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Abstract
Topological photonics is a hot topic in recent years. We combine it with the quantum optics and explore the dynamics of two quantum dots (QDs) separated by the finite coupled-cavity arrays (CCAs). The finite CCAs possessing the alternating hopping strengths will lead to the existence of the topological protected edge modes, also called zero energy modes, when the boundaries leave the weak hopping at two ends. Due to the two edge modes, i.e., symmetric and antisymmetric, with nearly degenerate frequencies, the dynamics of two QDs coupled to the cavities at both ends exhibit complicated behaviors. When the CCAs are composed of a large number of cavities, there are two kinds of phenomena: if the coupling between QDs and cavity is weak, two edge modes will cancel each other out and isolate two QDs deeply; if the coupling between QDs and cavities is large compared with hopping strength, the edge mode disappears and two QDs can be connected through extend modes. Importantly, when the CCAs are formed by a small number of cavities, energy can be transferred to each other between two QDs through the edge modes. Such energy transfer is topologically protected, and the period is long and easily controlled. We also investigate the effects of topologically protected quantum entangled states on such system and find that the quantum entanglement can be well kept or generated for appropriate choices of system parameters and initial states. The investigations enrich the manifestation of topological physics and are helpful to apply the topological protection to quantum computation and quantum communication.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past few years, considerable interest has been devoted to the topological photonics which aims to explore the effects related to topological protected phases in photonics systems [1–4]. Soon after that, various proposals for realizing the band structures with topological invariants can be found in different relevant photonic systems including photonic crystals [5], coupled waveguides or resonators [6–8], and metamaterials [9] from the perspective of various dimensions. Early attention has been paid to topological photonics of two-dimensional systems because they are optical correspondence of quantum Hall effect [5,10] and quantum spin Hall effect [11]. Besides, one-dimensional photonic topological structures have also been extensively investigated [12,13], and one of the most representative one-dimensional topological models is the Su-Schrieffer-Heeger (SSH) model. With a chiral symmetry, the SSH model has been discussed to describe the Zak phase [14] and associated topological edge states [15]. Importantly, many experimental tests based on optical topological chain have been performed, for example, a simple realization of Majorana edge state has been proposed in the zigzag chain of metallic nanodisks which is shown to be robust against distant interactions and disorder [16]. And double SSH chain structure composed of split ring resonator is implemented to study the controllable topological bound state and its exceptional point physics recently [17]. Moreover, interesting new schemes based on a photonic dimer chain composed of ultrasubwavelength resonators has been proposed to realize the long-range wireless power transfer with high transmission efficiency [18]. Topological properties in non-Hermitian systems have also been discussed in optical lattices [19], coupled optical waveguides [20], and acoustic system [21]. Recently, it has been studied in a finite optical lattice or topological dimer chain system to reveal the fact that robustness of topological edge state is weakened due to the near-field coupling effect [22,23]. Moreover, one-dimensional optical topologies of different wavebands [24–26] are predicted. And recently, based on the coupling between a two-lever atom with and topological states, the robustness of Rabi splitting of topological quantum optics has been studied in the X-ray region [27].
Nowadays, one of the current frontiers of the burgeoning field with many experimental and theoretical developments is the exploration of the interplay between topological photons and quantum emitters, i.e., atoms or quantum dots, as photon is the flying qubit which is used to transmit the information, meanwhile quantum dot is the local qubit which is used to deal with the information. At present, although the long-range atom-atom interaction has been realized based on metamaterials [28,29], there was considerable work concerning the interaction between the atoms (QDs) and the optical SSH chain in pursuit of better robustness interaction. For example, Ciccarello discussed the resonant atom-field interaction in large size CCAs [30]. When the number of cavities is odd, there is only one edge state which is localized on one side. Then they place one atom in each cavity and focus on the collective phenomena. More recently, the interaction of QDs with the topological waveguide has also been concerned [31]. In the research, they focus on the infinite chain and just consider the influence of topology of band on the dynamics of the QDs and nontrivial multibody phase.
Different from the above work we consider the interaction of two QDs mediated by the finite CCAs here. The CCAs correspond to a typical optical SSH model which provides the topological protected edge modes. We expect to control the interaction between two widely separated quantum dots mediated by the edge modes, so as to investigate the entanglement evolution and entanglement generation between the distant QDs, because the generation of entanglement has been a hot topic over decades due to its promising applications in quantum information and quantum computation [32]. In fact, entanglement not only has its roots in quantum information including teleportation of entanglement of EPR pairs [33] and quantum cryptography with the Bell theorem [34] but also plays a fundamental role in one-way quantum computing [35] which is based on subsequent measurements of qubits and uses up the advanced prepared multipartite state, and linear optics quantum computing [36] as quantum computing was generally assumed to achieve a universal set of quantum gates to generate entanglement between qubits in a deterministic and reversible fashion [37].
