Here we show how to generate a dark two-mode squeezed state of a trapped ion, employing a three-level ion in a V configuration with a strong decay of the excited states. The degree of squeezing can be manipulated by choosing the intensity of the driving fields. Our scheme is robust against the usual dissipation mechanism and could be implemented with present-day technology. The validity of the approximations employed in this work was tested by numerical calculations, which agreed completely with the analytical solutions.
We propose a method for generating a spin-squeezed state that is a symmetric Dicke state, with trapped ions using only global access. The eigenstates of the ions under a strong dressing field become symmetric Dicke states and the M$ø$lmer--S$ø$rensen interaction selectively couples one of them to an initially populated auxiliary state. A $\mid D_{2n}^n>$ state, which is maximally spin squeezed, can be generated with high fidelity using only square pulses. Using an adiabatic technique, the ideal maximally spin-squeezed state is generated.
We propose a method to create two-mode squeezed states and their superposition in the center-of-mass mode and breathing mode of two-trapped ions. Each ion is illuminated simultaneously by two standing waves. One of the fields is tuned to excite resonantly and simultaneously both upper sidebands of the two normal modes, while the other field tuned to the corresponding lower sidebands.
The marginal distribution of squeezed and rotated quadrature for two types of nonclassical states of trapped ion -- for squeezed and correlated states and for squeezed even and odd coherent states (squeezed Schrödinger cat states) is studied. The obtained marginal distribution for the two types of states is shown to satisfy classical dynamical equation equivalent to standard quantum evolution equation for density matrix (wave function) derived in symplectic tomography scheme.
The marginal distribution of squeezed, rotated and shifted quadrature for two types of nonclassical states of a trapped ion - squeezed correlated states and squeezed even and odd coherent states (squeezed Schrodinger cat states) - is studied.
Trapped ions offer a pristine platform for quantum computation and simulation, but improving their coherence remains a crucial challenge. Here, we propose and analyze a new strategy to enhance the coherent interactions in trapped ion systems via parametric amplification of the ions' motion--by squeezing the collective motional modes (phonons), the spin-spin interactions they mediate can be significantly enhanced. We illustrate the power of this approach by showing how it can enhance collective spin states useful for quantum metrology, and how it can improve the speed and fidelity of two-qubit …
In trapped-ion quantum information processing, interactions between spins (qubits) are mediated by collective modes of motion of an ion crystal. While there are many different experimental strategies to design such interactions, they all face both technical and fundamental limitations to the achievable coherent interaction strength. In general, obtaining strong interactions and fast gates is an ongoing challenge. Here, we extend previous work [Phys. Rev. Lett. 112, 030501 (2019)] and present a general strategy for enhancing the interaction strengths in trapped-ion systems via parametric amplif…
Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Marginal distributions of the quadrature for squeezed and correlated states and for nonlinear coherent states of a trapped ion are obtained and the density matrices in the Fock basis are expressed explicitly in terms of these marginal distributions.
A completely analytic description is given of the motion of a trapped ion which is in either an even or an odd squeezed state. Comparison is made to recent results on the even or odd coherent states, and possible experimental work is discussed.
Collective many-body dynamics for time-dependent quantum Hamiltonian functions is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group ${\cal {SL}}(2, \mathbb R)$, which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which…
Within the framework of the Gaussian-state theory, we show that the quantum many-body ground state of a trapped condensate with weakly attractive interaction is a single-mode squeezed vacuum state, as oppose to the coherent state under repulsive interaction. The spatial mode of the squeezed-state condensates satisfies a Gross-Pitaevskii like equation in which the interaction strength is augmented by a factor $3$ due to the large particle fluctuation of the squeezed state. We also study the collective excitations of the condensates by the tangential space projection, which leads to new two-part…
Quasiclassical dynamics of trapped ions is characterized by applying the time dependent variational principle (TDVP) on coherent state orbits, in case of quadrupole and octupole combined (Paul and Penning) and radiofrequency (RF) traps. A dequantization algorithm is proposed, by which the classical Hamilton (energy) function associated to the system results as the expectation value of the quantum Hamiltonian on squeezed coherent states. We develop such method and particularize the quantum Hamiltonian for a combined and for a RF trap, with axial symmetry and a RF anharmonic electric potential. …
We consider a trapped ion with a quantized center-of-mass motion in 2D trap potential. With external laser fields the effective (non)linear coupling of two orthogonal vibrational modes can be established via stimulated Raman transition. Nonclassical vibrational states such as squeezed states or two-mode entangled states (Schroedinger cat-like states) can be generated. When the vibrational modes are entangled with internal energy levels the Greenberger-Horne-Zeilinger (GHZ) states can be prepared.
