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arXiv:1604.08933 · Physical Review A

Quantum transitions and quantum entanglement from Dirac-like dynamics simulated by trapped ions

Victor A. S. V. Bittencourt, Alex E. Bernardini, Massimo Blasone A

Published May 17, 2016• 18 citations
arXivHTMLPDF

Abstract

Quantum transition probabilities and quantum entanglement for two-qubit states of a four-level trapped ion quantum system are computed for time-evolving ionic states driven by Jaynes-Cummings Hamiltonians with interactions mapped onto a $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ group structure. Using the correspondence of the method of simulating a $3+1$ dimensional Dirac-like Hamiltonian for bispinor particles into a single trapped ion, one preliminarily obtains the analytical tools for describing ionic state transition probabilities as a typical quantum oscillation feature. For Dirac-like structures driven by generalized Poincar\'e classes of coupling potentials, one also identifies the $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ internal degrees of freedom corresponding to intrinsic parity and spin polarization as an adaptive platform for computing the quantum entanglement between the internal quantum subsystems which define two-qubit ionic states. The obtained quantum correlational content is then translated into the quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the trapping magnetic field. Experimentally, the controllable parameters simulated by ion traps can be mapped into a Dirac-like system in the presence of an electrostatic field which, in this case, is associated to ionic carrier interactions. Besides exhibiting a complete analytical profile for ionic quantum transitions and quantum entanglement, our results indicate that carrier interactions actively drive an overall suppression of the quantum entanglement.