This paper is organized as follows. In Sec.2, we introduce the optical SSH model made of finite CCAs and show the main property of the edge modes. In Sec.3, we consider the dynamic evolution of two QDs interacting with such CCAs, in which the edge modes play an important role. Two kinds of contributions of edge modes to the dynamic behavior are discussed. In Sec.4 we consider the entanglement of two QDs induced by the edge modes. In Sec.5, the robustness of system is discussed when we introduce perturbations. Finally, in Sec.6 we sum up and make a conclusion.
2. Model and the edge modes of the CCAs
We consider the CCAs composed of N (even number) identical cavities with frequency ${\mathrm{\omega }_c}$ arranged in one-dimensional formation, as shown in Fig. 1. The hopping strength between adjacent cavities alternates as t1 and t2, which can be achieved by adjusting the spacing between the cavities. Because the number of the cavities is even, the outermost hopping strength is t1. The Hamiltonian of the coupled-cavity arrays takes the form (setting $\hbar $=1 throughout the paper)
Fig. 1. The scheme of the one-dimensional CCAs which are composed of even number (N=2 m, m is an arbitrary positive integer) of identical cavities. The hopping strengths between adjacent cavities are described by t1 and t2. Two quantum dots (A and B) interconnect with two outermost cavities, which are marked by the circles.
Fig. 2. (a) The eigenvalues ωn and (b) two edge modes of the CCAs with N=34, ${\mathrm{\omega }_c}$=665THz, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz.
As the edge modes connect the outermost cavities, one intuitive idea is that they can be coupled with two QDs at a distance when the QDs interact with the outermost cavities. Therefore, we consider placing two identical QDs (A and B) with frequency ${\mathrm{\omega }_0}\; $ separately in the outermost cavities, as shown by the yellow circles in Fig. 1. The coupling coefficient between QDs and the outermost cavities is g $.$ Then the total Hamiltonian of the system can be written in the form
${H_A}\; $ is the Hamiltonian of the QDs,Here, $\sigma _i^\dagger $=$|{e\rangle_i}\langle{g}|$ and $\sigma _i^ - $=$|{g\rangle_i}\langle{e}|\; ({i\in\{{A,B} \}} )$ are the dipole raising and lowering operators where $|{g\rangle_i}\; (|{e\rangle_i}$) is the ground (excited) state of the ith QD. In other words, QD A can only be coupled to the 1st cavity, while QD B is coupled to Nth cavity.
Our goal is to control the indirect coupling between two QDs mediated by the CCAs, especially by the edge modes.
However, there was rare similar work before. What is the problem? The key is that there always exist two nearly degenerate edge modes, shown in Fig. 2(b). If QD A resonantly interacts with the edge modes, it always excites these two edge modes simultaneously with nearly the same amplitude of probability. As two edge modes are symmetrically and antisymmetrically distributed, they will cancel each other out on the other side due to the interference. Finally, two QDs will decouple with each other.
One approach to the problem is to find a way to break the degeneracy of two edge modes, and enable the QD to excite only one of the edge modes. There are two factors affecting the degeneracy of the edge modes. One is the cavity number N, the other is the ratio of two hopping strengths ${t_2}/{t_1}$, which is related to the dimerization |${t_{2}}$-${t_1}$|.
As an example, we analyze the degeneracy of the edge modes in the case of N=34, ${t_1} = $ 1 MHz and numerically compute the frequency difference between two edge modes and the band gap as functions of ${t_{2}}$/${t_1}$ in Fig. 3(a). The result indicates that the frequency difference of two edge modes $\Delta \mathrm{\omega }$ is inversely proportional to ${t_{2}}$/${t_1}$ while the band gap is in proportion to ${t_{2}}$/${t_1}\; $ as the ratio varies from 1.1 to 3. In other words, with the decrease of dimerization, it’s easier to lift the degeneracy of these two edge modes. In particular, to compare $\Delta \mathrm{\omega }$ with the size of the band gap, here we set the initial ratio as 1.1 instead of 1 to avoid the size of the band gap goes to 0. More specifically, at the starting point when ${t_2}$=1.1 MHz, ${t_1}$=1 MHz, the value of the edge states energy difference is 0.084 MHz and the band gap is 0.2 MHz. With the ratio increasing, the edge states energy difference is always smaller than the band gap, as the edge states energy difference decreases while the band gap linearly increases.
Fig. 3. (a) the frequency difference $\Delta \mathrm{\omega }$ of two edge modes (blue line) and the band gap (red line) as functions of ${t_{2}}$/${t_1}$. (b) The frequency difference $\Delta \mathrm{\omega }$ between two edge modes as function of N. The corresponding distribution of edge modes with N=8, 34 are shown in the inset with ${t_1}$=1 MHz and ${t_2}$=2.5 MHz.
