Spectral properties of single photons from quantum emitters

Quantum networks require flying qubits that transfer information between the nodes. This may be implemented by means of single atoms (the nodes) that emit and absorb single photons (the flying qubits) and requires full control of photon absorption and emission by the individual emitters. In this paper, we theoretically characterize the wave packet of a photon emitted by a single atom undergoing a spontaneous Raman transition in a three-level scheme. We investigate several excitation schemes that are experimentally relevant and discuss control parameters that allow one to tailor the spectrum of the emitted photon wave packet.

Figure 1

An atom with a $\mathrm{\Lambda }$-shaped level configuration is prepared in state $|1\rangle$ and excited to state $|e\rangle$ by incident light. A single photon may be emitted along the transition $|e\rangle \to |2\rangle$. We determine the spectral properties of the emitted photon as a function of the atomic parameters and of the properties of the incident light.

Figure 2

Area-normalized power spectral density $\mathcal{S}\left({\mathrm{\Delta }}_2\right)$ of the emitted photon after excitation by a single photon of sinc (light gray), Gaussian (gray), or Lorentzian (black) spectra. The spectra are given for three different linewidths $\mathrm{\Delta }{\omega }_1$ and for resonant excitation in (a), (c), and (e), and off-resonant excitation in (b), (d), and (f).

Figure 3

(a) Effective linewidth $\mathrm{\Delta }{\omega }_2$ and (b) success probability $\mathcal{N}$ of the emitted single photon versus the linewidth of the incident photon. Shown are the three cases of excitation with a single photon of sinc- (light gray), Gaussian- (gray), or Lorentzian-shaped (black) spectra at resonance (${\mathrm{\Delta }}_1=0$) and interaction time $t\to \infty$. For all cases, the asymptote is $\mathrm{\Delta }{\omega }_2=\mathrm{\Gamma }$ corresponding to the bare atomic Lorentzian.

Figure 4

Three cases of laser-induced generation of a single photon on the transition $|e\rangle \to |2\rangle$ with $N=0$, 1, or 2 additional spontaneously emitted photons on the transition $|1\rangle \leftrightarrow |e\rangle$.

Figure 5

Area-normalized power spectral density ${\mathcal{S}}_0\left({\mathrm{\Delta }}_2\right)$ Eq. (22) of the emitted single photon after excitation by a laser for various values of Rabi frequency $\mathrm{\Omega }$ and laser detuning ${\mathrm{\Delta }}_1$.

Figure 6

Atomic level configuration that leads to quantum beats. The atom is initially in a superposition of states $|g_1\rangle$ and $|g_2\rangle$. A single photon is emitted on $|e\rangle \to |g_3\rangle$ after excitation by incident light.

Figure 7

Area-normalized power spectral density $\mathcal{S}\left({\mathrm{\Delta }}_3\right)$ for excitation of an initial superposition state by a single photon tuned to the center of the two transitions $\left({\mathrm{\Delta }}_1=-\frac{\delta }{2}\right)$. The incident wave packet is Lorentzian with width $\mathrm{\Delta }{\omega }_1$ as indicated; the splitting $\delta$ is $\mathrm{\Gamma }$ in (a), (c), and (e) and $2\mathrm{\Gamma }$ in (b), (d), and (f). The gray levels indicate the phase between the two transitions: black for $c_1=c_2$ and gray for $c_1=-c_2$.

Figure 8

Partial spectra ${\mathcal{S}}_N\left({\mathrm{\Delta }}_3\right)$ for excitation by a laser tuned to the center of the two transitions $\left({\mathrm{\Delta }}_1=-\frac{\delta }{2}\right)$. The Rabi frequency is $\mathrm{\Omega }=\mathrm{\Gamma }$. The splitting $\delta$ is $\mathrm{\Gamma }$ in (a) and (c) and $2\mathrm{\Gamma }$ in (b) and (d). Gray levels indicate the number $N$ of intermediate photons: dark, medium, and light gray for $N=0$, 1, and 2, respectively, and black for the sum.