Quantum simulation in Fock-state lattices

ABSTRACT

Fock-state lattices (FSLs) consist of the Fock states of photons and atoms, establishing a quantum photonic platform for simulating condensed matter physics. Remarkably, various topological phenomena, such as topological edge states, strain-induced Landau levels, the valley Hall effect, and the quantum anomalous Hall effect, are intricately linked to the quantum properties of light. Recent advancements in state-of-the-art superconducting circuits have enabled the observation of these topological quantum photonic phenomena. With scalable dimensions and flexible structure engineering, FSLs offer a novel tool for investigating high-dimensional topological physics and devising innovative devices for quantum information processing. This review delves into the latest theoretical and experimental developments in this emerging field, situated at the intersection of quantum optics, topological physics, and quantum information.

Keywords:

• Quantum simulation
• fock-state lattice
• jaynes-Cummings model
• topological physics
• landau level

1. Introduction

Topology, a longstanding subject in mathematics, has also emerged as a vital player in modern physics. The revelation of the quantum Hall effect [1,2] marked a turning point, propelling topological physics into a prominent position that spans diverse fields from condensed matter physics [3–6] and photonics [7,8] to cold atoms [9,10] and quantum computing [11]. Its potential applications extend to the development of robust electronic devices impervious to noise [12] and fault tolerant topological quantum computing [13]. The ubiquity of topology stems from its exploration of invariant properties during the continuous deformation of objects. In condensed matter physics, the object under scrutiny is the Brillouin zone of lattice electrons, while the invariant property is the integral of the Berry curvature of an energy band. Given that the Brillouin zone and energy band find their origins in the wave properties of electrons within periodic structures, these concepts seamlessly translate to photonic crystals due to the mathematical parallels between the Schrödinger equation and Maxwell equations. This mapping has fostered a reciprocal exchange of inspiration between photonic and electronic topological physics.

The high tunability, easy fabrication, and measurement of the electromagnetic systems enable the realization of many long-sought theoretical models of topological physics [14,15], which are difficult to be experimentally explored in electronic materials. This research direction of classical optical simulation of topological physics is coined the name topological photonics [7,8]. Beyond electronic analogue, featured topological phenomena such as the unidirectional chiral edge modes have boosted the exploration of functional optical devices including topological lasers [16,17] and nonreciprocal optical elements [18,19]. Historically speaking, the research field of topological photonics was motivated by the optical simulation of the Haldane model [20], which reveals that the quantum Hall effect is rooted in the band topology rather than the Landau levels. This important model lays the foundation for topological insulators. However, the experimental realization of this model in electronic systems has been challenging [21–25]. Therefore, Haldane and coworkers suggested to simulate such a model in optical systems [26], which have been realized in both microwave [27] and optical [14] frequencies. Since then, all classical degrees of freedom endorsed by the Maxwell equations, such as spatial modes [14], momenta [28], frequencies [29], polarizations [30] and orbital angular momenta [31], have been exploited to construct lattice sites and simulate topological physics, some of them in the so-called synthetic dimensions [32]. In addition to these classical degrees of freedom, the second quantization of light offers a quantum degree of freedom, namely the Fock space composed by the photon number states that contain an integer number of photons. It is natural to ask whether this Fock space embeds novel photonic topological states that cannot be explained by classical optics, and if yes, whether these quantum topological states can be used to design new devices for quantum information processing. Recent progress in answering these questions has yielded promising results. The major task of this review is to summarize these results and have an outlook for this new direction of topological quantum photonics.

2. Multi-mode JC model

The particle-wave duality of light is reconciled by its second quantization. The classical wave property is determined by the Maxwell equations, from which and with proper boundary conditions, we can obtain the light mode characterized by the spatial mode function, frequency, momentum, polarization, and orbital angular momentum. The particle nature of light is demonstrated by the quantized energy of each mode, i.e. the eigenstates are Fock states $|\mathit{n}⟩$ with discrete energy levels, $(\mathit{n}+1/2)\mathit{h\nu }$, with h the Planck constant, n the photon number and $\mathit{\nu }$ the light frequency. Therefore, for each light mode, there is a new dimension provided by the Fock space. In this review, we introduce the quantum simulation based on lattices composed by these Fock states, i.e. the Fock-state lattices (FSLs) [33,34], in which quantum topological states that cannot be explained by classical optics have been discovered and expeimentally realized. Such simulation is made possible by the property of the bosonic annihilation operator, $\mathit{a}|\mathit{n}⟩=\sqrt{\mathit{n}}|\mathit{n}-1⟩$, i.e. the coupling strength between Fock states is determined by the photon number in the states [35–37]. With spin-boson coupled Hamiltonians, we can build infinite lattice structures of Fock states in arbitrary dimensions.

In this review, we constrain our discussion in the Jaynes-Cummings (JC) model [38] in which multiple cavity modes are strongly coupled to a single two-level atom. In a one-dimensional (1D) model of such lattices, the varying coupling strengths across the lattice result in two topological phases of the Su-Schrieffer-Heeger (SSH) model [39], and a topological edge state resides at the boundary of these two phases. This topological edge state is interestingly connected to the dark state in atom-photon coupled systems, which can be used in topological state transfer [34,40,41]. In two-dimensional (2D) FSLs, the spatial variation of the coupling strengths introduces a pseudo-magnetic field (PMF), similar to the effect of a strain field in two-dimensional materials [42,43], which results in pseudo-Landau levels. This lays the basis to investigate topological phenomena such as Lifshitz transition [44], valley Hall effect (VHE) [45,46] and the Haldane model [20]. In particular, FSLs provide a unique perspective to reveal the relation between seemingly unrelated topological physics, such as the equivalence between the VHE and the Haldane edge current [47]. In this example, the Berry phase of a spin and the Aharonov-Bohm (AB) phase of an electron in a magnetic field can also be unified. Moreover, the topology of FSLs also fosters new techniques in quantum information processing, such as novel approaches to generating NOON states [48], realizing quantum routers [49], and chiral quantum state manipulations in plasmonic nanorings [50].

