We present a robust quantum optimal control framework for implementing fast entangling gates on ion-trap quantum processors. The framework leverages tailored laser pulses to drive the multiple vibrational sidebands of the ions to create phonon-mediated entangling gates and, unlike the state of the art, requires neither weak-coupling Lamb-Dicke approximation nor perturbation treatment. With the application of gradient-based optimal control, it enables finding amplitude- and phase-modulated laser control protocols that work beyond the Lamb-Dicke regime, promising gate speed at the order of microseconds comparable to the characteristic trap frequencies. Also, robustness requirements on the temperature of the ions and initial optical phase can be conveniently included to pursue high-quality fast gates against experimental imperfections. Our approach represents a step in speeding up quantum gates to achieve larger quantum circuits for quantum computation and simulation, and thus can find applications in near-future experiments.
A scheme for preparing Schrödinger cat (SC) states is proposed beyond the Lamb-Dicke limit in a Raman-$Λ$-type configuration. It is shown that SC states can be obtained more efficiently with our scheme than with the former ones.
First order Doppler effects are usually ignored in laser driven trapped ions when the recoil frequency is much smaller than the trapping frequency (Lamb-Dicke regime). This means that the central, carrier excitation band is supposed to be unaffected by vibronic transitions in which the vibrational number changes. While this is strictly true in the Lamb-Dicke limit (infinitely tight confinement), the vibronic transitions do play a role in the Lamb-Dicke regime. In this paper we quantify the asymptotic behaviour of their effect with respect to the Lamb-Dicke parameter. In particular, we give ana…
Based on the exact conditional quantum dynamics for a two-ion system, we propose an efficient {\it single-step} scheme for coherently manipulating quantum information of two trapped cold ions by using a pair of synchronous laser pulses. Neither the auxiliary atomic level nor the Lamb-Dicke approximation are needed.
A scheme to implement quantum logic gates by manipulating trapped ions through interaction with monochromatic external laser field and quantized cavity field, beyond the Lamb-Dicke regime, is presented. Characteristic times, for implementing ionic state transitions using non-resont laser pulse or quantized cavity field, shows a sharp decline for large Lamb-Dicke parameter value of $η_{L}=η_{c}=0.2$, and is seen to decrease further with increase in number of initial state vibrational quanta $m$.
We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasi-continuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of trapping states which may slow down the cooling is considerably reduced. We show that cooling…
We investigate the dynamics of a cold trapped ion coupled to the quantized field inside a high-finesse cavity. We have used an approach for preparing the SC states of motion of ion. This approach, based on unitary transformating the Hamiltonian, allows its exact diagonalization without performing the Lamb-Dicke aproximation. We show that is possible to generate a SC states having rather simple initial state preparation, e.g., the vacuum sate for both cavity field and the ion motion.
Based on the conditional quantum dynamics of laser-ion interaction, we propose an efficient theoretical scheme to deterministically generate quantum pure states of a single trapped cold ion without performing the Lamb-Dicke approximation. An arbitrary quantum state can be created by using a series of classical laser beams with selected frequencies, initial phases and durations. As special examples, we further show how to create or approximate several typical macroscopic quantum states, such as the phase state and (even/odd) coherent states. Unlike previous schemes operated in the Lamb-Dicke re…
We propose a laser cooling scheme that allows to cool a single atom confined in a harmonic potential to the trap ground state $|0>$. The scheme assumes strong confinement, where the oscillation frequency in the trap is larger than the effective spontaneous decay width, but is not restricted to the Lamb-Dicke limit, i.e. the size of the trap ground state can be larger than the optical wavelength. This cooling scheme may be useful in the context of quantum computations with ions and Bose-Einstein condensation.