Then we fix the hopping strengths with ${t_1}$=1 MHz, ${t_2}$=2.5 MHz and show the frequency difference between two edge modes as a function of the cavity number N in Fig. 3(b). Similarly, there is an inverse relationship between $\Delta \mathrm{\omega }$ and N. It means that the degeneracy of two edge modes is more likely to break as N decreases. The corresponding distributions of edge modes with N taking values as 8 and 34 are presented in the inset of Fig. 3(b). It is clear that with a larger number of cavities, for example, N=34, the system exhibits stronger field localization at boundaries compared to the case with N=8. In this sense, we can see the topological protection property of the system is weakened with fewer cavities.
Next, we will discuss the dynamics of the QDs mediated by the edge modes which are supported by the CCAs. We focus on two cases in light of above discussion: (1) two edge modes are closely degenerate; (2) the degeneracy of two edge modes is broken.
3. Dynamics of QDs mediated by the CCAs
Due to the unique characteristics of the edge modes, it is interesting to study the coherent coupling between two QDs mediated by them. We open up a more detailed discussion from two aspects: two closely degenerate edge modes in long arrays, and two low degeneracy edge modes in short arrays.
3.1 Long arrays with closely degenerate edge modes
We would like to start by considering a long cavity array with N=34, ${t_1}$=1 MHz and ${t_2}$=2.5 MHz. In this case, the frequency difference $\Delta \mathrm{\omega }$ between these two edge modes is found to be 6.7635×10−7MHz, which indicates the edge modes are closely degenerate. In the meantime, two QDs with frequency ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$ interact with the outermost cavities by coupling strength g, as shown in Fig. 1, then we investigate the influence of coupling strength on the evolution of the whole system.
Firstly, we prefer to define “the weak coupling regime” as g is less than ${t_1}$, e.g., g=0.5 MHz. In this case, two QDs can be regarded as perturbations, and do not change the main properties of edge modes of the CCAs. Assuming that QD A is excited, QD B is in the ground state and there is no photon in all cavities at the initial time, i.e., $|\psi (0 )\rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. In Fig. 4(a), we provide a global view of evolution of the whole system including the QDs and N cavities. It is obvious that the energy is transferred coherently between QD A and the first cavity in the main. In addition, whenever the 1st cavity is excited, the 3rd cavity will also be excited with a lower probability. This is the convincing evidence that QD A interacts with the edge modes, as the edge modes are mainly distributed in the 1st and 3rd cavities on both sides of the arrays, displayed in Fig. 2(b). The two edge modes can be excited with the same probability amplitude because both of them are nearly degenerate and resonant with the QDs. Due to the symmetric and antisymmetric spatial distribution of these two edge modes, their coherent interference makes the field completely cancel out on the other side of the cavity arrays. Therefore, energy cannot be transferred between two QDs through the edge modes, although each edge mode is mainly distributed in two outmost cavities, shown in Fig. 2(b).
Fig. 4. Time evolution of the system under different initial conditions (a). $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle $ and (b). $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. The parameters are chosen to be N=34, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, g=0.5 MHz, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz.
Then we investigate the evolution of the system under the initial condition that two QDs are prepared in the superposition state, i.e., $|\psi (0 )= ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, illustrated in Fig. 4(b). The figure shows there is a synchronous evolution between two QDs and energy is alternately transferred between the QDs and the edge modes. However, according to the previous discussion, two QDs evolve independently based on their respective initial states. Hence, the synchronous evolution also depends on QDs’ initial state: $|{|\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 $. The topological protected property makes the field distributed in the boundaries and makes the QDs mainly interact with the outermost cavities since the edge modes still hold a destructive interaction effect. Although each edge mode connects the two outermost cavities, the simultaneous excitation of the two edge modes results in the two cavities completely unconnected. Thus, there is no coupling between the QDs even though they are coupled with the common cavity arrays. Moreover, in the weak coupling regime, changing the value of g only changes the Rabi frequency between the QDs and the outermost cavities.
However, the circumstance is quite different when the coupling coefficient g is greater than ${t_1}\; $ and ${t_2},$ which we define as “the strong coupling regime.” In this case, the QDs cannot be regarded as perturbations of the cavity arrays, and will change the topological properties of the whole system. We set the coupling between QDs and the cavities as g=3 MHz here. The evolution of the whole system with initial condition $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$ is shown in Fig. 5(a). It is apparent that as time goes on, the energy of the QDs is transferred from the 1st cavity to the last cavity one by one. Afterwards, when t is about 17μs, the energy is continuously transferred to QD B and then goes back to QD A in turn. Since all QDs and cavities of the system are involved in such process, it can be concluded that the edge modes make no contribution to the energy transfer. For a better understanding of the dynamic properties, the QDs and cavity arrays should be considered as a whole. Two QDs can be regarded as the new borders for the composite chain. When the system ends with strong coupling, the topological phase of the system will change into trivial. Therefore, there are no edge modes but extended modes. As all the extended modes take part in the propagating process, the resulting superposition of these modes will weaken the coherence of the system. So, the probability distribution becomes wider over time. By comparing Fig. 5(a) with Fig. 4(a), the relative strong coupling g changes the original nontrivial CCAs into a trivial composite chain, and transforms the system from isolated to transmissible correspondingly. Mediated by the CCAs, two QDs can interact with each other through the extended modes.
