Abstract
It is of fundamental interest in controlling the light–matter interaction for a long time in the field of quantum information processing. Here, we explore a model by coupling a giant atom with the dynamicallymodulated coupledresonator waveguide and find the bound state, where the light shows the localization effect and the atomic decay into resonator modes is inhibited, excited by a propagating photon. An analytical treatment based on the separation of the propagating states and localized states of light has been proposed and provides inspiring explanation of our finding, i.e., there supports a quantum channel where the propagating photon can be converted to the localized state through the quantum interference from light–atom interactions in three resonators at different frequency detunings. Our work therefore shows the potential for actively localizing the photon in a modulated coupledresonator waveguide system interacting with the giant atom, and also points out a way to study the light–atom interaction in a synthetic frequency dimension that holds the similar Hamiltonian.
Introduction
It is of great importance in achieving flexible manipulations of photons in atomwaveguide systems and exploring fundamental physics associated with strong light–atom interactions and atommediated photon–photon interactions, which also shows potential applications towards quantum communications and quantum networks^{1,2,3,4,5,6,7,8,9,10,11,12,13}. Similar with but different from the continuum waveguide, the coupledresonator waveguide provides an alternative structure for manipulating the spatial and spectral properties of photons, where photon transport can be controlled by designing combinations of resonators with the nonlinearity of the resonator^{14,15,16,17,18,19} or by actively connecting resonators with dynamic modulations^{20,21,22}. In both cases, atoms (or quantum emitters) can be added into the coupledresonator waveguide and hence further possible controllability of photons has been discussed^{23,24,25,26,27,28}. Although theoretical models may be originally studied in photonic structures, such coupledresonator waveguide has also been discussed in the onchip platform of superconducting transmission line resonators^{29,30,31}, where microwave photons transport and can be interacting with the artificial superconducting qubit^{26,32,33,34}.
Recently, the atomwaveguide system has been generalized to studies of interactions between the photon in the waveguide and a giant atom, where an artificial atom (quantum emitter) is fabricated to couple multiple locations on the waveguide^{35}. Due to the fact that multipath quantum interferences are included in interactions between waveguide photons and giant atoms, a variety of interesting quantum optical phenomena have been explored, including bound states or dressed states^{28,36,37,38,39}, decoherencefree interaction^{40,41,42,43}, electromagneticallyinduced transparency^{44,45,46,47}, and many others^{48,49,50,51,52,53,54,55,56}. Relevant experiments have also been demonstrated that microwave photons or propagating phonons have been successfully coupled to an artificial gaint atom^{41,45,57}. Hence, explorations of different opportunities in seeking exotic manipulations of photons via quantum interferences from the photon–giantatom interaction in the coupledresonator waveguide trigger further theoretical interests.
In this work, we study a theoretical model of an artificial twolevel giant atom coupled with dynamicallymodulated coupledresonator waveguide (see Fig. 1), where each resonator (labeled by m) supports a resonant mode at the frequency ω_{m} = ω_{0} + mΩ with ω_{0} being the transition frequency of the atom. The giant atom couples to the middle three resonators (0th and ±1st) through three separate paths. We find that, for the choice of modulation parameters, the wavepacket of the photon that transports inside the waveguide exhibits the localization effect, together with the time of the photon–atom interaction lasting longer than the decay rate between the atom and the resonator, i.e., namely bound state. The key feature here is the excitation of a bound state corresponding to photon modes with group velocities close to zero by the traveling photon, which is counterintuitive. We provide the analytical analysis based on the separation of propagating states and localized states of light, and find the quantum transition channel from propagating states to localized states. Moreover, our studied Hamiltonian also describes an artificial lattice in the synthetic frequency dimension^{58,59,60,61}, which could trigger further research interest in the photon–atom interaction with synthetic dimensions. Therefore, this work shows the photon manipulation through quantum interferences in a system composed by the giant atom and the dynamicallymodulated coupledresonator waveguide, which shall find potential applications in the quantum information processing^{62,63}.
