Detecting Topological Phases of Microwave Photons in a Circuit Quantum Electrodynamics Lattice

Topology is an important degree of freedom in characterizing electronic systems. Recently, it also brings new theoretical frontiers and many potential applications in photonics. However, the verification of the topological nature is highly nontrivial in photonic systems as there is no direct analog of quantized Hall conductance for bosonic photons. Here we propose a scheme of investigating topological photonics in superconducting quantum circuits by a simple parametric coupling method, the flexibility of which can lead to the effective \textit{in situ} tunable artificial gauge field for photons on a square lattice. We further study the detection of the topological phases of the photons. Our idea employs the exotic properties of the edge state modes which result in novel steady states of the lattice under the driving-dissipation competition. Through the pumping and the photon-number measurements of merely few sites, not only the spatial and the spectral characters, but also the momentums and even the integer topological quantum numbers with arbitrary values of the edge state modes can be directly probed, which reveal unambiguously the topological nature of photons on the lattice.


INTRODUCTION
Charged particles in two dimension exhibit integer quantum Hall effect (IQHE) when exposed to a perpendicular magnetic field [1], characterized by the quantized transverse conductances in transport experiments. This novel effect can be explained by the integer topological Chern numbers describing the global behavior of the energy bands [2,3]. Such topological insulating IQHE phase is robust against disorder and defects because the band topology remains invariant as long as the band gaps are preserved. Therefore, in the connection between the topologically nontrivial material and the trivial vacuum, there exist unavoidably the edge state modes (ESMs) spatially confining at the boundary and spectrally traversing the band gaps [4]. The presence of these gapless ESMs thus serves as an unambiguous signature of the topological non-triviality of the bulk band structure.
As photons are charge neutral, there have been several proposals of synthesizing artificial magnetic fields on a TLR lattice, with predicted strengths much stronger than those in conventional electronic materials [22][23][24][25]. Nevertheless, the synthetic Abelian gauge field has not been implemented so far despite the extensive theoretical studies, partially due to the complicated circuit elements required in these schemes. In addition, the detection of the integer topological invariants has also been addressed in recent research [26][27][28], which is nontrivial in the sense that the Hall conductance measurement cannot be transferred to circuit QED due to the absence of fermionic statistics.
Here, we propose a theoretical scheme of implementing topological photonics in a twodimensional circuit QED lattice. The distinct merit of our proposal is that we couple the TLRs by parametric frequency conversion (PFC) method which is simple in experimental setup and feasible with state-of-the-art technology [29][30][31][32]. The lattice in our scheme is formed by TLRs connected to the ground through the superconducting interference devices (SQUIDs) [29,30,33,34], where the tunable photon hopping with nontrivial phases between TLRs can be induced through the dynamic modulation of the SQUIDs, allowing the arbitrary synthesization of time-and site-resolved gauge fields on the square lattice [11,33]. Moreover, with the driving-dissipation mechanism being employed, various quantities of the ESMs can be measured by the pumping and the steady-state photon number (SSPN) detection of only few sites on the lattice [32]. In particular, the integer topological winding numbers of the ESMs with arbitrary values can be directly probed through the realization of the adiabatic pumping process [4,27]. Such measurement is equivalent to the measurement of the Chern numbers of the bulk bands and thus clearly examine the topological nontriviality of the photons. Furthermore, our detailed discussions show unambiguously that our proposal is very robust against various potential imperfection sources in experimental realizations due to the topological nature of the ESMs, pinpointing the feasibility with current level of technology. Being flexible for the extension to more complicated lattice configurations and the incorporation of effective photon correlation, our scheme serves therefore as a promising and versatile platform for the future investigation of various photonic quantum Hall effects.

RESULTS
The lattice.-We start with a square lattice consisting of TLRs with four different lengths placed in an interlaced form, as shown in Fig. 1(a). At their ends, the TLRs are commonly grounded by SQUIDs with effective inductances much smaller than those of the TLRs [33][34][35][36]. Due to their very small inductances, the grounding SQUIDs impose lowvoltage shortcuts at the ends of the TLRs. Therefore the lowest eigenmodes of the lattice can be approximated by the λ/2 modes of the TLRs with their ends being the nodes, and the whole lattice can be described by with a † r /a r being the creation/annilhilation operators of the rth photonic mode and ω r being the eigenfrequency. We further specify the eigenfrequencies of the four kinds of TLRs as yellow-ω 0 , blue-ω 0 + ∆, green-ω 0 + 3∆, and red-ω 0 + 4∆, respectively, with ω 0 /2π ∈ [10,20] GHz and ∆/2π ∈ [1, 3/2] GHz. Such configuration is for the following application of the dynamic modulation method and can be achieved through the length selection of the TLRs in the millimeter range [30][31][32]35].
We then consider how to implement on the TLR lattice the effective tight-binding Hamil- in the rotating frame of H S . Here T is the uniform hopping amplitude, and is the r → r hopping phase manifesting the presence of a vector potential A(x) through Peierls substitution [3]. For each plaquette of the lattice, the summation of the hopping phases around its loop has the physical meaning i. e. the synthetic local magnetic field for the microwave photons.