The precursor of the full quantum FSL is the semiclassical superradiance lattice (SL) in which timed Dicke states are coupled by lasers in momentum space [51–60]. In SLs, the lasers are considered as classical fields with a constant field strength despite their exchange of energy with the atoms. Since the lattice structure is in momentum space, it is hard to introduce edges in the SLs. However, if we consider the light quantization, the coupling strength depends on the photon number, and reduces to zero when encountering the vacuum state, which is a natural edge due to the non-existence of negative photon number states. In order to harness such quantum property of light, we need to couple the atom to high finesse cavities in the strong coupling regime. The JC Hamiltonian of a two-level atom resonantly coupling to d bosonic modes is (we set $\mathit{\hslash }=1$),

(1) $\mathit{H}={\sigma }^{+}\sum _{\mathit{j}=1}^{\mathit{d}}{g}_j{a}_j+\text{H.c.},$(1)

where ${\sigma }^{+}$ is the raising operator for the two-level atom, g_j is the vacuum coupling strength, and a_j is the annihilation operator for the jth mode. Since the Hamiltonian commutes with the excitation number $\mathit{N}=\sum _j{a}_j^{†}{a}_j+{\sigma }^{+}{\sigma }^{-}$, the Hilbert space can be divided into subspaces with different N’s. In each subspace, the Fock states of the d bosonic modes and the atom form a $(\mathit{d}-1)$-dimensional tight-binding lattice, such as 1D SSH model (d = 2), 2D honeycomb lattice (d = 3), and three-dimensional (3D) diamond lattice (d = 4), as illustrated in Figure 1. The two sublattices corresponding to the two atomic states are drawn as balls and squares, respectively. The eigenstates of FSLs can be analytically obtained by introducing a new basis of bosonic modes $b_i=\sum _j{u}_{\mathit{ij}}{a}_j$. The transformed basis contains one bright mode $b_1=\sum _{\mathit{j}}{g}_j{a}_j/g_0$ with $g_0=\sqrt{\sum _j|g_j{|}^2}$ and d −1 dark modes. The Hamiltonian is thus simplified to $\mathit{H}=g_0{\sigma }^{+}{b}_1+\text{H.c.}$, which is a single-mode JC model. The eigenenergies are $E_m^{\pm }=\pm g_0\sqrt{\mathit{m}}$ and the eigenstates are $|{\psi }_m^{\pm }⟩=(|\downarrow ,\mathit{m},m_1,\cdots ,m_{\mathit{d}-1}{⟩}_b\pm |\uparrow ,\mathit{m}-1,m_1,\cdots ,m_{\mathit{d}-1}{⟩}_b)/\sqrt{2}$, where we have used the subscript in $|\cdots {⟩}_b$ to denote the b-mode basis. Therefore, the energy spectra are determined by the photon number m in the bright mode. For each eigenenergy $E_m^{\pm }$, there are eigenstates with N − m photons in the d −1 dark modes. The possible configurations to distrubte these photons in the dark modes determine the degeneracy of each eigenenergy level, which is the combinatorial number $C_{N-m+1}^{d-2}$. For d = 2 the energy levels are non-degenerate, but for $\mathit{d}\ge 3$, each energy level can have more than one degenerate eigenstates.

Figure 1. The FSLs of d-mode JC model [33]. (a) The FSL with d = 2 is a 1D SSH lattice (N = 5). (b) The FSL with d = 3 is a 2D honeycomb lattice (N = 5). (c) The FSL with d = 4 is a 3D diamond lattice (N = 3). The two sublattices are denoted by the balls and squares, and the numbers $n_1{n}_2\dots n_d$ labelling the sites are the photon numbers in the d modes. The coupling strengths are proportional to the widths of lines connecting neighboring sites. Here we set the vacuum coupling strengths of all d modes equal, i.e. $g_j=\mathit{g}$.

3. Topological transport in 1D Fock-state lattices

The two-mode JC model is equivalent to an SSH model with the Hamiltonian,

(2) $H_{\text{1D}}={\sigma }^{+}(g_1{a}_1+g_2{a}_2)+\text{H.c.}.$(2)

The equivalence is established through the photon number-dependent coupling strengths between Fock states, $t_{\mathit{i}}=g_i\sqrt{n_i}$, which vary across the FSL (see Figure 1(a)). The relative strength between the intra-cell (t₁) and inter-cell (t₂) coupling determines the topological phases. The boundary where $t_1=t_2$ divides the lattice in two different topological phases, characterized by $t_1<t_2$ and $t_1>t_2$. According to the bulk-edge correspondence, there is a topological edge state localized on the boundary, whose location is determined by the ratio between the two coupling strengths $g_1/g_2$ (Figure 2(a)).

Figure 2. The topological defect transfer in 1D FSLs. (a) The probability distribution of topological defect states (solid lines) localized at the topological edge (dashed lines). The vacuum coupling strengths are $g_1=g_0\text{sin}\mathit{\theta }$ and $g_2=g_0\text{cos}\mathit{\theta }$ [33]. (b) Transport fidelity as a function of the total transport duration T. The transfer can be accelerated in the non-adiabatic regime (marked by the empty circles) [40]. (c) Robustness of the topological transport with different noise strength $\mathit{\delta }$ [40]. (d) False-color circuit image of the experimental device [34]. The main elements of the circuit consist of the central qubit Q₀ (green line) and three resonators $R_1,R_2,R_3$ as twisting lines in cyan, red and yellow, respectively. The resonators and the qubit are coupled via couplers (two-circle elements in blue), which are used for tuning the coupling strengths. The purple lines indicate the ancilla qubit used for preparation and readout of the resonators. Inset: schematic of the main elements a central qubit Q₀ (green) coupled to three resonators R₁ (cyan), R₂ (red) and R₃ (yellow). (e) Experimental observation of the defect state transport for an initial state $|\downarrow ,0,5⟩$. The photons are transferred from R₂ to R₁ at around t = 600 ns [34].