Results
Model
As schematically shown in Fig. 1, we consider a onedimensional photonic resonator lattice, with each resonator supporting a single resonance at ω_{m}. The dynamic modulation can be applied inbetween two adjacent resonators by modulating two resonances in the auxiliary resonator with a sinusoid external source \(2\eta \cos {{\Omega }}t\) where Ω ≪ ω_{0} is the frequency and η is the modulation amplitude^{20,64}. A twolevel giant atom is designed to couple with the 0th and ±1st resonators with the coupling strength κ. By assuming ℏ = 1, the corresponding Hamiltonian is
Here, σ_{z} = [σ_{+}, σ_{−}]. \({\sigma }_{+}=\lefte\right\rangle \left\langle g\right\) (\({\sigma }_{}\,=\,\leftg\right\rangle \left\langle e\right\)) is the ladder operator that transits the atom from ground state \(\leftg\right\rangle\) to excited state \(\lefte\right\rangle\) (and vice versa), and \({a}_{m}^{{\dagger} }\) (a_{m}) is the creation (annihilation) operator for the photon in the mth resonator. One can rewrite the Hamiltonian in the interaction picture under the rotatingwave approximation (RWA)^{65}
To simulate the dynamics of photon transport, we write the singleexcitation wave function
where v_{m} is the probability amplitude for creating the photon from the vacuum state \(\left0\right\rangle\) in the mth resonator while the atom remains at the ground state \(\leftg\right\rangle\), and ξ is the probability amplitude for the atom being excited (to \(\lefte\right\rangle\)) by the propagating photon. By using the Schrödinger’s equation, we obtain
In simulations, we consider a coupledresonator waveguide composed by 401 resonators (m = −200, ⋯ , 200). A Gaussianshape pulse \(S\,=\,{{{{\rm{e}}}}}^{{(t\,\,{t}_{0})}^{2}/{\tau }^{2}}\) is used to excite the leftmost resonator, where \(\tau \,=\,5\sqrt{2}{\eta }^{1}\) and t_{0} = 25η^{−1}. We first consider the case that κ = 0.5η and Ω = 3η. In Fig. 2a, we plot the distribution of ∣v_{m}∣^{2} on different resonators and ∣ξ∣^{2} versus the time t. One sees that the photon is injected into the system from the left and then propagates towards the right. Once it interacts with the atom, a portion of the wavepacket of the photon is reflected while the atom is excited, which shows consistence with the propagating photon interacting with a resonant atom in a waveguide^{66,67}.
The striking feature of the system is found when we choose Ω = 2.05η while keeping other parameters unchanged, with simulation results plotted in Fig. 2b. One sees that, once the light interacts with the atom, the wavepacket of the photon is stored for a relative long time in the vicinity of middle resonators (~150η^{−1} ≫ κ^{−1}) and the excitation of the atom also gives a relative long decay tail. Such the phenomenon denotes a bound state of photon and atom where the light shows the localization effect near middle resonators and atom exhibits the inhibited decay. Besides the bound state, together with the transmission of a portion of wavepacket at the original group velocity, other small portions of wavepacket are transmitted and reflected at a smaller group velocity. By further studying cases for Ω nearby 2.05η, we find that the existence of the bound state is critically dependent on the choice of Ω (see Supplementary Note 1 for details), with detailed physical mechanism will be analyzed analytically in the following.
Analytical analysis
In order to understand the bound state in our proposed model, we next analyze the Hamiltonian (2) analytically in details. The first term in Eq. (2) gives couplings between resonators driven by an external source, while the second term describes atom–resonator interactions, where ±Ω represents the detuning between the ±1st resonator and the atom. Since the atom only couples with the middle three resonators, and we consider a finite number of resonators (−M ≤ m < M with M being a positive integer), the influence of the photon state in resonators at two boundaries is negligible for M ≫ 1. We hence can take the state \(\leftk\right\rangle \,=\,\mathop{\sum }\nolimits_{m\,=\,M}^{M\,\,1}{a}_{m}^{{\dagger} }\left0\right\rangle {{{{\rm{e}}}}}^{{{{\rm{i}}}}mk{{{\rm{\pi }}}}/M}/\sqrt{2M}\) (k = −M, ⋯ , M − 1 as an integer) in the momentum space which is the eigenstate of \(\mathop{\sum }\nolimits_{m\,=\,M}^{M\,\,1}\eta ({a}_{m}^{{\dagger} }{a}_{m\,+\,1}\,+\,{{{\rm{h}}}}.c.)\) and is regarded as the Bloch wave in the lattice with the frequency \({\omega }_{k}\,=\,2\eta \cos (k{{{\rm{\pi }}}}/M)\). Note here we set the spatial distance between two resonators as 1 for the simplicity. The corresponding group velocity of the wavepacket is \({v}_{k}\,=\,2\eta \sin (k{{{\rm{\pi }}}}/M)\). Obviously, when ω_{k} = ±2η, the wavepacket has the group velocity v_{k} = 0 and therefore does not move in the lattice. We will refer to them as localized modes in the following discussions.
