Robustness of Generated Geometric Phase of Quantum Wells in Two Open Waveguide-Coupled Optical Cavities

In this article, we study the geometric phase in a system formed by two spatially separated cavities interacting with the environment. Each cavity is filled by a linear optical medium and contains a quantum well. For different initial states, the robustness of the generated geometric phase is analyzed under the effects of the optical susceptibility, the dissipation of the cavities, the exciton-cavity and fiber-cavity couplings. Our results show that the geometric phase is extremely sensitive to the effects of the cavity-exciton and the fiber-cavity couplings as well as to the optical susceptibility. This opens new routes to understand the storage and manipulation of quantum data in a quantum network.


I. INTRODUCTION
The explorations of the system-environment interactions open the door to study more critical quantum phenomena, which are primary resources in quantum information and quantum computation [1]- [5]. The geometric phase (GP) is the basis of quantum computation [6]. For example, the geometric phase was used to improve the quantum gate protocol in a superconducting qubits, and to realize a multi-targetqubit phase gate [7]- [10]. Recently, Wang et al. [9] propose a scheme of circuit QED that opens the possibility of implementing ultrafast quantum gates holding noise-resistant merits based on the advantages of geometric phases. GP is defined as the phase obtained for a quantum system during an adiabatic evolution. If the final time-dependent state of a quantum system returns to its initial state, the evolution of the quantum system is called ''cyclic''. The GP was extended to the quantum states of nonadiabatic cyclic [11] and noncyclic [12], [13] evolutions. It has also contributed to the significant applications in quantum information and The associate editor coordinating the review of this manuscript and approving it for publication was Chi-Yuan Chen . the optical susceptibility and the dissipation. The paper is organized as follows: The physical model, along with the system Hamiltonian, is presented in Sec.2. The dynamics of the geometric phase for different initial correlated/uncorrelated are illustrated and analyzed in Sec. 3. A conclusion of the proposed work is presented in Sec.4.

II. PHYSICAL MODEL
We study here, two spatially distant open cavity systems which are linked by a mode waveguide (see Fig.(1)). The system Hamiltonian can be written as FIGURE 1. Two spatially distant cavity systems (C 1 and C 2 ) are linked by an optical waveguide. Each of them contains one semiconductor quantum well (QW) and filled with a linear optical medium.
The two first terms of Eq.1 describe the cavity 1 and 2 respectively. Each cavity is filled with a linear optical medium and contains a quantum well. The quantum well is placed in a position that corresponds to the maximum of the electromagnetic cavity field [31]- [33]. The HamiltonianĤ k , can be written aŝ whereâ k (â † k ), ω cv,k and χ k (k = 1, 2) are the annihilation (creation) operators, the frequency and the linear optical susceptibility (of the optical medium) linked to the cavity k. ω ex,k andb k design the frequency and the annihilation operator of the exciton, respectively. The forth term of the Hamiltonian describes the interaction between the k-exciton and the k-cavity with the coupling constants λ k . The last term is denoting the optical pumping of the cavity. ε k and ω L,k are the amplitude and the frequency of the pump laser.
TheĤ f describes the coupling between the two-cavity systems and the optical fiber mode, it is given bŷ The ω f ,k are the frequencies of the fiber modes. Wheref k designs the annihilation operator for the fiber k-mode, the ν k represent the coupling constants between the optical fiber mode and the cavity modes.
In this work we focus on the case where: (1) The photonic and excitonic modes are quantified along the normal direction to the cavity. The pumping is perpendicular to the cavity in the direction of the symmetric axes of the cavity and the fiber that allows the excitation of the cavity photonic mode with the same direction of the photonic and excitonic modes. (2) Our analysis is developed in the rotating frame with the frequency ω L,k and for the resonant (ω cv,k = ω ex,k = ω L,k ).
We also limit our study to the case of the short fiber interaction which is realized in usual experimental situations [30], [34]. The condition of the short fiber limit is defined by 2lν 2πc 1, where l represents the fiber length, c designs the light speed in the fiber,ν is the cavity dissipation. We consider here that only one resonant modef of the fiber is able to interact with the cavity mode. ν k = λ 3 denote the fibermicrocavity coupling constants. As mentioned above the frequencies of the microcavities, the excitons and the fiber modes are equal. In this case the interaction Hamiltonian can be written aŝ The master equation of the system can be written by considering the dissipation as, where γ k 2 and κ k identify the normalized excitonic spontaneous emission and the microcavity dissipation rates respectively.
The system is in the strong coupling regime between the exciton and the photon of the cavity. The coupling constant of the exciton-photon is higher than the cavity dissipation and the exciton emission rates. We also consider the weak pumping regime: ε k κ k , the off-diagonal terms 2â k ρâ † k and 2b k ρb † k can be ignored in Eq.(5) [35]- [37] and the purity of the system state is almost preserved. Therefore, we can obtain: where the non-Hermitian operator H eff is, From Eq.(6), the dynamics of the system is governed by: where |ψ(t) satisfies We consider the description of the exciton basis space as: and the two-cavity basis space as: (1) describes the cavity-exciton and the fiber-cavity interactions, therefore, the total number of the excitations is determined by the operatorf †f Based on the fact that the system is in the strong coupling regime between the exciton and the photon of the cavity, the coupling constant of the exciton-photon is higher than the cavity dissipation. In this case for a short time there is approximately a constant excitation number N e , we restrict N e to N e = 3 and consider only single photon processes. Therefore, the wave function |ψ(t) can be represented by with The time-dependent coefficients q n are driven from Eq.(9) as, − i(χ 1 + χ 2 )q 17 + ε 1 q 9 + ε 2 q 13 , q 18 = −iλ 1 q 12 − iλ 2 q 16 − iλ 3 q 11 − (κ 1 + κ 2 )q 18 , − i(χ 1 + χ 2 )q 18 + ε 1 q 11 + ε 2 q 15 .
whereγ k = γ k and λ k are the coupling rates between excitons and photons, and ε k are the laser pump amplitudes. Eqs.11 are numerically solved to determine the wave function |ψ(t) and the dynamics of the geometric phase.

