Quantum dynamics of resonant molecule formation in waveguides

We explore the quantum dynamics of heteronuclear atomic collisions in waveguides and demonstrate the existence of a novel mechanism for the resonant formation of polar molecules. The molecular formation probabilities can be tuned by changing the trap frequencies characterizing the transverse modes of the atomic species. The origin of this effect is the confinement-induced mixing of the relative and center of mass motions in the atomic collision process leading to a coupling of the diatomic continuum to center of mass excited molecular states in closed transverse channels.


Introduction
Dimensionality plays a crucial role for interacting ultracold quantum gases [1,2,3,4]. For a strong two-dimensional confinement, i.e. a tight waveguide, and a close to resonant scattering process the atoms can feel the trapping potential within an individual atomatom scattering process. A striking manifestation herefore is the confinement-induced resonance (CIR) effect [5]. The narrow transversal confinement leads to a singularity of the coupling constant for the corresponding effective longitudinal motion and as a result we encounter a 1D strongly interacting gas of impenetrable bosons that can be mapped at resonance on a gas of free fermions, the so-called Tonks-Girardeau gas [6]. The CIR has been interpreted as a Feshbach resonance involving the atom-atom continuum and a bound state of a transversally excited channel of the harmonic waveguide [7,8,9]. CIRs have subsequently been discovered for three-body [10,11] and four-body [12] scattering under confinement and for a pure p-wave scattering of fermions [13]. The CIR behaviour has been found experimentally for s-wave scattering bosons [14,15] and for p-wave interacting fermions [16]. Recently a dual-CIR, which leads to a confinement-induced transparency effect [17,8], has been demonstrated. It is characterized by a complete suppression of strong s-and p-wave heteronuclear atomic scattering in 3D because of the presence of the waveguide.
It is to be expected that waveguides with an anharmonic and/or anisotropic transversal confinement will lead to novel ultracold atomic collision properties and consequently to an intriguing many-body dynamics of the quantum gas. Very first results in this direction include the observation of a modified CIR due to a coupling of the center of mass (CM) and relative motion of the atoms [18,8]. This nonseparability can be either achieved by employing anharmonic waveguides or, in case of heteronuclear collisions, by the fact that the two species experience different harmonic frequencies.
In the present work we show that the quantum dynamics of the coupled CM and relative motion for two-atomic species feeling different confinement potentials, i.e. waveguides, exhibits a resonant formation process of ultracold molecules. The latter is analyzed in detail and its existence is found to be independent of the specifics of the atomic interaction. As a consequence it is anticipated that heteronuclear atomicmolecular 1D quantum gases in an anisotropic waveguide should explore a rich manybody quantum dynamics.

Wave-packet propagation method
We employ a wave-packet dynamical approach to study the atomic scattering in waveguides which has been developed recently [17,8]. The collisional dynamics of two (distinguishable) atoms with coordinates r 1 , r 2 and masses m 1 , m 2 moving in the harmonic waveguide with the transverse potential 1 with the transformed Hamiltonian Here and describe the CM and relative atomic motions, V (r) describes the atom-atom interaction, ρ R and r = r 1 − r 2 → (r, θ, φ) → (ρ, φ, z) are the polar radial CM and the relative coordinates and M = m 1 + m 2 , µ = m 1 m 2 /M. The transformed Hamiltonian (1) is given in a rotated frame which exploits the conservation of the component (L 1 + L 2 ) of the total angular momentum and allows to eliminate the angular dependence of φ R of the CM degrees of freedom [8,19]. Our investigation will focus on the case M φ R = 0, the later being the quantum number belonging to 1 in the Hamiltonian (1) leads for two distinguishable atoms that feel different confining frequencies ω 1 = ω 2 to a nonseparability of the CM and relative atomic motion. We integrate the Schrödinger equation from time t = 0 to the asymptotic region t → +∞ with the initial wave-packet representing two different noninteracting atoms in the transversal ground state of the waveguide with a i = (1/m i ω i ) 1/2 and the overall normalization constant N defined by < ψ(0)|ψ(0) >= 1. We choose z 0 → −∞ to be far from the origin z = 0 and a z → ∞ to obtain a narrow width in momentum and energy space for the initial wave-packet. Our wave-packet moves with a positive interatomic velocity v 0 = k 0 /µ = 2ε /µ thereby approaching the scattering region located at z = 0. In the course of the collision, the wave-packet splits up into two parts moving in opposite directions z → ±∞. We model the interatomic interaction V (r) via the Lennard-Jones 6-12 potential V (r) = C 12 /r 12 − C 6 /r 6 . Note, that the effects discussed in the following exist for even qualitatively very different shapes of the interatomic potential (like 6-12 and screened Coulomb [8,9]). In this sense, our results will be universal, although the molecular formation probabilities, to some limited extend, depend on the appearance of V (r).
