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arXiv:1409.2691 · Physical Review A

Slicing the Fock space for state production and protection

R. F. Rossetti, G. D. de Moraes Neto, F. O. Prado, F. Brito, M. H. Y. Moussa

Published Sep 24, 2014• 19 citations
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Abstract

In this paper we present a protocol to engineer interactions confined to subspaces of the Fock space in trapped ions: we show how to engineer upper-bounded, lower-bounded, and sliced Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) Hamiltonians. The upper-bounded (lower-bounded) interaction acting upon Fock subspaces ranges from $|0\ensuremath{\rangle}$ to $|M\ensuremath{\rangle}$ ($|N\ensuremath{\rangle}$ $\text{to}\phantom{\rule{4pt}{0ex}}\ensuremath{\infty}$), and the sliced one confined to Fock subspace ranges from $|M\ensuremath{\rangle}$ to $|N\ensuremath{\rangle}$, whatever $M<N$. Whereas the upper-bounded JC or AJC interaction is shown to drive any initial state to a steady Fock state $|N\ensuremath{\rangle}$, the sliced one is shown to produce steady superpositions of Fock states confined to the sliced subspace ${|N\ensuremath{\rangle}\text{,}|N+1\ensuremath{\rangle}}$.

DOI
10.1103/physreva.90.033840

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In this paper we present a protocol to engineer interactions confined to subspaces of the Fock space in trapped ions: we show how to engineer upper-bounded, lower-bounded, and sliced Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) Hamiltonians. The upper-bounded (lower-bounded) interaction acting upon Fock subspaces ranges from ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ to ∣M\ensuremath⟩|M\ensuremath{\rangle}∣M\ensuremath⟩ (∣N\ensuremath⟩|N\ensuremath{\rangle}∣N\ensuremath⟩ to\ensuremath∞\text{to}\phantom{\rule{4pt}{0ex}}\ensuremath{\infty}to\ensuremath∞), and the sliced one confined to Fock subspace ranges from ∣M\ensuremath⟩|M\ensuremath{\rangle}∣M\ensuremath⟩ to ∣N\ensuremath⟩|N\ensuremath{\rangle}∣N\ensuremath⟩, whatever M<NM<NM<N. Whereas the upper-bounded JC or AJC interaction is shown to drive any initial state to a steady Fock state ∣N\ensuremath⟩|N\ensuremath{\rangle}∣N\ensuremath⟩, the sliced one is shown to produce steady superpositions of Fock states confined to the sliced subspace ∣N\ensuremath⟩,∣N+1\ensuremath⟩{|N\ensuremath{\rangle}\text{,}|N+1\ensuremath{\rangle}}∣N\ensuremath⟩,∣N+1\ensuremath⟩.
We present a framework to engineer nonlinear selective Jaynes-Cummings interactions for one-, two- and three-photon transitions. Higher order transitions are also discussed. Numerical simulations are presented to prove the effectiveness of our scheme. We further analyse how to apply these selective interactions to deterministic step-by-step preparation of number states, even in the non-ideal situation where cavity losses and atomic spontaneous emission are considered. Finally, we present a scheme, derived from the engineered selective interactions, to delay the decoherence process of the prepared Fock state.
We propose the implementation of selective interactions of atom-motion subspaces in trapped ions. These interactions yield resonant exchange of population inside a selected subspace, leaving the others in a highly dispersive regime. Selectivity allows us to generate motional Fock (and other nonclassical) states with high purity out of a wide class of initial states, and becomes an unconventional cooling mechanism when the ground state is chosen. Individual population of number states can be distinctively measured, as well as the motional Wigner function. Furthermore, a protocol for implementing quantum logic through a suitable control of selective subspaces is presented.
In this paper, we present a protocol to engineer upper-bounded and sliced Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonians in cavity quantum electrodynamics. In the upper-bounded Hamiltonians, the atom-field interaction is confined to a subspace of Fock states ranging from $\left\vert 0\right\rangle $ up to $\left\vert 4\right\rangle $, while in the sliced interaction the Fock subspace ranges from $\left\vert M\right\rangle $ up to $\left\vert M+4\right\rangle $. We also show how to build upper-bounded and sliced Liouvillians irrespective of engineering Hamiltonians. The upper-bounded and sliced Hamiltonians and Liouvillians can be used, among other applications, to generate steady Fock states of a cavity mode and for the implementation of a quantum-scissors device for optical state truncation.
We study a circuit QED setup involving a single cavity mode and a cyclic qutrit whose parameters are time modulated externally. It is shown that in the dispersive regime this system behaves as a versatile platform to implement effective nnn-photon Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) models by suitably setting the modulation frequency. The atomic levels and the cavity Fock states involved in the effective Hamiltonians can be controlled through adjustment of the system parameters, and different JC and AJC interactions can be implemented simultaneously using multitone modulations. Moreover, one can implement some models that go beyond simple JC and AJC-like interaction, such as multiphoton coupling between certain entangled states. We estimate analytically the associated transition rates and study numerically the dynamics in the presence of Markovian dissipation, demonstrating that lower-order effective Hamiltonians can be implemented with current technology.