Fig. 5. Time evolution of the system under different initial conditions. (a) $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$ and (b) $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$ with N=34, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, g=3MHz, ${t_1}$=1MHz, ${t_2}$=2.5MHz.
We then show the evolution of the whole system under the initial condition $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{{0_i}} \rangle$ in Fig. 5(b). As two QDs are excited with the same probabilities at the initial time, the whole evolution displays an X-shaped profile. The two “waves” meet in the middle cavity when t is about 10μs, and then cross-propagate in their respective directions, and cause the energy transfers back and forth between two QDs. By comparison, the evolution of two QDs in Fig. 5(b) and Fig. 4(b) is quite different even with the same initial condition. In Fig. 4(b), due to the effect of the edge modes, the two QDs always evolve synchronously, and the populations of the QDs always periodically return to their initial values. However, in Fig. 5(b), the populations for the QDs decline and hardly keep the initial values, although still with a synchronous evolution. This is because a large number of extended modules are involved in the nontrivial dynamics, resulting in a reservoir-like effect.
3.2 Short arrays with low degeneracy edge modes
As mentioned above, the QDs can be seen as perturbations when the coupling between QDs and cavity is weak, which makes the edge modes contribute to the evolution of whole system. For long cavity arrays, the two edge modes are closely degenerate and evolve with the same phase, leading to the isolation of QDs. However, the situation will be different when the arrays are short with the degeneracy of two edge modes is lifted. For instance, if we set the eigenfrequencies of the symmetric and anti-symmetric edge modes as ${\omega _s}$ and ${\omega _a}$, their frequency difference will be $\mathrm{\Delta }\mathrm{\omega } = {\omega _s}$-${\omega _a}$. Assuming that only QD A is excited at the initial time, and it will excite the two edge modes simultaneously with the same probability amplitude. It will be found that the edge modes will undergo unitary evolution in the formation $|{{\mathrm{\psi }_s}(z ){e^{i{\omega_s}t}} + {\mathrm{\psi }_a}(z ){e^{i{\omega_a}t}}} |= |{{\mathrm{\psi }_s}(z ){e^{i\mathrm{\Delta }\omega t}} + {\mathrm{\psi }_a}(z )} |$, where ${\mathrm{\psi }_s}(z )\; $ and ${\mathrm{\psi }_a}(z )$ are the two edge modes shown in Fig. 2(b). If we set $\mathrm{\Delta }\omega t = \mathrm{\Delta }\phi $, the phase $\mathrm{\Delta }\phi $ will continuously change over time. Then we present the modulus of the superposition of two edge modes with different $\mathrm{\Delta }\phi $ in Fig. 6. when $\mathrm{\Delta }\phi $=0, the superposition of two edge modes leads to the field located in the left cavities. When $\mathrm{\Delta }\phi $=$\mathrm{\pi }/4$, the probabilities of the field in the left cavities decrease and that in the right cavity increase. When $\mathrm{\Delta }\phi $=$\mathrm{\pi }/2$, the probabilities are the same in both side cavities. When $\mathrm{\Delta }\phi $=$3\mathrm{\pi }/4$, the probabilities in the left cavities are lower than that in the right cavities. when $\mathrm{\Delta }\phi $=$\mathrm{\pi }$, the superposition of two edge modes leads to the field located in the right cavities. Therefore, energy can transfer from left cavities to the right cavities through the “beat” formed by the two edge modes. It is also the principle of robust and broadband optical coupling in topological waveguide arrays [39].
Fig. 6. The modulus of superposition of two edge modes $|{{\mathrm{\psi }_s}(z ){e^{i\mathrm{\Delta }\omega t}} + {\mathrm{\psi }_a}(z )} |$ with different $\mathrm{\Delta }\phi $, with N=12, ${t_1}$=1MHz, ${t_2}$=2.5MHz.
