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Abstract
We investigate the quantum walks of a single particle in a one-dimensional periodically kicked circuit quantum electrodynamics lattice. It is found that the dynamic process of the quantum walker is affected by the strength of incommensurate potentials and the driven periods of the system. We calculate the mean square displacement to illustrate the dynamic properties of the quantum walks, which shows that the localized process of the quantum walker presents the zero power-law index distribution. By calculating the mean information entropy, we find that the next-nearest-neighbor interactions have a remarkable deviation effects on the quantum walks and make a more stricter parameter condition for the localization of the quantum walker. Moreover, assisted by the lattice-based cavity input-output process, the localized features of circuit quantum electrodynamics lattice can be observed by measuring the average photon number of the cavity field in the steady state.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum walks (QWs) are the quantum extension of classical random walks, which include the continuous-time quantum walks (CTQWs) and the discrete-time quantum walks (DTQWs) [1–3]. Compared with the classical random walks, the distinct difference in QWs is probability distribution owing to the constraint of Heisenberg uncertainty principle. In addition, the diffusion velocity of a particle is greatly raised due to the superposition principle of wave function. As a result of these features, QWs have been widely applied to quantum algorithms [4,5], quantum computing [6–8], and quantum simulation [9,10]. Many theoretical schemes have been proposed to investigate the QWs with defect and disorder [11,12], interactions [13,14], and decoherence [15,16]. Experimentally, QWs have already been developed successfully in many physical systems. The earliest experiment for the CTQWs has been implemented in two-qubit nuclear magnetic resonance (NMR) quantum computer [17], and this work demonstrated that the time evolution of the QWs relies on the quantum entanglement. Then the DTQWs have been experimentally implemented with a three-qubit liquid-state NMR processor [18]. Other experimental schemes have also been proposed to realize QWs in optical lattice [19,20] and trapped ions [21,22]. Besides above mentioned progress, the important advances have been achieved in the field of $\mathcal {PT}$-symmetric DTQWs with single photon [23].
Due to the significantly high integrability and extendibility, circuit quantum electrodynamics (QED) systems become one of the most promising candidates in quantum information processing [24,25], quantum computation [26,27], and quantum simulation [28–30]. Moreover, being beneficial for the high coherence between the superconductor qubits and the superconductor transmission lines, circuit QED systems have flexibly adjustable parameters. Thus, a great number of theoretical studies have been performed in terms of quantum entanglement [31,32], quantum gate [33–36], $\mathcal {PT}$-symmetry [37,38], and quantum Zeno effects [39,40]. Recently, with the rapid development of experimental technique, the quality factor of superconducting transmission lines and the coherent time of superconducting qubits have been improved greatly, and the theoretical schemes for implementing the quantum gate and quantum entanglement have also been achieved in experiment [41]. The controllable coupling between non-neighbor superconductor qubits has been experimentally realized by via superconducting line acted as an intermediary [42,43]. We also note that the nonadiabatic holonomic quantum manipulation with a high fidelity has been experimentally demonstrated [44]. Reference [45] proposed an interface between coherent superconducting qubit and a semiconductor double quantum dot in a hybrid circuit-QED system. Moreover, waveguide QED and cavity QED systems also provide an excellent technology for the implementation of QWs, quantum entanglement, quantum computation [46–51].
In this paper, motivated by the above, we construct a conceptually simple and experimentally feasible 1D circuit QED lattice model consisting of an array transition line resonators, which has the following features: (i) 1D circuit QED lattice model possesses the simplest configuration spatially by orderly arranging high coherence superconducting qubits and transmission line resonators in one direction. (ii) Resorting to the external electric and magnetic fields, this model can be manipulated in a single-site level, which is significantly improved the tunability of system parameters. (iii) Based on the easily-tuned system parameters, current model can be mapped into high-dimensional topological model by synthetic dimension method, which significantly reduces the technical challenge in experiment. (iv) As the property of Bose-statistics for the model, the essential topological feature can be measured by using the lattice-based cavity input-output theory. In such a system, the time evolution operator can be expressed by the Floquet unitary propagator in one period. We find that the strength of incommensurate potentials and the driven periods have a joint effects on the CTQWs. The analysis of the mean-square displacement illustrates that the localized process of quantum walker corresponds to the wave packet propagation with zero power-law index. Furthermore, we also investigate the influence of the next-nearest-neighbors (NNN) interactions on CTQWs and find that the dynamical transition point becomes lager and the smaller driven period needs to be satisfied by calculating the mean information entropy. The average photon number of the cavity field is more complicated compared with the detection of edge modes.