The topological defect state can be analytically obtained by using the bright and dark modes, $b_1=(g_1{a}_1+g_2{a}_2)/g_0$ and $b_2=(g_2{a}_1-g_1{a}_2)/g_0$. In this new basis, the Hamiltonian is $H_{\text{1D}}=g_0{\sigma }^{+}{b}_1+\text{H.c.}$. We can obtain the eigenenergies $E_n^{\pm }=\pm g_0\sqrt{\mathit{n}}$ and the corresponding eigenstates $|{\psi }_n^{\pm }⟩=({|\downarrow ,\mathit{n},\mathit{N}-\mathit{n}⟩}_b\pm {|\uparrow ,\mathit{n}-1,\mathit{N}-\mathit{n}⟩}_b)/\sqrt{2}$. The topological zero-energy state is $|{\psi }_0⟩={|\downarrow ,0,\mathit{N}⟩}_b$, which can be rewritten as,

(3) $|{\psi }_0⟩=\sum _{n_1+n_2=\mathit{N}}\sqrt{\frac{\mathit{N}!}{n_1!n_2!}}{g}_2^{n_1}(-g_1{)}^{n_2}|\downarrow ,n_1,n_2⟩,$(3)

whose probability distribution is centered at the topological edge [33] (see Figure 2(a)).

Such a topological defect state can be used for topological pumping, i.e. to pump a quantum state from left to right, i.e. from the state $|\downarrow ,\mathit{N},0⟩$ to $|\downarrow ,0,\mathit{N}⟩$, in the lattice by adiabatically tuning the parameters $g_1(\mathit{t})/g_2(\mathit{t})$ from 0 to $\mathrm{\infty }$. During this process, we can maintain $g_1^2+g_2^2$ as a constant, such that there is a constant energy gap between the zero-energy state and other eigenstates. The energy gap guarantees that the adiabaticity can be achieved in a finite time during the state transfer. Interestingly, such a state transfer scheme has been proposed previously without noticing its topological nature [61]. Beyond the adabatic regime, it even allows for high fidelity non-adiabatic state transfer by taking advantage of the coherent oscillation between the zero-energy state and other states [40].

The performance of the transport can be evaluated with the fidelity $\mathit{F}=|⟨\downarrow ,0,\mathit{N}|\mathit{\psi }(\mathit{T})⟩{|}^2$, where $\mathit{T}$ is the time when the state is transferred to the right end of the lattice. Taking into account the Landau-Zener tunneling [40], we obtain

(4) $\mathit{F}=1-\frac{N{\pi }^2}{2(\mathit{gT}{)}^2}(1-\mathit{cos}(\mathit{gT})).$(4)

As shown in Figure 2(b), the coherent oscillation enables faster-than-adiabatic transport with high fidelity, which has advantageous performance in comparison to other schemes [62,63], especially in the long-chain limit [40].

Such 1D topological transport can also be simulated with photonic lattices of coupled waveguides [41], where the coupling strengths between waveguides vary according to the FSLs. While in FSLs the coupling strengths are accurately determined by the natural property of the bosonic annihilation operators, in photonic lattices inaccuracy exists in fabrication. However, the topological transport is robust against such perturbation in coupling strengths. In Figure 2(c), we numerically compute the average results of 100 sets of such random noise in the coupling strengths $t_i\to (1+\mathit{\eta })t_i$ where $\mathit{\eta }\in (-\mathit{\delta },\mathit{\delta })$. The topological zero-energy state can be transferred with satisfactory fidelity even for large noises.

The adiabatic topological quantum state transfer has been implemented in a superconducting circuit. The key elements of the sample contain a central qubit Q₀ and three resonators that are coupled to Q₀, as shown in Figure 2(d) [34,64]. The coupling strengths between the qubit and the three resonators are independently tunable. After initial state preparation in $|\downarrow ,0,5⟩$, the coupling strengths are sinusoidally modulated, $g_1=\mathit{g}|\mathit{cos}(2\mathit{\pi \nu t})|$ and $g_2=\mathit{g}|\mathit{sin}(2\mathit{\pi \nu t})|$. The experimental parameters satisfy the adiabatic condition, $\mathit{\nu }\ll \mathit{g}$. The topological edge states is accordingly transferred between the two edges (see Figure 2(e)).

4. 2D Fock-state lattice

The Hamiltonian of a three-mode JC model is

(5) $H_{\text{2D}}=\mathit{g}{\sigma }^{+}(a_1+a_2+a_3)+\text{H.c.}.$(5)

Such a Hamiltonian couples the Fock states with a fixed excitation number N in a honeycomb lattice with a triangular boundary, where vacuum states are met, as shown in Figure 1(b). To diagonalize the Hamiltonian, we construct a bright mode $b_0=(a_1+a_2+a_3)/\sqrt{3}$ and two dark modes $b_{\pm }=\sum _{\mathit{k}=1}^3{a}_k\mathit{exp}(\mp 2\mathit{i}\mathit{k\pi }/3)/\sqrt{3}$. The transformed Hamiltonian $\mathit{H}=\sqrt{3}\mathit{g}{\sigma }^{+}{b}_0+\text{H.c.}$ can be easily diagonalized with eigenenergies $E_n^{\pm }=\pm \mathit{g}\sqrt{3\mathit{m}}$, where $\mathit{m}=0,1,\cdots ,\mathit{N}$ are the photon numbers in the bright mode. The associated eigenstates are $|{\psi }_{m,C}^{\pm }⟩=({|\downarrow ,m,m_{+},m_{-}⟩}_{\mathit{b}}\pm {|\uparrow ,m-1,m_{+},m_{-}⟩}_{\mathit{b}})/\sqrt{2}$ for m ≠ 0 and $|{\psi }_{0,\mathit{C}}⟩=|\mathit{g},0,m_{+},m_{-}{⟩}_b$. The mth level has $\mathit{N}+1-\mathit{m}$ degenerate states, each of which can be characterized by the chirality operator,