Now we introduce “beat” into interaction between two QDs mediated by CCAs. To highlight the role of edge modes, the coupling strength between QDs and cavity is taken to be less than t1 here. The reason the beat cannot be seen in Fig. 4 is that the frequency difference $\mathrm{\Delta }\omega $ is too small to be distinguished when N is 34. It takes more time to observe the beat, which is longer than decoherence time of the system. We take arrays with 8 cavities as an example, the evolution of the whole system is shown in Fig. 7(a) with the initial condition set as $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle,$ and other parameters the same as those in Fig. 4. It is clear that during the period $\textrm{t} \in [{0,25\mathrm{\mu }\textrm{s}} ]$, energy exchange only takes place between QD A and the 1st cavity. And during $\textrm{t} \in [{25\mathrm{\mu }\textrm{s},45\mathrm{\mu }\textrm{s}} ]$, both QD A and QD B exchange energy with their neighbor cavities. Then in the period $\textrm{t} \in [{45\mathrm{\mu }\textrm{s},65\mathrm{\mu }\textrm{s}} ]$, energy only exchanges between QD B and the 8th cavity. At last, the system will periodically evolve in this way. Meanwhile, as long as the 1st and the 8th cavities are excited, the 3rd and the 6th cavities are always excited, which is a manifestation of edge modes.
Fig. 7. Time evolution of system with the initial states (a) $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, and (b) $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. The parameters of system are N=8, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, ${t_1}$=1MHz, ${t_2}$=2.5MHz, g=0.5MHz.
Take a closer look at the parameters in Fig. 7(a), the frequency difference $\mathrm{\Delta }\omega $ is 0.0595 MHz, and it is observed that QD A is excited with a fixed period T $= $ 105 $\mathrm{\mu }\textrm{s}$. By the numerical fitting, a formula to depict the relation between period T and frequency difference can be written as $\textrm{T}\Delta \omega = 2$, which is consistent with the period of the beat.
In Fig. 7(b), we consider the situation with initial state $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. The evolution of the whole system is similar to that in Fig. 4(b) except for the different cavity numbers. The beat phenomenon is not observed in this case. This is because both QDs are in the excited states with the same amplitudes of probability at the initial time, even though the beat effect may exist, energy transfers from one side to the other is compensated by the opposite process under the action of a superposition of these two edge modes. The synchronous evolution of the two QDs finally leads to the symmetric evolution of the whole system. (Note that the time scales in Figs. 7(a) and (b) are different.)
Therefore, beat plays an important role in energy transfer with initial state $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$.
If we choose different ${t_2}$ to change the frequency difference of two edge modes, shown in Fig. 3(a), the period of beat will change correspondingly. As a result, it provides a new way to control the transmission time for the composite system by varying the beat frequency.
By comparison, the similarity between Fig. 7(a) with Fig. 5(a) is that the QDs can interact with each other indirectly through the CCAs. However, it is necessary to emphasize the difference between them. In Fig. 5(a), mediated by the extended modes, two QDs can interact with each other, so the energy transfers from one cavity to another and eventually dissipates into the whole system. At last, the populations of QDs will decrease after several periods. While in Fig. 7(a), when two QDs interact with each other through the edge modes, except 1st, 3rd, 6th and 8th cavities, other cavities will not be excited. The coherence of the system maintains good in that only two modes are involved. After several cycles, the populations of the QDs hardly change.
Next, we give a brief discussion about the impact of the length of the arrays on the beat effect. According to the formula $\textrm{T}\Delta \omega = 2$, which depicts the relation between beat period T and frequency difference, we first show the frequency difference between two edge modes as a function of the cavity number N when considering the coherent coupling with two QDs in Fig. 8.
Fig. 8. The frequency difference $\Delta \mathrm{\omega }$ between two edge modes as function of N when considering the coherent coupling with two QDs with ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz and g=0.5 MHz.
In Fig. 8, it can be seen that $\Delta \omega $ still exhibits a reverse proportional relation with N. Comparing with Fig. 3(b), even with the same N, the value of $\Delta \omega $ decreases when we consider the interaction of QDs. N=10 and $\Delta \omega$=0.025 MHz corresponds to T=252$\mu s$ and N=12, $\Delta \omega $ = 0.01 MHz corresponds to T=630$\mu s$. Then, we show the evolution of the system with N=10 and 12 in Fig. 9 respectively. It is clear that the period of beats is indeed consistent with the calculation, in which T is 252$\mu s$ in the case of N=10, and T is 630$\mu s$ in the case of N=12. Theoretically, without considering the dissipation, if the number of the array increases, the beat effect may not be observed until we extend the time. However, when the dissipation of the system is considered, as we show in Fig. 15 of Section 5, even taking a small value of dissipation $\gamma = 0.5i$ with N=8, this beat effect will be weakened. When we lengthen the array until the period is longer than the decoherence time of the system, beat effect cannot be observed.
Fig. 9. Time evolution of system with the initial states $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle.(\textrm{a} )N = 10(\textrm{b} )N = 12.$ The parameters of system are ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, ${t_1}$=1MHz, ${t_2}$=2.5MHz, g=0.5MHz.