To be specific we consider the pair collisions of 40 K and 87 Rb atoms with mass ration m 1 /m 2 = 40/87 and the trapping frequency ω 2 = 2π × 200 kHz for Rb. Hereafter we use the units µ =h = ω 0 = 1 with ω 0 = 2π × 10 MHz. For the case of a decoupled CM motion, ω 1 = ω 2 = 0.02 in these units, whereas for the coupled case ω 1 varies in the limits ω 2 < ω 1 ≤ 2.2ω 2 . Our focus is the analysis of the population dynamics of molecular resonance states (for reasons of brevity we will simply refer to them as molecular bound states with respect to V (r)) for ultracold atom-atom collision in the waveguide.

Two-body bound states in a harmonic waveguide
First we analyze how the molecular spectrum changes under the action of the confining waveguide (see Fig.1). We allow C 12 to vary for fixed C 6 = 1.847 [20] and focus on the regime of the appearance of a bound state of V (r). In case of the absence of the confining potential the first bound molecular state ε 0 < 0 appears at C 12 ≃ 0.13 and becomes increasingly bound with decreasing C 12 . To explore the dependence of the binding energies of these states on C 12 in the presence of the waveguide we solve the corresponding eigenvalue problem of the four-dimensional Hamiltonian H(ρ R , r) (1) employing the spectral method elaborated in ref. [21]. It is based on the computation of the autocorrelation function < ψ(ρ R , r, t = 0)|ψ(ρ R , r, t) > where the initial state ψ(ρ R , r, t = 0) can be considered as a test function of the spectrum. The solution ψ(ρ R , r, t) of the timedependent Schrödinger equation is obtained via our wave-packet propagation method [8]. For the case ω 1 = ω 2 = ω, i.e. for a decoupled CM motion, every unconfined bound state with energy ε 0 (C 12 ) transforms by the action of the waveguide into the spectrum ε n 1 n 2 (C 12 ) which represents at C 12 ≥ 0.15 the spectrum of two independent identical two-dimensional oscillators with radial quantum numbers n 1 = 0, 1, ... and n 2 = 0, 1, ... characterizing the transverse excitations of two noninteracting atoms in the waveguide. We use n 1 , n 2 as labels of the molecular bound states ε n 1 ,n 2 (C 12 ) at fixed C 12 correlating to the corresponding dissociation limit (6) where they become exact quantum numbers. It is evident from (6) that for ω 1 = ω 2 the excited states (n 1 , n 2 ) become degenerate ε n 1 n 2 = ε n 2 n 1 . Since the coupling between the CM and the relative motion is absent the quantum numbers n and N of the harmonic spectrum of the relative and the CM transverse eigenstates could be equally used to classify the spectrum.
For ω 1 = ω 2 the presence of the coupling between the CM and the relative motion leads to a lifting of the above-mentioned degeneracy. In Fig.1 we show the corresponding splitting ε 01 = ε 10 of the first excited state and the shift of the ground state.