Along with the atomic states \(\lefte\right\rangle\) and \(\leftg\right\rangle\), now we can rewrite V in the kspace, which leads to \(\tilde{V}\,=\,{\tilde{V}}_{0}\,+\,{\tilde{V}}_{1}\). Here
In the interactionpicture Hamiltonian under the k representation, i.e.,
one clearly sees that each state \(\leftk\right\rangle\) interacts with the atom through the middle three resonators with \(m^{\prime} \,=\,0,\,\pm\! 1\), while \(m^{\prime} {{\Omega }}\,+\,{\omega }_{k}\) represents the detuning between the atom and the Blochwave state for the \(m^{\prime}\)th resonator. Following this argument, we can assume that each state \(\leftk\right\rangle\) is only coupled with the resonator with the smallest detuning. Hence, we obtain the Hamiltonian as
where we divide k into three regions. In region K_{0}, the photon state of the Bloch wave has \({\omega }_{k}\,\in\, (\sqrt{2}\eta ,\sqrt{2}\eta )\), and is only coupled with the 0th resonator. Similarly, in region K_{±1}, \({\omega }_{k}\,\in\, (\mp \sqrt{2}\eta ,\mp 2\eta ]\), and the photon state is coupled with the ±1st resonator. We emphasize that, according to the resonance condition (\(m^{\prime} {{\Omega }}\,+\,{\omega }_{k}\,=\,0\)), the photon state that make most contributions have frequencies ω_{k} = 0, ±2η. Therefore, except for the states near these three frequencies, other states in the kspace are negligible in our analytical analysis, and the choice of the limit of the abovementioned regions is for the convenience purpose.
The wave function of the photon state in the kspace has the form
and together with Eq. (9), we can get the corresponding dynamic evolution equations. However, we notice that, there are 4 corresponding states \(\leftk\right\rangle\) at each ∣ω_{k}∣. For the sake of simplicity, we hence define linear combinations of states for the same ∣ω_{k}∣:
with k ∈ [0, M/4), and obtain
Here J_{k,s±} (k ∈ [0, M/4)) denotes the modes of the photon state whose group velocity is in the range of \((\pm \sqrt{2}\eta ,0]\) (where J_{0,s±} corresponds to modes with v_{k} = 0, i.e., localized states), and J_{k,0±} denotes those modes whose group velocities \({v}_{k}\,\in\, (\pm \sqrt{2}\eta ,\pm 2\eta ]\) (where J_{0,0±} corresponds to modes with v_{k} = ±2η, i.e., propagating states). Eqs. (13)–(17) indicate that J_{k,s±} and J_{k,0±} form two sets of subsystems, connected by the atom [see Eq. (15)]. We then focus on the subsystem described by Eqs. (13)–(15), where localized modes are included. In order to erase the impact of the other subsystem, we discard the second term on the righthand side of Eq. (15). Then, by setting that \({J}_{k,s\pm }(t)\,=\,{j}_{k,s\pm }{{{{\rm{e}}}}}^{{{{\rm{i}}}}{\lambda }_{s}t}\) and \(\chi (t)\,=\,{x}_{s}{{{{\rm{e}}}}}^{{{{\rm{i}}}}{\lambda }_{s}t}\), we find the special solution with λ_{s} = 0, which gives
Equations (18), (19) are solvable with the normalization condition involved. Equation (18) suggests that when Ω > 2η, ∣j_{k,s−}∣ decreases as k increases, which implies that the mode distribution of the photon state is concentrated near ω_{k} = ±2η. Notice that ∣j_{k,s−}∣ is proportional to ∣x_{s}∣. Once the atom is excited, it is possible to observe the localized state with v_{k} = 0. This is a bound state in which the photon is stored and the atom keeps excited with the inhibited decay. However, we need to point out that when Ω becomes large, the proportion of the atomic excited state in the bound state is also increasing. Due to normalization condition, the probability that the system is in the localized state in the vicinity of k = 0 is suppressed. On the other hand, when Ω < 2η, ∣j_{k,s−}∣ increases as k increases, and hence the maximum value of ∣j_{k,s−}∣ is at k ≠ 0 (v_{k} ≠ 0). In such case, the localized state is nearly impossible to be generated. Therefore, we only have a very narrow window of Ω to obtain a significant localized photon state, which is consistent with our previous numerical results.