3.3 Explanation from the viewpoint of the whole system
In the previous analysis, we regard the system evolution as the interaction between quantum dots and SSH chains. It is considered that the existence of quantum dots does not affect the properties of SSH chain. However, due to the existence of interaction, the excited state of quantum dots is not the eigenstate of the system, resulting in dynamic evolution. Now we take the quantum dot and SSH chain as a whole. Since only single quantum excitation is considered, we can diagonalize the whole system including quantum dot and SSH chain, and obtain the eigenvalues and eigenstates, so that we can also explain the previous results, especially the results in Figs. 5 and 7.
We take the case of N=8 as example, the common parameters are ${t_1}$=1 MHz, ${t_2}$=2.5 MHz. Two QDs are coupled to the outside cavities with the coupling g.
At first, we set g=0.5 MHz, the eigenvalues and the corresponding eigenstate are show in Fig. 10. Compared with Fig. 2(a), there are four eigenstates within the gap but they are obviously separated in frequency, shown in Fig. 10(a). Previous literatures [40] pointed out that SSH chain is topologically protected from internal perturbation of site-to-site couplings but sensitive to perturbation of on-site frequency at the end of the structure. Here, if we regard the outermost cavity embedding the QD as a composite cavity, the interaction between the outmost cavity and QD will lead to Rabbi splitting of the frequency of the composite cavity. It can be seen as the perturbation of on-site frequency at the end of the structure. So, the property of the edge modes is changed. The wave function of the modes marked as 5th and 6th are shown in Fig. 10(b). They can still be seen as the edge modes, because they still locate in the boundary area and exhibit certain topological protection property in our system. But the wave function cannot be zero in any site now. However, the result of Fig. 10 also well explains the beat-like phenomenon in Fig. 7.
Fig. 10. (a) The eigenvalues of the whole system including SSH chain and two QDs. (b) the wave function of the modes marked as 5th and 6th. N=8, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c},$ g=0.5 MHz.
Recently, it is also found that the near-field coupling in the finite size will destroy the topological protection of edge states. [22,23]. We then set g=3 MHz, the eigenvalues of the whole system are show in Fig. 11. There is no eigenstate within the gap, which means the disappearance of edge mode of the whole system, due to the strong coupling of QD and the outmost cavity, i.e., g=3 MHz.
Fig. 11. The eigenvalues spectrum of the whole system including SSH chain and two QDs. $N = 8$, ${t_1}$=1MHz, ${t_2}$=2.5MHz, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c},$ g=3MHz.
4. Entanglement of two QDs mediated by the edge modes
The purpose of this work is to effectively control the interaction between QDs mediated by the edge modes, and realize the generation and maintenance of entanglement between two QDs, even if two QDs are far apart in space.
For two-body system, the criterion of entanglement is necessary and sufficient. A widely used criterion for entanglement is the concurrence C(t) [41], which is defined as follows
in whichGiven that the dynamic evolution has been discussed in previous sections, now we focus on the entanglement between two separated QDs. As mentioned above, we classify the interaction between QDs and CCAs into three categories: (1) The QDs interact with the cavities weakly in long cavity arrays; (2) The QDs have strong interaction with the cavities in long cavity arrays; (3) The QDs interact with the cavities weakly in short cavity arrays. In these three cases, the evolution of entanglement between QDs shows different characteristics which will be presented as follows:
For the first case, the evolution of two QDs and the CCAs is shown in Fig. 4. It can be seen that due to the coherent interference of edge modes, two QDs cannot interact with each other directly. The two QDs are independent of each other and Rabi oscillations take place in their neighboring cavities. Therefore, entanglement cannot be generated spontaneously, but can be maintained for a long time. The evolution of concurrence is shown in Fig. 12(a) by taking the initial condition as $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, which is the maximum entangled state of two QDs.
Fig. 12. Time evolution of concurrence C(t) of two QDs with initial condition $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. (a) g=0.5 MHz (red line), g=1 MHz (black line), and (b) g=3 MHz. The other parameters are ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, N=34, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz.
In a weak coupling regime, i.e., the coupling strength g=0.5 MHz, which is smaller than the hopping amplitudes ${t_1}$ and ${t_2}$. The entanglement of two QDs can be kept perfectly well for a long time, as shown by the red line in Fig. 12(a). Generally speaking, without considering the interaction with QDs, the cavity arrays can be treated as an open system. While the interaction of edge modes with QDs make the two QDs decouple, and the system can be treated as a perfect isolator for entanglement in the circumstances. Once the concurrence takes the maximal value at the beginning, it will periodically evolve and be well kept when dissipation is not considered. Thus, we can say the entanglement is protected by the topological property of the edge modes. The evolution of concurrence of two QDs can also be affected by the coupling strength g between QDs and the cavities, as illustrated by the black line in Fig. 12(a) with g=1 MHz. With the coupling strength g increasing, maximum value of entanglement is taken to be less than 1, which indicates the entanglement of two QDs is destroyed to some extent.
