Lamb-Dicke Parameter Research Articles

Toward a Mølmer Sørensen gate with .9999 fidelity

Abstract Realistic fault-tolerant quantum computing at reasonable overhead requires two-qubit gates with the highest possible fidelity. Typically, an infidelity of ≲ 10 − 4 is recommended in the literature. Focusing on the phase-sensitive architecture used in laboratories and by commercial companies to implement quantum computers, we show that even under noise-free, ideal conditions, neglecting the carrier term and linearizing the Lamb–Dicke term in the Hamiltonian used for control-pulse construction for generating Mølmer–Sørensen XX gates based on the Raman scheme are not justified if the goal is an infidelity target of < 10 − 4 . We obtain these results with a gate simulator code that, in addition to the computational space, explicitly takes the most relevant part of the phonon space into account. With the help of a Magnus expansion carried to the third order, keeping terms up to the fourth order in the Lamb–Dicke parameters, we identify the leading sources of coherent errors, which we show can be eliminated by adding a single linear equation to the phase-space closure conditions and subsequently adjusting the amplitude of the control pulse (calibration). This way, we obtain XX gates with infidelities < 10 − 4 .

Pulse optimization for high-precision motional-mode characterization in trapped-ion quantum computers

High-fidelity operation of quantum computers requires precise knowledge of the physical system through characterization. For motion-mediated entanglement generation in trapped ions, it is crucial to have precise knowledge of the motional-mode parameters such as the mode frequencies and the Lamb–Dicke parameters. Unfortunately, the state-of-the-art mode-characterization schemes do not easily render the mode parameters in a sufficiently accurate and efficient fashion for large-scale devices, due to the unwanted excitation of adjacent modes in the frequency space when targeting a single mode, an effect known as the cross-mode coupling. Here, we develop an alternative scheme that leverages the degrees of freedom in pulse design for the characterization experiment such that the effects of the cross-mode coupling is actively silenced. Further, we devise stabilization methods to accurately characterize the Lamb–Dicke parameters even when the mode frequencies are not precisely known due to experimental drifts or characterization inaccuracies. We extensively benchmark our scheme in simulations of a three-ion chain and discuss the parameter regimes in which the shaped pulses significantly outperform the traditional square pulses.

Continuous Raman sideband cooling beyond the Lamb-Dicke regime in a trapped ion chain

We report continuous Raman sideband cooling of a long ion chain close to the motional ground state beyond the Lamb-Dicke (LD) regime. By driving multiple sideband transitions simultaneously, we show that nearly all axial modes of a 24-ion chain are cooled to the ground state, with an LD parameter as large as $\ensuremath{\eta}=1.3$, spanning a frequency bandwidth of 4 MHz. Compared to traditional ground-state cooling methods such as pulsed sideband cooling or electromagnetically-induced-transparency cooling, our method offers two key advantages: disposal of pulse optimization, and a wide bandwidth covering the entire spectrum. This technique contributes as a crucial step for large-scale quantum information processing with linear ion chains and higher dimensions alike, and can be readily generalized to other atomic and molecular systems.

Photon scattering errors during stimulated Raman transitions in trapped-ion qubits

We study photon scattering errors in stimulated Raman driven quantum logic gates. For certain parameter regimes, we find that previous simplified models of the process significantly overestimate the gate error rate due to photon scattering. This overestimate is shown to be due to previous models neglecting the detuning dependence of the scattered photon frequency and Lamb-Dicke parameter, a second scattering process, interference effects on scattering rates to metastable manifolds, and the counterrotating contribution to the Raman transition rate. The resulting improved model shows that there is no fundamental limit on gate error due to photon scattering for electronic ground-state qubits in commonly used trapped-ion species when the Raman laser beams are red detuned from the main optical transition. Additionally, photon scattering errors are studied for qubits encoded in the metastable ${D}_{5/2}$ manifold, showing that gate errors below ${10}^{\ensuremath{-}4}$ are achievable for all commonly used trapped ions.

