State-recycling and time-resolved imaging in topological photonic lattices

Photonic lattices—arrays of optical waveguides—are powerful platforms for simulating a range of phenomena, including topological phases. While probing dynamics is possible in these systems, by reinterpreting the propagation direction as time, accessing long timescales constitutes a severe experimental challenge. Here, we overcome this limitation by placing the photonic lattice in a cavity, which allows the optical state to evolve through the lattice multiple times. The accompanying detection method, which exploits a multi-pixel single-photon detector array, offers quasi-real time-resolved measurements after each round trip. We apply the state-recycling scheme to intriguing photonic lattices emulating Dirac fermions and Floquet topological phases. We also realise a synthetic pulsed electric field, which can be used to drive transport within photonic lattices. This work opens an exciting route towards the detection of long timescale effects in engineered photonic lattices and the realisation of hybrid analogue-digital simulators.

Here, L 1,5 are convex lenses, M 1−6 are silvercoated mirrors, F is a bandpass filter at 780±5 nm wavelength, BS 1−3 are beam splitters and P is a polariser. M 2−5 form the ring cavity. For precise imaging, the input facet of the lattice is imaged on a CCD camera (see the blue line) to observe the lattice sites, input state, and the state after the first pass. To perform the state-recycling using a linear cavity, the lattice is replaced by another lattice with silver-coated facets (shown in the green-dashed inset), BS 3 is replaced by a mirror to reflect the output state onto the Megaframe (MF32) and F is replaced by another bandpass filter as required for the experiment. camera consisting of 32×32 single-photon avalanche detector array, scale-bar: 800 µm. b, Spatial information i.e. intensity distribution summed over four and a half round trips for the driven one-dimensional lattice presented in Figure 1 in the main text. To reduce spatial overlap of light intensity from two consecutive lattice sites, optical modes were imaged onto alternative pixels of the MF32 (represented by white circles). Each pixel contains temporal information. c, Temporal intensity profiles for four pixels indicated by the coloured arrows. These peaks are normalised such that the total detected optical power for each round trip is unity. Here, the time-step (horizontal axis) is ∼ 53 ps. coupler (i.e. two evanescently coupled identical waveguides) was placed inside the ring cavity, light was initially launched at waveguide-1 and the evolution of optical fields was measured in a timeresolved manner. The solid lines indicate the expected variation of light intensity at waveguide-1 (blue) and waveguide-2 (red) as a function of Jz. Two sets of experiments were performed using optical pulse trains at 780 nm (square) and 750 nm wavelength (circle) which corresponds to two sets of tunnelling strengths, J = 0.046 and 0.038 per mm, respectively. b-e, Time-resolved interference experiment at 750 nm wavelength. b-e show interference fringes (at four consecutive round trips) generated by allowing the modes at the output of the coupler to interfere on the singlephoton avalanche detector array in the far-field. The fringes are rotated at 45 • because the coupling axis between the waveguides was orientated at that angle with respect to the vertical axis. The dashed lines are guides to the eye. The π phase shift observed in d and e compared to b and c is a characteristic of a directional coupler -after the full transfer of light, i.e. Jz > π/2, the relative phase between the optical modes of the waveguides exhibit a phase shift of π. These experiments prove that both phase and intensity of optical fields is recycled in the ring cavity scheme. θ. In this case, alternating electric field pulses are realised. Floquet quasienergy spectrum as a function of the phase shift between adjacent sites (2φ = Φ). The band collapses at Φ = π exhibiting destruction of tunnelling. b-e, Numerically calculated evolution of the intensity distributions for Φ = 0, π/2, 3π/4 and π, respectively. 8

