Tunneling of two interacting atoms from excited states

Accepted Manuscript Tunneling of two interacting atoms from excited states I.S. Ishmukhamedov, A.S. Ishmukhamedov PII:

S1386-9477(18)31297-9

DOI:

https://doi.org/10.1016/j.physe.2018.12.026

Reference:

PHYSE 13414

To appear in:

Physica E: Low-dimensional Systems and Nanostructures

Received Date: 5 September 2018 Revised Date:

24 November 2018

Accepted Date: 21 December 2018

Please cite this article as: I.S. Ishmukhamedov, A.S. Ishmukhamedov, Tunneling of two interacting atoms from excited states, Physica E: Low-dimensional Systems and Nanostructures (2019), doi: https:// doi.org/10.1016/j.physe.2018.12.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Tunneling of two interacting atoms from excited states I.S. Ishmukhamedova,b,c , A.S. Ishmukhamedovb,d a Bogoliubov

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Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russian Federation b Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan c Institute of Nuclear Physics, the Ministry of Energy of the Republic of Kazakhstan, Almaty 050032, Republic of Kazakhstan d Institute of Hydrogeology and Geoecology, the Ministry of Education and Science, Almaty 050010, Republic of Kazakhstan

Abstract

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We consider a tunneling problem of two interacting cold atoms, with the even spatial symmetry, subject to an anharmonic optical trap and linear magnetic-field gradient. The atoms are initially, i.e. at t = 0, prepared in the two lowest excited states with respect to relative and center-of-mass motions. We calculate the energy spectrum for a wide range of the interatomic coupling strength g1D . In the limit of g1D → 0, an avoided crossing of the energy levels is revealed. For the dynamics, i.e. for t > 0, we observe monotonic and non-monotonic dependence of a tunneling rate as a function of g1D . We find a condition to observe a transition from uncorrelated to correlated pair tunneling as a function of g1D and a size of the external trap barrier. This system, although for lower energy levels, has been recently investigated in the deterministic Heidelberg experiment using two interacting 6 Li atoms. Keywords: Ultracold atoms; Energy spectrum; Tunneling rate; Anharmonic trap; One-dimensional geometry; Avoided crossing.

1. INTRODUCTION

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Understanding of quantum tunneling is a key question which arise in various fields of physics such as a decay of an α particle [1, 2, 3] from a nucleus, superfluidity of electrons in metals [4] and 3 He atoms [5]. It is still, however, challenging to include a correlation into an analysis of few-body systems [2, 6, 7, 8, 9, 10, 11, 12, 13]. Only recently it has become possible for the experimental realization of a pure two-body quantum system [14, 15, 16]. This has been done in the setup of the Heidelberg group in which they have prepared two cold 6 Li atoms in the ground and lowest excited states of a dipole trap. Due to tight confinement in the transverse direction this system can be considered as a one-dimensional one [17]. By imposing an additional linear magnetic field, the atoms start to escape from the trap, due to quantum tunneling mechanism, through the trap barrier. The interatomic coupling strength is varied by means of a magnetic Feshbach resonance [18] in which the interaction strength could be varied in a wide range and could be attractive or repulsive. Thus, it is now already possible to directly test a two-body quantum system experimentally. There were several theoretical methods [6, 7, 8, 9] to describe tunneling of this two-atom system. However, they mostly considered tunneling from the ground state and only from few levels of the lowest excited state. In the present paper we extend these analyses and add more excited states, which also consist of a center-of-mass excited state branch. Adding more excited states reveals an avoided crossing in the energy spectrum, which results in the population transfer between an atomic pair and a molecule in an excited center-of-mass state [19]. This is caused by anharmonic terms of the trap potential. In our one-dimensional case, an avoided crossing is a consequence Preprint submitted to Physica E

