Recent theoretical studies of a pair of atoms in a 1D waveguide find that the system responds asymmetrically to incident fields from opposing directions at low powers. Since there is no explicit time-reversal symmetry breaking elements in the device, this has caused some debate. Here we show that the asymmetry arises from the formation of a quasi-dark-state of the two atoms, which saturates at extremely low power. In this case the nonlinear saturability explicitly breaks the assumptions of the Lorentz reciprocity theorem. Moreover, we show that the statistics of the output field from the driven system can be explained by a very simple stochastic mirror model and that at steady state, the two atoms and the local field are driven to an entangled, tripartite |W state. Because of this, we argue that the device is better understood as a saturable Yagi-Uda antenna, a distributed system of differentially-tuned dipoles that couples asymmetrically to external fields.Nonreciprocal devices, such as isolators, circulators, and gyrators, are important components for optical and microwave technologies. They are typically used to route or isolate signals propagating in different directions. Recently, a unidirectional, two-atom device has been identified as potentially useful in quantum electronics [1][2][3][4][5][6][7], building on earlier analyses of distributed atomic systems [8][9][10][11][12]. Transmission through this device depends asymmetrically on the direction of the input field, hence it has been dubbed a quantum diode.The quantum diode consists of a pair of spatiallyseparated, nondegenerate atoms in a 1D waveguide, shown in Fig. 1a, tuned to discriminate between a coherent field α incident from the left, and a coherent field β incident from the right. Prima facie, this appears to violate reciprocity: the transmission coefficients of a passive, linear, time-reversal-symmetric scatterer should satisfy T ← = T → , so there is an interesting question as to the origin of the transmission asymmetry.Here, we derive a master equation for the driven twoatom system shown in Fig. 1a. We show that the twoatom dark state [7] responsible for the asymmetry arises from entanglement between the matter and the field [10]. This leads to non-reciprocal [13] and incoherent [7] scattering matrices, and we establish the maximum possible 'diode efficiency' [2] of 2/3, for which the steady state is inverted. Finally, we show that a toy-model of a randomly fluctuating mirror replicates the statistics of the scattered field and corresponds exactly to the rate equation model when adiabatically eliminating all coherences.The picture that emerges is that in the steady-state, under cw-driving from one direction, the two atoms become entangled with the local electromagnetic field in a tripartite |W state. In the atomic Hilbert space, this corresponds to a long-lived, probabilistic mixture of the ground and dark states. Since scattering arises from coherence between the ground and bright states, the dark * c.muller2@uq.edu.au † stace@physics.uq.edu...

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Binary-outcome measurements allow to determine whether a multi-level quantum system is in a certain state while preserving quantum coherence between all orthogonal states. In this paper, we explore different regimes of the dispersive readout of a three-level superconducting quantum system coupled to a microwave cavity in order to implement binary-outcome measurements. By designing identical cavity frequency shifts for the first and second excited states of the system, we realize strong projective binary-outcome measurements onto its ground state with a fidelity of 94.3%. Complemented with standard microwave control and low-noise parametric amplification, this scheme enables the quantum non-demolition detection of leakage errors and can be used to create sets of compatible measurements to reveal the contextual nature of superconducting circuits.