Coherent control of quantum topological states of light in Fock-state lattices

The quantum Hall effect [1] reveals new phases of matter classified by the topological invariants of energy bands [2]. For two-dimensional electrons in strong magnetic fields, the chiral edge states between Landau levels contribute to the quantized Hall conductivity, which is immune to local defects. This topological effect can also exist without Landau levels, such as in the Haldane model [3], which lays the basis for topological insulators [4]. The optical simulation of quantum Hall edge states [5] opens a new research area, topological photonics [6, 7, 8], which brings a wealth of applications in routing and generating electromagnetic waves, such as backscattering-free waveguides [9] and topological insulator lasers [10]. While quantum wave and fermionic statistics play a fundamental role in topological phases of electrons, the topological modes in photonic lattices are purely classical [8]. Intriguingly, novel topological states emerging from light quantization and bosonic statistics have been predicted beyond classical interpretation [11]. Recent development in circuit quantum electrodynamics (QED) [12] makes it possible to realize these intrinsic quantum topological states of light, which provide quantum degrees of freedom in engineering photonic topology [11, 13] and offer topological control knobs in bosonic quantum information processing [14, 15, 16, 17].

Instead of a lattice of resonators or waveguides in topological photonics, the topological states of quantized light are embedded in lattices of Fock states $\Pi_{i}|n_{i}\rangle$ with $n_{i}$ being the photon number in the $i$th mode. In the Fock-state lattice (FSL) a light mode provides a dimension [14, 11, 13], in contrast to a lattice site in photonic lattices. Therefore, FSLs have the advantage of building high dimensional lattices with only a few cavity modes. To sketch such dimensional scalability we use the famous Jaynes-Cummings (JC) model [18], which describes the interaction between a two-level atom with quantized light. However, here we use multiple quantized light modes to couple the atom. With two light modes the Fock states form one-dimensional (1D) lattices of the Su-Schrieffer-Heeger (SSH) model (see Fig. 1 (a)) [19]. By just adding another mode, we obtain two-dimensional (2D) strained honeycomb lattices (see Fig. 1 (b)) [20]. These lattices are featured by site-dependent coupling strengths, which originate from the property of the bosonic annihilation operator $a$,

For the vacuum state, $a|0\rangle=0$, which leads to natural edges of FSLs when the photon number in one of the cavities reduces to zero. More interestingly, FSLs also have topological edges which host zero-energy states, resulted from the competition between resonators in exchanging photons with the atom. Such a simple mechanism enables FSLs to realize several most important models in topological physics, in particular the seminal SSH and Haldane models, which have been the focus in various quantum platforms [21, 22, 23, 24, 25, 26, 27]. Here we demonstrate adiabatic transport of topological zero-energy states in 1D SSH FSLs, where Fock states are topologically transferred from one cavity to another while maintaining the quantumness in superposition states. In 2D FSLs we observe the valley Hall effect (VHE) [28] and the Haldane chiral edge current [29], which offer a topological route of engineering quantum states of multiple resonators.

Leveraging the advantageous integrability and tunability of the circuit QED platform [30, 31, 32, 33], we design and fabricate a superconducting circuit device to build and engineer the FSLs. The device contains a central gmon qubit [34] (Q₀) and three resonators (R_j with $j$ running from 1 to 3), all with tunable frequencies. Each resonator R_j is coupled to Q₀ through an inductive coupler (C_j), as illustrated in Fig. 2 (a). The coupling strengths, $g_{j}/2\pi$, can be continuously tuned from $-15$ MHz to 10 MHz by changing the magnetic flux in C_j. In addition, each resonator R_j is capacitively coupled to an ancilla qubit Q_j, which is used to prepare and readout the quantum state of the resonator, with a coupling strength of around $11$ MHz. Other characteristics of the resonators and qubits can be found in the Supplementary Information.

