Quantum Noise and Self-Sustained Radiation of $\mathcal{P}\mathcal{T}$-Symmetric Systems

The observation that $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians can have real-valued energy levels even if they are non-Hermitian has triggered intense activities, with experiments, in particular, focusing on optical systems, where Hermiticity can be broken by absorption and amplification. For classical waves, absorption and amplification are related by time-reversal symmetry. This work shows that microreversibility-breaking quantum noise turns $\mathcal{P}\mathcal{T}$-symmetric systems into self-sustained sources of radiation, which distinguishes them from ordinary, Hermitian quantum systems.

Figure 1

Illustration of the scattering input-output approach to non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric systems, defining the scattering input-output operators ${\widehat{a}}_{L,R}^{\mathrm{in},\mathrm{out}}$ and internal bosonic noise operators ${\widehat{b}}_{L,R}$ in different parts of the system. The operators ${\widehat{a}}_0^{\mathrm{right},\mathrm{left}}$ and ${\widehat{a}}_{1,2}^{\mathrm{in},\mathrm{out}}$ feature in the intermediate steps of the calculation (for details see the text). Semitransparent mirrors with transmission probability $\Gamma$ are introduced to study the limit $\Gamma \to 0$ of a closed (nonleaky) system. In the context of present experiments with optical fibers [5], this sketch relates to the transverse confinement.