The above discussion shows that through the interaction with the ±1st resonators and the atom, the photon may move slowly or even stay localized in the waveguide. The next question is how to excite such localized state, since the initial wavepacket of the photon is prepared centered at ω_{k} = 0 with spectral width ~1/τ nearly does not contain any component of ω_{k} = ±2η (v_{k} = 0). From Eq. (15), one finds that the coupling strengths of the two subsystems and the atom are similar, which means that the effects of the two subsystems on the atom are comparable. Consequently, we have the following physical picture: An initial state of the photon is prepared in one subsystem (corresponding to J_{k,0±}) where the center frequency of the initial wavepacket is ω_{k} = 0 and v_{k} ~ 2η for example. Such wavepacket of the photon propagates inside the waveguide and causes the excitation of the atom, which in turn leads to the excitation of another subsystem (corresponding to J_{k,s±}), i.e., creating the bound state with the localized photon state and an excited atom lasting for a long time. This picture is consistent with our simulation result in Fig. 2b.
Finally, we perform simulations in the kspace with the separation approximation that we analytically discussed above to verify our analytical analysis. By combining Eq. (9) and the Schrödinger’s equation, we simulate the evolution of the states in the kspace over time with Ω = 2.05η and plot the result in Fig. 3a. The initial state of the system is assumed to be a Gaussian wavepacket in real space, with its center position at m_{0} = − 250 and propagating toward the right at a group velocity 2η, i.e., \(\left{\psi }_{I}\right\rangle \,=\,\mathop{\sum }\nolimits_{m\,=\,M}^{M}{{{{\rm{e}}}}}^{{(m\,\,{m}_{0})}^{2}/2{\delta }_{m}^{2}}{{{{\rm{e}}}}}^{{{{\rm{i}}}}{{{\rm{\pi }}}}(m\,\,{m}_{0})/2}\leftm\right\rangle\), with δ_{m} = 10. (This choice of the initial wavepacket is consistent with the boundaryexcitation source in simulations for Fig. 2) Therefore, the Fourier transformation of the initial state (with taking M = 500) here gives the initial wave function in Eq. (10) for simulations, with the central momentum being kπ/M = −π/2. Figure 3a indicates that the mode separation approximation we used in the above discussion is feasible, because one can see that the energy distribution of the photon’s wavepacket is centered near kπ/M = − π, −π/2, 0, π/2 with clear separations. Next, we follow the mode separation approximation and use Eqs. (13)–(17) with the same parameters for Fig. 3a and numerically calculate the photon distribution in the kspace. After we Fouriertransform the simulation results back into the real space, we plot the resulting probability distribution of photon in Fig. 3b. One can see that evolutions of the photon in both Figs. 3b, 2b match quite well, indicating our analytical analysis is valid in understanding such light localization phenomenon.
Discussion
There are several notes we want to provide further discussions. Different from previous works studying bound states in atomwaveguide systems using either giant atoms^{36} or small atoms^{68,69}, the energy of bound states is explicitly out of the regime for the propagating band of the waveguide. Hence, it is fundamentally difficult to excite such photonic bound states through the propagating waveguide photon. On the contrary, in our model, we find that the coupling between the giant atom and the 0th resonator provides the excitation of giant atom and the couplings between the ±1st resonators converts the atomic excited state to a hybrid bound state, which shows the possibility for exciting a bound state by the propagating photon without using external drive fields (see Supplementary Note 2 for details). This unique feature therefore provides potential important applications in quantum storage. Moreover, in simulations, we use an initial Gaussianshape excitation to provide the input pulse of the incoming propagating photon, where a narrow spectrum of the input wavepacket is used to excite the giant atom. For seeing clear effects of the propagating photon and the light localization, we use the waveguide with hundreds of resonators. In reality, a shorter waveguide with tens of resonators is also feasible, as long as it can carry the entire singlephoton pulse before the photon interacting the giant atom. In this case, one can inject the waveguide at the end of resonator from outer connecting waveguide where the effect of boundary reflections should be negligible. As an example, in Supplementary Note 3, we study the case of exciting the waveguide consisting 21 resonators with the same Gaussianshape input photon source, and see the bound state still existing, which is experimentally feasible^{69}. The energy distribution of such bound state could be further enhanced by increasing the full width at half maximum of the Gaussianshape pulse, which corresponds to a narrower width in the spectrum and larger portion of the wavepacket can then interact with the giant atom to excite the bound state. In Supplementary Note 4, we showcase the result with \(\tau \,=\,10\sqrt{2}{\eta }^{1}\) where a significant increase in the energy distribution of the photonic localized state together with the extension of the inhibited atomic decay time exists.