(6) $\mathit{C}=b_{+}^{†}{b}_{+}-b_{-}^{†}{b}_{-},$(6)

which is the counterpart of the lattice momentum of electrons in solids.

In the FSLs, the varying coupling strengths $t_i=\mathit{g}\sqrt{n_i}$ effectively introduce a strain field, which drastically modifies the energy spectrum of the FSL. In particular, at the incircle of the triangular boundary, a semimetal-insulator Lifshitz topological transition [44] is induced by such a strain field. Outside of this incircle, the strain is large enough to create a band gap. Within the incircle, the strain shifts the Dirac points, which has the effect of a magnetic field, and thus induces discrete degenerate energy levels similar to the Landau levels. Different from a real magnetic field, the strain-induced magnetic field preserves the time-reversal symmetry, i.e. has opposite signs at the two valleys, and hence is called a pseudo-magnetic field. In particular, the zero-energy states are confined within the incircle. In the following, we review the investigation on pseudo-Landau levels, valley Hall effect and the Haldane model based on these zero-energy states.

4.1. Pseudo-magnetic field

Within the incircle of the 2D FSL, the coupling strengths induced by the three cavity modes satisfy $|t_1-t_2|<t_3<|t_1+t_2|$, so that the Dirac points are shifted from their original positions. The Hamiltonian in the two valleys (K and K^ʹ points) is thus,

(7) $\mathit{H}=\mathit{\xi }{v}_F(\mathbf{r})[\mathbf{p}-\mathit{e}{\mathbf{A}}^{\mathit{\xi }}(\mathbf{r})]\cdot \overrightarrow{\sigma },$(7)

where $v_F(\mathbf{r})$ is the Fermi velocity, $\overrightarrow{\sigma }$ is the Pauli matrix vector for the two sublattices, $\mathit{e}$ is the fictitious electric charge, ${\mathbf{A}}^{\xi }(\mathbf{r})$ is the strain-induced vector potential and $\mathit{\xi }=+$ and − for K and K^ʹ points. We set $\mathbf{r}$ as the coordinate in the lattice, with the lattice center being defined as the coordinate origin, where the Fermi velocity $v_F(0)=\mathit{gu}\sqrt{3\mathit{N}}/2$ and $\mathit{u}$ is the lattice constant. The PMF is obtained from the vector potential,

(8) $B^{\xi }(\mathbf{r})=\mathrm{\nabla }\times {\mathbf{A}}^{\xi }(\mathbf{r})=-\mathit{\xi }\frac{B_0}{\sqrt{1-r^2/R^2}},$(8)

where $B_0=2/\mathit{Ne}{u}^2$, $\mathit{R}\equiv \mathit{Nu}/2$ is the radius of the incircle. The PMF is approximately uniform near the center of the lattice, while it tends to diverge approaching to the incircle, where the two Dirac points merge due to the large strain. ξ labels the opposite signs of the strain-induced PMF in the two valleys.

4.2. Pseudo-Landau levels and JC splittings

It has been noticed that the Hamiltonians of graphene electrons near the Dirac points in a magnetic field have a similar form to the JC model, i.e. the Landau levels have energies proportional to $\sqrt{\mathit{n}}$ [65,66]. However, the physics behind such similarity was unclear. With the help of the three-mode JC model, we can establish this connection. The PMF induced Landau levels can be obtained from the cyclotron frequency of Dirac electrons, ${\omega }_c=v_F(0)\sqrt{2\mathit{e}{B}_0}$ [65]. Thus, the energies of the Landau levels are

(9) $E_n^{\pm }=\pm {\omega }_c\sqrt{\mathit{n}}=\pm \mathit{g}\sqrt{3\mathit{n}},$(9)

which agrees with the result obtained by directly diagonalizing the three-mode JC model.

In the experiment, such quantized pseudo-Landau levels can be measured from the spectra of the Rabi oscillations of the central qubit. After preparing the initial state $|\downarrow ,0,5,0⟩$ and then tuning the three resonators in resonance with the qubit, the population of the $|\uparrow ⟩$ state of the qubit is measured as a function of the evolution time (see the upper panel in Figure 3(b)). The fast Fourier transform of the evolution curve shows discrete peaks corresponding to the quantized pseudo-Landau levels (see the lower panel). The energies of the peaks are approximately $E_n=\sqrt{3\mathit{n}}\mathit{g}$, consistent with the theoretical prediction.

Figure 3. The PMFs, pseudo-Landau levels and geometric phases. (a) The distribution of the strain-induced PMF, $\mathit{B}(\mathbf{r})/B_0^{+}$. The PMF is induced by the inhomogeneous hopping strengths inside the incircle of the lattice. The Dirac points are shifted as if being coupled to a vector potential [33]. (b) Experimental observation of the pseudo-Landau levels. The upper panel shows the evolution of the excited state population. Green dots and black line represent the experimental data and numerical simulation. The frequency components are shown in the bottom left panel by performing the fast Fourier transform of the Rabi oscillation, which agrees with the theoretically predicted $\sqrt{\mathit{n}}$-energy spectra in the bottom right panel [34]. (c) The equivalence between the Berry phase of the pseudo-spin and the AB phase induced by the PMF in lattice [47].