The Hamiltonian of the coupled system of R_j’s and Q₀ can be described by a multimode JC model [18] in the rotating-wave approximation,

where $a_{j}$ is the annihilation operator of R_j with the transition frequency $\omega_{j}$, $\sigma^{+}\equiv\left|\uparrow\right\rangle\left\langle\downarrow\right|$ and $\sigma^{-}\equiv\left|\downarrow\right\rangle\left\langle\uparrow\right|$ are the raising and lowering operators of Q₀ with the transition frequency $\omega_{0}$, and $d$ is the number of resonator modes. The Hamiltonian conserves the total excitation number $N=\sum_{j}n_{j}+(\sigma_{z}+1)/2$, where $n_{j}$ is the photon number of R_j and $\sigma_{z}=\left|\uparrow\right\rangle\left\langle\uparrow\right|-\left|\downarrow\right\rangle\left\langle\downarrow\right|$.

In the experiment we select R₁ and R₂ to construct the SSH FSL, with R₃ being far detuned and effectively decoupled from the system. The experimental pulse sequences are sketched in Fig. 2 (b). We first prepare the initial state $\left|\downarrow;0,5\right\rangle$, which is the topological zero-energy state of the SSH FSL with $N=5$ and $\lambda_{1}=1$, by pumping five photons successively into R₂ via Q₂ (upper pannel of Fig. 2 (b)). Then we tune R₁, R₂ and Q₀ in resonance at the frequency $\omega_{\text{int}}/2\pi\approx 4.81$ GHz and sinusoidally modulate the coupling strengths where $g_{0}/2\pi\approx 9$ MHz, $\lambda_{1}=|\cos(2\pi\nu t)|$ and $\lambda_{2}=|\sin(2\pi\nu t)|$ with $\nu=416$ kHz $\ll g_{0}$ to satisfy the adiabatic condition, as shown in the lower panel of Fig. 2 (b). Finally the wave packet of the zero-energy state in the FSL is measured (see Methods), with the data shown in Fig. 2 (c). The adiabatic transport of the topological edge state is witnessed by the oscillation of the photons between R₁ and R₂ following Eq. (3) with time-dependent $\lambda_{1}$ and $\lambda_{2}$. The zero-energy state is topologically protected by the energy gap $g_{0}$ from other eigenstates of the FSL and maintains coherence during the transport [36]. To show this, we further measure the density matrix of the two resonators by quantum state tomography. As shown in Fig. 2 (d), the two-mode binomial state remains a Fock state in the combinational dark mode of the two resonators, $\lambda_{2}a_{1}-\lambda_{1}a_{2}$, and the quantumness of the states is evident from the negative values of the Wigner functions.

A Lifshitz topological edge [39] locates on the incircle of the FSLs (see the dashed line in Fig. 1 (b)). The expectation positions of the states in the zeroth Landau level are confined within the incircle by a band gap [11]. The strain-induced magnetic field has opposite signs at the $K$ and $K^{\prime}$ valleys (see Supplementary Information). By introducing a linear potential to mimic the effect of an electric field to electrons, we can observe the VHE (see Fig. 1 (b)), i.e., the Hall response has opposite signs at the two valleys. To experimentally demonstrate this effect, we first prepare an initial state $|\psi_{0}\rangle$ with $\lambda_{1}=\lambda_{2}=1/\sqrt{2}$ in Eq. (3) following the procedure in Fig. 2 (b). Such an initial state on the Lifshitz topological edge is a Gaussian wave function in the zeroth Landau level [11]. Then we bring R₃ in resonance with Q₀ and set $g_{j}/2\pi\approx 9$ MHz for $j=1,2,3$. The linear potential with horizontal gradient is introduced by slightly shifting the frequencies of R₁ and R₂,

where the detuning $\delta/2\pi\approx 1.8$ MHz. We then measure populations on each lattice site in the FSL and obtain the average photon numbers in the three resonators, as shown in Fig. 4 (b). The linear potential drives photons from $R_{1}$ and $R_{2}$ to $R_{3}$ while the qubit stays in the ground state. To visualize the evolution of the wave function, we draw the population distributions in the FSL at five different times (see Fig. 4 (c)). The wave function first moves upward perpendicular to the force direction (black arrow) until being reflected by the Lifshitz topological edge near the top vertex, and then moves downward back to the initial state (up to a phase factor). In particular, when the wave function is at the center of the lattice but in different valleys, e.g., at $t=150$ and $350$ ns, it moves in opposite directions, which is a signature of the VHE [11]. It is noteworthy that the qubit remains in the ground state during the evolution, which reflects a fundamental difference between classical and quantum optics (see Extended Data Fig. 1).