In the strong coupling regime, g is larger than both of ${t_1}$ and ${t_2}$, in which QDs cannot be seen as perturbation of CCAs and can change the topological character of the cavity arrays, shown in Fig. 5. When the initial condition is $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, energy can transfer from QD A to QD B through the extended modes while two QDs cannot be excited spontaneously over time. Hence, there is no entanglement in this case. When the initial condition is $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, the dynamic evolution of the whole system is shown in Fig. 5(b). The energy of two QDs transfers to the cavity arrays through extended modes, and will return back to QDs synchronously with small populations. The corresponding evolution of concurrence of two QDs is shown in Fig. 12(b).
It is clear that at the initial time the concurrence is 1, and then the oscillation quickly decreases with the energy dissipating into the cavity arrays through extended modes. Subsequently, the concurrence displays collapse and recovery properties owing to the propagation of the field. When t is about 20$\mathrm{\mu }\textrm{s}$, the energy can be transfer back to QDs, and C(t) reaches the extreme value of 0.25, which is determined by the dynamic evolution characteristics of the system.
Finally, we discuss the third case, in which the QDs interact with the cavities weakly in the short arrays. As the QDs still interact with two edge modes, the frequency difference $\mathrm{\Delta }\omega $ of two edge modes becomes so obvious that “beat” phenomena occurs. In this sense, the entanglement not only can be spontaneously induced, but also can be maintained for a long time. We take the evolution of system in Fig. 7(a) as example. Initially only QD A is excited, and then the beat formed by two edge modes mediates the interaction between two QDs. Therefore, an effective Rabi Oscillation between two QDs occurs even if these two QDs are separated far away. The corresponding concurrence of two QDs is shown in Fig. 13(a).
Fig. 13. The evolutions of the concurrence for the initial condition (a) $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. and (b) $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$ with N=8, ${t_1}$=1MHz, ${t_2}$=2.5MHz, ${\mathrm{\omega }_0} = {\mathrm{\omega }_c}$, g=0.5MHz.
It is shown from Fig. 13(a) that there is no entanglement at the beginning, for only QD A is in the excited state. As time goes on, the concurrence oscillates with a cycle of 55$\mathrm{\mu }\textrm{s}$, accompanied with many fast oscillations caused by the interaction between the QDs and the cavities. The long period refers to the beat formed by edge modes, and the short period refers to the Rabi oscillation between QDs and cavities. To understand the evolution of concurrence, we check the corresponding populations (represented by the color) of two QDs in Fig. 7(a), and find that when the populations of two QDs are the same, it corresponds to the maximum entangled state. Meanwhile, when only one QD is excited, there is no entanglement. In contrast with Fig. 12(b), the maximum entanglement appears repeatedly and does not weaken owing to the effect of the edge modes.
Besides, the entanglement will be kept for a long time when the initial state is the largest entangled state shown in Fig. 13(b), which corresponds to the evolution situation shown in Fig. 7(b). It is apparent that the profile in Fig. 13(b) is quite similar with that in Fig. 12(a) except for the different cavity numbers. From previous discussion, we may know that by taking different cavity numbers, the evolution profiles are quite different in a certain period of time, while the cavity numbers seem to make no difference with the entanglement.
5. Robustness of the system
It is worth emphasizing that the interaction between two QDs mediated by the cavity arrays is of strong robustness. The defect can be incorporated into the system in two ways. One is the off-diagonal defect, which is represented by the perturbation of t1 and t2. This defect does not affect the chiral symmetry of the whole system, so it has no effect on the interaction mediated by the edge states. The other one can be added to the diagonal terms of Hc in Eq. (1), i.e., varying the cavity frequencies.
For optical systems, the dissipation is an important decoherence mechanism and the impact of the dissipation of cavities on the system is of great significance. The chirality of the system will be broken under the diagonal defect, thereby affecting the robustness of the system.
For Fig. 4(b) and Fig. 7(b), they share the same initial state: $|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle,$ but have different cavity numbers: 34 and 8 respectively. We add the dissipation rate $\gamma $ to all cavities, and the frequency of each cavity is changed into ${\mathrm{\omega }_c} = {\mathrm{\omega }_0} + 0.5i\; ({MHz} )$ phenomenologically as $\gamma $ takes an imaginary number 0.5i. The evolution of whole system with dissipation is shown in Fig. 14. Figure 14(a) refers to the case with 34 cavities while Fig. 14(b) refers to the case with 8 cavities. Theoretically, when the dissipation is added to all cavities, the coherent energy exchange is destroyed and the populations of two QDs decay exponentially as the Rabi oscillations quickly disappear. Therefore, the cavity dissipation breaks the chiral symmetry and damages the robustness of interactions mediated by edge mode.
