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Physical Review A

Perfect state transfer and maximal entanglement in a trimer of three-level systems

Abuenameh Aiyejina, Ethan Wyke, Roger Andrews, Andrew D. Greentree

Published Sep 25, 2023• 3 citations
Publisher

Abstract

We analyze a quantum wire that consists of a linear chain of three qutrits with nearest-neighbor interactions and detuning on the central qutrit. We derive the detuning conditions for perfect state transfer of a double excitation and for maximal entanglement between the first and third qutrits. We also analytically determine the times for perfect state transfer and maximal entanglement, which are found to be periodic. We also show that perfect state transfer is a necessary condition for maximal entanglement, with the first occurrence of maximal entanglement at half the time for the first occurrence of perfect state transfer.

DOI
10.1103/physreva.108.032421

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We analyze a quantum wire that consists of a linear chain of three qutrits with nearest-neighbor interactions and detuning on the central qutrit. We derive the detuning conditions for perfect state transfer of a double excitation and for maximal entanglement between the first and third qutrits. We also analytically determine the times for perfect state transfer and maximal entanglement, which are found to be periodic. We also show that perfect state transfer is a necessary condition for maximal entanglement, with the first occurrence of maximal entanglement at half the time for the first occurrence of perfect state transfer.
We derive and simulate the wavefunctions and double-excitation probabilities for dimer and trimer three-level systems. Perfect state transfer occurs for pulse strengths that are odd multiples of π /2 when the parameter Jt equals odd multiples of π /2 and π /2 for the dimer and trimer, respectively. Near-perfect state transfer with a single photon occurred for photon coupling strength g =2 J when Jt=1.99,19.87 in the dimer and Jt=2.62,25.51 in the trimer. The nature of perfect state transfer is due to localization and transfer of double excitations for given pulse strengths and times.
Perfect quantum state mirroring in a chain of N spins is defined as the condition in which the state |i〉 of the chain is swapped into the state |N — i〉 within a time evolution interval τ. Such a phenomenon is an interesting way of transfering entanglement. An expressions for the perfect mirroring of a single qubit contained in a spin chain were proposed in the past. We exploit such an expressions for calculating the evolution times in chains of both two and three spins. In the case of a chain of two qubits, we derive conditions under which the associated four Bell states diagonalize the Hamiltonian. It is found that for the two Bell states |Φ + 〉 and |Φ − 〉, perfect mirroring does not occur (i.e. entanglement is not preserved under swapping). On the other hand, perfect single qubit mirror effect (entanglement preservation) indeed occurs for the other two Bell states |Ψ + 〉 and |Ψ − 〉 which are mapped into |Φ + 〉 and |Φ − 〉 respectively. For the case of a chain of three qubits, the effects of a perfect single qubit mirroring on a set of four maximally entangled three qubit states ψ 1 , ψ 2 , χ 1 , and χ 2 are studied. Due to the fact that quantum mirroring preserves maximal entanglement, the states ψ 1 and ψ 2 are not altered. However, quantum mirroring changes the states χ 1 and χ 2 only if we apply perfect quantum state mirroring in the site a = 1 of the three qubits spin chain. The above constrains the preservation of maximal entanglement under qubit mirroring of such a state. Due to the fact that swapping has already been experimentally tested, a posible experimental implementations of single qubit mirroring is possible.
The excitation probabilities, concurrence, and relative entropy of coherence for an array of three coupled two-level systems are studied analytically and numerically at various temperatures. We determine the excitation probabilities for each site, the concurrence between the outer sites, and the relative entropy of coherence of the system for three energy configurations of the trimer and various temperatures. We find that a well configuration (i.e., negative detuning of the inner site) at all temperatures localizes the excitation on the inner site, while a barrier configuration (i.e., positive detuning of the inner site) causes the excitation to be localized on the outer sites. We determine the decay times for the probabilities, concurrence, and relative entropy of coherence for all energy configurations at each temperature. The barrier configuration creates the largest concurrence at zero temperature; however, we find that the uniform configuration is able to resist the loss of quantum coherence and entanglement at higher temperatures more so than any of the other configurations.