Quantum Pyragas control: Selective control of individual photon probabilities

Pyragas control allows us to stabilize unstable states in applied nonlinear science. We propose to apply a quantum version of the Pyragas protocol to control individual photon probabilities in an otherwise only globally accessible photon-probability distribution of a quantum light emitter. The versatility of quantum Pyragas control is demonstrated for the case of a two-level emitter in a pulsed laser-driven half cavity. We show that one- and two-photon events respond in a qualitatively different way to the half-cavity-induced feedback signal. One-photon events are either enhanced or suppressed, depending on the choice of parameters. In contrast, two-photon events undergo exclusively an enhancement up to $50\%$ for the chosen pulse areas. We hereby propose an implementation of quantum Pyragas control via a time-delayed feedback setup.

Figure 1

Illustration of the system under pulsed excitation where a single TLS is placed inside a waveguide. A photon propagating to the right side is reflected and may excite the TLS again with delay $\tau$.

Figure 2

Computation of one integrand of $C_3$ from the matrix product state in diagrammatic form. Round edges represent orthogonality. For each time combination, the intensity operator is applied at the corresponding time bins $A^{[...]}$. The cost of this operation grows linearly with the difference $|k-m|$.

Figure 3

Normalized probabilities $\overline{p}\left(n\right) [\overline{p}\left(n\right)=1$ is the no-feedback case] for destructive feedback vs delay time $\tau$ for $A=2\pi$ and $\nu =\frac{1}{10\mathrm{\Gamma }}\frac{1}{\sqrt{2\text{ln}\left(2\right)}}$ (a). Two-photon emission $\overline{p}\left(2\right)\ge 1$ is enhanced with feedback. Inset: $p\left(n\right)$ for no feedback (red, left) and with time-delay $\tau \mathrm{\Gamma }=0.06$ (orange, right). $p\left(2\right)$ is increased by approximately $50\%$. For $A=4\pi$ (b), the delay $\tau$ gives more access to control the photon statistics, due to the additional Rabi oscillation.

Figure 4

Feedback-phase dependency of the normalized ratio $r=p\left(2\right)/p\left(1\right)$ at $A=2\pi$. $r/r_{\mathrm{nofeedback}}=1$ indicates the case without feedback (green/gray). $p\left(1\right)$ dominates for $\varphi =\pi$ (cyan/light gray) and $p\left(2\right)$ for $\varphi =0$ (red/dark gray). Inset: Sketch for the TLS decay, for no feedback (green, solid line), for constructive feedback (cyan, dotted line), and for destructive feedback (red, dashed line).