4.3. Spin Berry phase and AB phase

In the original paper of Berry’s [67], the geometric phases of a spin accumulated in adiabatic rotation and the AB phases of electrons were shown as independent examples for geometric phases. Here in FSLs these two seemingly unrelated phases (except for their geometric nature) can be mapped to each other. By transforming the position operators to b-mode representation and dropping the terms involving $b_0$ and $b_0^{†}$, we obtain the intraband position operators,

(10) $\begin{array}{ll}\mathit{X}& =\frac{\mathit{iu}}{2}(b_{-}^{†}{b}_{+}-b_{+}^{†}{b}_{-}),\end{array}$(10)

(11) $\begin{array}{ll}\mathit{Y}& =\frac{u}{2}(b_{-}^{†}{b}_{+}+b_{+}^{†}{b}_{-}).\end{array}$(11)

These two operators are the coordinates of the guiding center of the states in the zeroth Landau level. The AB phases can be obtained from the effective magnetic flux enclosed by the trace of the guiding center.

On the other hand, the guiding center coordinates are related to the spin operators by $\mathit{X}=\mathit{u}{\mathit{J}}_y$, $\mathit{Y}=\mathit{u}{\mathit{J}}_x$, and $\mathit{C}=J_z/2$, where the commutation relation obeys $[J_{\mu },J_{\nu }]=\mathit{i}{ϵ}_{\mathit{\mu \nu \kappa }}{J}_k$ with ${ϵ}_{\mathit{\mu \nu \kappa }}$ denoting the Levi-Civita symbol. Each eigenstate in the Landau level can be viewed as an eigenstate of a spin-$\mathit{N}/2$,

(12) $|{\psi }_{0,\mathit{C}}⟩=|\mathit{j}=\mathit{N}/2,\mathit{m}=\mathit{C}/2⟩.$(12)

Such a relation also allows us to represent the spin state as a superposition of eigenstates in the zeroth Landau levels. The relation between the pseudo-spin coherent state $|\mathit{\theta },\mathit{\varphi }⟩$ [68,69] and the wavefunction in the zeroth Landau level is

(13) $|\mathit{\theta },\mathit{\varphi }⟩=\sum _k\sqrt{\frac{\mathit{N}!}{\mathit{N}!(\mathit{N}-\mathit{k})!}}{e}^{\mathit{ik\varphi }}{\cos }^{\mathit{N}-\mathit{k}}\frac{\theta }{2}{\sin }^{\mathit{k}}\frac{\theta }{2}|{\psi }_{0,\mathit{N}-2\mathit{k}}⟩,$(13)

where $\mathit{\theta }$ and $\mathit{\varphi }$ are the polar and azimuthal angles. The guiding center position is $⟨\mathbf{r}⟩=\mathit{Rsin}\mathit{\theta sin}\mathit{\varphi }\hat{x}+\mathit{Rsin}\mathit{\theta cos}\mathit{\varphi }\hat{y}$ with $\hat{x}$ and $\hat{y}$ being the unit vectors. The relation between the spin coherent state and the guiding center can be viewed in Figure 3(c), i.e. $⟨\mathbf{r}⟩$ is the projected coordinate of $|\mathit{\theta },\mathit{\varphi }⟩$ on the spin Bloch sphere, whose equator overlaps with the incircle of the FSL. If the adiabatic motion of the spin coherent state $|\mathit{\theta },\mathit{\varphi }⟩$ encircles an infinitesimal loop that subtends a solid angle $\mathit{d}\mathrm{\Omega }$, the accumulated Berry phase is

(14) $\mathit{d\gamma }=-(\mathit{N}/2)\mathit{d}\mathrm{\Omega }.$(14)

The guiding center in the FSL correspondingly accumulates an AB phase,

(15) $\mathit{d\gamma }=(\mathit{e}/\mathit{\hslash })B^{\xi }\mathit{dS},$(15)

where the infinitesimal area is related to the solid angle by $\mathit{dS}=\mathit{\xi }{R}^2\mathit{cos}\mathit{\theta d}\Omega$ with ξ taking value + and − for the upper and lower halves of the Bloch sphere. Such an equivalence between the two geometric phases is consistent with the PMF strength in Equation 8(8) $B^{\xi }(\mathbf{r})=\mathrm{\nabla }\times {\mathbf{A}}^{\xi }(\mathbf{r})=-\mathit{\xi }\frac{B_0}{\sqrt{1-r^2/R^2}},$(8) , which is derived from the strain field [47].

5. The valley Hall effect

In an electric field, the electrons in a perfect solid undergo Bloch oscillations, which have been simulated in cold atoms [70,71] and photonic lattices [72–74]. Such Bloch oscillations are attributed to the periodic band structures in momentum space. A static electric field induces a linear increase of lattice momentum, which goes back to itself after a Bloch period. Here, in the FSLs of the three-mode JC model, the PMF group eigenstates in discrete pseudo-Landau levels. An effective electric field (i.e. a linear potential in the FSL) also induces periodic oscillations of the states within the Landau levels [33], if the effective field strength is small enough such that the intraband transition is inhibited. In the configuration of the FSL, the wavepacket drifts perpendicular to the PMF and the effective electric field, which is a signature of the Hall effect. In particular, the PMF has opposite signs in the two valleys, where the wavepackets drift in opposite directions, which is known as the VHE [45,46].