Before we conclude our paper, we point out that our studied model naturally supports a synthetic lattice along the frequency axis of light, which is constructed in a modulated ring resonator and is similarly described by the Hamiltonian \(\mathop{\sum}\nolimits_{m}{\omega }_{m}{a}_{m}^{{\dagger} }{a}_{m}\,+\,\mathop{\sum}\nolimits_{m}2\eta \cos ({{\Omega }}t)({a}_{m}^{{\dagger} }{a}_{m+1}\,+\,{a}_{m\,+\,1}^{{\dagger} }{a}_{m})\) (the same as with the waveguide part in Eq. (1) with ω_{m} being resonant frequencies and Ω being the frequency of the modulator^{61}. Once a twolevel atom at the transition frequency ω_{0} is added to couple with the ring, an effective giant atom coupled with the synthetic lattice with the lineargradient detuning (ω_{m} − ω_{0}) at each connection is built, as shown in Fig. 4. We find that the affects from farfromresonance couplings weakly affect the system, and our findings in this work shall also map to the dynamics in the synthetic dimension (see Supplementary Note 5 for details). However, detailed consequences in the output of the singlephoton spectrum desires further studies with the inputoutput formalism carefully included, which is beyond the scope of this paper. Yet, our work is still useful in the future understanding of the photontransport problem with the recentdeveloped synthetic dimensions^{70,71,72} coupled to a atom, which opens an avenue to study quantum optics in a synthetic waveguide coupled with an effective giant atom with coupling positions reaching the order of 10^{3} ^{73}.
In summary, we study the photon propagation problem inside a coupledresonator waveguide under the dynamic modulation with the middle three resonators coupled with the giant atom, where the dynamic modulation frequency precisely equals to the frequency difference between two nearby resonators. We find that, through multiresonator couplings, one can excite the Blochwave state in the system by a propagating input photon, so the light field can be localized for a long time and the excitation of the atom exhibits the inhibited decay. An analytical approach is built to understand the intrinsic quantum interference dynamics. Our model is valid for a variety of potential experimental platforms, including photoniccrystal waveguide^{20,74}, coupled cavities in free space^{75,76}, superconducting transmission line resonators^{77,78,79,80}. Moreover, such a model can also be useful in understanding a synthetic lattice along the frequency axis of light coupling to a twolevel atom that has the similar Hamiltonian. The bound states with near zero group velocity for photons localize the energy of information and has extensive applications in quantum storages^{81,82}. Our work therefore shows a theoretical perspective for studying photon–atom interactions in waveguide systems and seeks additional external control of the propagating photon, which can find applications in quantum manipulations of a single photon.
Note.—When we prepare our paper, we notice an independent preprint, which is related but different from our work^{83}.
Data availability
The data files used to prepare the figures shown in the paper are available from corresponding authors upon request.
Code availability
The codes that support the findings of this study are available from corresponding authors upon request.
References
Goban, A. et al. Superradiance for atoms trapped along a photonic crystal waveguide. Phys. Rev. Lett. 115, 063601 (2015).
Yuan, L., Xu, S. & Fan, S. Achieving nonreciprocal unidirectional singlephoton quantum transport using the photonic AharonovBohm effect. Opt. Lett. 40, 5140–5143 (2015).
Pichler, H. & Zoller, P. Photonic circuits with time delays and quantum feedback. Phys. Rev. Lett. 116, 093601 (2016).
FornDíaz, P. et al. Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime. Nat. Phys. 13, 39–43 (2017).
Cheng, M.T., Ma, X., Fan, J.W., Xu, J. & Zhu, C. Controllable singlephoton nonreciprocal propagation between two waveguides chirally coupled to a quantum emitter. Opt. Lett. 42, 2914–2917 (2017).
Kumlin, J., Hofferberth, S. & Büchler, H. P. Emergent universal dynamics for an atomic cloud coupled to an optical waveguide. Phys. Rev. Lett. 121, 013601 (2018).