of a rotational symmetry breaking of a 2D harmonic oscillator spectrum. The current research shows that this effect already emerges in a two-body one-dimensional geometry and the fact that we use a realistic trap shape and parameters suggests that this effect can be experimentally observed. A calculation of the energy spectrum of an interacting system is a quite non-trivial task. It becomes even challenging with increasing the correlation between particles. One of the method to determine the bound energies is to match the tunneling time of the tunneled particle with the WKB (Wentzel, Kramers and Brillouin) theory, which, however, does not take into account the interparticle correlation [8, 15, 16, 20, 21]. This is where our model could be well applied since it includes the interaction between the particles and has one-to-one correspondence between tunneling dynamics and the initial, bound state, energy of the system. We should note other similar works on quantum tunneling [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In the works [22, 23] confinement-induced resonances of two identical and distinguishable atoms in cylindrical waveguides is explored. The work [24] considers the energy spectrum of bound states, bound-bound and bound-continuum transitions of hydrogenlike atoms subject to strong external fields. The work [25] considers population transfer of a Bose-Einstein condensate between the wells of a double-well trap. The work [26] considers transmission probabilities of bound atoms, tunneling through potential barriers. Tunnelling dynamics of repulsive few-boson systems through a one-dimensional potential barrier is studied in [27]. A comprehensive analysis of two interacting bosons in a double-well trap, exploring spectral properties of the system, is made in [28]. The work [29] explores a way to distinguish between different decay channels of an open two-body bosonic November 25, 2018

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system as the interparticle interaction strength is changed. Fewboson tunneling in a one-dimensional double well is studied in [30]. A problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space is considered in [31]. The work [32] discusses the correlated tunneling dynamics of few-body fermionic mixtures in a doublewell. The present research considers two interacting atoms subject to external optical field plus a linear magnetic field used in the recent experiments [15, 16]. We focus on a tunneling dynamics of the initially two lowest excited state branches. Our findings confirm our recently predicted non-monotonic behavior of the two-body decay [33] and we also find a condition for the transition from uncorrelated to correlated pair tunneling. The paper is organized as follows. Sec.2 describes the model Hamiltonian and a numerical method to solve the timedependent Schr¨odinger equation. Sec.3 discusses the spectrum of the initial states and tunneling dynamics of upper and lower excited state branches. Sec.4 summarizes the results.

]

t>0

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∂ψ(x1 , x2 , t) = H(x1 , x2 )ψ(x1 , x2 , t), ∂t

where the two-body Hamiltonian reads

Here H j (x j ) = −

~2 ∂2 + V (at) (x j ), 2m ∂x2j

j = 1, 2.

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Figure 1: (Color online) Plot of the trapping potential V (at) (x), for p = 0.75: the trap for the initial state preparation (dashed line), where we put a hard wall at x = 11ℓ [represents the trap at t = 0] and the trap for t > 0 used for an analysis of the system’s dynamics (solid line)

(1)

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H(x1 , x2 ) = H1 (x1 ) + H2 (x2 ) + V (aa) (x1 − x2 ).

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Tunneling dynamics of the two-atom system is modeled by the time-dependent Schr¨odinger equation i~

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(at)

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trap barrier. We take p = 0.75 everywhere (if not stated otherwise). The highest energy levels (see Fig.3 below) approximately reaches the height of the two-dimensional trap barrier of the potential V (at) (x1 ) + V (at) (x2 ) (i.e. of the true potential, without the hard wall). So, we assume, that at this value of p, most of the considered lowest excited states can be considered as quasistationary states of the (true) trap potential. At t > 0 we turn off the hard wall and analyze the dynamics of the system. For the interatomic interaction potential V (aa) (x1 − x2 ) we choose the potential of the Gaussian shape [8, 34]       (x1 − x2 )2   (aa) V (x1 − x2 ) = −VG exp  − , (5)     2r2 

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2. MODEL AND THE NUMERICAL METHOD

t=0

(3)

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are the single-particle Hamiltonians, in which V (at) describe the atom-trap interaction and m is the atomic mass. The atom-trap potential is taken from [8, 15, 16] which represents an optical trap and magnetic field gradient (Fig.1): ] [ 1 V (at) (x) = pV0 1 − − µB Cx, (4) (x/xR )2 + 1 √ where µB is the Bohr magneton, xR = 8.548ℓ (ℓ = ~/mω) is the Rayleigh range, V0 = 56.16~ω is the maximum depth of the optical trap, p is a fraction of the optical trap depth, C depends on external magnetic-field strength, magnetic-field gradient and the hyperfine state of the atoms and we take the value, which equals to C = 1894.18 G/m. This value is used in [8] to model the experimental realization [15, 16] of the initial state of the system of two 6 Li atoms. For the trap frequency we take the value from the experiment ω = 2π × 1234 Hz [8, 15, 16]. We take these values for the whole analysis in the paper and expect that slight deviations of the actual values won’t alter the qualitative picture of the system’s dynamics. To compute the initial states we put a hard wall [8], at x = 11ℓ, which suppresses tunneling of the system through the