The valley Hall response due to the PMF can be investigated by introducing a linear potential, which has the effect of a static electric field to an electron. Here, we constrain our discussion to the perturbative potential that does not induce interband transitions between pseudo-Landau levels. The Hamiltonian including the potential due to an effective electric field is,

(16) $H_{\text{v}}=H_{\text{2D}}-\mathit{e}\mathbf{E}\cdot \mathbf{r}.$(16)

Such effective electric fields can be introduced by detuning the frequencies of the resonators. For example, we can detune the resonators R₁ and R₂ to introduce a linear potential $\mathit{\delta }(a_1^{†}{a}_1-a_2^{†}{a}_2)$ along the $\hat{x}$-direction. The detuning is related to the electric field strength by $\mathit{\delta }=(\sqrt{3}/2)\mathit{e}{E}_x\mathit{u}$. In the limit $\mathit{e}{E}_x\mathit{u}\ll \mathit{g}$, the evolution is confined within a Landau level. The dynamics is governed by the intraband terms and the potential becomes $H_{\text{v}}=-\mathit{e}{E}_x\mathit{u}{\mathit{J}}_y$. The Heisenberg equation of motion is $\dot{Y}=-\mathit{i}[\mathit{u}{\mathit{J}}_x,H_{\text{v}}]=-\mathit{e}{E}_x{u}^2{\mathit{J}}_z$. We can obtain the expectation value of $\mathit{Y}$,

(17) $⟨\mathit{Y}(\mathit{t})⟩=\mathit{Rsin}(\mathit{ue}{E}_x\mathit{t}),$(17)

for the initial state $|{\psi }_{0,-\mathit{N}}⟩$ at K^ʹ point. We also obtain $⟨\mathit{X}⟩=0$, i.e. the wavepacket drifts in the direction perpendicular to the applied electric field. At the K point, the wavepacket drifts in the opposite direction, which is a signature of the VHE.

The VHE in the FSLs has been demonstrated in a superconducting circuit. The initial state was prepared by adiabatically transferring the 1D topological zero-energy state to the incircle at the lower edge of the lattice (see Figure 4(a)). By detuning two of the resonators to introduce an horizontal electric field, the wavepacket was set in motion in the vertical direction. The wavepacket oscillated within the Lifshitz topological edge, i.e. the incircle of the lattice boundary. At the two valleys, the Hall responses were in opposite directions.

Figure 4. The valley Hall effect in 2D FSLs [34]. (a) The evolution of the wavefucntion distribution in the FSL. The directions of the trajectories are perpendicular to the effective electric fields (arrow) and opposite on two valleys, due to the opposite signs of the PMF. The size of the dots indicates the population of the states. (b) The evolution of the averaged photon numbers in resonators for the coherent initial state $|\downarrow ,\mathit{\alpha },0,-\mathit{\alpha }⟩$ ($\mathit{\alpha }\approx 1.8$). Photons are exchanged between resonators without exciting the qubit. (c) The measured Wigner functions of the three resonator states at time t = 100 and 290 ns, showing opposite chirality of the states in the two valleys.

Remarkably, the VHE can also be demonstrated with coherent fields, which are considered as classical. Treating the light field quantum mechanically, such a state can be decomposed in subspaces with different total excitation numbers. In each subspace, the wavefunction undergoes synchronized oscillation governed by Equation (17)(17) $⟨\mathit{Y}(\mathit{t})⟩=\mathit{Rsin}(\mathit{ue}{E}_x\mathit{t}),$(17) . Therefore, a coherent state as a superposition of them undergoes the same dynamics and remains to be a coherent state, which is a direct product of the coherent states in the three cavities. The energy detuning between the coherent fields induces an effective electric field for the VHE, during which the resonators exchange photons to maintain a total zero field strength without exciting the atom. However, from a semiclassical perspective, the detuning between the coherent fields induces modulated total field strengths and thus the atom can be excited. These two different predictions give another test ground for the quantum nature of light. The quantum prediction has been verified in the experiment [34]. The evolution of the average photon numbers follow similar curves as those in each subspace (see Figure 4(b)). The total field strength was maintained as zero during the evolution, with the atom staying in the ground state. The chirality of the coherent fields in the resonators can be characterized by measuring the Wigner functions of the resonators. The phase chiralities of the resonator fields are opposite in the K and K^ʹ valleys (see Figure 4(c)).

6. The Haldane model

The Haldane model [20] can be constructed in 2D FSLs by adding next-nearest-neighbor (NNN) hoppings, which can be induced by Floquet modulation of the resonator frequencies or the coupling strengths [33,48]. For instance, the Hamiltonian can be modulated as $\mathit{H}(\mathit{t})=\sum _{\mathit{j}=1}^3(\mathit{g}+g_d\mathit{cos}({\nu }_d\mathit{t}+{\varphi }_j))({\sigma }^{+}{a}_j+{\sigma }^{-}{a}_j^{†})$ with ${\varphi }_j=2\mathit{\pi j}/3$. We decompose the Hamiltonian in a summation of Fourier components, $\mathit{H}=H_0+\sum _{\mathit{j}=1}^{\mathrm{\infty }}(H_j{e}^{ij{\nu }_d\mathit{t}}+H_{-\mathit{j}}{e}^{-ij{\nu }_d\mathit{t}})$. According to the Floquet theory [75], the first order effective Hamiltonian is $\mathit{H}=H_0+\sum _{\mathit{j}=1}^{\mathrm{\infty }}[H_j,H_{-\mathit{j}}]/(\mathit{j}{\nu }_d)$, which is,

(18) $H_{\text{H}}=H_{\text{2D}}+(\frac{\sqrt{3}}{6}\mathit{i\kappa }{\sigma }_z\sum _{\mathit{j}=1}^3{a}_{j+1}^{†}{a}_j+\text{H.c.}).$(18)