Mahmoodian, S. et al. Strongly correlated photon transport in waveguide quantum electrodynamics with weakly coupled emitters. Phys. Rev. Lett. 121, 143601 (2018).
Corzo, N. V. et al. Waveguidecoupled single collective excitation of atomic arrays. Nature 566, 359–362 (2019).
Mirhosseini, M. et al. Cavity quantum electrodynamics with atomlike mirrors. Nature 569, 692–697 (2019).
Xiao, H., Wang, L., Yuan, L. & Chen, X. Frequency manipulations in singlephoton quantum transport under ultrastrong driving. ACS Photonics 7, 2010–2017 (2020).
Kim, E. et al. Quantum electrodynamics in a topological waveguide. Phys. Rev. X 11, 011015 (2021).
Poshakinskiy, A. V. et al. Quantum Hall phases emerging from atomphoton interactions. NPJ Quantum Inf. 7, 34 (2021).
Nefedkin, N., Cotrufo, M., Krasnok, A. & Alù, A. Darkstate induced quantum nonreciprocity. Adv. Quantum Technol. 5, 2100112 (2022).
MarinPalomo, P. et al. Microresonatorbased solitons for massively parallel coherent optical communications. Nature 546, 274–279 (2017).
Stern, B., Ji, X., Okawachi, Y., Gaeta, A. L. & Lipson, M. Batteryoperated integrated frequency comb generator. Nature 562, 401–405 (2018).
Zhang, M. et al. Broadband electrooptic frequency comb generation in a lithium niobate microring resonator. Nature 568, 373–377 (2019).
Lu, J. et al. Periodically poled thinfilm lithium niobate microring resonators with a secondharmonic generation efficiency of 250,000%/W. Optica 6, 1455–1460 (2019).
Szabados, J. et al. Frequency comb generation via cascaded secondorder nonlinearities in microresonators. Phys. Rev. Lett. 124, 203902 (2020).
Yu, S.P. et al. Spontaneous pulse formation in edgeless photonic crystal resonators. Nat. Photonics 15, 461–467 (2021).
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).
Fang, K. & Fan, S. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013).
Williamson, I. A. D. et al. Integrated nonreciprocal photonic devices with dynamic modulation. Proc. IEEE 108, 1759–1784 (2020).
Hoffman, A. J. et al. Dispersive photon blockade in a superconducting circuit. Phys. Rev. Lett. 107, 053602 (2011).
Zhou, L., Yang, L.P., Li, Y. & Sun, C. P. Quantum routing of single photons with a cyclic threelevel system. Phys. Rev. Lett. 111, 103604 (2013).
Lu, J., Zhou, L., Kuang, L.M. & Nori, F. Singlephoton router: coherent control of multichannel scattering for single photons with quantum interferences. Phys. Rev. A 89, 013805 (2014).
Fitzpatrick, M., Sundaresan, N. M., Li, A., Koch, J. & Houck, A. A. Observation of a dissipative phase transition in a onedimensional circuit QED lattice. Phys. Rev. X 7, 011016 (2017).
Wang, L., Yuan, L., Chen, X. & Fan, S. Singlephoton transport in a topological waveguide from a dynamically modulated photonic system. Phys. Rev. Appl. 14, 014063 (2020).
Wang, X., Liu, T., Kockum, A. F., Li, H.R. & Nori, F. Tunable chiral bound states with giant atoms. Phys. Rev. Lett. 126, 043602 (2021).
Devoret, M. H., Girvin, S. & Schoelkopf, R. CircuitQED: how strong can the coupling between a Josephson junction atom and a transmission line resonator be? Ann. Phys. 16, 767–779 (2007).
Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.x & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718–719, 1–102 (2017).
Blais, A., Grimsmo, A. L., Grivin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021).
Hoi, I.C. et al. Demonstration of a singlephoton router in the microwave regime. Phys. Rev. Lett. 107, 073601 (2011).
Stockklauser, A. et al. Strong coupling cavity QED with gatedefined double quantum dots enabled by a high impedance resonator. Phys. Rev. X 7, 011030 (2017).
Yan, Z. et al. Strongly correlated quantum walks with a 12qubit superconducting processor. Science 364, 753–756 (2019).
Kockum, A. F., Delsing, P. & Johansson, G. Designing frequencydependent relaxation rates and Lamb shifts for a giant artificial atom. Phys. Rev. A 90, 013837 (2014).