However, it is nontrivial to have complex hopping constants between TLRs because the physical coupling between two TLRs takes real coupling constants, no matter capacitive [14,15] or inductive [31,32]. We then consider the dynamic modulation method studied in recent experiments [30][31][32]. The grounding SQUIDs can be modeled as flux-tunable inductances and it is now experimentally possible to modulate the SQUIDs by a. c. magnetic flux oscillating at very high frequencies (the experiment-achieved range is typically 8 ∼ 10 GHz [29,36] which is much higher than the following-proposed 1 ∼ 6 GHz). Such a. c.
modulation introduces a small a. c. coupling can be created with in situ tunability. A further estimation demonstrates that the uniform hopping strength can be synthesized in the range |T /2π| ∈ [5,15] MHz [30,31,35].
For the investigation convenience of the ESM physics, in what follows we endow a nontrivial ring geometry to the TLR lattice, i. e. an N x × N y square lattice with an n x × n y vacancy at its middle, as shown in Fig. 1(b). Through the careful setting of the hopping phases, we penetrate a uniform effective magnetic flux φ in each plaquette of the lattice and an extra α at the central vacancy. In Fig. 2(a) the energy spectrum of a finite lattice is calculated with N x = N y = 24 and n x = n y = 6. In the rational situation φ/2π = p/q with p, q being co-prime integers, the unit-cell of the lattice is enlarged by q times, leading to q nearly flat magnetic bands and the fractal Hofstadter butterfly spectrum [37] (Fig. 2(a)). These q With the detailed modeling being discussed in Methods, we emphasize that the physics behind is that the exotic properties of the ESMs result in the novel steady states of the lattice, and the information of the ESMs can be extracted from the SSPNs of only few sites on the lattice versus the pumping frequency and the pumping sites.
Firstly let us consider the single-site driving of a particular site r p described by with P S being the pumping strength and Ω SP being the detuning in the rotating frame of H S . The SSPN on the pumping site n rp SP = a † rp a rp in the situation p/q = 1/4 and α = 0 is numerically simulated based on equation (16) and plotted in Fig. 3. In what follows we show that the spatial and spectral information of the ESMs can be distilled by measuring the dependence of the single-site SSPN a † rp a rp on Ω SP and r p . If we choose r p = r O = (1, 24) as an outer edge site (OES), significant n r O SP can be detected when Ω SP falls in the 1st and 3rd gaps, indicated by the highlighted spectrum comb in Fig. 3(a) (for the even q = 4, the 2nd gap is closed as a Dirac point form). This can be attributed to the excitation of the outer ESMs. However, if Ω SP is chosen deeply in the magnetic bands, n r O SP has bare value because in this situation H SP can only excite bulk state modes (BSMs) which spread over the whole lattice, i. e. the weight of r p in the mode function becomes diluted. The situation of pumping an inner edge site (IES) r p = r I = (9, 13) is similar, where the comb-like spectrum of n r I SP centralized in the band gaps can also be found in Fig. 3(b). Meanwhile, there are still several interesting differences.
As the number of the IESs is smaller than that of the OESs, Fig. 3 increase/decrease with increasing Ω SP . In contrast, when we set r p = r B = (5, 13) as a bulk site (BS), the lattice will have detectable n r B SP iff Ω SP falls in the magnetic bands. When we choose Ω SP in the band gaps, the lattice cannot be excited because no BSM spectrally populates in the band gaps and no ESM spatially populates in the bulk of the lattice (notice the marked window at the 1st and 3rd gaps in Fig. 3(c)).
The above illustration can be experimentally detected by the proposed measurement scheme sketched in Fig. 1(a): A particular pumping site r p is capacitively connected to an external coil with input/output ports for pumping/measurement. The steady state of the lattice can be prepared by injecting microwave pulses through the input port for a sufficiently long time. During the steady-state period, energy will leak out of the r p th TLR from the coupling capacitance, which is proportional ω rp a † rp a rp with the proportional constant being determined by the coupling capacitance. The target observable a † rp a rp can therefore be measured by simply integrating the energy flowing to the output port in a given time duration. Actually, this measurement scheme has already been used in a recent experiment in which both the amplitude and the phase of a coherent state of a TLR were measured [32]. Here we emphasize that what we want to measure is the expectation value a † rp a rp , while the detailed probability of the multi-mode coherent steady state projected to the Fock basis is nevertheless not needed. It is this weak requirement that greatly simplify our measurement.
and investigate the summed SSPN n MP = m j=1 a † r j a r j on the m pumping sites. Here P M is the homogeneous pumping strength, k P is the phase gradient of the pumping between neighboring sites, Ω MP is the frequency detuning in the rotating frame, and r j for j = 1, 2 . . . m denotes the jth of the m pumping sites.
