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W-to-GHZ state conversion in the Rydberg-blockade regime of neutral-atom systems: dynamical-symmetry-based approach

Thorsten Haase, G. Alber, Vladimir M. Stojanović

Published Dec 30, 2020• 0 citations
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Abstract

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.

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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.

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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.
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 [${T}_{\text{DS}}=(1.33\ensuremath{-}1.66)\phantom{\rule{0.222222em}{0ex}}{T}_{\text{QB}}$].
In the paper, we propose a protocol for the one-step interconversions between the Greenberger-Horne-Zeilinger (GHZ) states and the WWW states with deterministic success probabilities. The Rydberg interactions among three Rydberg atoms are utilized here to structure nonlocal operations for the interconversions. Additionally, we construct a Lewis-Riesenfeld invariant for a four-level system and afterwards use it to reversely design the Rabi frequencies of the effective Hamiltonian. The numerical simulations indicate that the protocol is robust against the spontaneous emission, the dephasing, the thermal noise, the systematic errors on Rabi frequencies, and the frequency mismatching. Therefore, the protocol may contribute to the research of the interconversions between entanglements.
Conversion between different types of entangled states is an interesting problem in quantum mechanics. But research on the conversion between Greenberger-Horne-Zeilinger (GHZ) state and Knill-Laflamme-Milburn (KLM) state in atomic system is absent. In this paper, we propose a scheme to realize the interconversion (one-step) between GHZ state and KLM state with Rydberg atoms. By utilizing Rydberg-mediated interactions, we simplify the system. By combining Lie-transform-based pulse design, the evolution path is built up to realize interconversion of GHZ state and KLM state. The numerical simulation result shows that the present scheme is robust against decoherence and operational imperfection, the analysis shows that the scheme is feasible with current experimental technology.
Abstract Conversion between different types of entangled states is an interesting problem in quantum mechanics. But research on the conversion between the Greenberger–Horne–Zeilinger (GHZ) state and Knill–Laflamme–Milburn (KLM) state in an atomic system has not been reported. In this paper, we propose a scheme to realize the interconversion (one-step) between the GHZ state and KLM state with Rydberg atoms. By utilizing Rydberg-mediated interactions, we simplify the system. By combining a Lie-transform-based pulse design, the evolution path is built up to realize interconversion of the GHZ state and KLM state. The numerical simulation result shows that the present scheme is robust against decoherence and operational imperfection.
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.