Surprisingly, the VHE can also be observed with initial classical states such as $|\psi_{c}\rangle=\left|\downarrow;\alpha,0,-\alpha\right\rangle$, i.e., $R_{1}$ and $R_{3}$ are in the coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ and $R_{2}$ is in the vacuum state. Such a state can be expanded as a superposition of two-mode binomial states similar to the initial state in Fig. 4 (b) with different total excitation numbers $N$ [11]. Since the dynamics different subspaces is synchronized, the fields in the three resonators remain as a direct product of coherent states and the evolution of the average photon numbers in the three resonators follow similar curves as those for the initial binomial state (see Fig. 4 (d) and Supplementary Information).

We can further measure the chirality of the states at the two valleys. Since the states of the three resonators are separable, we perform simultaneous quantum state tomography and obtain their Wigner functions (see Fig. 4 (e)). As expected, the phases are distributed in a counter-clockwise ($C>0$) and clockwise ($C<0$) manner at $t=100$ and $290$ ns when the wave function moves to the $K$ and $K^{\prime}$ points, respectively. Therefore, the VHE in FSLs can be used to coherently transport the wave function between two valleys and control the chirality of the quantum states of multiple resonators.

In summary, we have demonstrated the coherent control of topological zero-energy states in 1D and 2D FSLs. These states only occupy the sublattice where the qubit is in the $\left|\downarrow\right\rangle$ state and they are protected from other eigenstates by an energy gap of the vacuum Rabi frequency. Perturbations with energy smaller than this gap, such as slow modulation of coupling strengths and small detunings between the resonators, can be used to coherently control the zero-energy state to realize topological transport, VHE and Haldane chiral edge currents. The techniques we have developed in this study can also be applied to control other eigenstates in the FSL, such as the excited states in higher Landau levels, although these states suffer from the decoherence of the qubit. With more resonators, we can synthesize higher dimensitional lattices that are not available in real space. Our study paves the way of investigating topological phases in FSLs and developing novel control methods for quantum state engineering of bosonic modes.

As shown in Fig. 2 (also see details in Fig. S1), the superconduncting resonators (R₁, R₂, and R₃) are interconnected by the central qubit (Q₀), where the resonator-qubit coupling strengths are dynamically programmable with the corresponding inductive couplers (C₁, C₂, and C₃). Each superconducting resonator R_i ($i=1,2,3$) is a quarter-wavelength coplannar waveguide (CPW) resonator capacitively coupled to a standard frequency-tunable transmon qubit Q_i for resonator-state control [42, 43, 44, 45]. The center trace of R_i is connected to an inductance coil and then to a two-junction superconducting quantum interference device (SQUID) before shorting to ground. The inductance coil is coupled to a coupler loop C_i. The SQUID with its own flux line provides the frequency tunability for the resonator and also leads to a slight nonlinearity $\eta\approx-0.5$ MHz of the resonator. Q₀ is a gmon qubit with its junction inductance being connected to an inductor tail, which is coupled to three coupler loops.

The coupler loop consists of a Josephson junction and an inductance due to its loop geometry. The external flux across the coupler can be adjusted to change the effective inductance in order to tune the coupling strengths [34]. An excitation in Q₀ (R_i) generates a current in the loop of C_i which then excites the neighboring R_i (Q₀). The magnetic flux through the coupler loop determines the effective junction inductance, which controls the current flowing through the coupler loop and effectively tunes the Q₀-R_i coupling strengths.