Effects of three-body collisions in a two-mode Bose-Einstein condenstate

We study the effects of three-body collisions in the basic physical properties of a two-mode Bose-Einstein condensate. By finding the exact analytical solution of a model which includes two-body and three-body elastic and mode-exchange collisions, we show analytically that three-body interactions produce observable effects in the probability distribution of the ground state and the dynamics of the relative population. In particular, we find that three-body interactions under certain circumstances inhibit collapse of the relative population.

PACS numbers: 03.75.Gg, 42.50.Gy,03.75.Lm Most of our understanding of condensed matter is based on models which consider two-body collisions. However, in many situations three-and more-body collisions are relevant in the physical properties of such systems [1][2][3]. For example, three-body collisions are known to be important interactions in Hamiltonians which give rise to exotic quantum phases, such as topological phases [2] or spin liquids [3]. Moreover, many-body collisions are suspected to be important in the coldest phases of Bose-Einstein condensates where the dilute regime brakes down [4]. Microscopic calculations show that polar molecules driven by microwave fields undergo three-body interactions [5]. The interaction potentials of molecules trapped in an optical lattice give rise to Hubbard models with strong nearest-neighbor two-body and three-body interactions.
In this letter we find the exact analytical solution of a generalized two-mode Bose-Hubbard model which includes two-body and three-body elastic and modeexchange collisions. We then show that three-body collisions are relevant in the ground state properties of a two-mode Bose-Einstein condensate. The effects are also observable in the evolution of the relative population inhibiting, in some cases, quantum collapse. It is well known that three-body collisions produce particle loss in Bose-Einstein condensates by a process called three-body recombination [6]. During three-body collisions particles recombine to form a molecule which is not trapped by the potential. However, it has been shown that it is now possible to inhibit molecule three-body recombination in atomic Bose-Einstein condensates via the application of resonant 2π laser pulses [7]. In such situations, our model becomes of special interest since it considers three-body collisions where particles do not recombine and thus, remain trapped in the potential after the collision.
The model we introduce is applicable to describe the physics of a double-well Bose-Einstein condensate or a spin-1/2 Bose-Einstein condensate consisting of particles with two internal degrees of freedom trapped in a sin-gle well. In the context of the double-well Bose-Einstein condensate, the mode-exchange collisions we include are know as generalized nearest neighbor interactions [8] and give rise to coherent tunneling effects [9,10]. Recent analysis show that stronger two-body interactions are correlated with two-body coherent tunneling dynamics in which two particles simultaneously tunnel through the barrier [10]. This effect, also known as second order tunneling, has recently been observed in the laboratory [9]. Mode-exchange collisions are called inelastic collisions in the context of spin-1/2 condensates and occur when cold collisions take place in the presence of light fields. Such is the case in spin-1/2 condensates when a laser field is used to induce Josephson-type interactions which produce transitions among the spin degrees of freedom [11].
The modes a † , a and b † , b with frequencies λ a|a and λ b|b respectively, describe either atoms with two different hyperfine levels [12] or alternatively, two spatially separated condensates [13]. The Josephson-type interaction is induced by applying a laser [12] or a magnetic field gradient [13]. The Josephson-type term, in which one particle is annihilated in one mode and created in the other, has coupling constant λ a|b . The terms in H 2 which have four bosonic operators describe two-particle collisions. The two-body elastic scattering strengths are given by U aa|aa and U bb|bb for same mode collisions and U ab|ab when the particles colliding belong to different modes. Mode-exchange collisions have interaction strengths U aa|ab , U bb|ab when two particles collide and one of them is transformed into the other mode and interaction strength U aa|bb when the collision transforms two particles in one mode into the other mode. This process is also know as second order tunneling in the context of a double-well BEC [9,10].
The Hamiltonian H 2 = H 1 + H 2 has been studied in detail in [14]. This two-body interaction Hamiltonian coincides with the two-mode Bose-Hubbard model in the case where mode-exchange collisions are neglected. i.e. U aa|ab = U bb|ab = U aa|bb = 0. However, microscopical calculations show that such interactions, known as inelastic collisions in the context of spin-1/2 Bose-Einstein condensates, should be considered since they occur when particles collide in the presence of a laser field [11]. Surprisingly, including such collision allows for an exact analytical solution [14][15][16][17]. Here we include a three-body collision term given by H 3 , where threebody interactions consist of products of six operators (three creation and three annihilation). This term includes all possible three-body collisions where U aaa|aaa , U bbb|bbb , U aab|aab and U abb|abb correspond to elastic scattering lengths and U aaa|aab , U aaa|abb , U aaa|bbb correspond to mode-exchange collisions where one, two and three particles change mode, respectively.
We have found that the Hamiltonian H 3 has six families of exact analytical solutions. In this paper we present the solution which we consider of greatest physical interest. The other solutions will be presented elsewhere.
