We consider symmetry breaking in the simplest open quantum nonlinear systems such as dimer and plaquette of four nonlinear sites coupled with linear tight-binding wires. If the solution is stationary, the total Hilbert space can be projected into the inner states of the dimer or plaquette by the Feshbach procedure. That derives a nonlinear analogue of the Lippmann-Schwinger equation with injected wave as a source. By neglecting radiation shifts the Lippmann-Scwinger equation limits to the coupled mode theory equations widely used in optics. We show three scenarios for the transmission through the nonlinear quantum systems. The first one inherits the linear case and preserves the symmetry. In the second scenario the symmetry is broken because of different intensities at the dimer sites. In the third scenario the intensities at the sites are equaled but phases of complex wave function are different. That results in a vortical power flow between the nonlinear sites similar to the DC Josephson current. We show how the phenomenon of symmetry breaking can used for switching of outputs symmetrically coupled to the quantum dimer. Also we reveal a domain in the parameter space where none of stationary solutions exist. As a result injection of a monochromatic symmetric wave gives rise to emission of nonsymmetric satellite waves with energies different from the energy of the incident wave. Thus, the response exhibits non monochromatic behavior.
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