The rest of this paper is structured as follows. In Sec. 2, we present the physical model of 1D circuit QED lattice and the corresponding system Hamiltonian. In Sec. 3, we give the time evolution operators of periodically kicked system and analyze the effects of the strength of incommensurate potentials and the driven periods on CTQWs. And we also investigate the dynamic process of quantum walker from localizaion to delocalization by calculating the mean-square displacement and the mean information entropy. In Sec. 4, we discuss the average photon number of cavity field of the system based on the cavity input-output process. Finally, we present our conclusions in Sec. 5.
2. Model and Hamiltonian
As shown in Fig. 1, the model under consideration consists of an array of transmission line resonators, where each transmission line resonator is coupled to a two-level flux qubit denoted by one excited state $|e\rangle$ and one ground state $|g\rangle$, respectively. The corresponding Hamiltonian can be expressed as
Fig. 1. (a) The schematic diagram of 1D circuit QED lattice with NN interaction $J_{1}$ and NNN interaction $J_{2}$. (b) Each transmission line resonator is coupled to a two-level flux qubit and the coupling strength between the resonators can be tuned by a superconducting quantum interference device (SQUID).
The system dynamic process obeys the Schrödinger equation
where $|\Psi (t)\rangle =\sum _{j}C_{j}\left (t\right )c_{j}^{\dagger }|\textbf {0}\rangle$ and $|\textbf {0}\rangle$ is the vacuum state. Because of $[N,~H]=0$, the total particle number $N=\sum _{j}\hat {n}_{j}$ is conserved. Moreover, such a periodically kicked system is governed by the Floquet unitary propagator and the evolution operator for one period can be described by $U\left (T,~0\right )=e^{-iH_{0}T}e^{-i\sum _{j}^{L}V_{j}a_{j}^{\dagger }a_{j}}$, where $H_{0}=-\sum _{j}\left (J_{1}a_{j}^{\dagger }a_{j+1}+J_{2}a_{j}^{\dagger }a_{j+2}+{\rm H}.c\right )$. After $N$ periods, the state can be written as $|\Psi \left (NT\right )\rangle =\left [U\left (T\right )\right ]^{N}|\Psi \left (0\right )\rangle$.3. Single-particle continuous-time quantum walks
We now turn to investigate the CTQWs of single particle, which is initially located in the center of the circuit QED lattice ($j=200$). Here we have taken the length of the circuit QED lattice $L=400$. For convenience, we set $J_{1}=1$ as the unit of energy throughout this paper. We first take $J_{2}=0$ and the situation of nonzero $J_{2}$ will be discussed later. In this case, the system only involves the NN interaction between circuit QED lattices. We investigate the effects of the strength of incommensurate potentials on CTQWs for fixing the driven period $T=0.1$ with $\lambda =0.1$, $0.3$, $0.5$, and $0.7$, respectively. One can see that the quantum walker expands ballistically when the mild value of the incommensurate potential is chosen; see Fig. 2(a) with $\lambda =0.1$. Then with the increasing of the strengths of incommensurate potential, the quantum walker can be well localized in the center of circuit QED lattice ($j=200$) at last; see Fig. 2(d) with $\lambda$=$0.7$. It indicates that the dynamic localization process of quantum walker is affected by the strengths of incommensurate potential, i.e., bigger strengths are better.
Fig. 2. CTQWs of single particle for 1D periodically kicked circuit QED lattice with $L$=400. Here we choose the driven periods $T=0.1$, and the strength of incommensurate potential (a) $\lambda =0.1$, (b) $\lambda =0.3$, (c) $\lambda =0.5$, and (d) $\lambda =0.7$, respectively.
Next, we investigate the influence of the driven periods on CTQWs for fixing the strength of incommensurate potential $\lambda =0.9$. In Figs. 3(a) and (b), for the driven period $T=0.1$ and $0.3$, one can see that the quantum walker can be well localized in the center of circuit QED lattice ($j=200$). In this case, the system is kicked with high frequency, and each lattice of the system is continuously subjected to the incommensurate potential, which leads to the system being in Anderson localized phase. However, when choosing the driven period $T=0.5$ and $0.7$, the quantum walker expands ballistically, as shown in Figs. 3(c) and (d). This is because that, these parameter selections correspond to a low frequency, each lattice is subjected to the incommensurate potential by a large interval. Thus, the quantum walker always expands ballistically although the incommensurate potential is big enough. From above results, we can get that the dynamic process of quantum walker is simultaneously affected by the strength of incommensurate potentials and the driven periods. The exact threshold for dynamic localization transition will be discussed in the following paragraph.