The NNN term can be rewritten as $\mathit{\kappa }{\sigma }_z\mathit{C}/2$ with $\mathit{\kappa }=3g_d^2/(2{\nu }_d)$. The eigenenergies for n ≠ 0 are $E_{n,C}^{\pm }=\pm \sqrt{3\mathit{n}{g}^2+{\kappa }^2{C}^2/4}$, which form the upper and lower bands. It is particularly interesting to notice that the states in the original zeroth Landau level without the NNN terms become chiral edge states with eigenenergies $\mathit{E}=-\mathit{\kappa C}/2$, connecting the upper and lower bands, as shown in Figure 5(a). The chiral edge currents can be understood from the Heisenberg equation of the guiding center, $\dot{\mathbf{r}}=-\mathit{i}[\mathit{u}{\mathit{J}}_y\hat{x}+\mathit{u}{\mathit{J}}_x\hat{y},-\mathit{\kappa }{\mathit{J}}_z]=\mathit{u\kappa }(J_y\hat{y}-J_x\hat{x})$. With the initial state $|\mathit{\theta },\mathit{\varphi }⟩=|\mathit{\pi }/2,\mathit{\pi }⟩$ the solution is,

(19) $⟨\mathbf{r}(\mathit{t})⟩=-\mathit{Rsin}(\mathit{\kappa t})\hat{x}-\mathit{Rcos}(\mathit{\kappa t})\hat{y}.$(19)

Figure 5. The Haldane model in FSLs. (a) The spectrum of the FSL Haldane hamiltonian $H_{\text{H}}$. The color depicts the spin polarization $⟨{\sigma }_z⟩$. The edge modes connect the upper and lower bands [33]. (b) The evolution of the averaged photon number in resonators shows the chiral edge current, which is realized by periodically modulating the coupling strengths [34,48]. (c) The phase diagram of the generalized Haldane FSL with hamiltonian $H_{\text{H}}^{\prime }$. The color depicts the value of the topological marker near the lattice center [33].

The wavepacket rotates chirally on the Lifshitz edge (incircle), as shown in Figure 5(b). It is noted that the dynamics of the Haldane currents and the VHE are unified in the FSL [47]. In the experiment, the NNN terms are induced by periodically modulating the coupling strengths [34]. A coherent initial state $|\downarrow ,\mathit{\alpha },-\mathit{\alpha },0⟩$ prepared on the Lifshitz edge was observed in chiral rotation, as shown in Figure 5(b). The inward motion of the wavepacket to the lattice center was due to the decoherence of the qubit and resonators.

We can also introduce an energy offset between the two sublattices by detuning the atom from the resonance frequencies of the resonators. In general, the FSL Haldane model can be written as,

(20) $\begin{array}{l}{H}_{\text{H}}^{\prime }=\frac{1}{\sqrt{3}}{H}_{\text{2D}}+\frac{\mathit{N}\mathrm{\Delta }}{2}{\sigma }_z+(\frac{\sqrt{3}}{6}\mathit{\kappa }{e}^{i{\varphi }_h{\sigma }_z}\sum _{\mathit{j}=1}^3{a}_{j+1}^{†}{a}_j+\text{H.c.}),\end{array}$(20)

where ${\varphi }_h$ is the phase attached to the NNN hopping, $\mathrm{\Delta }$ is the detuning between the light modes and the atom. To characterize the topological phases of the FSL, which has no translational symmetry, we employ the local topological marker [76] near the center of the FSL and the corresponding phase diagram is shown in Figure 5(c), which agrees with the original Haldane model [20].

7. Quantum circulator

The NNN terms in the Haldane model can be regarded as a spin-orbit coupling Hamiltonian,

(21) $H_{\text{so}}=\mathit{i\kappa }{\sigma }_z\sum _{\mathit{j}=1}^3{a}_{j+1}^{†}{a}_j+\text{H.c.},$(21)

where the atom carries the spin degree of freedom and the photons carry the orbital degree of freedom in the FSL. Such a Hamiltonian allows us to design a quantum circulator, where the photons chirally rotate in the three cavities with chirality depending on the qubit states. Based on such a quantum circulator, we can prepare mesoscopical superposition states of photons [48,49]. With the spin–orbit interaction Hamiltonian $H_{\text{so}}$, the wavefunction rotates in opposite directions in the two sublattices, as shown in Figure 6(a). At a later time, the photons are totally transferred into different cavities depending on the qubit states. This scheme enables a quantum transistor by regarding one cavity as the input and the other two cavities as the output channels (see Figure 6(b)). The transmission of such a quantum transistor is controlled by the quantum state of the atom. It can be used as a building block for quantum computing and quantum information processing.

Figure 6. Generating NOON states through the quantum transistor. (a) The state evolution in the FSL with the spin-orbit coupling hamiltonian. The inset shows sublattice-dependent phases of the NNN hoppings. The blue (red) dots represent the probability amplitude on the $\uparrow$ ($\downarrow$) sublattice. $\mathit{T}$ is the time to complete photon exchange between cavities [48]. (b) Schematic configuration of the quantum transistor. The cavity 1 is the input and cavity 2 and 3 are the outputs. The output state is controlled by the quantum state of the qubit [49]. (c) The quantum circuit scheme for preparing NOON states with N = 10. Each line represents an inseparable state. n_j is the photon number of the jth cavity. The block ${\theta }_j$ denotes the $\mathit{\theta }$ Rabi rotation between the jth cavity and the qubit. $H_{\text{so}}$denotes the state evolution under the effective Haldane term for time $\mathit{T}$ [48].