We propose a scheme to generate nonclassical states of a quantum system, which is composed of the one-dimensional trapped ion motion and a single cavity field mode. We show that two-mode SU(2) Schrödinger-cat states, entangled coherent states, two-mode squeezed vacuum states and their superposition can be generated. If the vibration mode and the cavity mode are used to represent separately a qubit, a quantum phase gate can be implemented.
We show that the motion of a cold trapped ion can be squeezed by modulating the intensity of a phase-stable optical lattice placed inside the trap. As this method is reversible and state selective it effectively implements a controlled-squeeze gate. We show how to use this resource, that can be useful for quantum information processing with continuous variables, in order to prepare coherent superpositions of states which are squeezed along complementary quadratures. We show that these states, which we denote as "${\mathcal X}$-states", exhibit high sensitivity to small displacements along two …
We propose a method for scaling trapped ions for large-scale quantum computation and communication based on a probabilistic ion-photon mapping. Deterministic quantum gates between remotely located trapped ions can be achieved through detection of spontaneously-emitted photons, accompanied by the local Coulomb interaction between neighboring ions. We discuss gate speeds and tolerance to experimental noise for different probabilistic entanglement schemes.
The quantum behavior of a precooled cantilever can be probed highly efficiently by coupling to a trapped ultracold ion, in which a fast cooling of the cantilever down to the ground vibrational state is possible. We solve the dynamics of the coupling system by a squeezed-state expansion technique, and can in principle obtain the exact solution of the time-evolution of the coupling system. Compared to the treatment under rotating-wave approximation, we can present a more accurate description of the quantum behavior of the cantilever.
We describe the optimal realization of entangling quantum gates for trapped ions mediated by a dissipative bus mode. With suitably shaped control pulses one can substantially decrease ion-phonon entanglement while maintaining the mediated interaction.
Trapped ion hardware has made significant progress recently and is now one of the leading platforms for quantum computing. To construct two-qubit gates in trapped ions, experimental manipulation approaches for ion chains are becoming increasingly prevalent. Given the restricted control technology, how implementing high-fidelity quantum gate operations is crucial. Many works in current pulse design optimization focus on ion-phonon and effective ion-ion coupling while ignoring the higher-order expansion impacts of these two terms brought on by experiment defects. This paper proposed a novel robu…
We have experimentally observed the dynamics of a single polariton and two polaritons in a two-ion chain. By driving two trapped ions at a motional blue-sideband transition, we realize the anti-Jaynes-Cummings-Hubbard model. When a single polariton exists in a trapped-ion chain, the polariton hops between the ion sites. On the other hand, when there are single polaritons at each ion site, the hopping of the polaritons is suppressed because of the polariton-polariton interaction induced by the nonlinearity of the anti-Jaynes-Cummings interaction, thereby realizing the blockade of polariton hopp…
A new approach for reconstructing the vibrational quantum state of a trapped ion is proposed. The method rests upon the current ability of manipulating the trapped ion state and on the possibility of effectively measuring the scalar product of the two vibrational cofactors of a vibronic entangled state. The experimental feasibility of the method is briefly discussed.
There has been much interest in developing methods for transferring quantum information. We discuss a way to transfer quantum information between two trapped ions through a wire. The motion of a trapped ion induces oscillating charges in the trap electrodes. By sending this current to the electrodes of a nearby second trap, the motions of ions in the two traps are coupled. We investigate the electrostatics of a set-up where two separately trapped ions are coupled through an electrically floating wire. We also discuss experimental issues, including possible sources of decoherence.
In this tutorial we review the basic building blocks of Quantum Information Processing with cold trapped atomic-ions. We mainly focus on methods to implement single-qubit rotations and two-qubit entangling gates, which form a universal set of quantum gates. Different ion qubit choices and their respective gate implementations are described.
There are several known schemes for entangling trapped ion quantum bits for large-scale quantum computation. Most are based on an interaction between the ions and external optical fields, coupling internal qubit states of trapped-ions to their Coulomb-coupled motion. In this paper, we examine the sensitivity of these motional gate schemes to phase fluctuations introduced through noisy external control fields, and suggest techniques to suppress the resulting phase decoherence.
The decoherence in trapped ion induced by coupling the ion to the engineered reservoir is studied in this paper. The engineered reservoir is simulated by random variations in the trap frequency, and the trapped ion is treated as a two-level system driven by a far off-resonant plane wave laser field. The dependence of the decoherence rate on the amplitude of the superposition state is given.
We show how NOON states may be generated in ion traps. We use the individual interaction of light with each of two vibrational modes of the ion to entangle them. This allows us to generate NOON states with N=8.
Applying a magnetic field gradient to a trapped ion allows long-wavelength microwave radiation to produce a mechanical force on the ion's motion when internal transitions are driven. We demonstrate such a coupling using a single trapped \Yb{171}~ion, and use it to produce entanglement between the spin and motional state, an essential step towards using such a field gradient to implement multi-qubit operations.