0

where VG and r0 are the depth and range of the interatomic interaction (5). The value r0 = 0.1ℓ [8, 34] well represents the short-range interaction, which atoms experience at low energies. To parametrize the interaction (5) we use the 1D contact coupling strength g1D which has a simple relation with a 1D scattering length a1D as g1D = −2~2 /(ma1D ). To compute a1D , we solve the scattering problem for the relative motion x = x1 − x2 of the two-atom system in the absence of the trapping potential [33, 34]: [ 2 2 ] ~ d ~2 k2 (aa) − + V (x) ψsc (x) = ψsc (x), (6) 2 2µ dx 2µ √ where k = 2µE/~ is the wavenumber and µ = m/2 is the reduced mass. We use the boundary condition ψsc (x) −−−−−→ cos(k|x| + δ(k)), x→±∞

(7)

where δ(k) is the phase shift. In the low-energy limit, k → 0, the 1D scattering length a1D is computed as a1D = lim

k→0

cot(δ(k)) . k

(8)

We use the following range for VG : −43 . VG /(~ω) . 14. This range covers a wide range of the coupling strength values, g1D (inset of Fig.2). 2

ACCEPTED MANUSCRIPT put a hard wall at x = 11ℓ (Fig.1) and compute eigenstates and eigenenergies in such a trap. The corresponding stationary Sch¨odinger equation is solved by means of the method from [37]. Briefly, the method reduces the Sch¨odinger equation to a system of matrix equations which are solved by using the sweep method. Each solution, at every iteration, is then updated and it eventually converges to the desired one, according to the shifted inverse power method. The energy spectrum in such a trap is shown in Fig.3. We focus on the lowest excited state branches, which we label as upper and lower branches. These branches correspond to doubly excited states with respect to relative and center-of-mass motions [33]. The nodal patterns of the wave functions is shown in Fig.4. At negative coupling (g1D = −1 in Fig.4) one clearly identifies the two nodes with respect to relative (upper branch) and center-of-mass (lower branch) motion coordinates. When we cross the point g1D = 0, the nodal patterns of these states are mixed and at g1D = 1 the doubly excited relative motion state, observed for the upper branch at g = −1, turns into the doubly excited center-of-mass motion state. The same situation, only vice versa, occurs for the lower branch: the doubly excited center-of-mass state turns into the doubly excited relative motion state. The origin of such nodal pattern mixing is due to rotational symmetry breaking at g1D = 0: the anharmonic terms of the trap potential (4) lift the doubly degenerate harmonic energy level. A similar effect has been observed in [33, 38, 39]. With further increasing the coupling g1D , the picture of the nodal patterns, for both branches, becomes more complicated (g1D = 3 in Fig.4).

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Figure 2: (Color online) Dependence of the 1D coupling strength g1D on the depth VG of the finite-range potential (5) with fixed r0 = 0.1ℓ. The inset shows the range of the parameters VG and g1D which we use in the current paper.

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upper branch lower branch ground state

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W(x j ) = wc (|x j | − xc )2 θ(|x j | − xc ), j = 1, 2,

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In order to integrate (1) we use the split-operator method, which is based on ideas [35] and has been developed in the works [22, 23, 24] in application to confined ultracold atomatom collisions in waveguide-like traps: { } ∆t (aa) ψ(x1 , x2 , t + ∆t) = exp −i V (x1 − x2 ) 2~ { } { } i∆tH1 (x1 ) i∆tH2 (x2 ) exp − exp − ~ ~ { } ∆t (aa) exp −i V (x1 − x2 ) ψ(x1 , x2 , t) (9) 2~ { } The action of the operators exp −i∆tH j (x j ) is approximated by the Crank-Nicolson scheme, which maintains the accuracy order of the split-operator scheme (9) up to O(∆t3 ). The partial derivatives in (3) are approximated by the sixth-order finitedifferences. At large x1 and x2 the wave function should has a shape of an outgoing wave. This boundary condition can be modeled by introducing into the original Hamiltonian (2) a complex absorbing potential (CAP) iW(x j ) [27, 36], which absorbs the wavepacket at the edges of the simulation grid, of the form: where θ(x) is the Heaviside step function. Final results should not depend on a particular choice of the parameters wc and xc . Therefore, there is no unique choice of these parameters, but rather a domain where the final results do not significantly change under the variation of them. We find that wc = −1~ωℓ−2 and xc = 25ℓ lie in that domain. We compare our results for different number of spatial grid points and find that approximately 500 points on each variable ensures that the error of computed tunneling rates is less than few percents.

