We present a protocol that permits the generation of a subtle with superposition with 2^(l+1) displaced number states on a circle in phase space as target state for the center-of-mass motion of a trapped ion. Through a sequence of 'l' cycles involving the application of laser pulses and no-fluorescence measurements, explicit expressions for the total duration of laser pulses employed in the sequence and probability of getting the ion in the upper electronic state during the 'l' cycles are obtained and analyzed in detail. Furthermore, assuming that the effective relaxation process of a trapped ion can be described in the framework of the standard master equation for the damped harmonic oscillator, we investigate the degradation of the quantum interference effects inherent to superpositions via Wigner function.
In this paper we study the evolution of the two two-level atoms interacting with a single-mode quantized radiation field, namely, two-atom multiphoton Jaynes-Cummings model when the radiation field and atoms are initially prepared in the superpostion of displaced number states and excited atomic states, respectively. For this system we investigate the atomic inversion, Wigner function, phase distribution and entanglement.
We study the effects of counter rotating terms in the interaction of quantized light with a two-level atom, by using the method of small rotations. We give an expression for the wave function of the composed system atom plus field and point out one initial wave function that generates a quantum bit of the electromagnetic field with arbitrary amplitudes.
We show that a nonlinear asymmetric directional coupler composed of a linear waveguide and a nonlinear waveguide operating by nondegenerate parametric amplification is an effective source of single-mode squeezed light. This is has been demonstrated, under certain conditions and for specific modes, for incident coherent beams in terms of the quasiprobability functions, photon-number distribution and phase distribution.
Motivated by recent experiments, we consider a Schrödinger cat superposition of two widely separated coherent states in thermal equilibrium. The time development of our system is obtained using Wigner distribution functions. In contrast to our discussion for a two-Gaussian wave packet [Phys. Lett. A 286 (2001) 87], we find that, in the absence of dissipation, the interference term does not decay rapidly in time, but in common with the other two terms, it oscillates in time and persists for all times
In this paper, we study some quantum properties of a superposition of displaced squeezed two-mode vacuum and single-photon states, such as the second-order correlation function, the Cauchy-Schwartz inequality, quadrature squeezing, quasiprobability distribution functions and purity. This type of states includes two mechanisms, namely, interference in phase space and entanglement. We show that these states can exhibit sub-Piossonian statistics, squeezing and deviate from the classical Cauchy-Schwartz inequality. Moreover, the amount of entanglement in the system can be increased by increasing t…
We show that a cavity field can evolve from an initial displaced mixed thermal state to a macroscopic superpositions of displaced thermal states via resonant interaction with a two-level atom. As a macroscopic system (meter) is really in a mixed state before coupling with the microscopic system at some temperature, our result is important for studying the quantum measurement problem and decoherence under real conditions. For the two-mode case, entanglement of displaced thermal states between the modes can be obtained.
In this paper we discuss the quantum properties for superposition of squeezed displaced number states against multiphoton Jaynes-Cummings model (JCM). In particular, we investigate atomic inversion, photon-number distribution, purity, quadrature squeezing, Mandel $Q$ parameter and Wigner function. We show that the quadrature squeezing for three-photon absorption case can exhibit revivals and collapses typical to those occurring in the atomic inversion for one-photon absorption case. Also we prove that for odd number absorption parameter there is a connection between the evolution of the atomic…
Entangling gates in trapped-ion quantum computing have primarily targeted stationary ions with initial motional distributions that are thermal and close to the ground state. However, future systems will likely incur significant non-thermal excitation due to, e.g., ion transport, longer operational times, and increased spatial extent of the trap array. In this paper, we analyze the impact of such coherent motional excitation on entangling-gate error by performing simulations of Molmer-Sorenson (MS) gates on a pair of trapped-ion qubits with both thermal and coherent excitation present in a shar…
We propose a scheme for the generation and reconstruction of entangled states between the internal and external (motional) degrees of freedom of a trapped electron. Such states also exhibit quantum coherence at a mesoscopic level.
We propose a scalable procedure to generate entangled superpositions of motional coherent states and electronic states in N trapped ions. Beyond their fundamental importance, these states may be of interest for quantum information processing and may be used in experimental studies of decoherence.
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.
