Exceptional points of any order in a single, lossy waveguide beam splitter by photon-number-resolved detection

Abstract

Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e. the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remains an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here we analytically show that the number-resolved dynamics of a single, lossy, waveguide beamsplitter, excited by N indistinguishable photons and post-selected to the N -photon subspace, will exhibit an EP of order N + 1. By using the well-established mapping between a beamsplitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general N -photon state, and numerically simulate the system's evolution in the post-selected manifold. Our results pave the way towards realizing robust, arbitrary-order EPs on demand in a single device.

$${\mathscr{PT}}$$ -symmetry from Lindblad dynamics in a linearized optomechanical system

“…We note that the process of instantaneous renormalization is equivalent to restricting to a fixed excitation number N sector. In this sector, the Hamiltonian ĉ †d † H PT ĉd T is an N + 1 matrix in the photon-phonon number basis and post-selecting to this sector is equivalent to measuring the quantities n a (t), n b (t) and g (1) (t) [11,12].…”

“…where σ x , σ z are standard Pauli matrices and Γ = (γ a − γ b )/4. It follows that the decay rates of the two eigenmodes ofĤ L are equal (PT -symmetric phase) for |Γ| < g, they reach the maximum at |Γ| = g, and a slowly decaying eigenmode emerges for |Γ| > g [10,12,[29][30][31].…”

“…Such lossy systems are also a promising candidate for observing PT -symmetric quantum optics across EPs of arbitrary order with appropriate post-selection [12]. In systems with both gain and loss, the inclusion of non-classical light requires the introduction of (quantum) fluctuations induced by the linear media either by Langevin equation [13][14][15] or Lindblad master equation [16][17][18] formalism.…”

Photon-counting distribution for arrays of single-photon detectors

“…Assume that the single photon detectors click with a probability 1 − (1 − η) m when m photons hit the detector, that the probability per event for a dark count is p d and that the detector is memoryless. The probability for the single photon detector to click when m photons hit that detector is then given by (1) where k ∈ {0, 1} is the output corresponding to that the detector does not click or does click, respectively.…”

“…Photon-number-resolving detectors have applications in various optical fields, such as investigation of exceptional points in PT -symmetric systems [1], measurements in the number basis [2], quantum key exchange [3], photon-counting laser-radars [4], X-ray astronomy [5], evaluation of single-photon sources [6], and elementaryparticle detection [7]. These detectors can essentially be divided into two classes, inherent detectors and multiplexed detectors.…”