We propose a simple scheme for implementing quantum logic gates with a string of two-level trapped cold ions outside the Lamb-Dicke limit. Two internal states of each ion are used as one computational qubit (CQ) and the collective vibration of ions acts as the information bus, i.e., bus qubit (BQ). Using the quantum dynamics for the laser-ion interaction as described by a generalized Jaynes-Cummings model, we show that quantum entanglement between any one CQ and the BQ can be coherently manipulated by applying classical laser beams. As a result, universal quantum gates, i.e. the one-qubit rota…
Cavity-mediated cooling has the potential to become one of the most efficient techniques to cool molecular species down to very low temperatures. In this paper we analyse cavity cooling with single-laser driving for relatively large cavity decay rates kappa and relatively large phonon frequencies nu. It is shown that cavity cooling and ordinary laser cooling are essentially the same within the validity range of the Lamb-Dicke approximation. This is done by deriving a closed set of rate equations and calculating the corresponding stationary state phonon number and cooling rate. For example, whe…
We create entangled states of the spin and motion of a single $^{40}$Ca$^+$ ion in a linear ion trap. The motional part consists of coherent states of large separation and long coherence time. The states are created by driving the motion using counterpropagating laser beams. We theoretically study and experimentally observe the behaviour outside the Lamb-Dicke regime, where the trajectory in phase space is modified and the coherent states become squeezed. We directly observe the modification of the return time of the trajectory, and infer the squeezing. The mesoscopic entanglement is observed …
For certain {\it specific} (or {\it"magic"}) Lamb-Dicke (LD) parameters, Monroe {\it et al} showed [Phys. Rev. {\bf A 55}, R2489 (1997)] that a two-qubit quantum operation, between the external and internal degrees of freedom of a single trapped ion, could be implemented by applying a single carrier laser pulse. Here, we further show that, such a two-qubit operation (which is equivalent to the standard CNOT gate, only apart from certain phase factors) could also be significantly-well realized for {\it arbitrarily} selected LD parameters. Instead of the so-called "$π$-pulses" used in the previo…
A simple alternative scheme for implementing quantum gates with a single trapped cold two-level ion beyond the Lamb-Dicke (LD) limit is proposed. Basing on the quantum dynamics for the laser-ion interaction described by a generalized Jaynes-Cummings model, one can introduce two kinds of elementary quantum operations i.e., the simple rotation on the bare atomic state, generated by applying a resonant pulse, and the joint operation on the internal and external degrees of the ion, performed by using an off-resonant pulse. Several typical quantum gates, including Hadamard gate, controlled-Z and co…
We investigate the dynamics of cavity cooling of a single ion beyond the Lamb-Dicke regime and demonstrate a cooling limit of approximately 50% of the Doppler temperature using a high finesse cavity for the first time. We also study the statistical properties of the ion-cavity emission and present a model that maps the phonon states to the photon states. With this model, we explain the super-Poissonian photon distribution observed in our experiments and propose a method to estimate the ion temperature in a real-time manner based on the statistical behavior of the photon emission from the cavit…
We apply wave packet methods to study an ion-trap system in the strong excitation regime imposing neither the rotating wave nor the Lamb-Dicke approximations. By this approach we show the existence of states with restricted phase space evolution, as a genuine consequence of quantum interference between wave packet fractions. A particular instance of such a state oscillates between maximal entanglement and pure disentanglement between the constitute subsystems. The characteristic crossover time is very rapid making them suitable for state preparations of EPR or Schrodinger cat states. Over long…
We report continuous Raman sideband cooling (CRSC) of a long ion chain 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 $η= 1.3$, spanning a frequency bandwidth of 4 MHz. Compared to traditional ground-state cooling methods such as pulsed sideband cooling or electromagnetic-induced-transparency (EIT) cooling, our method offers two key advantages: robustness to timing errors; and an ultra-wide bandwidth unl…
We present the results of simulations of optical sideband cooling of atomic ions in a trap with a shallow potential well. In such traps, an ion cannot be Doppler cooled near to the Lamb-Dicke regime ($η^2(2n+1) \ll 1$). Outside the Lamb-Dicke regime, the sideband cooling dynamics are altered by the existence of various Fock states where the cooling becomes very slow. A calcium ion trapped in our Penning trap realises such a situation, hence single stage cooling is inefficient to prepare the ion in the motional ground state. For these systems, it is necessary to study the cooling dynamics in de…
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 mat…