DOI
10.1103/physreva.93.053823

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Quantum transition probabilities and quantum entanglement for two-qubit states of a four-level trapped ion quantum system are computed for time-evolving ionic states driven by Jaynes-Cummings Hamiltonians with interactions mapped onto a SU(2)\ensuremath⨂SU(2)\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)SU(2)\ensuremath⨂SU(2) group structure. Using the correspondence of the method of simulating a 3+13+13+1 dimensional Dirac-like Hamiltonian for bispinor particles into a single trapped ion, one preliminarily obtains the analytical tools for describing ionic state transition probabilities as a typical quantum oscillation feature. For Dirac-like structures driven by generalized Poincar'e classes of coupling potentials, one also identifies the SU(2)\ensuremath⨂SU(2)\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)SU(2)\ensuremath⨂SU(2) internal degrees of freedom corresponding to intrinsic parity and spin polarization as an adaptive platform for computing the quantum entanglement between the internal quantum subsystems which define two-qubit ionic states. The obtained quantum correlational content is then translated into the quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the trapping magnetic field. Experimentally, the controllable parameters simulated by ion traps can be mapped into a Dirac-like system in the presence of an electrostatic field which, in this case, is associated to ionic carrier interactions. Besides exhibiting a complete analytical profile for ionic quantum transitions and quantum entanglement, our results indicate that carrier interactions actively drive an overall suppression of the quantum entanglement.
A description of the effects of the local noise on the quantum entanglement constraining the internal degrees of freedom of Dirac bi-spinor structures driven by arbitrary Poincaré invariant potentials is proposed. Given that the Dirac equation dynamics including external potentials can be simulated by a suitable four level trapped ion setup, quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the external magnetic field (used for lift the ions degeneracy), are recovered by means of a suitable ansatz. This formalism allows the inclusion of noise effects, which leads to disentanglement in the four level trapped ion quantum system. Our results indicate the role of interactions in bi-spinor entanglement, as well as the description of disentanglement in ionic states under local noises. For a state prepared initially in one of the ionic levels, local noise induces entanglement sudden death followed by sudden revivals driven by the noiseless dynamics of the state. Residual quantum correlations are observed in the intervals where such state is separable. Schrödinger cat and Werner states partially loose their initial entanglement content due to the interaction with the noisy environment but presenting entanglement oscillations without sudden death. Because Dirac equation describes low energy excitations of mono layer and bi-layer graphene, the formalism can also be applied to compute, for instance, electron-hole or electron/electron entanglement in various circumstances.
Abstract Disentanglement and loss of quantum correlations due to one global collective noise effect are described for two- qubit Schrödinger cat and Werner states of a four level trapped ion quantum system. Once the Jaynes–Cummings ionic interactions are mapped onto a Dirac spinor structure, the elementary tools for computing quantum correlations of two- qubit ionic states are provided. With two- qubit quantum numbers related to the total angular momentum and to its projection onto the direction of an external magnetic field (which lifts the degeneracy of the ion’s internal levels), a complete analytical profile of entanglement for the Schrödinger cat and Werner states is obtained. Under vacuum noise (during spontaneous emission), the two- qubit entanglement in the Schrödinger cat states is shown to vanish asymptotically. Otherwise, the robustness of Werner states is concomitantly identified, with the entanglement content recovered by their noiseless-like evolution. Most importantly, our results point to a firstly reported sudden transition between classical and quantum decay regimes driven by a classical collective noise on the Schrödinger cat states, which has been quantified by the geometric discord.
We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.
We study the dynamics of Rydberg ions trapped in a linear Paul trap, and discuss the properties of ionic Rydberg states in the presence of the static and time-dependent electric fields constituting the trap. The interactions in a system of many ions are investigated and coupled equations of the internal electronic states and the external oscillator modes of a linear ion chain are derived. We show that strong dipole–dipole interactions among the ions can be achieved by microwave dressing fields. Using low-angular momentum states with large quantum defect, the internal dynamics can be mapped onto an effective spin model of a pair of dressed Rydberg states that describes the dynamics of Rydberg excitations in the ion crystal. We demonstrate that excitation transfer through the ion chain can be achieved on a nanosecond timescale and discuss the implementation of a fast two-qubit gate in the ion chain.
We study the dynamics of the 2+12+12+1 Dirac oscillator exactly and find spin oscillations due to a Zitterbewegung of purely relativistic origin. We find an exact mapping of this quantum-relativistic system onto a Jaynes-Cummings model, describing the interaction of a two-level atom with a quantized single-mode field. This equivalence allows us to map a series of quantum optical phenomena onto the relativistic oscillator and vice versa. We make a realistic experimental proposal, in reach with current technology, for studying the equivalence of both models using a single trapped ion.
Abstract The two-dimensional Dirac oscillator is one of the most transparent exactly solvable models in relativistic quantum mechanics. Its Hamiltonian is linear in both momentum and posi- tion and therefore has no direct elementary-particle realization in ordinary harmonic traps. This extended proposal develops two complementary laboratory routes. The first is an optical trapped-ion implementation in which the electronic two-level system encodes the Dirac bispinor and two transverse motional modes encode the two spatial degrees of freedom. Bichromatic red- and blue-sideband laser fields synthesize the Jaynes-Cummings and anti-Jaynes-Cummings cou- plings required by the oscillator. The second route is a microwave-based proposal. Two meanings of “microwave implementation” are distinguished: a microwave-driven trapped-ion implementa- tion using magnetic-gradient-induced spin-motion coupling, and a microwave-resonator/tight- binding analogue based on coupled dielectric resonators. The former remains a genuine single- ion quantum simulator and can, in principle, implement the same effective Hamiltonian with improved phase stability and reduced optical spontaneous-emission errors. The latter is experi- mentally simpler for spectral measurements and can serve as a second proposal, but it is a wave analogue rather than a single-particle trapped-ion quantum simulation. We do not claim the first mapping of the 2 + 1 Dirac oscillator onto trapped-ion dynamics, which is already established in the literature; rather, the contribution is a consolidated, experimentally oriented roadmap that compares the optical trapped-ion, microwave-driven trapped-ion, and microwave-resonator routes, with explicit feasibility conditions, an observable strategy, a corrected graphical map- ping, an explicit separation of the ideal Hamiltonian from its leading correction terms, a worked order-of-magnitude estimate for a 171 Yb+ ion, and an explicit quantitative cross-walk to the trapped-ion and microwave experiments on which the scheme builds. We identify the advan- tages and limitations of each route and propose a staged experimental program for measuring Zitterbewegung, the relativistic square-root spectrum, and chirality-dependent dynamics.