Efficient motional-mode characterization for high-fidelity trapped-ion quantum computing

To achieve high-fidelity operations on a large-scale quantum computer, the parameters of the physical system must be efficiently characterized with high accuracy. For trapped ions, the entanglement between qubits are mediated by the motional modes of the ion chain, and thus characterizing the motional-mode parameters becomes essential. In this paper, we develop and explore physical models that accurately predict both magnitude and sign of the Lamb–Dicke parameters when the modes are probed in parallel. We further devise an advanced characterization protocol that shortens the characterization time by more than an order of magnitude, when compared to that of the conventional method that only uses mode spectroscopy. We discuss potential ramifications of our results to the development of a scalable trapped-ion quantum computer, viewed through the lens of system-level resource trade offs.

Generation of Schrödinger Cat States in a Hybrid Cavity Optomechanical System.

We present an alternative scheme to achieve Schrödinger cat states in a strong coupling hybrid cavity optomechanical system. Under the single-photon strong-coupling regime, the interaction between the atom-cavity-oscillator system can induce the mesoscopic mechanical oscillator to Schrödinger cat states. Comparing to previous schemes, the proposed proposal consider the second order approximation on the Lamb-Dicke parameter, which is more universal in the experiment. Numerical simulations confirm the validity of our derivation.

Numerical Simulation of the Performance of Single Qubit Gates for Trapped Ions

Finite gate errors limit performance of modern quantum computers. In this work, we study single qubit gate fidelities for trapped ions. For this we have numerically solved Schrödinger equation using full Hamiltonian of the system for one, two, three and four ions. This approach allows us to analyze gate errors beyond the LambDicke approximation and to take into account not only a finite occupation of the phonon modes, but also the effects related to the ion–phonon entanglement. As a result, we show how infidelity of the global single qubit gates depend on the initial phonon mode occupations, the Lamb–Dicke parameter, Rabi frequency and the number of ions.

Motion-selective coherent population trapping for subrecoil cooling of optically trapped atoms outside the Lamb-Dicke regime

We propose a scheme that combines velocity-selective coherent population trapping (CPT) and Raman sideband cooling (RSC) for subrecoil cooling of optically trapped atoms outside the Lamb-Dicke regime. This scheme is based on an inverted $\mathsf{Y}$ configuration in an alkali-metal atom. It consists of a $\mathrm{\ensuremath{\Lambda}}$ formed by two Raman transitions between the ground hyperfine levels and the $D$ transition, allowing RSC along two paths and formation of a CPT dark state. Using the state-dependent difference in vibration frequency of the atom in a circularly polarized trap, we can tune the $\mathrm{\ensuremath{\Lambda}}$ to make only the motional ground state a CPT dark state. We call this scheme motion-selective coherent population trapping (MSCPT). We write the master equations for RSC and MSCPT and solve them numerically for a $^{87}\mathrm{Rb}$ atom in a one-dimensional optical lattice when the Lamb-Dicke parameter is 1. Although MSCPT reaches the steady state slowly compared with RSC, the former consistently produces colder atoms than the latter. The numerical results also show that subrecoil cooling by MSCPT outside the Lamb-Dicke regime is possible under a favorable, yet experimentally feasible, condition. We explain this performance quantitatively by calculating the relative darkness of each motional state. Finally, we discuss the application of the MSCPT scheme to an optically trapped diatomic polar molecule whose Stark shift and vibration frequency exhibit large variations depending on the rotational quantum number.

Motion-selective coherent population trapping by Raman sideband cooling along two paths in a Λ configuration

We report our experiment on sideband cooling with two Raman transitions in a $\Lambda$ configuration that allows selective coherent population trapping (CPT) of the motional ground state. The cooling method is applied to $^{87}$Rb atoms in a circularly-polarized one-dimensional optical lattice. Owing to the vector polarizability, the vibration frequency of a trapped atom depends on its Zeeman quantum number, and CPT resonance for a pair of bound states in the $\Lambda$ configuration depends on their vibrational quantum numbers. We call this scheme motion-selective coherent population trapping (MSCPT) and it is a trapped-atom analogue to the velocity-selective CPT developed for free He atoms. We observe a pronounced dip in temperature near a detuning for the Raman beams to satisfy the CPT resonance condition for the motional ground state. Although the lowest temperature we obtain is ten times the recoil limit owing to the large Lamb-Dicke parameter of 2.3 in our apparatus, the experiment demonstrates that MSCPT enhances the effectiveness of Raman sideband cooling and enlarges the range of its application. Discussions on design parameters optimized for MSCPT on $^{87}$Rb atoms and opportunities provided by diatomic polar molecules, whose Stark shift shows strong dependence on the rotational quantum number, are included.