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Figure 3: (Color online) Energy spectrum of the ”closed” trapping potential (with a hard wall at x = 11ℓ) (4) as a function of the coupling strength g1D . Upper and lower excited state branches correspond to the doubly excited states with respect to relative and center-of-mass motions (see text for details).

3.2. Tunneling dynamics For an analysis of the tunneling dynamics we consider the total probability P(t) ∫∫ P(t) = dx1 dx2 |ψ(x1 , x2 , t)|2 , (11)

3. RESULTS

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3.1. Initial solution and the energy spectum The first step to integrate Eq.(1) is to obtain its initial solution, i.e. ψ(x1 , x2 , t = 0). As was mentioned above, we

where, for the integration domain Ω, we use the simulation grid xk ∈ [−3, 40] [in units of ℓ]. 3

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g1D = 0

g1D = 1

g1D = 3

Upper branch

0.3 0.2 0.1 0 -0.1 -0.2 -0.3

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Figure 4: (Color online) Initial wave functions of the Hamiltonian (2) plotted at various coupling strength values, g1D [in units of ~ωℓ]. Here x = x1 − x2 [in units 2 of ℓ] is the relative motion coordinate and y = x1 +x [in units of ℓ] is the center-of-mass motion coordinate. 2

P(t) ∼ e−γt ,

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0,990

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Thus computed γ(t) converges to an approximately constant value over time, which can be noticed in Fig.5. This indicates the exponential decay law and the converged value of γ(t) is the tunneling rate of the two-body system. We omit the time dependence of γ in order to refer to this converged value. Another way to extract the tunneling rate is to fit the computed P(t) with the exponential function (12). We prefer the fitting over the formula (13) since it is more suitable and accurate. The result of the fitted tunneling rate γ as a function of the coupling strength g1D is presented in Fig.6.

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3.2.1. Upper branch From Fig.6 one can notice that the tunneling rate γ from the upper and lower branches behaves in non-monotonic and monotonic manners, respectively, as a function of the coupling strength g1D . Since in both cases the same Hamiltonian is used, this suggests that the origin of such non-monotonic or monotonic dependencies is due to the initial wave function distribution. This situation is similar to what has been observed in [33],

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t [ms] Figure 5: (Color online) Total probability P(t) and γ(t) [in units of s−1 ] for various coupling strength values, g1D [in units of ~ωℓ]. Initial state corresponds to the upper branch excited state. Notice the different scale of all the graphs.

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ACCEPTED MANUSCRIPT ¯ P2 (t). Hence, the mean atom number is calculated as N(t) ≈ ¯ is shown in Fig.8. P2 (t) + 1. The resulting N(t)

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Figure 6: (Color online) Tunneling rate γ [in units of s−1 ] as a function of the coupling strength g1D [in units of ~ωℓ]. Notice the logarithmic scale

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¯ for various Figure 8: (Color online) Mean number of trapped particles N(t) coupling strength g1D [in units of ~ωℓ].

where, however, the trap is unbound (in contrast to the situation considered here) in both directions in x. Here, for the realistic trap shape, used in the recent experiments [15, 16], we confirm the similar non-monotonic dependence. If we divide the whole configuration space into regions (Fig.7) and calculate the partial probabilities ∫∫ Pk (t) = dx1 dx2 |ψ(x1 , x2 , t)|2 , k = 0, 1, 2. (14)

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Since the tunneling goes only into the regions R1 , within the considered time domain, we can conclude that only the first particle tunnels out of the trap (Fig.8) while the second particle remains in the trap with small tunneling rate [8]. If we consider high repulsion case, g1D = 30 in Fig.8, we find approximately ¯ as for g1D = 1. This probably can the same decay law of N(t) be referred to the non-monotonic dependence of P(t) on g1D at positive values of g1D , as was observed in [33]. Reducing g1D (to lower values and attractive interactions) leads to a suppress¯ ing of a decay of N(t).