We start by considering the double-well potential shown in Fig.(1). Particles undergo two-and three-body collisions and we assume that first, second and third order tunneling events can occur. In second (third) order tunneling two (three) tunneling events can occur coherently. Therefore single particles can coherently tunnel two (three) times and two (three) particles can tunnel simultaneously during a collision. We consider that a particle in well A (or B) has probability amplitude A 1 cos θ (or −A 1 cos θ) of staying in well A (or B) and probability amplitude A 1 sin θ of tunneling to well B (or A). A 1 is the first order tunneling strength and θ is the tunneling phase. Note that the minus sign appears because we chose for simplicity well B to have negative energy corresponding to λ b|b = −λ a|a . We consider A 2 and A 3 to be second and third order tunneling strengths. Therefore A 2 sin 2 θ and A 3 sin 2 θ cos θ for example, are the second and third order probability amplitudes respectively, for a single particle in well A to tunnel back and forth.
The coefficients in the single particle Hamiltonian H 1 are found by considering all possible single particle events including second and third order tunneling. For example, + A 3 cos θ(cos 2 θ + sin 2 θ) = A 2 + (A 3 + A 1 ) cos θ is the probability amplitude for a single particle in well A to end in well A. The general two-body and threebody scattering lengths U ij|lm and U ijk|lmn are given by the product of the corresponding second and third order tunneling strengths times the appropriate tunneling phase amplitudes (sin θ if the particle tunnels during the collision and ± cos θ if the particle stays). For example, consider a three-body collision during which two particles change state. The total probability amplitude will be The factor 3 comes from the fact that there are three possible events that give rise to the same final outcome, according to the different time ordering of the events.
In the case of two-body collisions we consider that during a collision two and three tunneling events can occur. So collisions in which two particles in well A end up both in well B is given by U aa|bb = A 2 sin 2 θ + 3A 3 (cos θ sin 2 θ − sin 2 θ cos θ) = A 2 sin 2 θ, (4) where again the factor 3 comes from the time ordering.
Such considerations give rise to the parameters, Surprisingly, the Hamiltonian H 3 = H 1 + H 2 + H 3 has an exact analytical solution for this set of parameters. The solution is where 2J = N is the total number of particles, given by the operator:N and m the eigenvalue of the relative population The unitary operator e θ 2 (a † b−ab † ) is known as the twomode displacement operator with real displacement parameter θ. Since the number of particles in the system N is constant, m is restricted to values m = −J, ..., J.
Interestingly, the solution for m = 0 corresponds to the coherent state which has previously be found to describe appropriately several physical aspects of the two-mode Bose-Einstein condensate [19]. To verify that e −θ 2 (a † b−ab † ) |J, m is a solution of the Hamiltonian one must simply apply the two-mode displacement operator to the diagonal Hamiltonian The ground state of the system e −θ 2 (a † b−ab † ) |J, m 0 is found by minimizing the energy E = A 1 m + A 2 m 2 + A 3 m 3 . We obtain, If 3A 1 A 3 /A 2 2 > 1, m 0 is a complex number and the energy has no local minima. Therefore, the minimum energy will correspond to the extreme point m 0 = −N A 3 /|A 3 |. However, if 3A 1 A 3 /A 2 2 < 1 the minimum, which is given by Eq.(10), is m + 0 for A 3 > 0 and m − 0 when A 3 < 0. Another quantity of interest is the probability distribution of the relative population for the ground state, which is given by are the Wigner rotation matrix elements. Note that the sum must be done over k whenever none of the arguments of factorials in the denominator are negative. Different ground states parameterized by m 0 are obtained by changing the rate A 1 A 3 /A 2 2 . We plot in Fig.(2) an example for N = 100 particles with A 3 = 0 (i.e. assuming there are no third order tunneling and three-body collisions) and m 0 = A 1 /2A 2 = 100. Such a state has a single peak distribution. However, when A 3 = 1/600 we obtain an N peak distribution corresponding to m 0 = 0. This shows how three-body collisions and third order tunneling drastically change the structure of the ground state of the system.
We now analyze the effects of three-body collisions in the evolution of the average relative population m = a † a − b † b , for an initial condition |ψ(t = 0) . The evolution of the relative population is given by where the coefficients C m are defined by We can observe in Fig. (3) that three-body collisions have a noticeable effect on the behavior of the time evolution of the system, and in fact they tend to inhibit collapses in the relative population. In summary, we introduce a model of two-mode Bose-Einstein condensate which includes not only two-body but also three-body interactions. We find an analytical solution and provide the full spectrum of eigenvalues and the corresponding eigenvectors. This allows us to analyse the role of three-body interactions in physical quantities of interest, such as the probability distribution of the relative population or the time evolution of its expectation value. We find that three-body collisions have non-trivial effects, such as significant changes in the probability distribution of the ground state or the inhibition of collapses in the evolution of the relative population of the modes.
Our work provides insights on the effects of higher order collisions in the physics of a two component Bose-Einstein condensate. Following the formalism employed in this paper, higher-order collisions can also be included in the model [15]. We observe that the presence of three-body collisions changes the dynamics of the system and in fact inhibits quantum collapses.