Zhao, W. & Wang, Z. Singlephoton scattering and bound states in an atomwaveguide system with two or multiple coupling points. Phys. Rev. A 101, 053855 (2020).
Guo, L., Kockum, A. F., Marquardt, F. & Johansson, G. Oscillating bound states for a giant atom. Phys. Rev. Res. 2, 043014 (2020).
Cheng, W., Wang, Z. & Liu, Y.X. Boundary effect and dressed states of a giant atom in a topological waveguide. Preprint at https://arxiv.org/abs/2103.04542 (2021).
Vega, C., Porras, D. & GonzálezTudela, A. Qubitphoton bound states in topological waveguides with longrange hoppings. Phys. Rev. A 104, 053522 (2021).
Kockum, A. F., Johansson, G. & Nori, F. Decoherencefree interaction between giant atoms in waveguide quantum electrodynamics. Phys. Rev. Lett. 120, 140404 (2018).
Kannan, B. et al. Waveguide quantum electrodynamics with superconducting artificial giant atoms. Nature 583, 775–779 (2020).
Carollo, A., Cilluffo, D. & Ciccarello, F. Mechanism of decoherencefree coupling between giant atoms. Phys. Rev. Res. 2, 043184 (2020).
Soro, A. & Kockum, A. F. Chiral quantum optics with giant atoms. Phys. Rev. A 105, 023712 (2022).
Ask, A., Fang, Y.L. L. & Kockum, A.F. Synthesizing electromagnetically induced transparency without a control field in waveguide QED using small and giant atoms. Preprint at https://arxiv.org/abs/2011.15077 (2020).
Vadiraj, A. M. et al. Engineering the level structure of a giant artificial atom in waveguide quantum electrodynamics. Phys. Rev. A 103, 023710 (2021).
Zhu, Y., Wu, R. & Xue, S. Spatial nonlocality induced nonmarkovian EIT in a single giant atom. Preprint at https://arxiv.org/abs/2106.05020 (2021).
Zhao, W., Zhang, Y. & Wang, Z. Phasemodulated AutlerTownes splitting in a giantatom system within waveguide QED. Front. Phys. 17, 42506 (2022).
GonzálezTudela, A., Muñoz, C. S. & Cirac, J. I. Engineering and harnessing giant atoms in highdimensional baths: a proposal for implementation with cold atoms. Phys. Rev. Lett. 122, 203603 (2019).
Cilluffo, D. et al. Collisional picture of quantum optics with giant emitters. Phys. Rev. Res. 2, 043070 (2020).
Longhi, S. Photonic simulation of giant atom decay. Opt. Lett. 45, 3017–3020 (2020).
Du, L., Cai, M.R., Wu, J.H., Wang, Z. & Li, Y. Singlephoton nonreciprocal excitation transfer with nonMarkovian retarded effects. Phys. Rev. A 103, 053701 (2021).
Du, L. & Li, Y. Singlephoton frequency conversion via a giant Λtype atom. Phys. Rev. A 104, 023712 (2021).
Yu, H., Wang, Z. & Wu, J.H. Entanglement preparation and nonreciprocal excitation evolution in giant atoms by controllable dissipation and coupling. Phys. Rev. A 104, 013720 (2021).
Cai, Q. & Jia, W. Coherent singlephoton scattering spectra for a giantatom waveguideQED system beyond the dipole approximation. Phys. Rev. A 104, 033710 (2021).
Du, L., Chen, Y.T. & Li, Y. Nonreciprocal frequency conversion with chiral Λtype atoms. Phys. Rev. Res. 3, 043226 (2021).
Wang, X. & Li, H.R. Chiral quantum network with giant atoms. Quantum Sci. Technol. 7, 035007 (2022).
Andersson, G., Suri, B., Guo, L., Aref, T. & Delsing, P. Nonexponential decay of a giant artificial atom. Nat. Phys. 15, 1123–1127 (2019).
Yuan, L., Shi, Y. & Fan, S. Photonic gauge potential in a system with a synthetic frequency dimension. Opt. Lett. 41, 741–744 (2016).
Li, G. et al. Dynamic band structure measurement in the synthetic space. Sci. Adv. 7, eabe4335 (2021).
Yu, D., Peng, B., Chen, X., Liu, X.J. & Yuan, L. Topological holographic quench dynamics in a synthetic frequency dimension. Light Sci. Appl. 10, 209 (2021).