Molecular resonance states in waveguides
The above-discussed spectral structure of the diatomic molecule in the waveguide permits us to analyze the dynamics of atomic pair collisions with respect to the formation of molecular final states. We choose herefore C 12 = 0.109 and the energy of the colliding atoms to be ε = ω 1 + ω 2 + ε between the thresholds of the lowest ω 1 + ω 2 and the first excited ω 1 + 3ω 2 transverse channels (0 ≤ ε ≤ 2ω 2 ). Fig.1 indicates that for the chosen potential and collision energies one can observe one bound state ε 01 of the closed channel n 1 = 0, n 2 = 1 which becomes degenerate ε 01 = ε 10 for ω 1 = ω 2 . Fig.2 illustrates the time-evolution of the wave-packet in the course of the atomic collision for the case of CM nonseparability ω 1 = ω 2 (Fig.2a) and CM separation ω 1 = ω 2 (Fig.2b).
For ω 1 = ω 2 we observe that a considerable part of the scattered wave-packet (see the probability density distribution W (ρ R , r, t) at t = 8t 0 ) is located near r = 0 after the collision and corresponds to a molecular bound state ε 01 which in the (n, N)representation is a mixture of the two states n = 0,N = 1 and n = 1,N = 0 with dominating n = 0,N = 1 contribution. Note that during collision t ∼ 3t 0 the main part of the wave-packet is temporarily in the ground state ε 00 (n = N = 0), decays thereafter rapidly into the continuum but part of it goes into the excited molecular state ε 01 . In contrast to this the case ω 1 = ω 2 in Fig.2b (CM separation) shows an almost complete decay into the continuum: The remaining minor part near r = 0 is much smaller compared to the case ω 1 = ω 2 .
To quantitatively investigate the molecular formation process accompanied by the CM excitation we calculate the population probability P N (t) of the molecular bound states for N = 0 and N = 1 where Φ N are the two-dimensional oscillator states of the potential (1/2)(m 1 ω 2 1 + m 2 ω 2 2 )ρ 2 R and dΩ = sin θdθdφ. For the chosen interval of collision energies 0 ≤ ε ≤ 2ω 2 and C 12 = 0.109 the molecular states N = 0 and N = 1 are the ones which dominate during the collision process (see Fig.1). Note, that the integration in (7) over the interatomic distance r is limited to the region r ≤ r m = 10 of the action of the 6-12 potential. For sufficiently long times, i.e. after the collision process is concluded, all the unbound part of the scattered wave-packet resides outside this region. The complementary part is given by P N (t) and can be interpreted as a molecular bound state. For r m → ∞ we obtain P 0 (t) + P 1 (t) ≈ 1 for any time and any energy within the above-provided interval. Fig.3 shows (ω 1 = ω 2 ) that during the time interval of close collision the wave-packet temporarily occupies the molecular ground state N = 0 i.e. P 0 becomes large. However, with further increasing time P 0 decays to zero and P 1 increases rapidly which corresponds to the population of the excited state N = 1. P 1 decays very slowly for long enough times after the collision. We interprete this state as the first excited state n 1 = 0, n 2 = 1 of the molecular spectrum (see Fig.1). For ω 1 = ω 2 the coupling between the n 1 = n 2 = 0 (N = 0) and n 1 = 0, n 2 = 1(N = 1) molecular states is absent and the temporary population in the course of the collision of the molecular ground state decays back into the continuum after the collision (see behaviour of P 0 (ω 1 = ω 2 ). In particular P 1 (ω 1 = ω 2 ) is negligible always.
