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Universal Relations for Ultracold Reactive Molecules

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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 distri- bution 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 Schr¨odingerequation, 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.
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