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arXiv:2111.09718 · Physical Review Research

Dynamical generation of chiral W and Greenberger-Horne-Zeilinger states in laser-controlled Rydberg-atom trimers

Thorsten Haase, G. Alber, Vladimir M. Stojanović

Published Jul 29, 2022• 19 citations
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Abstract

Motivated by the significantly improved scalability of optically trapped neutral-atom systems, extensive efforts have been devoted in recent years to quantum-state engineering in Rydberg-atom ensembles. Here we investigate the problem of engineering generalized (``twisted'') $W$ states, as well as Greenberger-Horne-Zeilinger (GHZ) states, in the strongly interacting regime of a neutral-atom system. We assume that each atom in the envisioned system initially resides in its ground state and is subject to several external laser pulses that are close to being resonant with the same internal atomic transition. In particular, in the special case of a three-atom system (Rydberg-atom trimer) we determine configurations of field alignments and atomic positions that enable the realization of chiral $W$ states---a special type of twisted three-qubit $W$ states of interest for implementing noiseless-subsystem qubit encoding. Using chiral $W$ states as an example we also address the problem of deterministically converting twisted $W$ states into their GHZ counterparts in the same three-atom system, thus significantly generalizing recent works that involve only ordinary $W$ states. We show that starting from twisted---rather than ordinary---$W$ states is equivalent to renormalizing downward the relevant Rabi frequencies. While this leads to somewhat longer state-conversion times, we also demonstrate that those times are at least two orders of magnitude shorter than typical lifetimes of relevant Rydberg states.

