ) = dΩn(k) = 1 . (S18) | | k 2 Z | | Density correlations The density correlation function S(r) = dR n(R + r/2)n(R r/2) can be rewritten as h − i R r r 2 S(r) = N(N 1) drk Ψ r1, ..., ri = R + , ..., rj = R , ..., rN . (S19) − k=i,j 2 − 2 Z 6 1 Y In the regime, r k− , S(r) is written as F 2 (0) 2 (0) 2 (0) (1) (1) (0) (1) (0) (0) (1) S(r) = N(N 1) dR Y (ˆr) ϕ (r) g + ϕ (r)ϕ ∗(r)g ∗g + ϕ (r)ϕ ∗(r)g ∗g . (S20) − ij | 1m | | m | | m | m m m m m m m m m Z X h i where r = r . The integral over a shell allows us to obtain | | x+D x+D 1 2 (0) (0) (0) (1) 10 (1) (0) 01 P (x, D) = drS(r) = 2 r dr ϕm (r)ϕm ∗(r)C1m + ϕm (r)ϕm ∗(r)C1m + ϕm (r)ϕm ∗(r)C1m . x (4π) x Z m Z h i X (S21) Using Eqs. (4, 5) in the main text, we obtain ∂P (x, D) 1 1 1 1 1 1 x = 2 C1 2 + C2 + 2Re( )C1 + Re( )C2 Im( )C3 ∂D D 0 16π x 2 − vp re − re 3 → 3 4 2 1 1 x 1 1 1 x C2 6 + Re( )C2 Im( )C3 + C1 Re( )C2 + Im( )C3 x , −3 v − v 5 v 2 − v r v r 9 − 90 v 2 p p | p| p∗ e p∗ e | p| (S22) where the first line recovers Eq. (15) in the main text, and the second line includes higher order terms. When vp and re are known, the first line readily allows experimentalists to obtain C1,2,3 by fitting the experimental data. When vp and re are unknown, the second line is required to obtain Cν , vp and re.

Finite temperature results Homogenous systems We deﬁne the decay rate, β, using ∂ N = βN 2. (S23) t − For a two-body scattering state, we obtain 3 2 2 2 C1 v 0 = 12(4π) q vp 1 + 2Re(vp/re)q /V, (S24) | p→ | | 3 4 2 C2 v 0 = 24(4π) q vp /V, (S25) | p→ | | 3 2 2 C1 v = 12(4π) q− re /V, (S26) | p→∞ | | 3 2 2 C2 v = 24(4π) re 1 + 2Re(re/vp)q− /V. (S27) | p→∞ | | Thermal averaged results are written as

9 4 2 2 2 C1 T v 0 = (4π) Nn vp λT− 1 + 10πRe(vp/re)λT− , (S28) h i | p→ 2 | | 45 5 2 4 C2 T v 0 = (4π) Nn vp λT− , (S29) h i | p→ 4 | | 2 2 2 C1 T v = 24(4π) Nn re λT , (S30) h i | p→∞ | | 3 2 2 2 C2 T = 12(4π) Nn re 1 + Re(re/vp)λ . (S31) h i |vp | | π T →∞ 4

(a)5 (b)5 re = ( 0.01 0.01i)r0 re = ( 0.05 + 0.0001i)r0

re = ( 0.01 0.02i)r0 re = ( 0.05 + 0.0005i)r0

re = ( 0.01 0.05i)r0 re = ( 0.05 + 0.0010i)r0

4 re = (0.01 0.01i)r0 4 re = ( 0.05 + 0.0020i)r0

3 3

2 2

1 1

0 0 0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25

kBT kBT

3 2 2 2 2 FIG. 1: Decay rate β (in unit of [r0/(V h)][~ /(Mr0)]) in a homogenous system as a function of kB T (in unit of ~ /(Mr0)) at 3 different effective ranges. (a) For the limit vp 0, vp is fixed at (0.01 0.01i)r0. (b) For the limit vp , vp is fixed at (0.01 2000i)r3. → − → ∞ − 0

When the bound state is not occupied, C = 0, we obtain h 3iT 3 2 2 r0 2 vp 5 vp vp r0 Mr0 2 2 β (T ) = 144π Im kBT + Im k T , (S32) |vp 0 −V h r3 2 r3 v r 2 B → " 0 0 p∗ e ~ # 3 2 2 3 2 r ~ re r0 re r ~ 1 β (T ) = 0 96π2 Im + 2Im 0 . (S33) |vp −V h Mr2 r r −r v Mr2 k T →∞ 0 0 e e∗ p 0 B

In ﬁgure1, we plot the decay rate as a function of T for both small and large scattering volumes.