Sympathetic cooling of trapped ions has been established as a powerful technique for manipulation of non-laser-coolable ions (Raizen1992,Waki1992,Bowe1999,Barrett2003). For molecular ions, it promises vastly enhanced spectroscopic resolution and accuracy. However, this potential remains untapped so far, with the best resolution achieved being not better than $5\times10^{-8}$ fractionally, due to residual Doppler broadening being present in ion clusters even at the lowest achievable translational temperatures (Bressel2012). Here we introduce a general and accessible approach that enables Dopple…
Several sequential operations are usually needed for implementing controlled quantum gates and generating entanglement between a pair of quantum bits. Based on the conditional quantum dynamics for a two-ion system beyond the Lamb-Dicke limit, here we propose a theoretical scheme for manipulating two-qubit quantum information, i.e., implementing a universal two-qubit quantum gate and generating a two-qubit entangled state, by using a pair of simultaneous laser pulses. Neither the Lamb-Dicke approximation nor the auxiliary atomic level are required. The experimental realizability of this simple …
We propose and demonstrate a scheme for strongly radially confining a single neutral atom in a bichromatic far-off resonance optical dipole trap(BFORT) . BFORT is composed of a blue-detuned Laguerre-Gaussian $LG^1_ 0$ beam and a red-detuned Gaussian beam. The trapping radial dimension of a single atom trapped in the Gaussian FORT can be greatly compressed by imposing a blue-detuned Laguerre-Gaussian $LG^1_ 0$ beam with moderate potential depth. By modulating the potential depth of the Gaussian FORT we observed that the resonant and parametric excitation of the oscillatory motion of a single at…
We propose a fast and efficient technique to create classes of highly entangled states of trapped ions, such as arbitrary Dicke states and superpositions of them, e.g. NOON states. The ions are initialized in the phonon ground state and are addressed globally with a composite pulse that is resonant with the first motional sideband. The technique operates on comparatively short time scales, as resonant interactions allow one to use the minimum laser pulse area. The number of single pulses from the composite sequence is equal to the number of ions, thus the implementation complexity grows only l…
We have investigated the efficiency of pulsed Raman sideband cooling in the presence of multiple decay and excitation channels. By applying sum rules we identify parameter regimes in which multiple scattering of photons can be described by an effective wave vector. Using this method we determine the rate of heating caused by optical pumping inside and outside the Lamb-Dicke regime. On this basis we discuss also the efficiency of a recently proposed scheme for ground-state cooling outside the Lamb-Dicke regime [G. Morigi, J.I. Cirac, M. Lewenstein, and P. Zoller, Europhys. Lett. {\bf 39}, 13 (1…
We discuss continuous observation of the momentum of a single atom by employing the high velocity sensitivity of the index of refraction in a driven $Λ$-system based on electromagnetically induced transparency (EIT). In the ideal limit of unit collection efficiency this provides a quantum limited measurement with minimal backaction on the atomic motion. A feedback loop, which drives the atom with a force proportional to measured signal, provides a cooling mechanism for the atomic motion. We derive the master equation which describes the feedback cooling and show that in the Lamb-Dicke limit th…
We propose a scalable design of entangling quantum gates for large ion crystals with the following desirable features: 1) The gate design is universal and applicable for large ion crystals of arbitrary sizes; 2) The gate has no speed limitation and can work outside of the Lamb-Dicke region; 3) The gate operates by driving from either continuous-wave or pulsed laser beams; 4) The gate is insensitive to slow variation of the laser optical phase and works under a thermal state for the ions' motion; 5) The intrinsic gate infidelity can be reduced to a level well below the threshold for fault-toler…
Coherent quantum-state manipulation of trapped ions using classical laser fields is a trademark of modern quantum technologies. In this work, we study aspects of work statistics and irreversibility in a single trapped ion due to sudden interaction with the impinging laser. This is clearly an out-of-equilibrium process where work is performed through illumination of an ion by the laser. Starting with the explicit evaluation of the first moments of the work distribution, we proceed to a careful analysis of irreversibility as quantified by the nonequilibrium lag. The treatment employed here is no…
Trapped ion in the Lamb-Dicke regime with the Lamb-Dicke parameter $η\ll1$ can be cooled down to its motional ground state using sideband cooling. Standard sideband cooling works in the weak sideband coupling 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 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 …
Essentially all known quantum gates rely on a weak-coupling approximation resulting in linear dynamics. With the explicit example of trapped ions, we show how high-fidelity quantum gates can be achieved outside such an approximation, and we derive readily implementable driving fields to realize gates with extremely high fidelities for ions well outside the Lamb-Dicke regime with motional temperatures achievable by only Doppler cooling.