DOI
10.1103/physrevresearch.4.033087

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Advances in Theoretical & Computational Physics•Jun 5, 2022•0 citations
Motivated by the significantly improved scalability of optically trapped neutral-atom systems, extensive efforts have been devoted in recent years to quantum-state engineering in Rydberg-atom ensembles. Here we investigate the problem of engineering generalized (``twisted'') WWW states, as well as Greenberger-Horne-Zeilinger (GHZ) states, in the strongly interacting regime of a neutral-atom system. We assume that each atom in the envisioned system initially resides in its ground state and is subject to several external laser pulses that are close to being resonant with the same internal atomic transition. In particular, in the special case of a three-atom system (Rydberg-atom trimer) we determine configurations of field alignments and atomic positions that enable the realization of chiral WWW states---a special type of twisted three-qubit WWW states of interest for implementing noiseless-subsystem qubit encoding. Using chiral WWW states as an example we also address the problem of deterministically converting twisted WWW states into their GHZ counterparts in the same three-atom system, thus significantly generalizing recent works that involve only ordinary WWW states. We show that starting from twisted---rather than ordinary---WWW states is equivalent to renormalizing downward the relevant Rabi frequencies. While this leads to somewhat longer state-conversion times, we also demonstrate that those times are at least two orders of magnitude shorter than typical lifetimes of relevant Rydberg states.
We investigate the possibilities for a deterministic conversion between two important types of maximally entangled multiqubit states, namely, WWW and Greenberger-Horne-Zeilinger (GHZ) states, in the Rydberg-blockade regime of a neutral-atom system where each atom is subject to four external laser pulses. Such interconversions between WWW states and their GHZ counterparts have quite recently been addressed using the method of shortcuts to adiabaticity, more precisely techniques based on Lewis-Riesenfeld invariants [R.-H. Zheng et al., Phys. Rev. A 101, 012345 (2020)]. Motivated in part by this recent work, we revisit the WWW to GHZ state-conversion problem using a fundamentally different approach, which is based on the dynamical symmetries of the system and a Lie-algebraic parametrization of its permissible evolutions. In contrast to the previously used invariant-based approach, which leads to a state-conversion protocol characterized by strongly time-dependent Rabi frequencies of external lasers, ours can also yield one with time-independent Rabi frequencies. This feature makes our protocol more easily applicable experimentally, with the added advantage that it allows the desired state conversion to be carried out in a significantly shorter time with the same total laser pulse energy used.
Using the quantum-brachistochrone formalism, we address the problem of finding the fastest possible (time-optimal) deterministic conversion between WWW and Greenberger-Horne-Zeilinger (GHZ) states in a system of three identical and equidistant neutral atoms that are acted upon by four external laser pulses. Assuming that all four pulses are close to being resonant with the same internal (atomic) transition---the one between the atomic ground state and a high-lying Rydberg state---each atom can be treated as an effective two-level system (grgrgr-type qubit). Starting from an effective system Hamiltonian, which is valid in the Rydberg-blockade regime and defined on a four-state manifold, we derive the quantum-brachistochrone equations pertaining to the fastest possible WWW-to-GHZ state conversion. By numerically solving these equations, we determine the time-dependent Rabi frequencies of external laser pulses that correspond to the time-optimal state conversion. In particular, we show that the shortest possible WWW-to-GHZ state-conversion time is given by ${T}_{\text{QB}}=6.8\phantom{\rule{0.222222em}{0ex}}\ensuremath{\hbar}/E$, where EEE is the total laser-pulse energy used, this last time being significantly shorter than the state-conversion times previously found using a dynamical-symmetry-based approach [${T}_{\text{DS}}=(1.33\ensuremath{-}1.66)\phantom{\rule{0.222222em}{0ex}}{T}_{\text{QB}}$].
We investigate the possibilities for a deterministic conversion between two important types of maximally-entangled multiqubit states -- namely, W and Greenberger-Horne-Zeilinger (GHZ) states -- in the Rydberg-blockade regime of a neutral-atom system where each atom is subject to four external laser pulses. Such interconversions between W states and their GHZ counterparts have quite recently been addressed using the method of shortcuts to adiabaticity, more precisely techniques based on Lewis-Riesenfeld invariants [R.-H. Zheng {\em et al.}, Phys. Rev. A {\bf 101}, 012345 (2020)]. Motivated in part by this recent work, we revisit the W-to-GHZ state-conversion problem using a fundamentally different approach, which is based on the dynamical symmetries of the system and a Lie-algebraic parametrization of its permissible evolutions. In contrast to the previously used invariant-based approach, which leads to a state-conversion protocol characterized by strongly time-dependent Rabi frequencies of external lasers, ours can also yield one with time-independent Rabi frequencies. This feature makes our protocol more easily applicable experimentally, with the added advantage that it allows the desired state conversion to be carried out in a significantly shorter time.
While the Rydberg-blockade regime provides the natural setting for creating WWW-type entanglement with cold neutral atoms, it is demonstrated here that a scalable entanglement resource of this type can even be obtained under completely different physical circumstances. To be more precise, a special instance of twisted WWW states---namely, $\ensuremath{\pi}$-twisted ones---can be engineered in one-dimensional arrays of cold neutral atoms with Rydberg-dressed resonant dipole-dipole interaction. In particular, it is shown here that this is possible even when a (dressed) Rydberg excitation is coupled to the motional degrees of freedom of atoms in their respective, nearly harmonic optical-dipole microtraps, which are quantized into dispersionless (zero-dimensional) bosons. For a specially chosen (``sweet-spot'') detuning of the off-resonant dressing lasers from the relevant internal atomic transitions, the desired $\ensuremath{\pi}$-twisted WWW state of Rydberg-dressed qubits is the ground state of the effective excitation---boson Hamiltonian of the system in a broad window of the relevant parameters. Being at the same time separated from the other eigenstates by a gap equal to the single-boson energy, this WWW state can be prepared using a Rabi-type driving protocol. The corresponding preparation times are independent of the system size and several orders of magnitude shorter than the effective lifetimes of the relevant atomic states.
We propose a scalable protocol in the Rydberg atomic system to generate the Greenberger-Horne-Zeilinger (GHZ) states, where the qubits are encoded on the hyperfine ground levels. Our system is featured with off-resonant driving fields rather than strong Rydberg interaction sufficient for blockade. Closely it can follow the desired nonadiabatic passage during the time evolution and avoid the unwanted transition by imposing the passage-dependent and fast-varying global phase, that serves as an error correction to the universal quantum control. In our protocol, an NNN-qubit GHZ state is prepared in N−1N-1N−1 steps, which is found to be robust against both environmental noise and systematic deviation. Our protocol therefore provides an avenue toward large-scale entanglement, which is essential for quantum information processing and computation based on neutral atoms.
In this work, we present a quantum control methodology to create multi-particle entangled states of two typical classes, the W and the Greenberger-Horne-Zeilinger (GHZ). The methodology is demonstrated on a generation of the W and GHZ three-atomic states via the mechanism of two-photon passage using overlapping chirped pulses and the interplay of the chirp rate, the one-photon and two-photon detuning, the peak Rabi frequency and the strength of the Rydberg-Rydberg interactions. A chain of N alkali 87Rb atoms in an optical lattice interacting with two laser fields is modeled within a semi-classical theory. The strategy to create different classes of multiparticle entangled states was revealed through performed dressed state analysis