We propose an architecture and methodology for large-scale quantum simulations using hyperfine states of trapped-ions in an arbitrary-layout microtrap array with laserless interactions. An ion is trapped at each site, and the electrode structure provides for the application of single and pairwise evolution operators using only locally created microwave and radio-frequency fields. The avoidance of short-lived atomic levels during evolution effectively eliminates errors due to spontaneous scattering; this may allow scaling of quantum simulators based on trapped ions to much larger systems than c…
Trapped-ion quantum information processors store information in atomic ions maintained in position in free space via electric fields. Quantum logic is enacted via manipulation of the ions' internal and shared motional quantum states using optical and microwave signals. While trapped ions show great promise for quantum-enhanced computation, sensing, and communication, materials research is needed to design traps that allow for improved performance by means of integration of system components, including optics and electronics for ion-qubit control, while minimizing the near-ubiquitous electric-f…
We demonstrate tunable spin-spin couplings between trapped atomic ions, mediated by laser forces on multiple transverse collective modes of motion. A $σ_x σ_x$-type Ising interaction is realized between quantum bits stored in the ground hyperfine clock states of Yb ions. We demonstrate entangling gates and tailor the spin-spin couplings with two and three trapped ions. The use of closely-spaced transverse modes provides a new class of interactions relevant to quantum computing and simulation with large collections of ions in a single crystal.
We propose a scheme to perform basic gates of quantum computing and prepare entangled states in a system with cold trapped ions located in a single mode optical cavity. General quantum computing can be made with both motional state of the trapped ion and cavity state being qubits. We can also generate different kinds of entangled states in such a system without state reduction, and can transfer quantum states from the ion in one trap to the ion in another trap. Experimental requirement for achieving our scheme is discussed.
A k-quantum nonlinear Jaynes-Cummings model for two trapped ions interacting with laser beams resonant to k-th red side-band of center-of-mass mode, far from Lamb-Dicke regime, has been obtained. The exact analytic solution showed the existence of quantum collapses and revivals of the occupation of two atoms.
We show that a multi-step quantum walk can be realized for a single trapped ion with interpolation between quantum and random walk achieved by randomizing the generalized Hadamard coin flip phase. The signature of the quantum walk is manifested not only in the ion's position but also its phonon number, which makes an ion trap implementation of the quantum walk feasible.
The mode-mode entanglement between trapped ions and cavity fields is investigated in the dispersive regime. We show how a simple initial preparation of Gaussian coherent states and a postselection may be used to generate motional non-local mesoscopic states (NLMS) involving ions in different traps. We also present a study of the entanglement induced by dynamical Stark-shifts considering a cluster of N-trapped ions. In this case, all entanglement is due to the dependence of the Stark-shifts on the ions' state of motion manifested as a cross-Kerr interaction between each ion and the field.
We propose a physical implementation of time and spatial parity transformations, as well as Galilean boosts, in a trapped-ion quantum simulator. By embedding the simulated model into an enlarged simulating Hilbert space, these fundamental symmetry operations can be fully realized and measured with ion traps. We illustrate our proposal with analytical and numerical techniques of prototypical examples with state-of-the-art trapped-ion platforms. These results pave the way for the realization of time and spatial parity transformations in other models and quantum platforms.
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.
The interaction of a trapped ion with a laser beam in the strong excitation regime is analyzed. In this regime, a variety of non--classical states of motion can be prepared either by using laser pulses of well defined area, or by an adiabatic passage scheme based on the variation of the laser frequency. We show how these states can be used to investigate fundamental properties of quantum mechanics. We also study possible applications of this system to build an ion interferometer.
We show how to produce a fast quantum Rabi model with trapped ions. Its importance resides not only in the acceleration of the phenomena that may be achieved with these systems, from quantum gates to the generation of nonclassical states of the vibrational motion of the ion, but also in reducing unwanted effects such as the decay of coherences that may appear in such systems.
We consider interactions that generate a universal set of quantum gates on logical qubits encoded in a collective-dephasing-free subspace, and discuss their implementations with trapped ions. This allows for the removal of the by-far largest source of decoherence in current trapped-ion experiments, collective dephasing. In addition, an explicit parametrization of all two-body Hamiltonians able to generate such gates without the system's state ever exiting the protected subspace is provided.
We propose a method for generating maximally entangled states of N two-level trapped ions. The method is deterministic and independent of the number of ions in the trap. It involves a controlled-NOT acting simultaneously on all the ions through a dispersive interaction. We explore the potential application of our scheme for high precision frequency standards.
