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arXiv:1609.09775 · Physical Review A

Measurement-induced chaos and quantum state discrimination in an iterated Tavis-Cummings scheme

Juan Mauricio Torres, József Zsolt Bernád, G. Alber, Orsolya Kálmán, T. Kiss

Published Feb 14, 2017• 18 citations
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Abstract

A cavity quantum electrodynamical scenario is proposed for implementing a Schr\"odinger microscope capable of amplifying differences between nonorthogonal atomic quantum states. The scheme involves an ensemble of identically prepared two-level atoms interacting pairwise with a single mode of the radiation field as described by the Tavis-Cummings model. By repeated measurements of the cavity field and of one atom within each pair a measurement-induced nonlinear quantum transformation of the relevant atomic states can be realized. The intricate dynamical properties of this nonlinear quantum transformation, which exhibits measurement-induced chaos, allow approximate orthogonalization of atomic states by purification after a few iterations of the protocol and thus the application of the scheme for quantum state discrimination.

DOI
10.1103/physreva.95.023828

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•Aug 5, 2021•0 citations
A cavity quantum electrodynamical scenario is proposed for implementing a Schr"odinger microscope capable of amplifying differences between nonorthogonal atomic quantum states. The scheme involves an ensemble of identically prepared two-level atoms interacting pairwise with a single mode of the radiation field as described by the Tavis-Cummings model. By repeated measurements of the cavity field and of one atom within each pair a measurement-induced nonlinear quantum transformation of the relevant atomic states can be realized. The intricate dynamical properties of this nonlinear quantum transformation, which exhibits measurement-induced chaos, allow approximate orthogonalization of atomic states by purification after a few iterations of the protocol and thus the application of the scheme for quantum state discrimination.
We introduce a complete Bell measurement on atomic qubits based on two-photon interactions with optical cavities and discrimination of coherent states of light. The dynamical system is described by the Dicke model for two three-level atoms interacting in two-photon resonance with a single mode of the radiation field, which is known to effectively generate a nonlinear two-photon interaction between the field and two states of each atom. For initial coherent states with a large mean photon number, the field state is well represented by two coherent states at half-revival time. For certain product states of the atoms, we prove the coherent generation of Greenberger-Horne-Zeilinger states with two atomic qubits and two orthogonal Schr\"odinger cat states as a third qubit. For arbitrary atomic states, we show that discriminating the two states of the field corresponds to different operations in the Bell basis of the atoms. By repeating this process with a second cavity prepared in a phase-shifted coherent state, we demonstrate the implementation of a complete Bell measurement. The experimental feasibility of our protocols is discussed for cavity-QED, circuit-QED, and trapped-ion setups.
We consider the task of deciding whether an unknown qubit state falls in a prescribed neighborhood of a reference state. We assume that several copies of the unknown state are given and apply a unitary operation pairwise on them combined with a postselection scheme conditioned on the measurement result obtained on one of the qubits of the pair. The resulting transformation is a deterministic, nonlinear, chaotic map in the Hilbert space. We derive a class of these transformations capable of orthogonalizing nonorthogonal qubit states after a few iterations. These nonlinear maps orthogonalize states which correspond to the two different convergence regions of the nonlinear map. Based on the analysis of the border (the so-called Julia set) between the two regions of convergence, we show that it is always possible to find a map capable of deciding whether an unknown state is within a neighborhood of fixed radius around a desired quantum state. We analyze which one- and two-qubit operations would physically realize the scheme. It is possible to find a single two-qubit unitary gate for each map or, alternatively, a universal special two-qubit gate together with single-qubit gates in order to carry out the task. We note that it is enough to have a single physical realization of the required gates due to the iterative nature of the scheme.
We derive an analytical approximate solution of the time-dependent state vector in terms of material Bell states and coherent states of the field for a generalized two-atom Tavis-Cummings model with nonlinear intensity-dependent matter-field interaction. Using this solution, we obtain simple expressions for the atomic concurrence and purity in order to study the entanglement in the system at specific interaction times. We show how to implement entangling atomic operations through measurement of the field. We illustrate how these operations can lead to a complete Bell measurement. Furthermore, when considering two orthogonal states of the field as levels of a third qubit, it is possible to implement a unitary three-qubit gate capable of generating authentic tripartite entangled states such as the Greenberger-Horne-Zeilinger state and the WWW state. As an example of the generic model, we present an ion-trap setting employing the quantized mode of the center-of-mass motion instead of the photonic field, showing that the implementation of realistic entangling operations from intrinsic nonlinear matter-field interactions is indeed possible.
The realization of a Jaynes-Cummings model in the optical domain is proposed for an atomic ensemble. The scheme exploits the collective coupling of the atoms to a quantized cavity mode and the nonlinearity introduced by coupling to high-lying Rydberg states. A two-photon transition resonantly couples the single-atom ground state $|g\ensuremath{\rangle}$ to a Rydberg state $|e\ensuremath{\rangle}$ via a nonresonant intermediate state $|i\ensuremath{\rangle}$, but due to the interaction between Rydberg atoms only a single atom can be resonantly excited in the ensemble. This restricts the state space of the ensemble to the collective ground state $|G\ensuremath{\rangle}$ and the collectively excited state $|E\ensuremath{\rangle}$ with a single Rydberg excitation distributed evenly on all atoms. The collectively enhanced coupling of all atoms to the cavity field with coherent coupling strengths which are much larger than the decay rates in the system leads to the strong coupling regime of the resulting effective Jaynes-Cummings model. We use numerical simulations to show that the cavity transmission can be used to reveal detailed properties of the Jaynes-Cummings ladder of excited states and that the atomic nonlinearity gives rise to highly nontrivial photon emission from the cavity. Finally, we suggest that the absence of interactions between remote Rydberg atoms may, due to a combinatorial effect, induce a cavity-assisted excitation blockade whose range is larger than the typical Rydberg dipole-dipole interaction length.
We experimentally realize a nonlinear quantum protocol for single-photon qubits with linear optical elements and appropriate measurements. Quantum nonlinearity is induced by postselecting the polarization qubit based on a measurement result obtained for the spatial degree of freedom of the single photon which plays the role of a second qubit. Initially, both qubits are prepared in the same quantum state and an appropriate two-qubit unitary transformation entangles them before the measurement of the spatial part. We analyze the result by quantum state tomography of the polarization degree of freedom. We then demonstrate the usefulness of the protocol for quantum state discrimination by iteratively applying it to either of two slightly different quantum states which rapidly converge to different orthogonal states by the iterative dynamics. Our work creates an opportunity to employ effective quantum nonlinear evolution in quantum information processing.
We derive an analytical approximate solution of the time-dependent state vector in terms of material Bell states and coherent states of the field for a generalized two-atom Tavis-Cummings model with nonlinear intensity dependent matter-field interaction. Using this solution, we obtain simple expressions for the atomic concurrence and purity in order to study the entanglement in the system at specific interaction times. We show how to implement entangling atomic operations through measurement of the field. We illustrate how these operations can lead to a complete Bell measurement. Furthermore, when considering two orthogonal states of the field as levels of a third qubit, it is possible to implement a unitary three-qubit gate capable of generating authentic tripartite entangled states such as the Greenberger-Horne-Zeilinger (GHZ) state and the W-state. As an example of the generic model, we present an ion-trap setting employing the quantized mode of the center of mass motion instead the photonic field, showing that the implementation of realistic entangling operations from intrinsic nonlinear matter-field interactions is indeed possible.