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2The 30th International Conference on Advanced Laser Technologies P-O-8
ALT'23
Investigation of the lifetime of entangled states of interacting qubits in an electromagnetic field by the path integration method
A. Biryukov1, M. Shleenkov2
1- Samara State Transport University 2- Samara National Research University Main author email address: biryukov_1@mail.ru
The entangled states of two identical qubits are super important in quantum informatics. They have lots of potential applications like quantum computers, cryptography, and quantum teleportation. But here's the catch: in real models, qubits can only stay highly entangled for a limited time. That's why there's been a ton of research, both theoretical and experimental, to understand and extend the duration of quantum entanglement for qubits over the last few decades.
In this case, we'll focus on a mathematical model where two qubits interact with an external electromagnetic field. The interaction between their dipole moments is described by a specific time-dependent function. Our goal is to study how the degree of entanglement depends on the parameters that characterize the qubit interaction and external electromagnetic field. By finding the most optimal values for these parameters, we can maximize the degree of entanglement for the qubits.
Hamiltonian of this model has the following form Hfuii = hq + v(t), V(r) = vqf(t) + vqq(t), where hq is hamiltonian of two non-interacting qubits; vqf(t) is operator which describe interaction between qubits and one mode electromagnetic field; vqq (t) is operator which describe dipole-dipole interaction between qubits.
We describe the investigated system by statistical operator p(t) in the energy representation using the path integration [1]. To quantify the quantum entanglement of two qubits we use the Peres-Horodecki criterion [2,3] with the measure £. The entanglement is maximal, when s = 1, and minimal when s = 0.
The proposed system of equations allows using numerical methods to construct graphs of dependence £ on system parameters and time. As an example of calculating the qubit entanglement parameter, let's consider the case when there is no external field and the qubits interact with each other. The graph of £ is represented by a dashed curve in Fig. 1.
fip=o.8-:jp ::
0 2 4 5 3 1C 12 14 15 18 20 22 74 25 20 30 32 34 35 35 40 42 44 46 43 50 52 54 55 Dimensionier time. Opt
The case is considered when the frequency of the external monochromatic field coincides with the transition frequency of the qubits. In the second numerical experiment, the qubits are affected by the field. The graph of the dependence of £ on the dimensionless time fiRt in Fig. 1 is represented by a solid curve. From the analysis of the graphs, it can be inferred that the external field stabilizes the entangled state of the qubits.
The proposed mathematical model allows investigating the qubit entanglement parameter £ under various system parameter changes within a wide range.
[1] A. Biryukov, V. Derbov and M. Shleenkov "Phase-difference-dependent laser-induced quantum entanglement in a pair of cubits," Proc. of SPIE 9448 (2015).
[2] A. Peres "Separability criterion for density matrices," Phys. Rev. Lett. 77, 1413-1415 (1996)
[3] M. Horodecki, P. Horodecki and R. Horodecki "Separability criterion for density matrices," Phys. Rev. A 232, 333-338 (1997).