III. GEOMETRIC PHASE
Here, we explore the the geometric phase. If the final state |ψ(t) cannot be obtained from an unitary transformation of the initial state |ψ(0) , the evolution is then considered as a noncyclic. In this case the phase is not trivial, we use the Pancharatnam phase [38]- [43] approach to prescribe the total phase. In the open dynamics of the hybrid system, a pure state convert to a mixed state that often is described via its nonunitary evolution, say ρ(t). For this case, Uhlmann [44] and Tong et al. [45] extended the geometric phase to the mixed states. But, with the considered approximations, the effective description of the open system is governed by the master equation that is derived by neglecting the non-diagonal terms, then the dynamics of the system is described by the non-Hermitian HamiltonianĤ eff . Therefore, the evolution of the initial state |ψ(0) is governed by the Schrödinger equation as Our system in the strong regime that means the dissipation is small compared to the interaction coupling, i.e., we neglect the dissipation for a short time, so that the wave function conserves approximately the norm for a short time other than that the wave function can be renormalized. Pancharatnam's relative phase between the states |ψ(0) and |ψ(t) of Eq.10 is expressed by [38], [39] G P (t) = arg ψ(0)|ψ(t) .
Here, we will analyze the robustness of the generated geometric phase for the correlated and uncorrelated initial states by considering three different cases of the initial system state. First one, when the system is initially in the maximally entangled state and the other two cases are considered for the system in unentangled states. The wave function for the the first case can be written as and for the two other unentangled states, we consider: We focus here on the case where the phase angle ϕ = π 4 . The type of entanglement of the initial states of Eq.14 is very useful for distributed quantum information [30].
In the numerical calculations, the time t represents a unitless time normalized to τ c , where τ c is the round trip time for a photon inside the cavity. We normalize also the parameters of the system to 1/τ c as:
In the case of the absence of the optical linear medium inside the cavity χ = 0.0, we noted that: (1) The PGP function has rectangular oscillations verifying the inequality −π < G(t) < π. (2) The number of rectangular oscillations is enhanced with the increase of the coupling constants λ k (k = 1, 2, 3) and uncertain intervals [17] appear, in which the GP values can not be unequivocally determined.
(3) Dashed curve of the Fig.3 shows the dynamics of the PGP for the the dissipation rates κ k = γ k = 0.4. These dissipations lead to the reduction of the oscillation frequency. Fig.2b. shown that the generated GP oscillations are affected by increasing the optical susceptibility, χ k = 2. We observed that: (1) The generated GP has more smoother oscillations (GP) unlike the case of χ k = 0 that achieved rectangular GP oscillations. (2) The phenomena of the collapses and revivals disappear entirely. (3) The optical susceptibility enhances the robustness of the generated GP against the dissipation rates.
In Fig.3, we demonstrated the same when the cavities are filled with an optical medium, χ = 0, 2, and the coupling strength of the fiber-cavity coupling is reduced to λ 3 = 0.5. We find that the amplitude of the GP is reduced compared to the previous case where λ 3 = 2. We also observe that the GP is fragile against the dissipation rates. The ability of the optical susceptibility to maintain the robustness of the GP is vanished for λ 3 = 0.5.   4 illustrates the sensitivity of the generated GP oscillations to the exciton-microcavity coupling constants λ k = 0.5. By employing the same values of the coupling constant to the environment κ k = γ k = 0.4, we note that the oscillation frequency of the GP is increased. If the cavities are filled with an optical medium, the effect of the optical susceptibility, χ k = 2 for λ k = 0.5, leads to more GP oscillations. The effect of the excitonic and the microcavity dissipation is smoothing the shape of the GP which shows a regular dynamics. By increasing the Coupling constants λ k , the generated GP oscillations become more more robust.
Finally, we deduce that the generated geometric phase is controlled by the external environment, the coupling constants of the exciton-microcavity and the fiber-microcavity as well as by the susceptibility of the optical medium.