Suppose Ω MP matches the eigenfrequency of a particular outer ESM, there arises an interesting question that, how does n MP depend on k P ? For a photon in that ESM, we can imagine its propagation around the edge with its ESM momentum k 0 (this can be verified by the discussion of the coherent dynamics in Discussion). Therefore, the reduced ESM mode function on the m pumping sites can be represented by a vector where the e ik 0 factor denotes the phase delay between two consecutive sites and the equalweighting character of k 0 reflects the uniform spatial distribution of the ESM on the confined edge (see Figs. 3(d) and 3(e)). It is this form of k 0 that inspires the inhomogenous multi-site pumping H MP , which can be represented by another vector Based on the above physical picture, we can conjecture that the maximum of n MP will emerge at the point where the excitations from the m pumping sites constructively interfere with each other.
The dependence of n MP on k P and m is plotted in Fig. 4, where the positions of the peaks infer the value of k 0 . In addition, the full width of half maximum (FWHM) of the peaks decreases with the increase of m. This can be understood by considering the two extreme cases: If m = 1, there is certainly no peak because the steady state is independent on k P .
Meanwhile, when all the OESs participate in the inhomogeneous pumping and the lattice size grows up, the inner product |k † P k 0 | 2 describing the interference between the pumping sites becomes nonzero iff k P = k 0 . In this situation the peaks in Fig. 4 approach a δ-like function. As implied in Fig. 4, for a moderate m = 5 the FWHM is already sharp enough to discriminate the ESM momentums with a satisfactory resolution.
Probing the ESMs: The integer topological invariants.-We further consider the measurement of the integer topological quantum numbers of the system. The topological property of a electronic Bloch band is captured by the quantized Hall conductance which turns out to be its Chern number [3]. This transport measurement is nevertheless inaccessible in circuit QED systems due to the absence of Fermi statistics. Meanwhile, the presence of the ESMs provides an alternative way of probing the topological invariants according to the bulk-edge correspondence [4]. In the rational situation φ/2π = p/q, the eigenenergies of the where h is the gap index and t h , s h are integers. Especially, t h is the topological winding number of the hth gap which is related to the Chern number C h of the hth band as Measuring the winding numbers of the ESMs is thus equivalent to measuring the Chern numbers of the magnetic bands. For q = 4 we have C 1 = C 3 = 1 and t 1 = −t 3 = 1, while for q = 5 we have C 3 = −4 and C j = 1 for the other four bands, and t 1 = −t 4 = 1,t 2 = −t 3 = 2 (see Figs. 2(b) and 2(c)).
As the spatial configuration of the proposed lattice is equivalent to the Laughlin cylinder

Robustness against imperfection factors.-The imperfection in realistic experiments
accompanies inescapably with the ideal scheme proposed above, including the diagonal and off-diagonal disorders ω r → ω r + δω r , T → T + δT r r added into H T by the fabrication errors of the circuit and the low-frequency 1/f background noises [40], and the non-nearestneighbor hoppings induced the residual long-range coupling between TLRs which are not presented in the proposed H T . Understanding these effects is thus crucial for our scheme.
As briefly summarized below and detailedly studied in ref.
[35], these imperfection factors lead to unwanted terms which are all much smaller than the band gaps of the lattice (∼ T ). In actual experimental circuits, the 1/f noise at low frequencies far exceeds the thermodynamic noise [40].
where the detailed evaluation can be found in ref. alternative way to the study of photonic graphene [44]. In addition, while the cross-talk between the diagonal next-nearest-neighbor TLRs is suppressed in our scheme due to frequency mismatch, it can indeed be opened by adding another tone to the modulating pulses of the SQUIDs. This may offer potential facilities in the future study of anomalous quantum Hall effect in the checkerboard lattice configuration [25].
In recent research, lattice configurations supporting a dispersionless flat band have been investigated extensively, including the Lieb and the Kagomé lattices [25]. These band structures provide an idea platform of achieving strongly-correlated phases as the kinetic energy is quenched [45,46]. These lattice configurations can also be synthesized by the variation of the proposed square lattice. While the photonic topological insulator considered in this paper can be understood in the single-particle picture, the introduction of interaction significantly complicates the problem and may lead to much richer but less explored physics. On the other hand, with the demonstrated strong coupling between TLRs and superconducting qubits [7,8] (and also atomic system, see ref. [47]), the Bose-Hubbard [16,18] and Jaynes-Cummings-Hubbard nonlinearities [17,19] can be incorporated into the proposed lattice.
A further research direction should therefore be the implementation of photonic fractional Chern insulators and the understanding of strong correlation in the proposed architecture and its potential hybrid-system generalizations which may utilize the advantages of different physical systems [47,48].
where ρ is the density matrix of the lattice, κ r is the decay rate of the rth TLR, and the matrix B is defined by a † Ba = H T . As a linear system (i. e. there is no photon-photon interaction), the lattice can be described in the picture of multi-mode coherent state, and its steady state can thus be determined by where K is the diagonal matrix of the TLRs' decay rates. From equation (16) our idea emerges that, Ω can be used to select the mode we are interested in, and the information of that mode can be extracted from the dependence of a on P. In addition, as the pumping sites are coupled to external coil, they suffer more severe decoherence than the other conventional sites. Therefore, we set the decay rates of the pumping sites as uniformly κ rp /2π = 2 MHz and those of the conventional sites as uniformly κ r /2π = 100 kHz throughout the numerical simulation of this paper, i. e. there is a 20 times difference between them [30][31][32].