Creation of nonclassical states of light in a chiral waveguide

Creating nonclassical states of light from simple quantum systems together with classical resources is a challenging problem. We show how chiral emitters under a coherent drive can generate nonclassical photon states. For our analysis, we select a specific temporal mode in the transmitted light field, resulting in a coupled master equation for the relevant mode and the chiral emitters. We characterize the mode's state by its Wigner function and show that the emission from the system predominantly produces mixtures of few-photon-added coherent states. We argue that these nonclassical states are experimentally accessible and show their application for quantum metrology.

Figure 1

Scattering coherent light with amplitude $\alpha$ on a chiral emitter chain produces light in multiple orthogonal modes. The light in a specific mode $v\left(t\right)$ is studied by placing a virtual cavity behind the emitters, with a time-dependent coupling rate $g_v\left(t\right)$ tuned to capture only photons in mode $v\left(t\right)$. The coherent light drives both the cavity and the $|G\rangle \leftrightarrow |W\rangle$ transition of the emitters. Our theory includes emission of $|W\rangle$ out of the waveguide at rate $\mathrm{\Gamma }$ and an induced transfer with rate ${\gamma }_D$ of $|W\rangle$ into a nonradiating excited state $|D\rangle$.

Figure 2

(a) A single emitter's excited-state population for $\alpha =0.9\sqrt{\kappa }$, ${\gamma }_D=0=\mathrm{\Gamma }$. The vertical markers indicate the different choices of binning intervals studied in (b). (b) Wigner functions of ${\rho }_v$ for square-mode pulses on the different time intervals. For short time bins (left) the Wigner function closely resembles the coherent state $|\sqrt{\tau }\alpha \rangle$, while for $\kappa \tau \gg 1$ the Wigner function becomes positive as the emitter's emission close to its steady state weakly impacts the coherent background (right). Bins centered around the first Rabi peak provide the largest Wigner negativities for suitable $\alpha$ (middle), with the negative region indicated by the black contour line.

Figure 3

Wigner functions of ${\rho }_v$ at different $\alpha$ and ${\gamma }_D=0=\mathrm{\Gamma }$. The binning interval at given $\alpha$ is chosen to increase the Wigner negativity at ${\gamma }_D=0$ and always includes the first Rabi peak of the excited-state population. Black contour lines indicate the negative areas.

Figure 4

Wigner functions of ${\rho }_v$ at $\alpha =0.5\sqrt{\kappa }$ for different $\mathrm{\Gamma }$ and ${\gamma }_D$. The binning interval $(t_0,t_0+\tau )$ is the same for all examples and was chosen to maximize the Wigner negativity at $\mathrm{\Gamma }=0={\gamma }_D$. Black contour lines indicate the negative areas.

Figure 5

(a) Excited-state population of the $i\mathrm{th}$ emitter in a six-emitter chain at $\alpha =0.5\sqrt{\kappa }$. The gray shaded region indicates the region which generally provides good bins for large Wigner negativities. The vertical dashed lines indicate the bin for the top row of (b), which shows the alternating pattern in the Wigner function, when binning in the emitters' steady state. (b) The bottom row shows Wigner functions for one to six emitters. The binning interval was numerically optimized for each set of emitters and lies within the gray shaded region in (a). Black contour lines indicate the negative areas.

Figure 6

(a) Sketch of the Mach-Zehnder experiment. Light on the two input ports, $a$ and $b$, interferes at the beam splitters, and measurements on the output ports, $a^{\prime }$ and $b^{\prime }$, are used to estimate the unknown phase $\phi$. (b) and (c) We use ${\rho }_v$, generated from one and two emitters, respectively, at the input port $a$ and a coherent state $|\sqrt{N_b}\rangle$ at port $b$ to estimate $\phi$. The plots show the improvement of the $\mathrm{\Delta }\phi$ sensitivity and Cramér-Rao bound compared to shot noise $\mathrm{\Delta }{\phi }_{\mathrm{SN}}=1/\sqrt{N_a+N_b}$ at multiple values ${\gamma }_D$. For any finite ${\gamma }_D$ the optimal driving strength $\alpha$ and bin $(t_0,t_0+\tau )$ depend little on $N_b$. However, this is not the case for a single emitter at ${\gamma }_D=0$, and we choose to optimize at $N_b=100$, after which the sensitivities typically approach their asymptotic values. (d) At high decay ${\gamma }_D=2\kappa$ multiple emitters provide more noise-resilient $\mathrm{\Delta }\phi$-sensitivity improvements.