Deterministic Shaping and Reshaping of Single-Photon Temporal Wave Functions

Thorough control of the optical mode of a single photon is essential for quantum information applications. We present a comprehensive experimental and theoretical study of a light-matter interface based on cavity quantum electrodynamics. We identify key parameters like the phases of the involved light fields and demonstrate absolute, flexible, and accurate control of the time-dependent complex-valued wave function of a single photon over several orders of magnitude. This capability will be an important tool for the development of distributed quantum systems with multiple components that interact via photons.

Figure 1

Model of the experimental setup. (a) A single atom of ${}^{87}\mathrm{Rb}$ is trapped in a high finesse optical cavity via a 2D optical lattice (not shown). The control field of Rabi frequency $\mathrm{\Omega }(t)$ and $\pi$-polarization impinges from the side of the cavity. The incoming photon is a weak coherent pulse of temporal shape $e(t)$. The atom is coupled to the cavity field $\mathcal{E}(t)$ with the light-matter coupling constant $g$. (b) We use a $\mathrm{\Lambda }$ scheme with three excited states. The Zeeman states involved are $m_F=0$ for $F=1$ and $m_F=-1$ for $F=2$ and $F^{\prime }=1$, 2, 3.

Figure 2

Efficiency as a function of the single-photon detuning. (a) The three curves correspond to three theoretical models: the dotted blue line for one excited state with $F^{\prime }=1$, the orange dashed line for two excited states with $F^{\prime }=1$, 2, and the plain green line for three excited states. The dots correspond to experimental measurements. The statistical errors are smaller. Data can be reproduced to within about 10% (error bars). (b)–(d) Normalized temporal amplitudes for different detunings. The black lines correspond to the targeted shapes. The blue histogram corresponds to the detected photons with right circular polarization ($R$) and the red with left circular polarization ($L$). The amount of left polarized light compared to the expectation of only right polarized light witnesses the increase of incoherent processes for large detunings.

Figure 3

Measurement of the complex temporal shape. The real and imaginary parts of $e(t)$ reconstructed by temporal mode analysis are plotted in arbitrary units but identical scales. (a) Corresponds to the case without compensation of the phase chirp induced by light shift and (b) to the case with phase compensation. The plain lines correspond to the experimental measurements and the dashed lines to the theoretical shapes. When the phase chirp is compensated, the fidelity between the target shape and the measured one is 90%. The dotted lines in plot (b) correspond to the shape with a residual detuning of 180 kHz with which the fidelity goes up to 98%. In contrast, without phase compensation, the fidelity is 55%. For these measurements, the detuning is $\mathrm{\Delta }/2\pi =-20\mathrm{MHz}$ and the targeted temporal shape $e(t)$ of the single photon is a hyperbolic secant with the characteristic time $T=0.5\mu \mathrm{s}$.

Figure 4

Temporal mode selection. (a) In black is plotted the amplitude of the input photon (experimentally a weak coherent state), which at time $t=0$ has a phase jump of $\mathrm{\Delta }\varphi$. The two other plots correspond to the amplitude of the control field, in plain blue without a phase jump, in red dashed with a phase jump of $\pi$. (b) The global efficiency (absorption and emission) as a function of $\mathrm{\Delta }\varphi$ for the two different control phase profiles. The curves are fits of the form $A{\sin }^2(\mathrm{\Delta }\varphi +{\varphi }_0)+B$.

Figure 5

Photon conversion: a single photon (experimentally a weak coherent state input) is stored and re-emitted with a different temporal shape with constant efficiency. (a),(b) illustrate two different photon shape conversions $T=0.5\mu \mathrm{s}\leftrightarrow T=500\mu \mathrm{s}$. The histograms correspond to the detection event of the photon counters, and the plain lines correspond to the targeted temporal shapes. In (b), we have subtracted the dark counts of the photon counters.