We show the procedure of preparing NOON states in Figure 6(c). We first prepare an initial state $|{\psi }_0⟩=|\downarrow ,\mathit{N},0,0⟩$. We then couple the cavity 1 to the atom and after a $\mathit{\pi }/2$ Rabi rotation, we obtain $|{\psi }_1⟩=(1/\sqrt{2})(|\downarrow ,\mathit{N},0,0⟩-|\uparrow ,\mathit{N}-1,0,0⟩)$. The free evolution under $H_{\text{so}}$ leads to $|{\psi }_2⟩=(1/\sqrt{2})(|\downarrow ,0,0,\mathit{N}⟩-|\uparrow ,0,\mathit{N}-1,0⟩)$. At the end, we couple the qubit to cavity 2 for a π Rabi rotation, which results in the final state $|{\psi }_3⟩=(1/\sqrt{2})|\downarrow ⟩\otimes (|0,0,\mathit{N}⟩-|0,\mathit{N},0⟩)$. This protocol can be generalized to prepare Schrödinger cat states.

8. Summary and outlook

The strategy of quantum simulation in FSLs is to represent a many-body (although simple) atom-photon coupled Hamiltonian in the form of coupled Fock states, which are used to compare with the single-particle tight-binding Hamiltonian of electrons. The challenge in such a mapping lies in the absence of discrete translational symmetry in the FSLs, since the coupling strengths change locally across the lattices, which also have boundaries due to the existence of the vacuum state. We cannot directly use band theory to investigate the problems. However, as we have shown in this review, such inhomogeneity of the coupling strengths bring new opportunities to induce a topological edge or synthesize a PMF, which is a crucial ingredient for simulating topological physics. The robustness of topological physics against local perturbations, which guarantees that the site-varying coupling strengths do not destroy the topological properties, is also important for the validity of such an approach. The FSLs also provide a new perspective to solve other problems in cold atoms [77] and quantum optics [78].

Although we have only reviewed the multi-mode JC models, the configuration and dimension of the FSLs have infinite possibilities. We can add more qubits and cavity modes to extend the number of sublattices and dimensions of the lattice. Common lattice structures such as triangular, square, Lieb, kagome and dice lattices can be straightforwardly constructed [78]. For example, by adding another atom in the lattice with Hamiltonian $\mathit{H}=\mathit{g}\sum _{\mathit{j}=1}^3({\sigma }_1^{+}+{\sigma }_2^{+})a_j+\text{H.c.}$, a dice lattice is constructed as shown in Figure 7(a), where three sublattices are identified by the triplet states $|\downarrow \downarrow ⟩$, $\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)$, and $|\uparrow \uparrow ⟩$. The zeroth energy level of dice lattice is highly degenerate and the high levels are also flat as induced by the lattice strain (see Figure 7(b)). A triangular lattice can be constructed from honeycomb FSL in the large detuning limit, where two sublattices are effectively decoupled [78]. It can be also constructed with cavity arrays [79], where synthetic magnetic field and chiral transfer have been studied. A square lattice can be constructed from Schwinger bosons with Hamiltonian $\mathit{H}=g_1(a_1^{†}{a}_2+a_2^{†}{a}_1)+g_2(b_1^{†}{b}_2+b_2^{†}{b}_1)+\text{H.c.}$ in a subspace with fixed excitation numbers [77]. More FSLs can be designed (see Figure 7(c)) within the current capability of experiments, even beyond the existing condensed matter models, with richness both in dimensions and lattice configurations.

Figure 7. Extensions of the multi-mode JC model. (a) Fock-state lattice of JC models with three cavities being coupled to two atoms. The three types of sites correspond to $|\downarrow \downarrow ⟩$ (squares), $(|\downarrow \uparrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ (circles) and $|\uparrow \uparrow ⟩$ (diamonds). The total excitation number N = 4. (b) The energy spectrum of the FSL in (a). Compared to the FSL with only one atom, the zero-energy level has a much higher degeneracy, containing a flat band due to the dice lattice configuration. (c) Schematic design of an atom-resonator lattice with a high dimensional FSL for unexplored topological states.

The quantum topological states can be used to design fault-tolerant quantum algorithms for quantum computing. A promising direction is to perform quantum simulation for high-dimensional topological physics [80], such as the 4D quantum Hall effect [81,82]. Exotic phenomena can be found involving multi-boson dynamics [83–85]. Compared with the works in synthetic dimensions [14,29,60], where complex designs of the lattice structures are needed to simulate the topological models, in FSLs the topological phases naturally emerge from an intrinsic property of bosonic statistics without artificial intervention. This fundamental connection between cavity QED and topological models in condensed matter physics is indeed surprising. In traditional platforms of synthetic dimension, classical degrees of freedom of light have been used to introduce a collection of modes to form new dimensions, such as the modes with different frequencies in a single resonator. The internal spin or motional degrees of freedom of cold atoms, although being considered as quantum, play a similar role to introduce new modes. The FSLs featured by many-body quantum statistics exploit the quantum states in the infinite Hilbert space of a light mode, further exploring topological states involving quantum complexities.

The FSLs also bridge important physics and concepts that are seemingly unrelated, such as the relation between different Berry phases, the unification of the VHE and the Haldane model, the relation between the dark states in the atom-photon coupled systems and the topological zero-energy states. Such correspondence reveals deep connections between remote phenomena. The FSLs provide new insight for photonic engineering, such as the adiabatic transfer of topological edge states in waveguide lattices [41] and the all-band-flat lattices of microwave resonators [86]. The all-band-flat spectra have prominent applications in multi-mode transport [87], enhancement of nonlinear effect [88] and the study of Anderson transitions [89]. Future work may extend to non-Hermitian effect [90] by engineering the decay ratios of the cavities, and nonlinear Kerr effect [91].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grants No. 12325412, 11934011, 12374335), the National Key Research and Development Program of China (Grants No. 2019YFA0308100), Zhejiang Province Key Research and Development Program (Grant No. 2020C01019) and the Fundamental Research Funds for the Central Universities (Grant No. 2023QZJH13 and 226-2023-00037).