Topological phases of quantized light

Abstract Topological photonics is an emerging research area that focuses on the topological states of classical light. Here we reveal the topological phases that are intrinsic to the quantum nature of light, i.e. solely related to the quantized Fock states and the inhomogeneous coupling strengths between them. The Hamiltonian of two cavities coupled with a two-level atom is an intrinsic one-dimensional Su-Schriefer-Heeger model of Fock states. By adding another cavity, the Fock-state lattice is extended to two dimensions with a honeycomb structure, where the strain due to the inhomogeneous coupling strengths of the annihilation operator induces a Lifshitz topological phase transition between a semimetal and three band insulators within the lattice. In the semimetallic phase, the strain is equivalent to a pseudomagnetic field, which results in the quantization of the Landau levels and the valley Hall effect. We further construct an inhomogeneous Fock-state Haldane model where the topological phases can be characterized by the topological markers. With d cavities being coupled to the atom, the lattice is extended to d − 1 dimensions without an upper limit. In this study we demonstrate a fundamental distinction between the topological phases in quantum and classical optics and provide a novel platform for studying topological physics in dimensions higher than three.

By using the Stirling's formula for N 1 and 1 n 1 N , we obtain the condition of the distribution maxima in the lattice, ∂ ∂n ln | ↓, n 1 , n 2 |ψ s | 2 ∝ ln u 2 2 u 2 1 n 2 n 1 = 0, which results in i.e., the state is centered at the point where the two neighboring coupling strengths are equal, and the photon number in a 1 mode is n 1 = u 2 2 N .

The eigen wavefunction in the zeroth Landau level
Here we compare the wavefunction in the zeroth Landau level near K point |ψ 0,−N (see Fig. 3 (d)) with that in the Landau level of a real magnetic field in the symmetric gauge, ψ 0LL (r) ∝ exp(−r 2 /4l 2 B ) [1]. We will show that |ψ 0,−N is more localized in the Fock-state lattice than ψ 0LL (r) due to the inhomogeneity of the magnetic field (see Fig. 3 (b) and Eq. (15)). The probability distribution of |ψ 0,−N is which has rotational symmetry. Therefore, we only need to consider the distribution along a radial direction from the center of the lattice to vertex 1 in Fig. 3 (a), e.g., in states |p ≡ |↓, N/3 − 2p, N/3 + p, N/3 + p where −N/3 p N/6 being an integer. By using the Stirling's formula for N 1 and p N , we obtain where the radius r is related to number p through r = 3pq/2. Comparing with |ψ 0LL (r)| 2 ∝ exp(−r 2 /2l 2 B ), |ψ 0,−N has a smaller variance.
It has been shown that in graphene the wavefunctions at K and K points have phase winding of 0, 2π/3, −2π/3 for three neighboring lattice sites in the same sublattices [2]. We show that this phase winding also exists in |ψ 0,±N (see Fig. S1).

Bosonic chirality operator
The spin chirality operator is defined for three spins, Figure S1 The phase distribution of the wavefunctions at the K and K points for a FSL with N = 20. The phase winding of the state |ψ 0,N on the K point is opposite to the one of |ψ 0,−N at the K point. tor of the jth spin. To generalize the chirality operator to bosons, we need to notice that the spin chirality operator breaks both the parity P and time-reversal T symmetry, but conserves the PT symmetry [3]. Another key feature of the spin chirality operator is that its evolution operator chirally rotates the spin states, |s 1 s 2 s 3 → |s 2 s 3 s 1 → |s 3 s 1 s 2 [4]. The chirality operator defined in Eq. (9) has these two properties. It is easy to verify that PCP −1 = −C, T CT −1 = −C and PT CT −1 P −1 = C. The evolution operator of C rotates photons chirally among the three modes a 1 → a 2 → a 3 [5].

The valley Hall response
In the limit of a small effective electric field, δ g, the evolution of an initial state in the zeroth Landau level is confined in that level. We can study the evolution by projecting the Hamiltonian H 3 in Eq. (11) to the subspace of the zeroth Landau level, where P 0 = C |ψ 0,C ψ 0,C | is the projection operator in the zeroth Landau level. The Heisenberg equations of the operators are We obtain the state evolution The evolution of the state |↓, 0, 0, N b at K point is determined by (S10) where |c n,N −n (τ )| ∝ |(cos δ √ 3 τ ) n (sin δ √ 3 τ ) N −n | can be obtained through Eq. (S9) and the distribution is shown in Fig. 4 (a) The x and y coordinates in the Fock-state lattice are (S12) Since x commutes with H eff , it does not change with time. This is a signature of the Hall response considering that the force is along the x direction. Using Eq. (S12) and the inverse relation of Eq. (S9) and considering the initial state |↓, 0, 0, N b , we obtain the evolution in the y direction, T . (S13) The Landau-Zener tunneling and the wavefunctions beyond the zeroth Landau level The Landau-Zener tunneling appears when the potential difference between neighboring lattice sites δ is comparable with the bandgap g, as shown in Fig. S2. While oscillating between the K and K points, the state tunnels to other Landau levels. In the FSL, the Landau-Zener tunneling add new features such as the splitting of the wavepackets into multiple components, demonstrating interference between states in different Landau levels.
The eigenstates in the zeroth Landau level of the 2D FSL is a two-dimensional extension of the topological zero-energy states in the 1D FSL. In particular, the weight of a wavepacket centered at the state |ψ 0,0 locates on the incircle, as shown by Fig. 5. Therefore, we can use |ψ 0,0 to set the boundary between the eigenstates in the strained semimetal and the band insulator. In Fig. S3, the eigenstates with variations r 2 smaller than that of |ψ 0,0 are plotted red, otherwise blue, which can be regarded as in two different topological phases. Several typical eigen wavefunctions are also plotted. The one in the blue area occupies more sites outside of the incircle.

Topological marker
We calculate the local topological marker in the filled hexagon in Fig. S4 according to ref. [6], where |s are the Fock states in the unit cell colored in Fig. S4, with tan θ = κC/2g √ m, x and y are position operators in Eq. (S12). The topological marker M is plotted in Fig. 6 for the phase diagram of the FSL Haldane model. Figure S3 The eigenstates in (red) and out of (blue) the incircle. Four typical wavefunctions in the first (diamond), third (pentagon), fourth (hexagon) and sixth (star) Landau levels are plotted in the FSL. Figure S4 The Fock-state lattice (with N = 20) where the topological marker is calculated in the colored unit cell with the six vertices filled with black color.