arXiv:1102.3256v1 [quant-ph] 16 Feb 2011 Robust optical delay lines via topological protection Mohammad Hafezi ∗ , 1 Eugene E. Demler, 2 Mikhail D. Lukin, 2 and Jacob M. Taylor 1 1 Joint Quantum Institute, University of Maryland and NIST, College Park, MD 20742 2 Physics Department, Harvard University, Cambridge, MA 02138 Phenomena associated with topological properties of physical systems are naturally robust against perturbations. This robustness is exemplified by quantized conductance and edge state transport in the quantum Hall and quantum spin Hall effects. Here we show how exploiting topological properties of optical systems can be used to implement robust photonic devices. We demonstrate how quantum spin Hall Hamiltonians can be created with linear optical elements using a network of coupled resonator optical waveguides (CROW) in two dimensions. We find that key features of quantum Hall systems, including the characteristic Hofstadter butterfly and robust edge state transport, can be obtained in such systems. As a specific application, we show that the topological protection can be used to dramatically improve the performance of optical delay lines and to overcome limitations related to disorder in photonic technologies. Particles in two-dimensional structures with a magnetic field exhibit a remarkable variety of macroscopic quantum phenomena, including integer [1] and fractional [2] quantum Hall and quantum spin Hall effects [3], and predicted regimes of fractional or non-abelian statistics [4, 5]. A hallmark of these systems is the presence of edge states, whose transport properties are robust against disorder and scattering, leading to quantized conductance sufficient to provide a resistance standard [6, 7]. Natural robustness of topological states is actively explored in quantum computation [8, 9]. Recently, approaches to observing similar quantum Hall behavior in bosonic systems including ultra-cold gases (for a review see Ref.[10]) and photons [11–16] have been suggested. Our method for realization of topological protected photonic devices makes use of two dimensional arrays of coupled resonator optical waveguides (CROW) to simulate a 2D magnetic tight-binding Hamiltonian with degenerate clockwise and counter-clockwise modes. This approach does not require explicit time-reversal symmetry breaking [11–15], but the degenerate modes —time-reversed pairs— behave analogously to spins with spin-orbit coupling in the electronic quantum spin Hall effect (QSHE) [17–19], and experience a spin-dependent magnetic field (Fig. 1). When the clockwise and counter-clockwise modes are decoupled, we can selectively drive each mode and observe quantum Hall behaviors without breaking the time-reversal symmetry in the tight binding Hamiltonian. In a direct analogy to the electronic integer quantum Hall effect, we show that photonic edge states carry light at the perimeter of the system, while being insensitive to disorder, and therefore forms a basis for robust photonic devices. In particular, in comparison to state-of-the-art 1-D CROW systems, our approach can be dramatically more resistant to scattering disorders and fabrication errors. 2D Photonic System and Quantum Spin Hall Hamiltonian As illustrated in Fig. 1, our system comprises optical ring microresonators that support degenerate clockwise and counter-clockwise modes, restricted to one pair per resonator. We consider these modes as two components of a pseudo-spin, i.e., clockwise (σ = −1, or psuedo-spin down) and counter-clockwise (σ = +1, pseudo-spin up) circulation. Resonators are evanescently coupled to each other and have been studied in the context of 1D CROW [20], where the coupling leads to a tight-binding model for photons and the corresponding photonic band structure. By coupling these modes in a two-dimensional arrangement, as we show below under appropriate conditions, the dynamics of such photonic system is described by a Hamiltonian for charged bosons on a square lattice (tight-binding), but with the addition of a perpendicular, spin-dependent effective magnetic field: ∗ email: hafezi@umd.edu