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at each of them, then it is possible to extract the mean atom number that remains in the trap during the decay of the system. Region R2 approximately covers the size of the two-atom trap (we take it as x j ∈ [−3, 11] [in units of ℓ]) and therefore P2 defines the probability to find the two atoms in the trap, P1 defines the probability to find one atom in the trap and P0 defines the probability that the two atoms escape from the trap [8]. The mean atom number is defined as ¯ = 2P2 (t) + P1 (t). N(t)

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3.2.2. Lower branch The tunneling rate from the lower branch of the excited states exhibits a monotonic dependence on the coupling strength g1D (Fig.6). This is more or less expected behavior: with increasing the coupling strength g1D , due to increasing the energy level (Fig.3), the effective barrier size decreases. This, in turn, leads to increasing the tunneling rate γ. It is remarkable that this excited state branch allows us to observe a transition from correlated pair tunneling to uncorrelated one by decreasing the size of the trap barrier, by using lower values of p in (4). This transition takes place when the initial energy level of the excited bound state is close enough to the barrier level and there is strong enough attractive coupling g1D . We find that the trap value p = 0.68 is sufficient to observe such a transition. We identify this transition by computing the probability current ( ) ∂ψ ∂ψ∗ ~ ψ∗ −ψ , k = 1, 2. (16) jk (x1 , x2 , t) = 2mi ∂xk ∂xk

In Fig.9, with p = 0.68, one can notice that |j(x1 , x2 , t)| at g1D = −2~ωℓ predominantly goes along the x1 = x2 axis, which represents the tunneling of the two-atom system as a bound object, whereas at higher values of g1D , g1D = 0, the tunneling goes along the axes x1 and x2 , which indicates a tunneling of the first particle, i.e. a sequential tunneling scenario. For higher values of p, p = 0.75, the tunneling for the attractive interaction is significantly suppressed and the system starts to decay when we set g1D = 0. Thus, reducing the size of the trap barrier leads to a possibility of observing the correlated pair tunneling.

Figure 7: (Color online) Partition of the configuration space into the regions: R2 (the region that covers the trap size), R1 and R0 . xc is the position at which the CAP starts.

An analysis of the partial probabilities for the upper branch shows that the tunneling predominately goes into the regions R1 . Due to the normalization condition, P2 (t)+P1 (t)+P0 (t) = 1, and that P0 (t) is small, we approximate P1 (t) as P1 (t) ≈ 1 − 5

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Figure 9: (Color online) Probability current |j(x1 , x2 , t)| of the lower excited state branch for the coupling strength values g1D = −2, 0 [in units of ~ωℓ] computed at t = 100 ms and different values of p, which defines a size of the trap. The values of the probability current shown in the legends are in ω/ℓ units (notice the different scale of all the graphs).

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We model the two-atom system with a realistic trap potential used in the recent experiments [15, 16]. We have extended the analyses made in [6, 7, 8, 9] by considering tunneling from higher energy levels and other excited states. We have calculated the energy spectrum of the lowest excited states. In the limit of weak coupling strength g1D → 0 the avoided crossing in the energy spectrum is observed. The nodal patterns of the wave functions of the center-of-mass and relative motion states mixes when the energy levels cross the point g1D = 0. It has been found that the decay from the upper excited state branch goes in a non-monotonic manner as a function of the coupling strength g1D , whereas the tunneling from the lower excited state branch exhibits monotonic dependence as a function of g1D . Since in both cases the same Hamiltonian is used, this suggests that the origin of such dependencies is due to the initial wave function distribution. We find a condition to observe a transition from uncorrelated to correlated pair tunneling. This transition is found by decreasing a size of the trap barrier. The correlated pair tunneling can be observed at strong enough attractive coupling strength g1D . With increasing g1D the two particles decay in a sequential manner. This transition is found to be much suppressed for larger sizes of the trap barrier, for which only the sequential tunneling scenario remains. The particles from the upper excited state branch, in the considered range of g1D , decay in the sequential way. The analysis made in the paper can be extended to problems with more spatial degrees of freedom, which can include transverse optical confinement [8]. It is also interesting to add more particles into the consideration. The concept of adiabaticity, which in our system occurs by slowly changing the interatomic coupling strength, leading to an adiabatic population transfer between an atomic pair and a molecule in a center-of-mass excited state, is important both from fundamental and practical point of views. This includes equilibrium thermodynamics and preparation of complex ground or excited states in interacting isolated systems, used in quantum computation and quantum annealing [40]. These subjects requires complete knowledge of the interparticle correlation impact as well as an influence of the external field on the system’s properties, which can be successfully included in our model. Another promising research can be found by applying an external periodic force. Driven quantum systems exhibit numerous fascinating phenomena, e.g. dynamic localization [41] and in-phase oscillations and dipole-like modes [20, 42, 12]. We expect that our model can also serve as a testing ground for approximate methods such as Floquet theory. ACKNOWLEDGMENTS We thank V. S. Melezhik and K. Hagino for helpful discussions. The authors acknowledge the support by the JINR Young Scientists and Specialists Grant (the grant number 18-302-04).