We investigate theoretically a hybrid system consisting of a Bose-Einstein condensate (BEC) trapped inside a laser driven membrane-in-the-middle optomechanical cavity assisted with squeezed vacuum injection whose moving membrane interacts both linearly and quadratically with the radiation pressure of the cavity. It is shown that such a hybrid system is very suitable for generating strong quadrature squeezing in the mechanical mode of the membrane and the Bogoliubov mode of the BEC in the unresolved sideband regime. More interestingly, by choosing a suitable sign for the quadratic optomechanica…
We propose a scheme to perform a fundamental two-qubit gate between two trapped ions using ideas from atom interferometry. As opposed to the scheme considered by J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995), it does not require laser cooling to the motional ground state.
We present a single step scheme to generate, experimentally, maximally entangled tripartite GHZ state in a single ion, using trapped ion interacting simultaneously with a resonant external laser field and red sideband tuned quantized cavity field. Besides the simplicity of execution and short operation time, GHZ state generation is reduction free. Pacs: 03.67.-a, 42.50.-p, 03.67.Dd
We improve a previously proposed scheme (Phys. Rev. A 66 (2002) 065401) for generating vibrational trio coherent states of a trapped ion. The improved version is shown to gain a double advantage: (i) it uses only five, instead of eight, lasers and (ii) the generation process can be made remarkably faster.
We present a selective vibronic interaction for manipulating motional states in two trapped ions, acting resonantly on a previously chosen vibronic subspace and dispersively on all others. This is done respecting technical limitations on ionic laser individual addressing. We discuss the generation of Fock states and entanglement in the ionic collective motional degrees of freedom, among other applications.
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.
The resonant heating of a harmonically trapped ion by a standing-wave light field is described as a quantum stochastic process combining a coherent Schroedinger evolution with Bohr-Einstein quantum jumps. Quantum and semi-quantum treatments are compared.
A single laser-cooled and trapped 9Be+ ion is used to investigate methods of coherent quantum-state synthesis and quantum logic. We create and characterize nonclassical states of motion including "Schroedinger-cat" states. A fundamental quantum logic gate is realized which uses two states of the quantized ion motion and two ion internal states as qubits. We explore some of the applications for, and problems in realizing, quantum computation based on multiple trapped ions.
We present a way of treating the problem of the interaction of a single trapped ion with laser beams based on successive aplications of unitary transformations onto the Hamiltonian. This allows the diagonalization of the Hamiltonian, by means of recursive relations, without performing the Lamb-Dicke approximation.
Unsharp POVM measurements allow a variety of measurement applications which minimally disrupt the state of the quantum system. Experimental schemes are proposed for implementing unsharp measurements on the qubit levels of a trapped ion. The schemes rely on introducing weak entanglement between the state of a target ion, and that of an auxiliary ion, using standard ion trap quantum logic operations, and then realizing an unsharp measurement through projective measurement on the auxiliary atom. We analyze common sources of error and their effect on different applications of unsharp measurements.
We study two trapped ions coupled to the axial phonon modes of a one-dimensional Coulomb crystal. This system is formally equivalent to the "two spin-boson" model. We propose a scheme to dynamically generate a maximally entangled state of two ions within a decoherence-free subspace. Here the phononic environment of the trapped ions, whatever its temperature and number of modes, serves as the entangling bus. The efficient production of the pure singlet state can be exploited to perform short-ranged quantum communication which is essential in building up a large-scale quantum computer.
We describe a simplified scheme for quantum logic with a collection of laser-cooled trapped atomic ions. Building on the scheme of Cirac and Zoller, we show how the fundamental controlled-NOT gate between a collective mode of ion motion and the internal states of a single ion can be reduced to a single laser pulse, and the need for a third auxiliary internal electronic state can be eliminated.
We present a scheme utilizing position-dependent ac Stark shifts for doing quantum logic with trapped ions. By a proper choice of direction, position and size, as well as power and frequency of a far-off-resonant Gaussian laser beam, specific ac Stark shifts can be assigned to the individual ions, making them distinguishable in frequency-space. In contrast to previous all-optical based quantum gates with trapped ions, the present scheme enables individual addressing of single ions and selective addressing of any pair of ions for two-ion quantum gates, without using tightly focused laser beams.…
The theoretical foundations of a new general approach to the measurement problem of vibrational observables in trapped ion systems is reported. The method rests upon the introduction of a simple vibronic coupling structure appropriately conceived to link the internal ionic state measurement outcomes to the mean value of a motional variable of interest. Some applications are provided and discussed in detail, bringing to light the feasibility and the wide potentiality of the proposal.
We examine the validity of the harmonic approximation, where the radio-frequency ion trap is treated as a harmonic trap, in the problem regarding the controlled collision of a trapped atom and a single trapped ion. This is equivalent to studying the effect of the micromotion since this motion must be neglected for the trapped ion to be considered as a harmonic oscillator. By applying the transformation of Cook and Shankland we find that the micromotion can be represented by two periodically oscillating operators. In order to investigate the effect of the micromotion on the dynamics of a trappe…