We propose three schemes to engineer 2^M and M+1 circular states for the motion of the center of mass of a trapped ion, $M$ being the number of laser pulses. Since the ion is subjected to several laser pulses, we analyze the necessary duration of each one for generating the circular states, and from these, the Fock states and superposition of two-Fock states. We also calculate the probability for obtaining the required states.
Coherently displaced harmonic oscillator number states of a harmonically bound ion can be coupled to two internal states of the ion by a laser-induced motional sideband interaction. The internal states can subsequently be read out in a projective measurement via state-dependent fluorescence, with near-unit fidelity. This leads to a rich set of line shapes when recording the internal-state excitation probability after a sideband excitation, as a function of the frequency detuning of the displacement drive with respect to the ion's motional frequency. We precisely characterize the coherent displ…
We develop a new method of representation of quantum states in terms of the displaced number states. We call it representation, where is an amplitude of the base displaced states. In particular, representation was obtained for set of the displaced number states with different amplitude of the displacement, two-mode squeezed vacuum (TMSV) and superposition of vacuum and single photon. The treatment is employed for building of elementary linear operators, in particular, two-qubit control-sign gate, Hadamard matrix between two two-dimensional Hilbert spaces. Main idea underlying the method is ext…
We analyze collisional decoherence of atoms or molecules prepared in a coherent superposition of nondegenerate internal states at ultralow temperatures and placed in an ultracold buffer gas. Our analysis is applicable for an arbitrary bath particle/tracer particle mass ratio. Both elastic and inelastic collisions contribute to decoherence. We obtain an expression relating the observable decoherence rate to pairwise scattering properties, specifically the low-temperature scattering amplitudes. We consider the dependence on the bath particle/tracer particle mass ratio for the case of light bath …
We investigate how superpositions of motional coherent states naturally arise in the dynamics of a two-level trapped ion coupled to the quantized field inside a cavity. We extend our considerations including a more realistic set up where the cavity is not ideal and photons may leak through its mirrors. We found that a detection of a photon outside the cavity would leave the ion in a pure state. The statistics of the ionic state still keeps some interference effects that might be observed in the weak coupling regime.
We propose a scheme to generate a superposition with arbitrary coefficients on a line in phase space for the center-of-mass vibrational mode of N ions by means of isolating all other spectator vibrational modes from the center-of-mass mode. It can be viewed as the generation of previous methods for preparing motional states of one ion. For large number of ions, we need only one cyclic operatin to generate such a superposition of many coherent states.
We investigate the feasibility of observing matter-wave interference of a micron-sized charged particle by putting it into a quantum superposition of states with a distinguishable separation. In the proposed method, an atomic ion is confined in a linear Paul trap along with the massive charged particle so that we can make use of the extensive toolbox of experimental techniques developed to control quantum states of trapped ions, and to manipulate their motions with high fidelity operations. This approach provides a stringent test of the predictions of dynamical reduction models of delocalised …
Ultracold atoms are trapped circumferentially on a ring that is pierced at its center by a flux tube arising from a light-induced gauge potential due to applied Laguerre-Gaussian fields. We show that by using optical coherent state superpositions to produce light-induced gauge potentials, we can create a situation in which the trapped atoms are simultaneously exposed to two distinct flux tubes, thereby creating superpositions in atomic quantum rings. We consider the examples of both a ring geometry and harmonic trapping, and in both cases the ground state of the quantum system is shown to be a…
We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their disp…
Quantum coherence is one of the clearest departures from classical physics, exhibited when a system is in a superposition of different basis states. Here the coherent superposition of three motional Fock states of a single trapped ion is experimentally certified, with a procedure provably robust against imperfect operation. As the motional state cannot be directly interrogated, our scheme uses an interference pattern generated by projective measurement of the coupled qubit state. The minimum number of coherently superposed states is inferred from a series of threshold values based on analysis …
We describe a novel method to measure the surface charge densities on optical fibers placed in the vicinity of a trapped ion, where the ion itself acts as the probe. Surface charges distort the trapping potential, and when the fibers are displaced, the ion's equilibrium position and secular motional frequencies are altered. We measure the latter quantities for different positions of the fibers and compare these measurements to simulations in which unknown charge densities on the fibers are adjustable parameters. Values ranging from $-10$ to $+50$ e/$μ$m$^2$ were determined. Our results will be…
We present a straightforward scheme for creating macroscopic superpositions of different superfluid flow states of Bose-Einstein condensates trapped in optical lattices. This scheme has the great advantage that all the techniques required are achievable with current experiments. Furthermore, the relative difficulty of creating cats scales favorably with the size of the cat. This means that this scheme may be well-suited to creating superpositions involving large numbers of particles. Such states may have interesting technological applications such as making quantum-limited measurements of angu…
We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Using suitably chosen internal electronic state-dependent displacements based on bi-chromatic laser fields we map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out using fluorescence detection. Combining the two readout results provides full information about the symmetric characteristic function. We characterize the performance of the technique by applying it to a rang…
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.
