Topological photonics originates from ideas and models that were first developed to understand topological phases of matter in solid-state physics. Based on the exploration of the topological structure of photonic states in suitable parameter spaces, one of its main objectives is to topologically characterise nontrivial photonic lattices by measuring their topological invariant. As an example, it has been demonstrated that the topological winding number of conservative one-dimensional systems emerges in the long time limit of the observable known as mean chiral displacement, measured over initially localised states. In realistic photonic systems, it is however not possible to fully avoid or neglect losses. In our paper, we propose an extension of the mean-chiral displacement method to a driven-dissipative context, under coherent illumination. Our proposed method allows to directly detect the topological winding number of one-dimensional chiral photonic lattices by integrating over the pump light frequency, with a correction of the order of the loss rate squared. Furthermore, we demonstrate that this method can be successfully applied to lattices along synthetic frequency dimensions.
Context
Topological photonics [1] provides powerful methods to engineer exotic photonic band structures and robust boundary modes. Consider a quantum particle in a periodic potential: as Bloch’s theorem states, the particle’s eigenstates are organised into energy bands, separated by energy gaps, presenting a geometrical structure. Crucially, the latter is reflected in integer-valued topological invariants associated to each band. When these topological invariants attain non-trivial values, new phenomena arise, an example of which is the emergence of topologically protected edge states localised on the physical boundaries of the system. Indeed, the experimental confirmation of a non-trivial topology is often obtained through the detection of these edge-localised modes.
Frequency synthetic dimensions
A platform attracting a growing interest in the photonic community for topological band engineering is based on the so-called synthetic frequency dimensions scheme. The key idea of this approach is to use the different modes of an optical cavity as an extra dimension, so that the different sites along the synthetic dimensions can be selectively addressed via their frequency (Fig.3). As a key advantage, the synthetic frequency framework allows to realise systems with a very large number of lattice sites. Finite size effects are, thus, strongly suppressed and the only remaining source of discrepancy is due to photon losses.
The edges of these systems are, however, not well defined, rendering it impossible to directly access the edge-localised states. This problem can be solved by harnessing a general principle called the bulk-edge correspondence, which guarantees a one-to-one correspondence between the phenomena occurring at the edges of the system and those relative to the bulk.
SSH
In our work, we consider the Su-Schrieffer-Heeger (SSH) model [2], a one-dimensional lattice with a chiral symmetry. The bulk eigenmodes of such chiral lattice are characterised by an integer-valued topological invariant called the winding number W. From the bulk-edge correspondence, we know there exist W edge-localised modes under the open boundary condition (Fig.1).
Motivation
A powerful method to obtain W is through the intensity profiles, measuring the mean-chiral displacement [3,4]: this is the expectation value of the operator Γx, where Γ is the chiral operator and x is the position operator. For an initial state localized on the central unit cell, ⟨Γx⟩ converges to W/2 in the long-time limit of a conservative evolution.
Measurements of the mean chiral displacement have been successfully implemented in several photonic platforms. In all these works, a reduced amount of losses allows to accurately describe the propagation of the light field in terms of a unitary evolution and to apply the original formulation of the mean chiral displacement method. In real experimental setups, however, the dynamics is not conservative, but suffers instead from significant photon losses. A pioneering step in this direction was made by the experiment in Ref. [5] where the winding number of polaritonic lattices was measured under a continuous-wave incoherent illumination.
Results
Here we make a further step by theoretically considering systems under a coherent monochromatic illumination. In specific, we show that a frequency-integration of the mean-chiral displacement measured on the steady-state at a given pump frequency provides an accurate estimate of the winding number in realistic cases where losses are comparable or smaller than the characteristic bandwidth of the photonic states. The basic idea behind our proposal is the following. The mean-chiral displacement method can be regarded as being based on the time evolution in response to a pulsed source. On this premise, we show that the topological winding number may be extracted as an integral over the coherent drive frequency. When the loss γ is smaller than the typical energy scale for the hopping amplitude, we demonstrate that, as in the long-time limit of the mean-chiral displacement in conservative systems, the leading term exactly recovers the winding number W/2, with the next correction being roughly proportional to γ2 over the squared photonic lattice bandwidth. This result is independent of the choice of the source profile within the central unit cell. This is the central result of our work (Fig.2).
Frequency synthetic dimensions
A straightforward way to extract the mean chiral displacement in the synthetic frequency dimension frame-work is based on measuring the intensities of the different spectral components of the out-put signal and then translating them into the usual definition of mean chiral displacement to our context. A key advantage of synthetic frequency dimensions is the ability to selectively pump a given site of the lattice with a coherent pump. In particular, we demonstrate that the frequency distribution of the output light in the synthetic frequency lattice is exactly the same as the steady state distribution of a model defined in the ordinary, spatial, dimension. This equivalence implies that the method of integrated mean chiral displacement can be used to directly detect the topological winding number of chiral Hamiltonians in the frequency synthetic dimension. [fig.4]
Our method keeps being accurate for relatively small lattice sizes, providing a versatile tool to measure the topological winding number in generic driven-dissipative photonic systems.
[1] Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
[2] Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).
[3] Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Commun. 8, 15516 (2017).
[4] Maffei, M., Dauphin, A., Cardano, F., Lewenstein, M. & Massignan, P. Topological characterization of chiral models through their long time dynamics. N. J. Phys. 20, 013023 (2018).
[5] St-Jean, P. et al. Measuring topological invariants in a polaritonic analog of graphene. Phys. Rev. Lett. 126, 127403 (2021).
Mean-chiral displacement in coherently driven photonic lattices and its application to synthetic frequency dimensions