Yuan, L., Dutt, A. & Fan, S. Synthetic frequency dimensions in dynamically modulated ring resonators. APL Photonics 6, 071102 (2021).
Wendin, G. Quantum information processing with superconducting circuits: a review. Rep. Prog. Phys. 80, 106001 (2017).
Blais, A., Girvin, S. M. & Oliver, W. D. Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat. Phys. 16, 247–256 (2020).
Yuan, L. & Fan, S. Topologically nontrivial Floquet band structure in a system undergoing photonic transitions in the ultrastrongcoupling regime. Phys. Rev. A 92, 053822 (2015).
Scully, M. O. & Zubairy, M. S. Quantum Optics (Cambridge Univ. Press, 1997).
Shen, J.T. & Fan, S. Coherent single photon transport in a onedimensional waveguide coupled with superconducting quantum bits. Phys. Rev. Lett. 95, 213001 (2005).
Shen, J.T. & Fan, S. Strongly correlated multiparticle transport in one dimension through a quantum impurity. Phys. Rev. A 76, 062709 (2007).
Calajó, G., Ciccarello, F., Chang, D. & Rabl, P. Atomfield dressed states in slowlight waveguide QED. Phys. Rev. A 93, 033833 (2016).
Scigliuzzo, M. et al. Extensible quantum simulation architecture based on atomphoton bound states in an array of highimpedance resonators. Preprint at https://arxiv.org/abs/2107.06852 (2021).
Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).
Lustig, E. & Segev, M. Topological photonics in synthetic dimensions. Adv. Opt. Photon. 13, 426–461 (2021).
Lee, N. R. A. et al. Propagation of microwave photons along a synthetic dimension. Phys. Rev. A 101, 053807 (2020).
Chen, L.H., Chen, G., Liu, R. & Wang, X.H. Dynamically tunable multifunctional QED platform. Sci. China Phys. Mech. Astron 62, 974211 (2019).
Peng, B. et al. Paritytimesymmetric whisperinggallery microcavities. Nat. Phys. 10, 394–398 (2014).
Jiang, X.F., Zou, C.L., Wang, L., Gong, Q. & Xiao, Y.F. Whisperinggallery microcavities with unidirectional laser emission. Laser Photon. Rev. 10, 40–61 (2016).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Yin, Y. et al. Catch and release of microwave photon states. Phys. Rev. Lett. 110, 107001 (2013).
Pechal, M. et al. Microwavecontrolled generation of shaped single photons in circuit quantum electrodynamics. Phys. Rev. X 4, 041010 (2014).
Kuzmin, R., Mehta, N., Grabon, N., Mencia, R. & Manucharyan, V. E. Superstrong coupling in circuit quantum electrodynamics. NPJ Quantum Inf. 5, 20 (2019).
Phillips, D. F., Fleischhauer, A., Mair, A., Walsworth, R. L. & Lukin, M. D. Storage of Light in Atomic Vapor. Phys. Rev. Lett. 86, 783 (2001).
Karpa, L. & Weitz, M. A SternGerlach experiment for slow light. Nat. Phys. 2, 332–335 (2006).
Du, L., Zhang, Y., Wu, J.H., Kockum, A. F. & Li, Y. Giant atoms in a synthetic frequency dimension. Phys. Rev. Lett. 128, 223602 (2022).
Acknowledgements
The research is supported by National Natural Science Foundation of China (12122407, 11704241, and 11974245), National Key R&D Program of China (2017YFA0303701), Shanghai Municipal Science and Technology Major Project (2019SHZDZX01), and Natural Science Foundation of Shanghai (19ZR1475700). L.Y. thanks the sponsorship from Yangyang Development Fund and the support from the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. X.C. also acknowledges the support from Shandong Quancheng Scholarship (00242019024).
Author information
Authors and Affiliations
Contributions
H.X. and L.W. contributed equally to this work. Z.L. and L.Y. conceived the idea and developed an analytical model. H.X., L.W., and Z.L. performed the numerical calculations. X.C. and L.Y. supervised the project. All authors contributed to discussion of the results and writing the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Xiao, H., Wang, L., Li, ZH. et al. Bound state in a giant atommodulated resonators system. npj Quantum Inf 8, 80 (2022). https://doi.org/10.1038/s41534022005917
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41534022005917
This article is cited by

A giant atom with modulated transition frequency
Frontiers of Physics (2023)

Simulating topological materials with photonic synthetic dimensions in cavities
Quantum Frontiers (2022)