The observation of genuine quantum features of nano-mechanical motion is a key goal for both fundamental and applied quantum science. To this end, a promising approach is the stabilization of nonclassical features in the presence of dissipation, by means of the tunable coupling with a photonic environment. Here we present a scheme that combines dissipative squeezing with a mechanical nonlinearity to stabilize arbitrary approximations of (displaced) mechanical Fock state of any number. We consider an optomechanical system driven by three control lasers--at the cavity resonance and at the two me…
Using a single, harmonically trapped $^9$Be$^+$ ion, we experimentally demonstrate a technique for generation of arbitrary states of a two-level particle confined by a harmonic potential. Rather than engineering a single Hamiltonian that evolves the system to a desired final sate, we implement a technique that applies a sequence of simple operations to synthesize the state.
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.
The scheme recently proposed in [M. Scala et al., Phys Rev Lett 111, 180403 (2013)], where a gravity-dependent phase shift is induced on the spin of a nitrogen-vacancy (NV) center in a trapped nanodiamond by the interaction between its magnetic moment and the quantized motion of the particle, provides a way to detect spatial quantum superpositions by means of spin measurements only. Here, the effect of unwanted coupling with other motional degrees of freedom is considered and we show that it does not affect the validity of the scheme. Both this coupling and the additional error source due to m…
Generating Einstein$\unicode{x2013}$Podolsky$\unicode{x2013}$Rosen correlations for teleporting collective spin states in a two dimensional trapped ion crystal
We propose protocols to prepare highly excited energy eigenstates of a trapped ion in a harmonic trap which do not require laser pulses to induce transitions among internal levels. Instead the protocols rely on smoothly deforming the trapping potential between single and double well configurations. The speed of the changes is set to minimize non-adiabatic transitions by keeping the adiabaticity parameter constant. High fidelities are found for times more than two orders of magnitude smaller than with linear ramps of the control parameter. Deformation protocols are also devised to prepare super…
According to the Gottesmann-Knill theorem the non-Gaussian states are necessary component for a nontrivial quantum computation. We show two efficient and deterministic methods of $χ^{(3)}$ non-Gaussian state generation for a cavity mode using a single trapped ion. Both require ion motional state transfer to the cavity field. The methods are experimentally feasible. The first is based on the well-known protocol for an ion finite motional superposition state generation. It allows for an arbitrary good approximation of $χ^{(3)}$ non-Gaussian states. We give criteria based on the Wigner function w…
We propose an experimental scheme to simulate and observe the Klein tunneling of relativistic Maxwell particles with trapped ions. We explore the scattering dynamics of the pseudospin-one Maxwell particles and demonstrate that the scattered state should be a superposition of a reflection state, a localization state, and a transmission state. The probabilities of these states can be analytically obtained by the approach of Landau-Zener transition. We further show that the Maxwell Hamiltonian and the associated scattering dynamics can be mimicked with two trapped ions. The Maxwell spinors are en…
We demonstrate a coherence time of 2.1(1)~s for electron spin superposition states of a single trapped $^{40}$Ca$^+$ ion. The coherence time, measured with a spin-echo experiment, corresponds to residual rms magnetic field fluctuations $\leq$~2.7$\times$10$^{-12}$~T. The suppression of decoherence induced by fluctuating magnetic fields is achieved by combining a two-layer $μ$-metal shield, which reduces external magnetic noise by 20 to 30~dB for frequencies of 50~Hz to 100~kHz, with Sm$_2$Co$_{17}$ permanent magnets for generating a quantizing magnetic field of 0.37~mT. Our results extend the …
We present a study of the coherence properties of a variety of motional states of a single ion confined in a Penning ion trap. We demonstrate that the motion of the ion has a coherence time of the order of one second, using Ramsey interferometry. We introduce a technique for preparing the ion in an incoherent superposition of highly-excited motional states using a simple modification of optical sideband cooling. Coherent manipulation of these states allow measurements of optical and motional coherence to be carried out. We show that these highly-excited motional state superpositions have long …
A trapped-ion quantum tunneling rotor (QTR) is in a quantum superposition of two different Wigner crystal orientations. In a QTR system, quantum tunneling drives the coherent transition between the two different Wigner crystal orientations. We theoretically study the quantum dynamics of a QTR, particularly when the spin state of one of the ions is flipped. We show that the quantum dynamics of an $\it{N}$-ion QTR can be described by continuous-time cyclic quantum walks. We also investigate the quantum dynamics of the QTR in a magnetic field. Flipping the spin state breaks the rotational symmetr…
We investigate the Nth order amplitude squeezing in the fan-state |ξ;2k,f>_F which is a linear superposition of the 2k-quantum nonlinear coherent states. Unlike in usual states where an ellipse is the symbol of squeezing, a 4k-winged flower results in the fan state. We first derive the analytical expression of squeezing for arbitrary k, N, f and then study in detail the case of a laser-driven trapped ion characterized by a specific form of the nonlinear function f. We show that the lowest order in which squeezing may appear and the number of directions along which the amplitude may be squeezed…
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…
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of large number of states. The mesoscopic superposition of the generalized cat state is correlated with the corresponding state overlap function, controlled by the sub-Planck structures arising from phase space interference. The asymptotic expression of this overlap function is evaluated and the validity of large phase space support and distinguishability of the constituent states, in which context the asymptotic limit is achieved, are discussed in detail. …
A simple technique for robust generation of vibrational Fock states in a chain of trapped ions is proposed. The method is fast and easy to implement, since only a single chirped laser pulse, simultaneously addressing all of the ions, is required. Furthermore, because the approach uses collective adiabatic passage, significant fluctuations in the intensity or frequency of the laser pulse can be tolerated, and the technique performs well even on the border of the Lamb-Dicke regime. We also demonstrate how this technique may be extended in order to create non-classical superposition states of the…
Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by…
We propose a scheme to generate number states (and specific superpositions of them) of the vibrational motion of a trapped ion. In particular, we show that robust to noise qubits can be generated with arbitrary amplitudes.
We present a proposal for protecting states against decoherence, based on the engineering of pointer states. We apply this procedure to the vibrational motion of a trapped ion, and show how to protect qubits, squeezed states, approximate phase eigenstates and superpositions of coherent states.
Faraday rotation of a laser field induced by a single atom is demonstrated by tightly focussing a linearly polarized laser beam onto a laser-cooled ion held in a harmonic Paul trap. The polarization rotation signal is further used to measure the phase-shift associated with electromagnetically-induced-transparency and to demonstrate read-out of the internal state on the qubit transition with a detection fidelity of 98 $\pm$ 1%. These results have direct implications for single atom magnetometery and dispersive read-out of atomic superpositions.
We present an effective method of coherent state superposition (cat state) generation using single trapped ion in a Paul trap. The method is experimentally feasible for coherent states with amplitude $α\le 2$ using available technology. It works both in and beyond the Lamb-Dicke regime.
Quantum computing gates are proposed to apply on trapped ions in decoherence-free states. As phase changes due to time evolution of components with different eigenenergies of quantum superposition are completely frozen, quantum computing based on this model would be perfect. Possible application of our scheme in future ion-trap quantum computer is discussed.
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 …
The stable operation of quantum computers will rely on error-correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator. A powerful encoding is formed from a periodically spaced superposition of position eigenstates. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach. Here, we demonstra…
We demonstrate the preparation and coherent control of the angular momentum state of a two-ion crystal. The ions are prepared with an average angular momentum of $7780\hbar$ freely rotating at 100~kHz in a circularly symmetric potential, allowing us to address rotational sidebands. By coherently exciting these motional sidebands, we create superpositions of states separated by up to four angular momentum quanta. Ramsey experiments show the expected dephasing of the superposition which is dependent on the number of quanta separating the states. These results demonstrate coherent control of a co…
We present an efficient scheme to generate two-mode SU(2) macroscopic quantum superposition (Schrödinger cat) states, entangled number states and entangled coherent states for the vibrational motion of an ion trapped in a two-dimensional harmonic potential well. We also show that the same scheme can be used to realize a Fredkin gate operation.