Fig. 14. Time evolution of system with (a) N=34. (b) N=8. The initial condition $\textrm{is\; }|{\psi (0 )} \rangle = ({|{{e_A}{g_B}} \rangle + |{{g_A}{e_B}} \rangle } )/\sqrt 2 \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz, g=0.5 MHz, with dissipation is added to all the cavities, i.e., ${\mathrm{\omega }_c}$=${\mathrm{\omega }_0} + 0.5i({MHz} )$.
However, by checking the profile of edge modes in Fig. 2(b), we find the edge modes are rarely distributed in even-site cavities of the left part and the odd-site cavities of the right part. Thus, if the dissipation is only added to the specific cavities with no distribution of the edge modes, such diagonal defect will have less impact on the evolution of the system. We take the case of Fig. 7(a) as example: the cavity number is 8 and the initial state is $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. Here we add the dissipation to the specific cavities, which are 2nd,4th,5th and 7th, so that the frequencies of these cavities are $\mathrm{\omega }_c^{\prime} = {\mathrm{\omega }_0} + 0.5i\; ({MHz} ),$ while the other cavities frequencies keep unchanged. The evolution of the whole system is shown in Fig. 15. After comparing Fig. 15 with Fig. 7(a), we find the dissipation has less impact on the evolution process, the edge modes mediated beat phenomenon and the populations of the QDs are still obvious. The energy transfers between quantum dots A and B under the action of the beat effect.
Fig. 15. Time evolution of system with the initial states $|{\psi (0 )} \rangle = |{{e_A}{g_B}} \rangle \otimes \mathop \prod \nolimits_{n = 1}^N |{0_i}\rangle$. The parameters of system are N=8, ${t_1}$=1 MHz, ${t_2}$=2.5 MHz, g=0.5 MHz. with dissipation is added to specific cavities, i.e., ${\mathrm{\omega }_2} = {\mathrm{\omega }_4}$=${\mathrm{\omega }_5} = {\mathrm{\omega }_7} = {\mathrm{\omega }_0} + 0.5i({MHz} )$.
Therefore, although the CCAs has no topological protection for the diagonal defects, interaction mediated by the edge mode is still topologically protected by only considering partially occupied cavities dissipation due to the special distribution of the edge modes.
Here we need to explain for the parameters used during the calculation. We take the microcavity array as an example (In fact, any resonator or waveguide array can be used). In Ref. [31], they considered a topological waveguide system which is a photonic analog of the SSH model, with the alternating nearest-neighbor hopping in their paper is taken as 0.3J, while the coupling coefficient between the emitters and the chain can be taken to be a larger value 0.4J, where J represents the uniform hopping strength. In Ref. [42], the cavity-qubit coupling g is taken to be 0.01, with hopping strength J1=0.001 and J2 varying from 0 to 0.01. The topological induced defect state in coupled dielectric microwave resonators has been realized in Ref. [43]. In their experiment, the bare frequency of the cavity is taken to be 6.65 GHz and the distance between adjacent resonators are separated with d1 = 12 mm and d2 = 15 mm, which correspond to coupling t1 = 37.1 MHz and t2 = 14.8 MHz, respectively. We only take a set of data as an example because we are doing theoretical calculation, but the phenomenon we reveal is applicable to all similar practical structures.
6. Conclusion
In summary, we have investigated the quantum effects on a system consists of two QDs interacting with the one-dimensional cavity arrays. The dynamics evolution of the system is numerically investigated and we discussed the entanglement property of the system. By varying the coupling strength between the QDs and the cavity arrays, the quantum properties of the system can be significantly modified as the topological property for the system has been changed. When the composite system ends with weak coupling strength, we show that with a nontrivial topological property, although under different initial conditions, the evolution of the whole system is always localized in the boundary areas under the combined effect of the symmetric and antisymmetric edge modes. In the meanwhile, entanglement can be kept well and the system can be treated as a perfect isolator. While keep increasing the coupling strength to be larger than the alternating hopping strength, our system can be regarded as a transmitter as the topological of the system is trivial, and the transmission property can be attributed to the extended modes, which take part in the propagating process and weaken the coherence of the system. It should be pointed out that, for the case of less cavity numbers with low contrast of hopping strength, the entanglement of the system can also be generated as the symmetric mode and antisymmetric mode construct a beat effect. The delayed coherent interaction situation is quite different from the other cases and by varying the beat frequency, the transmission time for the composite system can be easily controlled. It is also found that under specific conditions, the interaction mediated by the edge mode is still topologically protected when considering the dissipation. Our work is not limited to the cavity arrays system, but also applicable to other optical SSH model.
Funding
National Natural Science Foundation of China (11874287, 12174288, 11774262, 61975154).
Acknowledgements
We sincerely thank Prof. Hong Chen for constructive suggestion.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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