Fig. 3. CTQWs of single particle for 1D periodically kicked circuit QED lattice with $L$=400. Here we choose the strength of incommensurate potential $\lambda =0.9$ and the driven period (a) $T=0.1$, (b) $T=0.3$, (c) $T=0.5$, and (d) $T=0.7$, respectively.
To further describe the dynamical localization transition, we calculate the mean-square displacement $\sigma ^{2}(t)=\sum _{j=1}^{L}(j-L/2)^{2}|C_{j}(t)|^2$, which is used to characterize the propagation nature of the wave packet. We calculate the mean-square displacement for different driven periods $T=0.3$ and $0.5$ with fixing the strength of incommensurate potential at $\lambda =0.9$. After a brief numerical analysis, we find that the mean-square displacement obeys the power-law distribution with $T=0.5$, i.e., $\sigma ^{2}(t)~\propto ~\tau ^{1.75}$ and this kind of dynamical expansion belongs to the superdiffusion process for the quasiperiodic system because of the power-law index $1<\gamma <2$ [52]. On the contrary, there is a zero power-law index for distribution of the mean-square displacement with $T=0.3$, which corresponds to the localization process of the system.
In the case of $J_{2}\not =0$, we investigate how the NNN interactions influence the CTQWs. Figure 4 shows the dynamic process of quantum walker with different NNN hopping amplitudes. We find that the quantum walker expands ballistically and dose not occur the localization phenomenon with the introduction of NNN hopping amplitude. To explain this phenomenon clearly, we calculate the mean information entropy described as $\overline {\mathcal {S}}=L^{-1}\sum _{i=1}^{L}\mathcal {S}_{i}$ and $\mathcal {S}_{i}=-\sum _{j=1}^{L}|C_{j}\left (E_{i}\right )|^{2}\rm {ln}|C_{j}\left (E_{i}\right )|^{2}$, which can be used to characterize the dynamical transition. For the case of $J_1\neq 0$ and $J_2=0$, the dynamical transition occurs at $\lambda /T=2$ in high-frequency regime as reported in Ref. [52]. Figure 5 shows the mean information entropy versus both the strength of incommensurate potential strength $\lambda$ and the driven period $T$ with the NNN hopping amplitudes $J_{2}=0.9$. We can get following results: $\left (\rm {i}\right )$ The dynamical transition point will be larger than the critical value $\lambda /T$ with the NNN hopping amplitudes increasing. $\left (\rm {ii}\right )$ The requirement for the scope of high-frequency regime, i.e., the size of driven period $T$, is more stricter. This is because that the transmission paths of photons increase accordingly with the introduction of NNN interactions.
Fig. 4. CTQWs of single particle for 1D periodically kicked circuit QED lattice with $L$=400. Here we set $T=0.2$, $\lambda =0.9$, and the NNN hopping amplitudes (a) $J_{2}=0.1$ and (b) $J_{2}=0.9$.
Fig. 5. The mean information entropy versus both the strength of incommensurate potential $\lambda$ and the driven periods $T$ with $L$=400 and $J_{2}=0.9$.