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[28] S. Hunn, K. Zimmermann, M. Hiller, and A. Buchleitner, Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach, Phys. Rev. A 87, 043626 (2013). [29] J. Dobrzyniecki and T. Sowi´nski, Dynamics of a few interacting bosons escaping from an open well, Phys. Rev. A 98, 013634 (2018). [30] S. Z¨ollner, H.-D. Meyer, and P. Schmelcher, Tunneling dynamics of a few bosons in a double well, Phys. Rev. A 78, 013621 (2008). [31] E. Fasshauer and A. U. J. Lode, Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and fewfermion tunneling to open space, Phys. Rev. A 93, 033635 (2016) [32] J. Erdmann, S. I. Mistakidis, and P. Schmelcher, Correlated Tunneling Dynamics of an Ultracold Fermi-Fermi Mixture Confined in a DoubleWell, arXiv:1809.00908 (2018) [33] I. S. Ishmukhamedov and V. S. Melezhik, Tunneling of two bosonic atoms from a one-dimensional anharmonic trap, Phys. Rev. A 95, 062701 (2017). [34] I. S. Ishmukhamedov, D. S. Valiolda, and S. A. Zhaugasheva, Description of ultracold atoms in a one-dimensional geometry of a harmonic trap with a realistic interaction, Phys. Part. Nuclei Lett. 11: 238 (2014). [35] G. I. Marchuk, Methods of Numerical Mathematics, (Springer-Verlag, New York,1975) Sec. 4.3.3 [36] U. V. Riss and H.-D. Meyer, Calculation of resonance energies and widths using the complex absorbing potential method, J. Phys. B 26, 4503 (1993). [37] I. S. Ishmukhamedov, D. T. Aznabayev, and S. A. Zhaugasheva, Twobody atomic system in a one-dimensional anharmonic trap: The energy spectrum, Phys. Part. Nuclei Lett. 12: 680 (2015). [38] S. Sala and A. Saenz, Theory of inelastic confinement-induced resonances due to the coupling of center-of-mass and relative motion, Phys. Rev. A 94, 022713 (2016). [39] D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Comparison of quantal, classical, and semiclassical behavior at an isolated avoided crossing, J. Chem. Phys. 78, 4018 (1983). [40] P. Weinberg, M. Bukov, L. DAlessio, A. Polkovnikov, S. Vajna, and M. Kolodrubetz, Adiabatic perturbation theory and geometry of periodicallydriven systems, Phys. Rep. 688, 1-35 (2017). [41] A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, (2017). [42] G. M. Koutentakis, S. I. Mistakidis, and P. Schmelcher, Quench-induced resonant tunneling mechanisms of bosons in an optical lattice with harmonic confinement, Phys. Rev. A 95, 013617 (2017).

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ACCEPTED MANUSCRIPT Two interacting atoms subject to anharmonic laser field are initially prepared in the lowest excited quantum states

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The energy levels of the atoms in the limit of weak interatomic interaction become avoided crossings, the origin of which is due to rotational symmetry breaking.

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Tunneling rate of the atoms from the excited states through the optical (laser) trap barrier increases either monotonically or non-monotonically as the interatomic coupling strength increases, depending on the excited state under the consideration.

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Transition from uncorrelated to correlated pair tunneling can be observed by reducing the size of the optical trap barrier and increasing the attraction of the interatomic coupling strength.