Fast multi-qubit global-entangling gates without individual addressing of trapped ions

We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb–Dicke approximation by including up to second order terms of the Lamb–Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.

Two measures elaborated for entangled states: Quantum entropy and fidelity using Schmidt coefficients of the reduced density matrix of full TRI

In the present study, we determined quantum entanglement in a full trapped ion (TRI)-coherent system and its dependence on the LambDicke parameter (LDP). We investigated the entanglement in view of two elaborated measurements of the family: entropy and fidelity. We selected three values of the deep LDP to demonstrate the benefits of these two critical measures. The findings obtained in this study showed that the maximum value of fidelity for entangled states is quantified approximately to be 0.35, and the long lifetime is also observed with entropy measurement. The findings suggest that three coupling parameters play a significant role in developing quantum entanglement.

Entanglement and atomic inversion for two two-level atoms interacting with an intensity-dependent quantized field

Our aim is to present an analytical solution for the quantum optics model that has two two-level atoms interacting together and interacting with an intensity-dependent quantized field. The quantized field is initially in binomial state and based on the Lamb–Dicke parameter and associated Laguerre polynomials. While two atoms are symmetric and initially in superposition state. Besides, the quantum entanglement between the quantum system components under the effect of the parameters of detuning, coupling, binomial, Lamb–Dicke and the initial state of the model is discussed. Several phenomena have been clearly observed and analyzed, such as entanglement sudden death (ESD) followed by entanglement sudden birth (ESB) in the concurrence and the negativity curves, and long-lived quantum coherence in the von Neumann entropy curve. In addition, the presence of revivals and collapses in the atomic population inversion curves has been figured out. Our analysis aims to help the scientists to determine the critical values required of quantum system parameters to control entanglement dynamics.

Nonlinear interaction of a three-level atom with a two-mode field in an optomechanical cavity: Field and mechanical mode dissipations

In this paper we investigate the dynamics of f-deformed interaction (nonlinear atom-field coupling) of a three-level atom in V-configuration with a two-mode quantized field in the presence and absence of Kerr medium as well as the phonon and photon dissipations in an optomechanical cavity. We solve the associated time-dependent Schrödinger equation and arrive at the corresponding state vector at arbitrary time. In the continuation, we evaluate several nonclassical properties including atomic von Neumann entropy, atomic information entropy squeezing, sub-Poissonian statistics and atomic squeezing, numerically. Our numerical results show that, with the parameters at our disposal, we can adjust the above-mentioned properties according to our purposes. For instance, significant amount of von Neumann entropy may be achieved, in the linear and nonlinear atom-field coupling, without or even with considered dissipations. However, atomic information entropy squeezing can be observed only in the presence of nonlinear atom-field coupling by tuning the proper values of Lamb-Dicke parameter. Atomic squeezing can be seen in the presence or absence of linear and nonlinear atom-field coupling. Negative values of Mandel parameter, showing the sub-Poissonian statistics as the nonclassicality of the field can be observed with and without nonlinear atom-field coupling, however, the presence of Lamb-Dicke nonlinearity can make this effect more visible. Unless the atomic information entropy squeezing, in all above-mentioned physical phenomena stability of the evaluated parameters can be accessible via choosing appropriate parameters containing Lamb-Dicke parameter, Kerr effects, dissipation parameters, laser pumps and photon-phonon coupling.

Fidelity and concurrence of entangled qudits between a trapped ion and two laser beams

A quantum scheme is presented by which a three-level trapped ion interacts with a two laser beams in the absence and presence of the e®ect of classical field. We analyze the impact of the classical ¯eld and the Lamb-Dicke parameter (LDP) on the dynamical behavior of entanglement quanti¯er, population probabilities and the geometric phase. Based on four different variations of these two effects, LDP = 0.1, LDP=0.01 and ¯ = 0.0, ¯ = 0.49, the time dependence of geometric phase and populations probabilities are shown. The ¯nding emphasizes that both the time-dependent and LDP play an important role in the development of the entanglement, the geometric phase, ¯delity, and populations probabilities. This in-sight may be very useful in various applications in quantum optics and information processing.

