Lamb-Dicke Parameter Research Articles

Chaotic Behavior in the Raman Interaction of a Trapped Ultracold Ion with Two Traveling Wave Lasers**The project supported by National Natural Science Foundation of China

By means of the Bloch–Maxwell equation, the Raman interaction of a trapped ultracold ion with two traveling wave lasers is treated semiclassically. As the model works without limitation of the Lamb–Dicke limit and the weak excitation regime, we study chaotic behavior of the system in the wide range of the parameters. It is shown that the chaotic behavior is more and more pronounced with the increase of the Lamb–Dicke parameter and Rabi frequency, and can be exhibited experimentally by using quantum jump technique.

Universal quantum computation with two-level trapped ions

Although the initial proposal for ion trap quantum computation made use of an auxiliary internal level to perform logic between ions, this resource is not necessary in principle. Instead, one may perform such operations directly using sideband laser pulses, operating with an arbitrary (sufficiently small) Lamb-Dicke parameter. We explore the potential of this technique, showing how to perform logical operations between the internal state of an ion and the collective motional state and giving explicit constructions for a controlled-not gate between ions.

Trapped ions in laser fields: A benchmark for deformed quantum oscillators

Some properties of the nonlinear coherent states (NCS), recognized by Vogel and de Matos Filho as dark states of a trapped ion, are extended to NCS on a circle, for which the Wigner functions are presented. These states are obtained by applying a suitable displacement operator ${D}_{h}(\ensuremath{\alpha})$ to the vacuum state. The unity resolutions in terms of the projectors $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha}{,h}^{\ensuremath{-}1}|,|\ensuremath{\alpha}{,h}^{\ensuremath{-}1}〉〈\ensuremath{\alpha},h|$ are presented together with a measure allowing a resolution in terms of $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha},h|.$ ${D}_{h}(\ensuremath{\alpha})$ is also used for introducing the probability distribution funtion ${\ensuremath{\rho}}_{A,h}(z)$ while the existence of a measure is exploited for extending the P representation to these states. The weight of the $n\mathrm{th}\mathrm{}$ Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter $\ensuremath{\eta},$ oscillates so wildly as n grows up to infinity that the normalized NCS fill the open circle ${\ensuremath{\eta}}^{\ensuremath{-}1}$ in the complex $\ensuremath{\alpha}$ plane. In addition, this prevents the existence of a measure including normalizable states only. This difficulty is overcome by introducing a family of deformations that are rational functions of n, each of them admitting a measure. By increasing the degree of these rational approximations, the deformation of a trapped ion can be approximated with any degree of accuracy and the formalism of the P representation can be applied.

Realization of Quantum Logic Gates in Ion Traps Without Any Requirement on the Lamb-Dicke Parameter

A scheme is presented for realizing quantum logic gate in an ion trap. In the scheme the state of the mth ion is first replicated onto the collective motion. Then a transformation, conditional on the motional state, is performed on the nth ion. Finally the motional state is replicated back onto the mth ion. By this way the quantum logic gate between the mth and the nth ions is realized. The scheme does not put any requirement on the Lamb-Dicke parameter.

Quantum logic operation with a single trapped ion using the nonlinear interaction beyond the Lamb-Dicke limit

Using the nonlinear interaction between internal and external degrees of a trapped ion assisted by a laser beam, we present simplified methods for two-qubit quantum logic operation. The nonlinear interaction arises when the Lamb-Dicke parameter is not small enough. Two special cases, one with the laser tuned to the first red sideband and the other tuned to the carrier, are discussed in detail.

Quantum logic gate operation between different ions in a trap

We suggest a scheme for realizing a universal two-quantum-bit (qubit) operation. The scheme uses two laser beams with different intensities to illuminate two different ions in a harmonic trap. Both beams are tuned to the red motional sideband. We find that, under certain conditions, the interaction will realize a universal two-qubit operation, the controlled-rotation operation, which is to rotate the phase of the state by $\ensuremath{\pi}$ only when the two states are both in the $|1〉$ state. The scheme requires careful control of laser intensity and Lamb-Dicke parameters. But after we reach the required parameters, the logic operation is very simple.

High-order nonlinearities in the motion of a trapped atom

We study the counterpart to the multi-photon down conversion in the quantised motion of a trapped atom. The Lamb-Dicke approximation leads to a divergence of the mean motional excitation in a finite interaction time for k-quantum down conversions with k>=3, analogous to the situation in the parametric approximation of nonlinear optics. We show that, in contrast to the Lamb-Dicke approximation, the correct treatment of the overlap of the atomic center-of-mass wave function and the driving laser waves leads to a proper dynamics without any divergence problem. That is, the wavy nature of both matter and light is an important physical property which cannot be neglected for describing the motional dynamics of a trapped atom, even for small Lamb-Dicke parameters.

Exact solution of a trapped ultracold ion interacting with a standing wave laser without rotating wave approximation

In coherent state representation and in the absence of rotating wave approximation, the interaction between a standing wave laser and a trapped ultracold ion within the Lamb-Dicke limit, expanded to the second-order term of the Lamb-Dicke parameter, can be solved analytically by series expansion technique. It can be seen that, for some particular detunings, the eigenenergies of the system can be exactly obtained and the eigenfunctions, presented as series forms, are a superposition of displaced Fock states and displaced squeezed states. Making use of our solution, some particular nonclassical states could be prepared.

The master equation for laser cooling of trapped particles

The authors consider the laser cooling of a trapped two-level system during its final approach to equilibrium. Then it moves only within one optical wavelength and an expansion in the Lamb-Dicke parameter is possible. This allows an adiabatic elimination of the internal degrees of freedom. There remains a slow time evolution on a new time scale related to the Lamb-Dicke parameter. This scale is determined by an effective time evolution operator, which the authors derive using the method of degenerate perturbation theory. The ensuing master equation can be completely solved both for its time evolution and the ultimate steady state. In addition to providing a complete description of the final stages of the laser cooling, the calculation can be seen to add one more soluble case to the discussion of non-equilibrium statistical mechanics.