Supplementary Note 1: A one-dimensional driven lattice and the Dirac Hamiltonian
In this section, we present the driving protocol related to the 1D lattice illustrated in Figure 1b (in the main text) as well as its photonic implementation ( Figure 1c). Let us first consider a general situation, a 1D tight-binding lattice with staggered hopping amplitudes (J 1,2 ). In the static case, the k-space Hamiltonian can be written aŝ where d is the inter-site separation,σ x,y,z are Pauli matrices and the Brillouin zone spans 0 ≤ k ≤ π/d. Now consider that J 1,2 are varying in a time periodic manner with a period T . The driving protocol is the following: Note that the probability for a particle to hop to its nearest site is unity when the tunnelling/hopping is allowed. The Floquet operator and the effective Hamiltonian for this two-step driving protocol can be written as 1Û whereĤ 1,2 are the Hamiltonians for the two driving steps and1 is a 2 × 2 identity matrix. Equation (5) is the 1D Dirac Hamiltonian, which describes pseudo-relativistic particles in linearly-dispersive bands, with a speed of light v D = 2d/T . The Floquet spectrum associated with Supplementary Equation (5) is shown in Figure 1d (main text). Now let us discuss the photonic implementation of the driving protocol in Supplementary Equation (2) and (3). To perform the state-recycling using a linear cavity, we consider a driven photonic lattice of length L (as shown in Figure 1c) where each bond is a 50 : 50 directional coupler. Note that light travels along the +z direction for the first half of the complete driving period (T ≡ 2L) and then almost 90% of light reflects back and travels along the −z direction. For the time-correlated single photon counting (TCSPC) measurement, the transmitted light is imaged onto the SPAD array. The four-step driving protocol for the photonic lattice can be written as follows: Light propagation along +z where and after reflection (i.e light propagation along −z) Note that the propagation distance is the analogous time (z ↔ t). The Floquet operator for the four-step driving is given byÛ where T indicates the time ordering and n = 1, 2, 3, 4. The effective Hamiltonian then becomeŝ H eff = (2d/T )k(sin(kd)σ x − cos(kd)σ y ) + (π/T )1.
It can be shown that the effective Hamiltonian in Supplementary Equation (11)  When this lattice is placed inside a ring cavity, the complete driving period, T = L, consists of the following two steps: (1) time evolution determined by the coupling strength, J for 0 ≤ z ≤ L; (2) tunnelling is frozen and an analogous static field is applied at z = L.
Formally, the time-evolution operator over each period T can be written in the two-step form, wherex = d s s|s s| is the position operator on the lattice andĤ 0 = J s |s + 1 s| + h.c.
is the hopping Hamiltonian with amplitude J. For weak synthetic electric fields, φ 1, this 13 time-evolution operator can be simplified according to the Trotter formula, which describes the motion of a particle hopping on a lattice in the presence of an effective electric field E = φ/dT . We note that a similar form can be obtained in the large-field regime φ ∼ 1, using the full Baker-Campbell-Hausdorff formula.
For the purpose of numerical calculations, a photonic lattice with 41 waveguides and φ = π/7.5 (1) time evolution determined by the coupling strength, J for 0 ≤ z ≤ L; (2) tunnelling is frozen and an analogous static field (characterised by 2φ) is applied at z = L; (3) time evolution determined by the coupling strength, J for L ≤ z ≤ 0; (4) tunnelling is frozen and an analogous static field (characterised by −2φ) is applied at z = 0.
Formally, the time-evolution operator over each period T can be written in the four-step form, where we used the same notations as above. Using the following expression 1 , which indicates that the pulsed and alternating electric field simply renormalizes the hopping amplitude J → J cos(φ). We note that the effective Hamiltonian appearing in Supplementary Equation (17), namely,Ĥ eff = cos(φ)Ĥ 0 , is valid for any φ. In particular, for φ = π/2, we find that the Floquet operator is trivial,Û (T ) =1, which indicates that the hopping is effectively annihilated by the pulsed electric field.
It should be mentioned that this particular four-step driving protocol can also be realised by fabricating the photonic lattice such that the waveguide axes are tilted at an angle with respect to the length of the glass sample. In that case, both facets will be at equal angles with respect to the waveguide axes without the requirement of angle polishing. For this driving protocol, the Floquet quasienergy spectrum as a function of the phase shift between adjacent sites (2φ = Φ) is presented in Supplementary Figure 7a signs, which means that the associated synthetic pulsed electric fields will be along the same direction. In this experiment, the total driving period, T = 2L, consists of four driving steps: (1) tunnelling is frozen and an analogous static field (characterised by +2φ 1 ) is applied at z = 0; (2) time evolution determined by the coupling strength, J for 0 ≤ z ≤ L; (3) tunnelling is frozen and an analogous static field (characterised by +2φ 2 ) is applied at z = L; (4) time evolution determined by the coupling strength, J for L ≤ z ≤ 0; Formally, the time-evolution operator over each period T can be written in the four-step form, whereĤ 0 describes the hopping in the 1D array, see Supplementary Equation (14). Supplementary Equation (19) describes the motion of a particle hopping on a lattice in the presence of an effective electric field E = 2(φ 1 + φ 2 )/(dT ). In our experiment, the tunnelling strength J = 0.029 mm −1 and the expected inter-waveguide phase shifts are φ 1,2 ≈ π/12 which means that the associated Bloch period is ≈ 12L = 360 mm. In Figure 2 (main text), we probed the dynamics up to 210 mm which is approximately half of this period.
In conclusion, the state-recycling technique introduced in this work indeed allows one to engineer synthetic electric fields, which effectively act on top of the dynamics associated with the engineered photonic lattice. For constant electric field [Supplementary Equation (14)], this could be used to perform transport experiments in view of probing response functions (e.g. the conductivity tensor) or geometric effects through Bloch oscillations 5 . For alternating electric fields, this could be used to control the properties (e.g. tunnelling) of the engineered photonic lattice.