Steady-state phonon occupation of electromagnetically-induced-transparency cooling: Higher-order calculations

Electromagnetically induced transparency (EIT) cooling has established itself as one of the most widely used cooling schemes for trapped ions during the past twenty years. Compared to its alternatives, EIT cooling possesses important advantages such as a tunable effective linewidth, a very low steady state phonon occupation, and applicability for multiple ions. However, existing analytic expression for the steady state phonon occupation of EIT cooling is limited to the zeroth order of the Lamb-Dicke parameter. Here we extend such calculations and present the explicit expression to the second order of the Lamb-Dicke parameter. We discuss several implications of our refined formula and are able to resolve certain difficulties in existing results.

Nonclassical effects in a nonlinear two trapped-particles system under intrinsic decoherence

Abstract In this paper, we analytically explore the dynamics of a nonlinear two-qubit system derived from a physical model that describes laser-irradiated two trapped particles in the Lamb-Dicke regime under appropriate resonance conditions and the intrinsic decoherence. The dynamics of the particle population inversion, the entanglement between the two trapped-particles and their center-of-mass modes, and the entanglement between the two trapped-particles are investigated under the Lamb-Dicke parameter and the intrinsic decoherence. In is found that, in the absence of decoherence, the generated particle-field and two trapped-particles entanglement can be enhanced by increasing the Lamb-Dicke parameter. The Lamb-Dicke nonlinearity effect on the generated stationary entanglement, the sudden death and sudden birth of the entanglement, and the intrinsic decoherence effect become more pronounced with its large values. For the high Lamb-Dicke non-linearity, the decoherence effect on the generated nonclassical effects can be weakened.

Fast cooling of trapped ion in strong sideband coupling regime

Trapped ion in the Lamb–Dicke (LD) regime with the LD parameter η ≪ 1 can be cooled down to its motional ground state using sideband cooling. Standard sideband cooling works in the weak sideband coupling (WSC) limit, where the sideband coupling strength is small compared to the natural linewidth γ of the internal excited state, with a cooling rate much less than γ. Here we consider cooling schemes in the strong sideband coupling (SSC) regime, where the sideband coupling strength is comparable or even greater than γ. We derive analytic expressions for the cooling rate and the average occupation of the motional steady state in this regime, based on which we show that one can reach a cooling rate which is proportional to γ, while at the same time the steady state occupation increases by a correction term proportional to η 2 compared to the WSC limit. We demonstrate with numerical simulations that our analytic expressions faithfully recover the exact dynamics in the SSC regime.

NOON state generation beyond the Lamb–Dicke limit in trapped-ion systems

In this paper, we show that multi-phonon NOON states can be generated in a two-dimensional anisotropic trapped-ion system. In the proposal, two laser pulses are applied to an ion along different directions in the ion trap plane to exchange the information between the external and internal states of the ion. Different from the previous frameworks, the proposal is outside the Lamb–Dicke regime. The distinct advantage of the proposed scheme is that the entanglement generation is deterministic and no measurement on the system is required. Numerical simulations show that the fidelity of the prepared entangled states is strongly affected by Lamb–Dicke parameters.

Analyzing of Quantum Entanglement with Concurrence in the Deep Lamb-Dicke Regime

The entangled states of three-level of trapped ion and two phonons (into coherent state) in Lambda configuration forming a Hilbert space of 12-dimensional (D) are analyzed. The concurrence is considered "such as a quantum measure" in trapped ion-coherent state system. Four values of Lamb-Dicke parameter (LDP), eta=0.005, 0.07, 0.08$ and $0.09 are investigated by the worths of concurrence. We demonstrate that as eta is increased, sudden birth of entangled states in the system is increased. Thus, establishing of entangled states can be tuned by LDP.

Quantifying of quantum entanglement in Schrödinger cat states with the trapped ion-coherent system for the deep Lamb-Dick regime

The entangled qudits of three-level trapped ion and two phonons (in coherent state) in $$\Lambda $$ configuration forming a Hilbert space of 12-D are investigated. The quantum entropy is analyzed “such as an elaborated measure” in trapped ion-coherent state system. Four values of Lamb-Dicke parameter (LDP), $$\eta =0.005, 0.07, 0.08$$ and 0.09 are probed for deep Lamb-Dick regime. We elucidate that as $$\eta $$ is increased, sudden death of entangled state in the trapped ion-coherent state system is decreased or vice versa. By this way, sudden birth of entangled state can be tuned by LDP. All graphs in this study are plotted with aid of the Wolfram Mathematica 9.