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arXiv:0901.3850 · Physical Review A

Efficient creation of maximally entangled states by modulation of tunneling rates

Gentaro Watanabe A

Published Feb 22, 2010• 24 citations
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Abstract

For systems described by the two-site Bose-Hubbard Hamiltonian, I show that a sinusoidal modulation of the tunneling matrix element assists higher-order cotunneling processes. Using this mechanism, I propose an efficient scheme for creating a coherent superposition of states in which all particles are either on one site or all on the other site, the so-called NOON state. This scheme yields an almost perfect NOON state periodically. For larger numbers of particles, further reduction of the time to create the state is possible if more than one modulation frequency is employed. With this scheme, NOON states with a larger number of particles could be realized with state-of-the-art techniques for cold Bose gases in a double-well potential.

DOI
10.1103/physreva.81.021604

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For systems described by the two-site Bose-Hubbard Hamiltonian, I show that a sinusoidal modulation of the tunneling matrix element assists higher-order cotunneling processes. Using this mechanism, I propose an efficient scheme for creating a coherent superposition of states in which all particles are either on one site or all on the other site, the so-called NOON state. This scheme yields an almost perfect NOON state periodically. For larger numbers of particles, further reduction of the time to create the state is possible if more than one modulation frequency is employed. With this scheme, NOON states with a larger number of particles could be realized with state-of-the-art techniques for cold Bose gases in a double-well potential.
We investigate theoretically the generation of microscopic atomic NOON states, corresponding to the coherent $|N,0\ensuremath{\rangle}+|0,N\ensuremath{\rangle}$ superposition with $N\ensuremath{\sim}5$ particles, via collective tunneling of interacting ultracold bosonic atoms within a symmetric double-well potential in the self-trapping regime. We show that a periodic driving of the double well with suitably tuned amplitude and frequency parameters allows one to substantially boost this tunneling process without altering its collective character. The timescale to generate the NOON superposition, which corresponds to half the tunneling time and would be prohibitively large in the undriven double well for the atomic populations considered, can thereby be drastically reduced, which renders the realization of NOON states through this protocol experimentally feasible. Resonance- and chaos-assisted tunneling are identified as key mechanisms in this context. A quantitative semiclassical evaluation of their impact on the collective tunneling process allows one to determine the optimal choice for the driving parameters in order to generate those NOON states as fast as possible.
We suggest a simple scheme for creating a NOON state of repulsively interacting Bose atoms in the double-well potential. The protocol consists of two steps. First, by setting atom-atom interactions to zero, the system is driven to the upper excited state. Second, the interactions are slowly increased and, simultaneously, the interwell tunneling is decreased to zero. We analyze fidelity of the final state to the NOON state depending on the number of atoms, ramp rate, and fluctuations of the system parameters. It is shown that for a given fidelity the ramp rate scales algebraically with the number of atoms.
To generate a NOON state with a large photon number NNN, the number of operational steps could be large and the fidelity will decrease rapidly with NNN. Here we propose a method to generate a type of quantum entangled states, $(|NN00\ensuremath{\rangle}+|00NN\ensuremath{\rangle})/\sqrt{2}$, called ``double NOON'' states, with a setup of two superconducting flux qutrits and five circuit cavities. This scheme operates essentially by employing a two-photon process; i.e., two photons are simultaneously and separately emitted into two cavities when each coupler qutrit is initially in a higher-energy excited state. As a consequence, the ``double'' NOON state creation needs only N+2N+2N+2 operational steps. One application of double NOON states is to get a phase error of 1/(2N)1/(2N)1/(2N) in phase measurement. In comparison, to achieve the same error, a normal NOON state of the form $(|2N,0\ensuremath{\rangle}+|0,2N\ensuremath{\rangle})/\sqrt{2}$ is needed, which requires at least 2N2N2N operational steps to prepare by using the existing schemes. Our numerical simulation demonstrates that high-fidelity generation of the double NOON states with $N\ensuremath{\le}10$ even for imperfect devices is feasible with the present circuit QED technique.
We study the tunneling dynamics in a time-periodically modulated two-mode Bose-Hubbard model using Floquet theory. We consider situations where the system is in the self-trapping regime and either the tunneling amplitude, the interaction strength, or the energy difference between the modes is modulated. In the former two cases, the tunneling is enhanced in a wide range of modulation frequencies, while in the latter case the resonance is narrow. We explain this difference with the help of Floquet analysis. If the modulation amplitude is weak, the locations of the resonances can be found using the spectrum of the nonmodulated Hamiltonian. Furthermore, we use Floquet analysis to explain the coherent destruction of tunneling (CDT) occurring in a large-amplitude modulated system. Finally, we present two ways to create a NOON state (a superposition of NNN particles in mode 1 with zero particles in mode 2 and vice versa). One is based on a coherent oscillation caused by detuning from a partial CDT. The other makes use of an adiabatic variation of the modulation frequency. This results in a Landau-Zener type of transition between the ground state and a NOON-like state.
The generation, manipulation and fundamental understanding of entanglement lies at very heart of quantum mechanics. Among various types of entangled states, the NOON states are a kind of special quantum entangled states with two orthogonal component states in maximal superposition, which have a wide range of potential applications in quantum communication and quantum information processing. Here, we propose a fast and simple scheme for generating NOON states of photons in two superconducting resonators by using a single superconducting transmon qutrit. Because only one superconducting qutrit and two resonators are used, the experimental setup for this scheme is much simplified when compared with the previous proposals requiring a setup of two superconducting qutrits and three cavities. In addition, this scheme is easier and faster to implement than the previous proposals, which require using a complex microwave pulse, or a small pulse Rabi frequency in order to avoid nonresonant transitions.