In Fig.4 we illustrate the mechanism of molecule formation for the ground state n 1 = n 2 = 0(N = 0) with transversal ω 1 + ω 2 and relative longitudinal ε = µv 2 0 /2 energies. Under the action of the attractive tail −C 6 /r 6 of interatomic interaction the atoms accelerate up to the energy µv 2 (t)/2 = ε + ω 1 + ω 2 − ε 00 at the instant of collision. The occurence of the molecular state with energy ε 00 (bound molecular state with respect to the ground transversal channel) as an intermediate in the course of the collision process can be seen by inspecting the intermediate probability density shown in Fig.2a and Fig.3. The released energy subsequently transfers to the closed channel n 1 = 0, n 2 = 1(N = 1) via formation of a molecule in a CM excited state since this closed channel is coupled to the N = 0 by the term W (ρ R , r) = µ(ω 2 1 − ω 2 2 )rρ R sin θ cos φ (4) of the total Hamiltonian (1). This mechanism also explains the delay of the onset of the collisional interaction observed in Fig.3 in the absence of the coupling term ω 1 = ω 2 compared to the case ω 1 = ω 2 . For ω 1 = ω 2 the collisional interaction happens later than for ω 1 = ω 2 because the binding energy in the channel ε 00 (ω 1 = ω 2 ) is considerably lower than the binding energy in the channel ε 00 (ω 1 = ω 2 ) leading to a stronger atomic attraction and acceleration of the colliding atoms for ω 1 = ω 2 . A simple semiclassical estimate of the time difference between the onsets of collisional interactions for the two  cases is in good agreement with the numerical result of Fig.3. From the above scheme it also follows that if the energy release ε + ω 1 + ω 2 − ε 00 is equal to the excitation energy ε 01 − ε 00 we have to expect a resonant enhancement of the molecular formation process i.e. we encounter the resonance condition To analyze the energy dependence of the atom-molecule reaction we inspect the population probabilities P N (ε ) for the molecular states N = 0 and 1 as a function of the longitudinal energies ε (see Fig.5). The values P N (ε ) = P N (ε , t → ∞) were calculated after the collision at t = 8t 0 , in the asymptotic region where P 1 (ε , t) is very slowly decaying if ω 1 = ω 2 . P 1 (ε ) exhibits a pronounced resonant behaviour whereas P 0 (ε ) shows a very weak energy dependence which is in agreement with the abovediscussed reaction mechanism. The calculated position ε = ε r ∼ 0.01 of the maximum of the population P 1 (ε ) is shifted considerably compared to ε r (ω 1 = ω 2 ) = ε 01 (ω 1 = ω 2 ) − ω 1 − ω 2 = 0.026 following from the resonance condition (8). We attribute this difference to the fact that we are using for the calculation of P 1 (ε ) in eq.(7) the oscillator wave function Φ N (ρ R ) instead of the exact transversal CM part of the molecular wave function n 1 = 0, n 2 = 1(N = 1). The resulting deviation from the exact resonance conditions (8) is particularly due to the missing effects of the CM coupling which provides the shift ε 01 (ω 1 = ω 2 ) − ε 01 (ω 1 = ω 2 ) = 0.014 resulting in a significantly better agreement of the calculated position of the resonance with the resonant energy from eq.(8). The molecular formation probability P 1 depends also on ∆ω = ω 1 − ω 2 and reaches a maximum at a value corresponding to the resonance condition (8). This probability can therefore be controlled by changing the transversal confinement frequencies for the different atomic species. By nature the process we observed is a molecular resonance in the atomic scattering and back decay of the molecule to the atom-atom continuum will occur. However, this resonance possesses a long lifetime due to its two-step character which is clearly demonstrated in Fig.3. Finally we note that P 1 amounting to several percent should not obscure the fact that substantial molecular formation rates λ = n A v 0 P 1 could be achieved by integrating over many collisions even for low linear atomic densities n A . Thus, for n A = 10 4 cm −1 and T = 100nK we obtain the estimate λ ∼ 10 3 s −1 .

Conclusions
Perspectives concerning applications of the molecular formation mechanism are based on the fact that it represents a key ingredient for the preparation and following investigation of the dynamics of mixed atomic and dipolar molecular quantum gases in waveguides. The latter are expected to possess a novel collective excitation dynamics particularly in the ultracold regime discussed here for which only a few transversal channels are accessible. If wanted, one could stabilize the molecular gas and prevent back decay into atoms by a corresponding switch of the trapping fields such that the CM decouples from the interatomic motion finally. Open questions include the impact of inelastic atommolecule and molecule-molecule collisions, which however goes beyond the scope of the present investigation.