We show how to create quantum gates of arbitrary speed between trapped ions, using a laser walking wave, with complete insensitivity to drift of the optical phase, and requiring cooling only to the Lamb-Dicke regime. We present pulse sequences that satisfy the requirements and are easy to produce in the laboratory.
We propose a scheme to perform probabilistic quantum gates on remote trapped atom qubits through interference of optical frequency qubits. The method does not require localization of the atoms to the Lamb-Dicke limit, and is not sensitive to interferometer phase instabilities. Such probabilistic gates can be used for scalable quantum computation.
We propose a new concept for a two-qubit gate operating on a pair of trapped ions based on laser coherent control techniques. The gate is insensitive to the temperature of the ions, works also outside the Lamb-Dicke regime, requires no individual addressing by lasers, and can be orders of magnitude faster than the trap period.
We show that in the trapped ion-laser interaction all the regimes may be considered analytically. We may solve not only for different laser intensities, but also away from resonance and from the Lamb-Dicke regime. It is found a dispersive Hamiltonian for the high intensity regime, that, being diagonal, its evolution operator may be easily calculated.
In this tutorial we review physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated wit…
We present a robust and fast laser cooling scheme suitable for trapped atoms and ions. Based on quantum interference, generated by a special laser configuration, it is able to rapidly cool the system such that the final phonon occupation vanishes to zeroth order in the Lamb-Dicke parameter in contrast to existing cooling schemes. Furthermore, it is robust under conditions of fluctuating laser intensity and frequency, thus making it a viable candidate for experimental applications.
The Raman interaction of a trapped ultracold ion with two travelling wave lasers is studied analytically with series solutions, in the absence of the rotating wave approximation (RWA) and the restriction of both the Lamb-Dicke limit and the weak excitation regime. The comparison is made between our solutions and those under the RWA to demonstrate the validity region of the RWA. As a practical example, the preparation of Schrödinger-cat states with our solutions is proposed beyond the weak excitation regime.
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.
Raman interaction of a trapped ultracold ion with two traveling wave lasers has been used extensively in the ion trap experiments. We solve this interaction in the absence of the rotating wave approximation by a continued fraction, without considering the restriction of the Lamb-Dicke limit and the weak excitation regime. Some interesting characteristics of the ion-trap system, particularly for the ion outside the weak excitation regime are found. Finally, a comparison of our results with the solution under the rotating wave approximation is made.