B. DYNAMICS OF GP WITH UNCORRELATED STATES
For initial uncorrelated states |ψ(0) 2 and |ψ(0) 3 , we investigate the effect of the couplings λ k (i = 1, 2, 3) to generate GP oscillations under the presence of the optical susceptibility χ i and the dissipation rates κ k = γ k . The PGP of the initial uncorrelated states |ψ(0) i (i = 2, 3) is given by  Fig.5a, shows the dynamics of the geometric phase G(t) for the initial state |ψ(0) 2 when the interactions of the two cavity-excitons and the fiber-cavity are strong λ k = 2(k = 1 − 3) (as Fig.2) for different values of the cavity dissipation rates κ k = γ k and in the absence of the optical linear medium χ k = 0. We observed that the geometric phase has rectangular oscillations which represent the collapses/revivals phenomena. The geometric phase identified by G(t), verifies the inequality 0 < G(t) < π. Dashed curve of Fig.5a illustrates the robustness of the generated PGP oscillations against the cavity dissipations.
By comparing the dynamics of GP for the case χ k = 0 of Fig.5a, and for the case χ k = 2 of Fig.5b, we observe that: (1) The effect of the optical susceptibility leads to smoothing the shape of the oscillations. (2) The GP verifies the inequality −π < G(t) < π, and it is robust against the cavity dissipation rates. In Fig.6, the geometric phase for another initial uncorrelated state |ψ(0) 3 is depicted with λ k = 2(k = 1 − 3) and κ k = γ k = 0, 0.4. From Figs.6 and Figs.6, we observe that the initial state |ψ(0) 3 realizes PGP with more oscillations than the one with |ψ(0) 2 . The amplitudes and the number of the GP oscillations increase for the initial state |ψ(0) 3 . Dashed curves of the Fig.6 show the robustness of the generate GP for the initial state |ψ(0) 3 against the cavity dissipations rates. We deduce that the Geometric phase is more robust than the case with the initial uncorrelated state |ψ(0) 2 . Finally, we can deduce that the robustness of the generated PGP against the cavity dissipations depends on susceptibility of the optical medium and the coupling constants of the excitonmicrocavity and the fiber-microcavity interactions as well as on the initial states.

IV. CONCLUSION
In this work, we have investigated a system formed by two spatially separated open microcavities linked by a waveguide. VOLUME 8, 2020 Each microcavity contains one-exciton and is filled by a linear optical medium. The robustness of the generated geometric phase of the entire system is investigated. It is shown that the generation and robustness of the GP oscillations are very sensitive to three parameters; first, the effects of the cavityexciton; second, the fiber-cavity coupling rates and finally to the susceptibility of the linear medium. These parameters control the regularity, the amplitudes, and the frequencies of the GP oscillations. The dissipations are responsible of the GP vanishing. These effects can be delayed by the optical susceptibility and the coupling constants of the cavityexciton and the fiber-cavity. These results could strengthen the understanding of the geometric phase and its impact on implementing a distributed quantum network.
The qubit-cavity systems linked by a waveguide present several potential applications for the quantum-gate realizations [7], the quantum correlations [27], distributed quantum computation [30] and quantum network [46].