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arXiv:quant-ph/0304093 · Physical Review Letters

Quantum State Control via Trap-Induced Shape Resonance in Ultracold Atomic Collisions

R. Stock, Ivan Deutsch, Eric L. Bolda

Published Oct 31, 2003• 55 citations
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Abstract

We investigate controlled collisions between trapped but separated ultracold atoms. The interaction between atoms is treated self-consistently using an energy-dependent delta-function pseudopotential model, whose validity we establish. At a critical separation, a "trap-induced shape resonance" between molecular bound states and a vibrational eigenstate of the trap can occur. This resonance leads to an avoided crossing in the eigenspectrum as a function of separation. We investigate how this new resonance can be employed for quantum control.

DOI
10.1103/physrevlett.91.183201

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We investigate controlled collisions between trapped but separated ultracold atoms. The interaction between atoms is treated self-consistently using an energy-dependent delta-function pseudopotential model, whose validity we establish. At a critical separation, a "trap-induced shape resonance" between molecular bound states and a vibrational eigenstate of the trap can occur. This resonance leads to an avoided crossing in the eigenspectrum as a function of separation. We investigate how this new resonance can be employed for quantum control.
We consider controlled collisions between two ultracold atoms guided by external harmonic potentials. We derive analytical solutions of the Schr\"odinger equation for this system and investigate the properties of eigenergies and eigenstates for different trap geometries as a function of a trap separation and of the scattering length. When varying the trap separation the energy spectrum exhibits avoided crossings, corresponding to trap-induced shape resonances. Introducing an energy-dependent scattering length we investigate the behavior of the system in the vicinity of a magnetic Feshbach resonance. Finally, we illustrate our analytical results with two examples: the quantum phase gate controlled by the external magnetic field and a scheme for a coherent transport of atoms in optical lattices into higher Bloch bands.
We consider the feasibility of observing a trap-induced resonance [Stock et al., Phys. Rev. Lett. 91, 183201 (2003)] for the case of two 133Cs^{133}\mathrm{Cs}133Cs atoms, trapped in separated wells of a polarization-gradient optical lattice, and interacting through a multichannel scattering process. Due to the anomalously large scattering length of cesium dimers, a strong coupling can occur between vibrational states of the trap and a weakly bound molecular state that is made resonant by the ac Stark shift of the lattice. We calculate the energy spectrum of the two-atom system as a function of the distance between two potential wells by connecting the solutions of the Schr\"odinger equation for the short-range molecular potential to that of the long-range trap in a self-consistent manner. The short-range potential is treated through a multichannel pseudopotential, parametrized by the KKK matrix, calculated numerically for atoms in free space in a close-coupling approximation. This captures both the bound molecular spectrum as well as the energy-dependent scattering for all partial waves. We establish realistic operating conditions under which the trap-induced resonance could be observed and show that this strong and coherent interaction could be used as a basis for high-fidelity two-qubit quantum logic operations.
We consider a system composed of a trapped atom and a trapped ion. The ion charge induces in the atom an electric dipole moment, which attracts it with an ${r}^{\ensuremath{-}4}$ dependence at large distances. In the regime considered here, the characteristic range of the atom-ion interaction is comparable or larger than the characteristic size of the trapping potential, which excludes the application of the contact pseudopotential. The short-range part of the interaction is described in the framework of quantum-defect theory, by introducing some short-range parameters, which can be related to the sss-wave scattering length. When the separation between traps is changed we observe trap-induced shape resonances between molecular bound states and vibrational states of the external trapping potential. Our analysis is extended to quasi-one-dimensional geometries, when the scattering exhibit confinement-induced resonances, similar to the ones studied before for short-range interactions. For quasi-one-dimensional systems we investigate the effects of coupling between the center of mass and relative motion, which occurs for different trapping frequencies of atom and ion traps. Finally, we show how the two types of resonances can be employed for quantum state control and spectroscopy of atom-ion molecules.
A (diatomic) shape resonance is a metastable state of a pair of colliding atoms quasi-bound by the centrifugal barrier imposed by the angular momentum involved in the collision. The temporary trapping of the atoms' scattering wavefunction corresponds to an enhanced atom pair density at low interatomic separations. This leads to larger overlap of the wavefunctions involved in a molecule formation process such as photoassociation, rendering the process more efficient. However, for an ensemble of atoms, the atom pair density will only be enhanced if the energy of the resonance comes close to the temperature of the atomic ensemble. Herein we explore the possibility of controlling the energy of a shape resonance by shifting it toward the temperature of atoms confined in a trap. The shifts are imparted by the interaction of non-resonant light with the anisotropic polarizability of the atom pair, which affects both the centrifugal barrier and the pair's rotational and vibrational levels. We find that at laser intensities of up to 5×1095\times 10^{9}5×109 W/cm2^22 the pair density is increased by one order of magnitude for 87^{87}87Rb atoms at 100μ100 μ100μK and by two orders of magnitude for 88^{88}88Sr atoms at 20μ20 μ20μK.
We study the out-of-equilibrium dynamics of noninteracting atoms confined within a one-dimensional harmonic trap triggered by dragging an external long-range potential through the system. The symmetry-breaking nature of this moving potential couples adjacent eigenstates in the atoms' effective potential, leading to an energy landscape reminiscent of systems exhibiting trap-induced shape resonances. These couplings may be exploited to selectively excite the atoms into higher vibrational states of the harmonic trap by controlling the motion of the dragged potential. To this end, we consider two protocols designs: The first protocol strives to maintain adiabaticity at critical points during the atoms' dynamics, while the second protocol utilizes the fast tunneling of the atoms within their effective double-well potential. These protocols take place in the few to many millisecond regime and achieve high-fidelity excitation of the atoms into pure vibrational states and superpositions thereof. Overall, our study highlights the significance of dragged potentials for controlling and manipulating atom dynamics and offers intuitive protocols for achieving desired excitations.
In quantum mechanics, collisions between two particles are captured by a scattering matrix which describes the transfer from an initial entrance state to an outgoing final state. Analyticity of the elements of this SSS matrix enables their continuation onto the complex energy plane and opens up a powerful and widely used framework in scattering theory, where bound states and scattering resonances for a physical system are ascribed to SSS-matrix poles. In the Gedankenexperiment of gradually changing the potential parameters of the system, the complex energy poles will begin to move, and in their ensuing flow, two poles approaching will interact. An actual observation of this intriguing interaction between scattering poles in a collision experiment has, however, been elusive. Here, we expose the interplay between two scattering poles relating to a shape resonance and a magnetically tunable Feshbach resonance by studying ultracold atoms with a laser-based collider. We exploit the tunability of the Feshbach resonance to observe a compelling avoided crossing of the poles in their energies which is the hallmark of a strongly coupled system.