Ultracold atomic gases provide a novel platform with which to study spin-orbit coupling, a mechanism that plays a central role in the nuclear shell model, atomic fine structure and two-dimensional electron gases. This paper introduces a theoretical framework that allows for the efficient determination of the eigenenergies and eigenstates of a harmonically trapped two-atom system with short-range interaction subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. Energy spectra for experimentally relevant parameter combinations are presented and future extensions of the approach are discussed. PACS numbers:Over the past decade, much progress has been made in preparing isolated ultracold few-atom systems experimentally [1][2][3][4]. Moreover, a variety of tools for manipulating and probing such systems have been developed. On the theoretical side, a number of analytical and numerical approaches have been developed [5][6][7][8][9][10][11][12][13]. A large number of analytical treatments approximate the true alkali atom-alkali atom potential by a zero-range potential [8,[14][15][16]. This replacement yields reliable results in the low-energy regime where the de Broglie wave length is larger than the van der Waals length. For example, using zero-range contact interactions, the energy spectrum of two harmonically trapped atoms has been determined analytically [8][9][10]. These two-body solutions are available in 1D, 2D and 3D [8], and have played a vital role in guiding and interpreting experiments [17][18][19] as well as in theoretical studies of the two-body dynamics [20,21] and of larger harmonically trapped systems [11][12][13][22][23][24].Recently, synthetic gauge fields, which allow for the realization of Hamiltonians that contain spin-orbit coupling terms, have been realized experimentally [25][26][27][28][29][30][31][32][33][34][35]. The purpose of this paper is to address how the trapped two-particle spectrum, obtained by modeling the twobody interaction via a zero-range δ-function, changes in the presence of spin-orbit and Raman coupling. While the two-particle system with spin-orbit coupling in free space [36][37][38][39] as well as the trapped single-particle system with spin-orbit coupling [40,41] have received considerable attention, little is known about the trapped twoparticle system with spin-orbit coupling and two-body interaction [42,43]. In going from the trapped single-atom to the trapped two-atom system, a new length scale, i.e., the atom-atom scattering length, comes into play. Thus, an interesting question concerns the interplay between the interaction energy and the energy scales associated with the spin-orbit and Raman coupling strengths.Our framework applies to the situation where the spinorbit (or more precisely, spin-momentum) coupling term acts, as in recent experiments [29][30][31][32][33][34][35], along one spatial direction, say the x-direction. This corresponds to an equal mixture of Rashba and Dresselhaus spi...

Double-well systems loaded with one, two, or many quantum particles give rise to intriguing dynamics, ranging from Josephson oscillation to self-trapping. This work presents theoretical and experimental results for two distinct double-well systems, both created using dilute rubidium Bose-Einstein condensates with particular emphasis placed on the role of interaction in the systems. The first is realized by creating an effective two-level system through Raman coupling of hyperfine states. The second is an effective two-level system in momentum space generated through the coupling by an optical lattice. Even though the non-interacting systems can, for a wide parameter range, be described by the same model Hamiltonian, the dynamics for these two realizations differ in the presence of interactions. The difference is attributed to scattering diagrams that contribute in the lattice coupled system but vanish in the Raman coupled system. The internal dynamics of the Bose-Einstein condensates for both coupling scenarios is probed through a Ramsey-type interference pulse sequence, which constitutes a key building block of atom interferometers. These results have important implications in a variety of contexts including lattice calibration experiments and momentum space lattices used for quantum analog simulations.

Most interferometers operate with photons or dilute, non-condensed cold atom clouds in which collisions are strongly suppressed. Spinor Bose-Einstein condensates (BECs) provide an alternative route toward realizing three-mode interferometers; in this realization, spin-changing collisions provide a resource that generates mode entanglement. Working in the regime where the pump mode, i.e., the m = 0 hyperfine state, has a much larger population than the side or probe modes (m = ±1 hyperfine states), f = 1 spinor BECs approximate SU(1,1) interferometers. We derive analytical expressions within the undepleted pump approximation for the phase sensitivity of such an SU(1,1) interferometer for two classes of initial states: pure Fock states and coherent spin states. The interferometer performance is analyzed for initial states without seeding, with single-sided seeding, and with double-sided seeding. The validity regime of the undepleted pump approximation is assessed by performing quantum calculations for the full spin Hamiltonian. Our analytical results and the associated dynamics are expected to guide experiments as well as numerical studies that explore regimes where the undepleted pump approximation makes quantitatively or qualitatively incorrect predictions.

Previous work developed a K-matrix formalism applicable to positive energies for the scattering between two s-wave interacting particles with two internal states, isotropic spin-orbit coupling and vanishing center-of-mass momentum [H. Duan, L. You and B. Gao, Phys. Rev A 87, 052708 (2013)]. This work extends the formalism to the entire energy regime. Explicit solutions are obtained for the total angular momentum J = 0 and 1 channels. The behavior of the partial cross sections in the negative energy regime is analyzed in detail. We find that the leading contributions to the partial cross sections at the negative energy thresholds are governed by the spin-orbit coupling strength kso and the mass ratio. The fact that these contributions are independent of the two-body scattering length as is a direct consequence of the effective reduction of the dimensionality, and hence of the density of states, near the scattering thresholds due to the single-particle spin-orbit coupling terms. The results are analytically continued to the energy regime where bound states exist. It is shown that our results are consistent with results obtained by alternative approaches. Our formulation, which can be regarded as an extension of the standard textbook partial wave decomposition, can be generalized to two-body systems with other types of spin-orbit coupling, including cases where the center-of-mass momentum does not vanish. PACS numbers: