Article

Information entropy of a time-dependent three-level trapped ion interacting with a laser field

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  • Applied Sciences University, Bahrain
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Abstract

Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, frequency standard applications and quantum information processing. Therefore, in this communication we present a comprehensive analysis of the pattern of information entropy arising in the time evolution of an ion interacting with a laser field. A general analytic approach is proposed for a three-level trapped-ion system in the presence of the time-dependent couplings. By working out an exact analytic solution, we conclusively analyse the general properties of the von Neumann entropy and quantum information entropy. It is shown that the information entropy is affected strongly by the time-dependent coupling and exhibits long time periodic oscillations. This feature attributed to the fact that in the time-dependent region Rabi oscillation is time dependent. Using parameters corresponding to a specific three-level ionic system, a single beryllium ion in a RF-(Paul) trap, we obtain illustrative examples of some novel aspects of this system in the dynamical evolution. Our results establish an explicit relation between the exact information entropy and the entanglement between the multi-level ion and the laser field. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion/field and on the values of Lamb-Dicke parameter η.

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... Hence, our model can be (is) a qutrit-quadrit entangled system. For the high degree of entanglement, we report the quantum dynamics using quantum entropy (Abdel-Aty 2005) in the presence of the coupling parameters. ...
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... 6. The upper bound of S H (t) in this manuscript 6.1 and by comparing this result with [112,113], we may conclude that the upper bound of Shannon entropy is related to the initial number of photonsn by the relation S M ax ≤ ln(n) = =1 lnn s.t.n > 1 and is the number of field modes. So, we may reformulate the definition of Shannon information entropy as S H (t) = 1 =1 lnn ln P(n 1 , n 2 , t) P(n 1 ,n 2 ,t) ...
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Higher dimensional quantum entanglement in a trapped three-level ion interacting with two laser beams in Λ scheme is investigated beyond the Lamb–Dicke limit. It is shown that higher dimensional entanglement can be established in a single step, with a tunable dimensionality and duration via the Lamb–Dicke parameter.
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We study the interaction of a V-type three-level atom with a two-mode field through the two-photon interaction when the atom is initially in the upper state and the initial field is in the two-mode squeezed vacuum state. The influence of the atomic motion on the evolution of the atomic Q-function and atomic Wehrl entropy is examined. The results show that the atomic motion and the mode structure play important roles in the evolution of the atomic Q-function, atomic Wehrl entropy, and marginal atomic Wehrl entropy.
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Motivated by recent developments in quantum entanglement, we study the relations among concurrence and phase entropy of a three-level atom interacting with a bimodal cavity field. Analytical results are presented when the photonic band gap is exhibited by the presence of photonic crystals. The evolution of the atomic inversion with the field initially in a coherent state is examined, and different nonclassical effects in its dynamics are discussed. An extension of the notion of concurrence introduced by Wooters is used to quantify the entanglement. We conclusively calculate the phase entropy and entanglement using the Pegg-Barnett phase formalism. Evidence has been found to support the idea that phase entropy and concurrence are correlated in this particular model. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the presence of the detuning and the photonic band gap, which provides insight into the difference in the nature of the concurrence function for atom-field coupling, mode frequency, and different cavity parameters. We demonstrate how fluctuations in the concurrence and phase entropy are affected by the presence of the photonic band gap. Explicit results with numerical simulations applied to GaAs are obtained.
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Quantum entangled states in a system of trapped three-level ion interacting with two laser beams in Λ (Lambda) configuration is investigated. We have characterized a typical family of initial conditions for their potential to generate quantum entanglement of internal and external degrees of freedom of the ion. It is found that entangled qudits, specifially qutrits and quadrits, can be optimally for a certain preparation of the ionic system. Analytical results, describing the quantum entangled state explicity, are presented. The amount of quantum entanglement is quantified directly by calculating the generalized concurrence for arbitrary qudits. It is obtained that higher dimensional entanglement can be established with the Lamb-Dicke parameter (LDP). The LDP dependence of Schmidt coefficients is shown.
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In the language of quantum information theory we study the entropy squeezing of a two-level atom in a Kerr-like medium. A definition of squeezing is presented for this system, based on information theory. The utility of the definition is illustrated by examining the entropy squeezing of a two-level atom with a Kerr-like medium. The influence of the non-linear interaction of the Kerr medium, the atomic coherence and the detuning parameter on the properties of the entropy and squeezing of the atomic variables is examined.
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We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.
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A new class of uncertainty relations is derived for pairs of observables in a finite-dimensional Hilbert space which do not have any common eigenvector. This class contains an ``entropic'' uncertainty relation which improves a previous result of Deutsch and confirms a recent conjecture by Kraus. Some comments are made on the extension of these relations to the case where the Hilbert space is infinite dimensional.
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Among the numerous types of architecture being explored for quantum computers are systems utilizing ion traps, in which quantum bits (qubits) are formed from the electronic states of trapped ions and coupled through the Coulomb interaction. Although the elementary requirements for quantum computation have been demonstrated in this system, there exist theoretical and technical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated on using quantum communication to link a number of small ion-trap quantum systems. Developing the array-based approach, we show how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques already demonstrated for manipulating small quantum registers. The use of decoherence-free subspaces significantly reduces decoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.
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Experiments directed towards the development of a quantum computer based on trapped atomic ions are described briefly. We discuss the implementation of single-qubit operations and gates between qubits. A geometric phase gate between two ion qubits is described. Limitations of the trapped-ion method such as those caused by Stark shifts and spontaneous emission are addressed. Finally, we describe a strategy to realize a large-scale device.
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The phenomenon of atomic population trapping in the Jaynes-Cummings Model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a `weighted dressedness' distribution characterising the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalisation of the Poisson Summation Formula method for analytical description of revivals developed by Fleischhauer and Schleich [Phys. Rev. A {\bf 47}, 4258 (1993)].
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Methods for, and limitations to, the generation of entangled states of trapped atomic ions are examined. As much as possible, state manipulations are described in terms of quantum logic operations since the conditional dynamics implicit in quantum logic is central to the creation of entanglement. Keeping with current interest, some experimental issues in the proposal for trapped-ion quantum computation by I. Cirac and P. Zoller (University of Innsbruck) are discussed. Several possible decoherence mechanisms are examined and what may be the more important of these are identified. Some potential applications for entangled states of trapped-ions which lie outside the immediate realm of quantum computation are also discussed.
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This communication is an enquiry into the circumstances under which concurrence and phase entropy methods can give an answer to the question of quantum entanglement in the composite state when the photonic band gap is exhibited by the presence of photonic crystals in a three-level system. An analytic approach is proposed for any three-level system in the presence of photonic band gap. Using this analytic solution, we conclusively calculate the concurrence and phase entropy, focusing particularly on the entanglement phenomena. Specifically, we use concurrence as a measure of entanglement for dipole emitters situated in the thin slab region between two semi-infinite one-dimensionally periodic photonic crystals, a situation reminiscent of planar cavity laser structures. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the presence of the detuning and the photonic band gap, which provides insight into the difference in the nature of the concurrence function for atom-field coupling, mode frequency and different cavity parameters. We demonstrate how fluctuations in the phase and number entropies effected by the presence of the photonic-band-gap. The outcomes are illustrated with numerical simulations applied to GaAs. Finally, we relate the obtained results to instances of any three-level system for which the entanglement cost can be calculated. Potential experimental observations in solid-state systems are discussed and found to be promising. Comment: 28 pages, 10 figures: Accepted in Applied Physics B: Laser and Optics
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An approach to the seperability problem of a bipartite system based on the entropic uncertainty relations was proposed. The proposed strategy took advantage of the geometrical structure of the tensor product Hilbert space of the system and underlines the connections between uncertainty relations and entanglement. To replace the statistical variance with the Shannoy entropy as an estimator of the uncertainities associated with the measurement process was the basic idea of the proposed approach. It was shown that entropic uncertainty relations can be derived for more than two observables at a time.
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We have investigated motional heating of laser-cooled 9Be+ ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes. The results show that heating is due to electric-field noise from the trap electrodes which exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating. Comment: With minor changes. 24 pages, including 7 figures. Submitted by Phys. Rev. A
Book
The author provides, in nine chapters, a compact and up-to-date coverage of the entire range of applications of the two most important ion traps: the Paul('radiofrequency') trap and the Penning('dc) trap. The book begins with full details of the ion confinement principles of both these traps; this is followed by a presentation of the basic experimental techniques, including details of a few actual traps. There is then a chapter on the methods of ion cooling, now an essential integral part of all trap-based experiments. The next four chapters provide a comprehensive coverage of applications in four major areas, broadly classified as: atomic physics, frequency standards, collision studies, and analytical mass spectrometry. The text is appended by a set of more than 600 fully titled chronologically arranged references which mirror the growth of the field as well as providing a comprehensive guide to original research papers. The text should be useful to students both at the senior undergraduate and beginning graduate level as a general reader for professionals in atomic physics, chemical physics, mass spectometry and related fields.
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The usual definition of squeezing, based on the Heisenberg uncertainty principle, measures uncertainty in terms of the standard deviation. It can run into difficulties when applied to squeezing in the two-level atom. An alternative definition of squeezing is presented for this system, based on information entropy theory, which overcomes the disadvantages of the definition based on the Heisenberg uncertainty relation. The utility of this definition is illustrated by examining squeezing in the information entropy of a two-level atom in the Jaynes-Cummings model, and in resonance fluorescence.
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Generalized models are presented for the interaction between an N-level atom and (N−1) modes. Spin-1 operators are used to describe the 3-level atom, and the generators of the U(N) group are used in the general case. Different statistical quantities related to the photons and the atmoic system are calculated. Multiphoton processes are discussed. A model is presented in the 3-level atom and 2-mode system, in which infinite series tend to closed forms, for thermal and coherent distributions for the modes.
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A generalized Jaynes-Cummings model is presented to discuss the interaction between an N-level atom and (N − 1) modes of the radiation field. The level to which the rest of the levels are connected lies in the middle. A detuning parameter is introduced to make the model solvable. Distribution and characteristic functions and different statistical averages are calculated. Multiphoton processes are discussed. The system of a three-level atom and two modes with its three different configurations, and a configuration of the system of a five-level atom and four modes are considered under specified initial atomic conditions. The interaction with squeezed modes of light is investigated. The dependence of the features of the phenomenon of collapses and revivals on the configuration of the system, the atomic initial conditions, the detuning parameter and the statistical aspects of the interacting squeezed modes are shown.
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We show that, if one combines the Jaynes-Cummings and anti-Jaynes-Cummings dynamics in a trapped-ion system driven by a laser, additional series of collapses and revivals of the vibrational state of the ion can be generated.
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The object of this Letter is to show that except in the case of canonically conjugate observables, the generalized Heisenberg inequality does not properly express the quantum uncertainty principle. It is, in general, too weak. An inequality is obtained which does express the principle.
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This Letter reports on the existence of periodic spontaneous collapse and revival of coherence in the dynamics of a simple quantum model. Also given are the first accurate expressions for the intermediate-time and long-time dynamical behavior of the model.
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In this work, the time evolution of the entropy of a single-mode field interacting with a Λ-type degenerate quantum beat three-level atom is investigated in the strong field limit. The field is assumed to be prepared in a coherent state. The results show that the initial atomic state and the detuning of the field play important roles in the evolution of the field entropy and the generation of a Schrödinger cat state at the half-revival time of the atomic inversion needs strict conditions. The squeezing properties of the field are also explored, and some new conclusions are obtained.
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A theorem is proven for quantum information theory that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy S of the density operator describing an ensemble of pure quantum signal states is equal to the number of spin-1/2 systems (‘‘quantum bits’’ or ‘‘qubits’’) necessary to represent the signal faithfully. The theorem holds whether or not the signal states are orthogonal. Related results are also presented about the fidelity of quantum coding and about representing entangled quantum states.
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We investigate quantum dynamical properties of a trapped three-level ion interacting with two laser beams in Λ configuration. A unitary transformation method is developed to study the interaction of the ion with its vibrational phonons, quanta of ion’s own center-of-mass motion. Under certain conditions on laser parameters, this interaction is shown to be unitarily equivalent to two-phonon cascade transitions. Complicated temporal behaviors of level populations and mean number of phonons are described clearly by identifying dynamical variables of the cascade model as building blocks. Furthermore, analyzing quantum states of vibrational phonons by Husimi-Q function, we find that at times, determined by the underlying cascade dynamics, two- and three-component macroscopic quantum superposition states can be obtained depending on the Lamb-Dicke parameter and the initial conditions of the system. A wide range of initial conditions and experimental parameters are discussed using both exact and analytical solutions. Alternative routes to reach the target states are found.
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In this paper, we use the quantum field entropy to measure the degree of entanglement in the time development of a three-level atom interacting with two-mode fields including all acceptable kinds of nonlinearities of the two-mode fields. This model describes a very general situation in which the system admits two detuning parameters and includes an arbitrary form of the nonlinearity of the two modes. It is shown that when the individual modes of the field are detuned far from the intermediate atomic level, the dynamic Stark shift is induced by the nonlinear medium. The results show that the non-linearity effect yields the superstructure of atomic Rabi oscillations and changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.
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We have investigated the evolution of the field quantum entropy and the entanglement of the atom field in a three-level atom, with an additional Kerr-like medium for one mode. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. A factorization of the initial density operator is assumed, where the privileged field mode is in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The effect of a Kerr-like medium on the entropy is analysed. It is shown that the addition of the Kerr medium has an important effect on the properties of the entropy and the entanglement. The results show that the effect of the Kerr medium changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.
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From a quantum information point of view we study the entropy squeezing of a two-level atom interacting with two modes with intensity-dependent coupling. The Hamiltonian we consider consists of all acceptable forms of nonlinearities. A definition of squeezing is presented for this system, based on information theory. The utility of the definition is illustrated by examining squeezing in the information entropy of a two-level atom in the presence of a nonlinear medium. We examine the influence of the nonlinear interaction, the atomic coherence and the detuning parameter on the properties of the entropy and squeezing of the atomic variables. It is shown that features of the quantum entropy are influenced significantly by the kinds of intensity-dependent atom-field coupling and the nonlinearities of the two-mode fields.
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The number-phase entropic uncertainty relation for the multiphoton coherent state (MCS) and nonlinear coherent state (NCS) are studied and compared with an ordinary coherent state (CS). We show that the MCS has higher (lower) number (phase) entropy while the NCS has lower (higher) number (phase) entropy in comparision to the CS. We also discuss the number-phase Wigner function for these states which gives a graphical representation of the complimentary nature of the number and phase properties for these states. With the help of this Wigner representation a photon number cat formation in the NCS is discussed.
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Analytic expressions have been found for transition probabilities in a degenerate n-level atom interacting with a strong external field that gives a common time dependence to all of the transition matrix elements. Except for solving a simple nth-order equation to determine eigenvalues of dressed states, the method is entirely analytic. These expressions may be used to control electron populations in degenerate n-level atoms. Examples are given for n = 2 and 3.
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In the present communication we investigate the usual Jaynes–Cummings Hamiltonian model, describing two-level atom interacting with an electromagnetic field, in the presence of the second harmonic generation (degenerate parametric amplifier). Exact solutions of the wave function in the Schrödinger picture have been obtained for two different cases. In the first case the field frequency ω is not equal to the splitting photon frequency ε, where the canonical transformation has been invoked to obtain the solution of the wave function. In the second case, we considered both frequencies are equal (ε = ω) and the system is taken to be at exact resonance. Both solutions have been used to discuss the atomic inversion as well as the entropy squeezing. It has been shown that the system is sensitive to any change in the coupling parameter responses of the second harmonic generation as well as to the atomic phase angle.
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We present theoretical and experimental studies of the center-of-mass c.m. stability of ions in a Penning trap with a quadrupole rotating electric field. The rotation frequency of an ion cloud in a Penning trap determines the cloud density and shape, and it can be precisely controlled by a rotating electric field. The quadrupole rotating-field scheme can control pure single-species plasmas in contrast to the dipole field, which is effective only for plasmas composed of two or more species of ions. However, the quadrupole field can modify the trap stability because of the spatial dependence of the electric field. In this study, we theoretically and experimentally determine the c.m. stability condition for ions in a Penning trap with a rotating quadrupole field. The experimental results agree well with the theoretical prediction. In the limit of zero magnetic field we obtain a type of rf trap which uses a rotating quadrupole field and in which the c.m. motion is analytically solvable.
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The exact upper and lower bounds on the sum of the information entropies of three spin-1/2 operators SX, SY, SZ are derived. They show that, for a set of more than two observables, entropic uncertainty and certainty relations can exist which do not reduce to those satisfied by the pairs in the set. This result is generalized to sets of N+1 complementary observables existing in N-dimensional Hilbert spaces. Publicado
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The exact lower bound on the sum of the information entropies is obtained for arbitrary pairs of observables in two-dimensional Hilbert space. The result coincides with that given by Garrett and Gull for the particular case of real transformation matrices and state vectors. A weaker analytical bound is also obtained. Publicado
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The entropic uncertainty relation for sets of N+1N+1 complementary observables {Ak}\{A_k\} in NN-dimensional Hilbert space, kH(Ak)(N+1)ln[12(N+1)]\sum_kH(A_k)\geq (N+1)\ln[\frac12(N+1)], is sharpened to kH(Ak)12Nln(12N)+(12N+1) ⁣ln(12N+1)\sum_kH(A_k)\geq\frac12N\, \ln(\frac12N)+(\frac12 N+1)\!\ln(\frac12N+1) for even NN. A nontrivial upper bound on the entropy sum (entropic certainty relation) is also obtained for not completely mixed states, while a previously given expression for this bound is proved to hold only when N=2N=2. Publicado
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Considering a system consisting of a two-level atom, initially prepared in a coherent superposition of upper and lower levels, interacting with a coherent state of the field, we show that the dynamics of the atom as well as the spectrum of the field are sensitive to the relative phase between the atomic dipole and the cavity field. It is shown that, for a certain choice of this phase, ``coherent trapping'' occurs in two-level atoms. In the case of spectra, for the same choice of the phase, instead of a three-peaked symmetric spectrum, we have an asymmetric two-peaked spectrum.
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We investigate the phenomenon of quantum revivals in the Jaynes-Cummings model for an arbitrary quantized field mode. With the help of the Poisson summation formula, we cast the infinite sum determining the atomic inversion into an infinite sum of integrals. Each integral, when evaluated using the method of stationary phase, yields under appropriate conditions one revival. We present simple approximate analytical expressions for these revivals and illustrate this general technique by the examples of a coherent and a highly squeezed state. The oscillatory photon distribution of the latter creates slightly different Rabi frequencies which give rise to a beat note; that is, echos in the revivals. We obtain the photon statistics of the quantized field by ``measuring'' the atomic collapse of a single revival-a technique which might be applicable in the realm of the one-atom maser.
Article
We propose a technique to generate nonclassical vibrational states of the quantized center-of-mass motion of an ion in a harmonic trap, based on the quantum conversion between the quantum cavity field and the quantized center-of-mass motion. It is shown that when an ion trap system interacts with the eigenmode of a single-mode Fabry-Pérot cavity, where the trap is set in, and with an external classical driving electromagnetic field through Raman transitions, the interchange of the quantum features between the quantum cavity field and the quantized trap occurs. This kind of quantum conversion can be used to prepare some nonclassical trap states for the ion trap as well as to measure the quantum statistics of an initial nonclassical vibrational state.
Article
A quantum computer can be implemented with cold ions confined in a linear trap and interacting with laser beams. Quantum gates involving any pair, triplet, or subset of ions can be realized by coupling the ions through the collective quantized motion. In this system decoherence is negligible, and the measurement (readout of the quantum register) can be carried out with a high efficiency.
Article
We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}
Article
We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound {sup 9}Be{sup +} ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.
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Quantum computers require the storage of quantum information in a set of two-level systems (called qubits), the processing of this information using quantum gates and a means of final readout. So far, only a few systems have been identified as potentially viable quantum computer models--accurate quantum control of the coherent evolution is required in order to realize gate operations, while at the same time decoherence must be avoided. Examples include quantum optical systems (such as those utilizing trapped ions or neutral atoms, cavity quantum electrodynamics and nuclear magnetic resonance) and solid state systems (using nuclear spins, quantum dots and Josephson junctions). The most advanced candidates are the quantum optical and nuclear magnetic resonance systems, and we expect that they will allow quantum computing with about ten qubits within the next few years. This is still far from the numbers required for useful applications: for example, the factorization of a 200-digit number requires about 3,500 qubits, rising to 100,000 if error correction is implemented. Scalability of proposed quantum computer architectures to many qubits is thus of central importance. Here we propose a model for an ion trap quantum computer that combines scalability (a feature usually associated with solid state proposals) with the advantages of quantum optical systems (in particular, quantum control and long decoherence times).
Article
I propose a scheme which allows for reliable transfer of quantum information between two atoms via an optical fibre in the presence of decoherence. The scheme is based on performing an adiabatic passage through two cavities which remain in their respective vacuum states during the whole operation. The scheme may be useful for networking several ion-trap quantum computers, thereby increasing the number of quantum bits involved in a computation. Comment: 4 pages, 3 figures, RevTeX, submitted to PRL
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We propose a scheme to utilize photons for ideal quantum transmission between atoms located at spatially-separated nodes of a quantum network. The transmission protocol employs special laser pulses which excite an atom inside an optical cavity at the sending node so that its state is mapped into a time-symmetric photon wavepacket that will enter a cavity at the receiving node and be absorbed by an atom there with unit probability. Implementation of our scheme would enable reliable transfer or sharing of entanglement among spatially distant atoms. Comment: 4 pages, 3 postscript figures
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We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. Especially we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits. Comment: 8 pages, 3 figures, v2: small changes
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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 with trapped ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke limit, sympathetic cooling and laser cooling using electromagnetically induced transparency. After that we discuss the problem of state detection using the electron shelving method. Then quantum gates are described. We introduce single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also comment on more advanced multi-qubit logic gates. We describe how quantum logic networks may be used for the synthesis of arbitrary pure quantum states. Finally, we discuss the speed of quantum gates and we also give some numerical estimations for them. A discussion of dynamics on off-resonant transitions associated with a qualitative estimation of the weak coupling regime and of the Lamb-Dicke regime is included in Appendix.
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We first consider the basic requirements for a quantum computer, arguing for the attractiveness of nuclear spins as information-bearing entities, and light for the coupling which allows quantum gates. We then survey the strengths of and immediate prospects for quantum information processing in ion traps. We discuss decoherence and gate rates in ion traps, comparing methods based on the vibrational motion with a method based on exchange of photons in cavity QED. We then sketch the main features of a quantum computer designed to allow an algorithm needing 10^6 Toffoli gates on 100 logical qubits. We find that around 200 ion traps linked by optical fibres and high-finesse cavities could perform such an algorithm in a week to a month, using components at or near current levels of technology.