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CHARGING PERFORMANCE OF THE ... PHYSICAL REVIEW RESEARCH 4, 013172 (2022) 5.90 5.88 5.86 5.84 5.82 5.80 -1.0 -0.5 0.0 0.5 13.5 12.0 10.5 9.0 1.0 -1.0 -0.5 0.0 0.5 1.0 (a) J=0.3 9@(c) rB(c) qq FIG. 5. The maximum energy and ergotropy of a QB with spin number N = 6. (a) and (b) are the energies and ergotropies for J = 0.3 and J = 2.5, respectively. are no longer symmetrical with the dimerization parameters. There is one more pair of spins having an interaction of strength J(1 + δ) than that of J(1 − δ) in an even-spin chain. The ground state energy is lower when δ < 0 and thus the charged energy E is higher. As for the fully nondegenerate ground state regime case with N = 6, the dimerization parameter significantly influ- ences the energy and ergotropy. The larger dimerization parameter will bring a much higher energy and ergotropy than the nondimerization situation. The δ > 0 that corresponds to a much higher energy and ergotropy is also due to the positive δ having one more pair of spins, as shown in Fig. 1(b). We find that in both the odd- and even-spin chains, the dimerization parameter has the same effect. The energy and ergotropy have significantly increased in the fully nonde- generate ground state regime and slightly decreased in the degenerate ground state regime. For the extreme dimerization situation (δ = ±1), the energy and ergotropy have been in- creased by about 15%–40%. In the degenerate ground state regime, the energy and ergotropy have been only decreased by about 0.3%. We have already discussed the dimerization parameter in the degenerate and fully nondegenerate ground state regimes. Now we will calculate the maximum energy and ergotropy versus J and δ, as shown in Fig. 6. When the QB is in the degenerate ground state regime, the maximum energy and ergotropy hardly change with the dimerization parameter and the hopping interaction strength. However, in the fully nonde- generate ground state regime, the dimerization parameter will significantly improve the SSH QB’s maximum energy and ergotropy. In this region, when there are more spin pairs with hopping interaction strengths larger than J (δ > 0), the energy and ergotropy become higher, as seen in the upper right-hand corners in Figs. 6(c) and 6(d). In the other regions (the undiscussed quantum phase area), the SSH QB’s ground state will frequently cross with other excited states. After each crossing point, the dimerization may influence the energy and ergotropy of the QB. The numbers of ground state crossing points increase with spin numbers, which makes it difficult to discuss the dimerization param- eter in these regions in detail. Fortunately, the energy and ergotropy are not significantly influenced by the dimerization parameter in these regions. B. Quantum advantage of dimerization spin pairs In the previous section, we have already discussed how the dimerization parameter influences the SSH QB. We have con- firmed that the dimerization parameter will improve the QB’s energy and ergotropy in the energy’s fully nondegenerate area. Now we further investigate how the dimerization parameter improves the maximum energy and ergotropy of a QB. Now we calculate the occupation of each spin. The occu- pation of the ith spin is defined as Oi = ⟨σ+i σ−i ⟩max. ⟨· · · ⟩max is averaged on the charged QB state. First, we calculate the occupation of each spin for the odd-spin chain (N = 5), as shown in Fig. 7. The occupation also slightly influences the degenerate ground state regime. Each dimerization spins in pairs, which corresponds to a nearly full occupation. Due to the degenerate ground state regime, the dimerization pa- rameter will suppress the energy and ergotropy, making the dimerization spin pairs correspond to lower occupations. For the fully nondegenerate ground state regime, the dimerization parameter has a significant influence on the occupation. Each dimerization spins in pairs, which corresponds to a higher occupation. The minus and plus dimerization parameters ap- pear in the numbers of the same spin pairs, which makes the occupation symmetrical with the dimerization parameter. Then we calculate the occupation of the even-spin chain (N = 6), as shown in Fig. 8. The occupation also has a slight influence on the degenerate ground state regime. Each dimerization spins in pairs, which corresponds to a lower oc- cupation. For the fully nondegenerate ground state regime, the occupation has a significant influence, as shown in Fig. 8(b). It is also different from the degenerate ground state regime. Each dimerization spins in pairs, which corresponds to a higher occupation. From comparing the odd- and even-chain occupation, we have found that the occupation has the same influence as the dimerization parameters. For the degenerate ground state regime, the dimerization spin couples have a smaller oc- cupation. The smaller energy and ergotropy mean that the 013172-5 (b) J=2.5 9@(c) rB(c) 9E(c), rB(c)PDF Image | Charging performance of the Su-Schrieffer-Heeger quantum battery
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