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 study the nonlinear dynamics of trapped-ion models far away from the Lamb-Dicke regime. This nonlinearity induces a sideband cooling blockade, stopping the propagation of quantum information along the Hilbert space of the Jaynes-Cummings and quantum Rabi models. We compare the linear and nonlinear cases of these models in the ultrastrong and deep strong coupling regimes. Moreover, we propose a scheme that simulates the nonlinear quantum Rabi model in all coupling regimes. This can be done via off-resonant nonlinear red and blue sideband interactions, yielding applications as a dynamical qua…
Recent experiments [K. R. Brown, et al., Nature 471, 196 (2011); and M. Harlander, et al., Nature 471, 200 (2011)] have demonstrated the coherent manipulations on the external vibrations of two ions, confined individually in the separated ion traps. Using these recently developed techniques, we propose here an approach to realize the coherent operations, e.g., the universal quantum gates, between the separated ion-trap qubits encoded by two internal atomic states of the trapped ions. Our proposal operates beyond the usual Lamb-Dicke limits, and could be applied to the scalable ion traps couple…
We investigate laser-driven vibronic transitions of a single two-level atomic ion in harmonic and hard wall traps. In the Lamb-Dicke regime, for tuned or detuned lasers with respect to the internal frequency of the ion, and weak or strong laser intensities, the vibronic transitions occur at well isolated "Rabi Resonances", where the detuning-adapted Rabi frequency coincides with the level spacing of the vibrational modes. These vibronic resonances are characterized as avoided crossings of the dressed levels (eigenvalues of the full Hamiltonian). Their peculiarities due to symmetry constraints …
We propose a scheme to achieve quantum computation with neutral atoms whose interactions are catalyzed by single photons. Conditional quantum gates, including an $N$-atom Toffoli gate and nonlocal gates on remote atoms, are obtained through cavity-assisted photon scattering in a manner that is robust to random variation in the atom-photon coupling rate and which does not require localization in the Lamb-Dicke regime. The dominant noise in our scheme is automatically detected for each gate operation, leading to signalled errors which do not preclude efficient quantum computation even if the err…
Two techniques are described that simplify the experimental requirements for measuring and manipulating quantum information stored in trapped ions. The first is a new technique using electron shelving to measure the populations of the Zeeman sublevels of the ground state, in an ion for which no cycling transition exists from any of these sublevels. The second technique is laser cooling to the vibrational ground state, without the need for a trap operating in the Lamb-Dicke limit. This requires sideband cooling in a sub-recoil regime. We present a thorough analysis of sideband cooling on one or…
We propose a scheme to implement quantum gates on any pair of trapped ions immersed in a large linear crystal, using interaction mediated by the transverse phonon modes. Compared with the conventional approaches based on the longitudinal phonon modes, this scheme is much less sensitive to ion heating and thermal motion outside of the Lamb-Dicke limit thanks to the stronger confinement in the transverse direction. The cost for such a gain is only a moderate increase of the laser power to achieve the same gate speed. We also show how to realize arbitrary-speed quantum gates with transverse phono…
We present a scalable uniform technique for construction of highly conditional C$^n$-NOT quantum gates of trapped ion qubits, such as the Toffoli gate, without using ancilla states and circuits of an exorbitant number of concatenated one- and two-qubit gates. Apart from the initial dressing of the internal qubit states with vibrational phonons and the final restoration of the phonon ground state, our technique requires the application of just a single composite pulse on the target qubit and is applicable both in and outside the Lamb-Dicke regime. We design special narrowband composite pulses, …
The cooling rate for cavity mediated laser cooling scales as the Lamb-Dicke parameter eta squared. A proper analysis of the cooling process hence needs to take terms up to eta^2 in the system dynamics into account. In this paper, we present such an analysis for a standard scenario of cavity mediated laser cooling with eta << 1. Our results confirm that there are many similarities between ordinary and cavity mediated laser cooling. However, for a weakly confined particle inside a strongly coupled cavity, which is the most interesting case for the cooling of molecules, numerical results indicate…
In this contribution we investigate the interaction of a single ion in a trap with laser beams. Our approach, based on unitary transformating the Hamiltonian, allows its exact diagonalization without performing the Lamb-Dicke approximation. We obtain a transformed Jaynes-Cummings type Hamiltonian, and we demonstrate the existence of super-revivals in that system.
We study the laser cooling of one atom in an harmonic trap beyond the Lamb-Dicke regime. By using sequences of laser pulses of different detunings we show that the atom can be confined into just one state of the trap, either the ground state or an excited state of the harmonic potential. The last can be achieved because under certain conditions an excited state becomes a dark state. We study the problem in one and two dimensions. For the latter case a new cooling mechanism is possible, based on the destructive interference between the effects of laser fields in different directions, which allo…
Two or three dimensional Paul traps can confine a large number of ions forming a Wigner crystal, which would provide an ideal architecture for scalable quantum computation except for the micromotion, an issue that is widely believed to be the killer for high fidelity quantum gates. Surprisingly, here we show that the micromotion is not an obstacle at all for design of high fidelity quantum gates, even though the magnitude of the micromotion is significantly beyond the requirement of the Lamb-Dicke condition. Through exact solution of the quantum Mathieu equations, we demonstrate the principle …
