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CHARGING PERFORMANCE OF THE ... PHYSICAL REVIEW RESEARCH 4, 013172 (2022) 6 55 44 33 22 11 6 (a) N=5 9@t r7(t) (b) N=6 9@t r7(t) 00 2 4 6 8 10 00 2 4 6 8 10 gt gt FIG. 2. The dynamic energy and ergotropy of a QB with different hopping strengths. (a) and (b) are the energies and ergotropies of a QB withspinnumbersN=5andN=6,respectively.Otherparametersareωa =ωc =g=J=1,δ=0. discuss how to determine the charging time to ensure the QB obtains the maximum energy and ergotropy. Except for the dimerization parameter δ, the hopping strength J still brings on the quantum phase (QPT) in the QB. Hence we also discuss the hopping parameter, and the ground state quantum phase influences the energy and the QB ergotropy. We discuss the odd and even chains, respectively, due to the SSH model spin chain’s symmetry. In our QB charging model, ωa and ωc must be equal to ensure maximum energy transfer. The parameter g mainly influences the energy oscillator frequency. In all calculations, we take ωa as a dimensionless parameter and let ωa = 1. For simplicity, other parameters are taken as ωc = g = ωa = 1. Thus, we only focus on the critical parameters J and δ. Owing to the different structures from the SSH spin chain with odd and even spins, we consider N = 5 and N = 6, respectively. To ensure the QB acquires enough energy and ergotropy, the source cavity photon number should be Nc > 2N. The real photon in the cavity source is the multiphoton state. The min- imum photon number Nc is required to simulate the charging throughout the cavity source to obtain accurate numerical calculation results. We assume the source cavity is in the Fock states. The numerical results of the charging process of a QB with time are shown in Fig. 2. The energy and ergotropy oscillate between the spin and the outside cavity source. The dynamics of SSH QBs for the odd and even chains have similar properties. The maximum energy and ergotropy of a QB appear at the first peak during the charging process. For the maximum energy charging of a tinuous points for particular hopping strengths. For further discussion, we calculate the energy spectrum, as shown in Figs. 3(c) and 3(d). Each discontinuous point of the energy and ergotropy corresponds to a ground energy state crossing point. These discontinuous points on the energy spectrum indicate a QPT. The SSH spin chain model may also lead to a topological phase transition [53]. However, in our QB charging model, the system’s energy is mainly decided by the bulk state of the whole spin chain. Hence we only discuss the classical QPT. To study the quantum phase here, we introduce two order- ing parameters. One of the common ordering parameters is the z component of the averaged magnetization Mz, ⟨Sz ⟩g Mz = N , (9) where ⟨· · · ⟩g represents taking the average on the ground state, and the total spin operator is Sz = Ni=1 σzi. Here, we define another ordering parameter ξz as S2 ξz= zg, (10) N2 These parameters characterize the magnetic fluctuations in the spins along the z axes. The ordering parameters for odd and even chains are shown in Figs. 3(e) and 3(f). Before the first ground state energy crossing point, where the QB gets the maximum energy, the QB’s ground state is in the ferromagnetic phase. After the first ground state energy crossing point, the energy band becomes split. Until the last ground state energy crossing point, the ground state energy band is fully split. In the next section, we focus on discussing the QB’s maximum energy and er- gotropy in the degenerate and fully nondegenerate ground state regimes. IV. CHARGING PROPERTIES WITH DIMERIZATION PARAMETER A. Dimerization effects Owing to the difference between odd- and even-spin chains when considering the dimerization parameters, we here dis- cuss them separately. For the odd-spin chain, the maximum QB, we have to take this time as the charging time τ . Thus the c charging time to be controlled is τc ∝ 1/ N . In the following discussions, we consider only the QB’s maximum energies and ergotropies that appear at the charging time t = τc. Now we focus on the hopping interaction in a QB. Here, we only discuss the maximum energy and ergotropy of a QB during the charging process. For the nondimerization situation (δ = 0), the calculated maximum energies and ergotropies of a SSH QB with different hopping interactions are shown in Figs. 3(a) and 3(b). Without considering the dimerization parameter, both the odd and even chains have similar behavior with the hopping interaction. The energy and ergotropy have some discon- √ 013172-3 9E(t), rB(t)PDF Image | Charging performance of the Su-Schrieffer-Heeger quantum battery
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