4. Detections of bulk and edge sites resorting to cavity input-output process
In this section, we investigate the average photon number of cavity field for the periodically kicked system by utilizing lattice-based cavity input-output process. Under the conditions of weak disorder, high-frequency limit with $1/T \gg$ 1 and $\lambda \ll$ 1, and further ignoring the high-order terms, we can derive the following effective Hamiltonian
Figure 6 presents the energy distributions of the system and the photonic probability distributions of lattice sites in the case of two sets of parameters, i.e., $J_{2}=0$, $\lambda =0.3$, $T=0.1$ and $J_{2}=0.1$, $\lambda =0.5$, $T=0.1$, respectively. We investigate the properties of 200th lattice site (bulk site with eigenvalue of index X=86) and the 400th lattice site (edge site with eigenvalue of index X=248), respectively. For the commensurate off-diagonal Aubry-André Harper (AAH), Su-Schrieffer-Heeger (SSH), and Kitaev models, it has a maximal average photon number of the cavity field at the edge states with zero energy if the energy matching is satisfied between the lattice site and the external driving field. Contrarily, a small value is shown at the edge state by using the same external microwave field to drive the other sites (the bulk sites), and almost all of photons decay into the vacuum due to the non-resonance interaction [30,53,54]. However, a different cavity input-output process occurs in the present system. In order to detect the properties of the center lattice ($j=200$) and edge lattice ($j=400$), respectively, we introduce an external driving field to excite sites with $\vec {\Omega }=\left (0, 0,\ldots , \Omega _{200},\ldots , 0, 0\right )$ and $\vec {\Omega }=\left (0, 0,\ldots , 0, \Omega _{400}\right )$. The purpose is to make the frequency of the driving field resonant with the detected circuit QED lattice site.
Fig. 6. Eigenenergy distributions of $H_\textrm{eff}$ in ascending order and probability distribution of corresponding lattice site for 1D periodically kicked circuit QED lattice with $L$=400. Here we set the parameters (a), (c), and (d) $\lambda =0.3$, $T=0.1$, and $J_{2}=0$, (b), (e), and (f) $\lambda =0.5$, $T=0.1$, and $J_{2}=0.1$.
For the case of $J_{2}=0$, in Figs. 7(a)–7(c), we excite the 200th lattice site driven by the near-resonance, resonance, and non-resonance energies, i.e., $\Delta _{c}=-3.344$, $-2.912$, and $3.239$, which are marked in Fig. 6(a) (points A, B, and E), and plot the average photon number of the cavity field in the steady state. We can note that the photons are localized in the 199th lattice site but not in the 200th lattice site when it is driven, as shown in Figs. 7(a) and 7(b); unlike the detection of topologically localized edge mode, there is not a maximal average photon number at the detected lattice site although the system is in Anderson localized phase. It indicates that the detections of the bulk site are failed with above mentioned technique because it is hard to distinguish these two energies. However, Fig. 7(c) shows that the photons are localized in 200th lattice site when we choose the non-resonance energy. The result means that the non-resonance energy makes the lattice site occupy the large energy, and continuously accumulates in the lattice site. Subsequently, we excite the 400th lattice site (edge site) driven by the near-resonant, resonance, and non-resonance energies, i.e., $\Delta _{c}=0.253$, $0.941$, and $3.239$, which are marked in Fig. 6(a) (points C, D, and E). We find that the average photon number has a maximal distribution in the edge site driven by either near-resonance or resonance energy, as shown in Figs. 7(d) and 7(e). In this case, although these two energies have a mild difference, we can detect the properties of edge site via driving it with the resonance energy. Nevertheless, if we choose the non-resonance energy (point D) to drive the 400th lattice site, the photons are not localized at the edge site, which results from the non-resonance interaction, as shown in Fig. 7(f). Therefore, the detections of edge site are efficient. While for the case of $J_{2}\not =0$, the results are similar with discussions above. Hence, we do not make discussions in detail again.
Fig. 7. Average photon number of cavity field in the steady state for 1D periodically kicked circuit QED lattice with $L=400$. The parameters selections are the same as in Fig. 6(a). We choose the driving energies (a) $\Delta _{c}=-3.344$, (b) $\Delta _{c}=-2.912$, (c) $\Delta _{c}=3.239$, (d) $\Delta _{c}=0.253$, (e) $\Delta _{c}=0.941$, and (f) $\Delta _{c}=3.2392$.
5. Conclusions
In conclusion, we have investigated the CTQWs of single particle in two cases based on 1D periodically kicked circuit QED lattice. Firstly, we consider the effects of strength of incommensurate potentials and the driven periods on CTQWs only involving the NN interaction. We find that the quantum walker can be well localized in the regime of strong strength of incommensurate potentials and short driven periods, which can be characterized by the mean-square displacement. Then, we investigate the dynamic process of quantum walker in the case of containing the NNN interactions. The results show that NNN interactions significantly deviate dynamical transition point by calculating the mean information entropy. Furthermore, assisted by the lattice-based cavity input-output process, we plotted the average photon number of cavity field to describe the localized features of the circuit QED lattice.
Funding
National Natural Science Foundation of China (11874132, 61575055, 61822114); Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (20160519022JH).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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