Harmonic traps 3 In the high temperature regime, the density can be well approximated by n = exp[µ/(kBT )]λT− . Applying the local density approximation in a harmonic trap, µ is replaced by the local chemical potential µ(r) = µ(0) Vext(r), 2 2 2 2 3 − Vext(r) = (1/2)Mω (x1 + x2 + x3), ω is the harmonic frequency, and n(r) = exp[µ(r)/(kBT )]λT− where µ(0) is the chemical potential at the center of the trap and is ﬁxed by the total particle number, d3rn(r) = N trap. Within this trap 2 3/2 trap framework, the density at the center of the trap is determined by N and T , n(0) = [(2πkBT )/(Mω )]− N . The total contacts are obtained from integrating the local contacts in the trap, R

3/2 trap 2V (r)/(k T ) πkBT C = (0) e− ext B dr = (0). (S34) h ν iT Cν Mω2 Cν Z where ν = Cν /V T is the contact density of νth contact at the center of the trap. In theC weakh interactioni limit and without the bound state, one has Ctrap = 0, and h 3 iT 1/2 trap 3/2 2 5 trap 2 1 TF vp dh 2 2 T C = 72π v d− (N ) 1 + 5Re k d , (S35) h 1 iT | p| h k2 d2 T d3 r F h T F h h e F 1/2 trap 3/2 2 7 trap 2 2 2 T C = 360π v d− (N ) k d , (S36) h 2 iT | p| h F h T F 1/2 trap 1/6 trap 1/3 1/2 where kF = (2Mω/~) (3N ) , TF = (~ω/kB)(3N ) , and dh = [~/(Mω)] . The decay rate is then

2 3 ~ω vp dh 1 TF 5 vp dh T β (T ) = 18√π Im Im kF dh . (S37) |vp 0 h d3 v k d T − 2 v r T → h " p F h r p∗ e r F #

2 2 FIG. 2: Decay rate β (in unit of (~ω)/h) in a harmonic trap as a function of T/TF (in unit of kF− dh− ) at the limit vp 0 for 3 → different effective ranges, (a) For the limit vp 0, vp is fixed at (0.0001 0.0001i)dh. (b) For the limit vp , vp is fixed at (0.01 20i)d3 . → − → ∞ − h

˜ 3 2 2 FIG. 3: Decay rate β (in unit of (dh(~ω/h)) for n(0) at the center of a harmonic trap as a function of T/TF (in unit of kF− dh− ) 3 at the limit vp 0 for different effective ranges, (a) For the limit vp 0, vp is fixed at (0.0001 0.0001i)dh. (b) For the limit → 3 → − vp , vp is fixed at (0.01 20i)d . → ∞ − h

In the strong interaction limit, one has Ctrap = 0, and h 3 iT 5/2 trap 3/2 2 1 trap 2 1 TF C = 96π r d− (N ) , (S38) h 1 iT | e| h k2 d2 T F h 3/2 3 trap 3/2 2 3 trap 2 1 TF dh re 1 TF C = 96π r d− (N ) 1 + 4Re . (S39) h 2 iT | e| h k2 d2 T v d k2 d2 T F h p h F h The decay rate is then

2 3/2 3 5/2 ~ω re dh 1 TF re d 1 TF β (T ) = 12√π Im + 2Im h . (S40) |vp − h d r k3 d3 T −r v k5 d5 T →∞ h " e F h e∗ p F h #

Figure2 shows the decay rate as a function of T in a trap. If the decay rate at the center of the trap β˜, which satisﬁes

∂ n(0) = βn˜ 2(0), (S41) t − 6 is considered, one has

ω v 2 d3 T 5 v d T 2 ˜ √ 2 3 ~ p h 2 2 p h 4 4 β (T ) = 36 2π dh 3 Im kF dh Im kF dh 2 , (S42) vp 0 h d vp TF − 2 v re T → h p∗ F 2 ω r d r d3 1 T ˜ √ 2 3 ~ e h e h F β (T ) = 24 2π dh Im + 2Im 2 2 . (S43) vp − h dh re −r vp k d T →∞ e∗ F h

The decay rate as a function of T at the center of the trap is shown in ﬁgure3.

Measure κ1,2,3 in a two-body system In a two-body system, is a good quantum number such that the three contacts are not independent. In fact, 2 2 C2 = 2C1Re(q ) and C3 = 2C1Im(q ). Therefore, the decay rate is determined by

~C1 ∂ N = [κ + 2Re(q2)κ + 2Im(q2)κ ]. (S44) t −8π2M 1 2 3

1 q k r0− C can be measured from the momentum distribution using n( k ) | || | C / k 2. In the above equation, 1 | | −−−−−−−−→ 1 | | except κ1,2,3, all other quantities are measurable. Therefore, κ1,2,3 can be extracted from the experimental data. To be more explicit, in a two-body system, ∂tN is determined by Im(). Equation (S44) is rewritten as 2 2 2 ∂tN = [(~C1)/(8π M)][κ1 + 2Re(q )κ2]/[1 + C1κ3/(16π )]. Thus, from the decay rate, the momentum distribution and the− real part of the energy that can be obtained from a spectroscopy measurement, all three microscopic parameters, κ1,2,3 can be accessed in experiments.

The decay rate of the density at the center of the trap 2 3/2 trap If we consider the density at the center of the trap, n(0) = [(2πkBT )/(Mω )]− N , in the weakly interacting regime, based on Eq. (S42), the diﬀerential equation satisﬁed by n(0) is written as

2 2 2 36√2π 2 90√2π vp 1 M vp 2 2 2 ∂tn(0) = Im(vp)n (0)kBT + Im( re− ) | 2 | n (0)kBT . (S45) h h vp∗ ~

We see that the ﬁrst term becomes linearly dependent on T . This is what has been observed in the recent experiment, which has deﬁned an average density proportional to n(0) [6]. Taking into account the second term, there will be a deviation from the linear dependence on T .

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