Optimal mode configuration for multiple phase-matched four-wave-mixing processes

We demonstrate an unseeded multimode four-wave-mixing process in hot ${}^{85}\mathrm{Rb}$ vapor using two pump beams of the same frequency that cross at a small angle. This results in the simultaneous fulfillment of multiple phase-matching conditions that reinforce one another to produce four intensity-stabilized bright output modes at two different frequencies. Each generated photon is directly correlated to exactly two others, resulting in the preferred four-mode output, in contrast to other multimode four-wave-mixing experiments. This provides significant insight into the optimal configuration of the output multimode squeezed and entangled states generated in such four-wave-mixing systems. We examine the power, temperature, and frequency dependence of this output and compare it to the conical four-wave-mixing emission from a single pump beam. The generated beams are spatially separated, allowing a natural distribution for potential use in quantum communication and secret-sharing protocols.

Figure 1

Diagram of the rubidium cell with pump beams crossing at an exaggerated angle. Whereas each individual pump generates a four-wave-mixing cone, the combination of both pumps creates four localized modes of four-wave mixing (not shown). The inset: Diagram showing interactions present in this configuration. The four generated modes are labeled 1–4 clockwise starting from top left throughout this paper. The light blue dotted line is a single-pump FWM interaction, whereas the dark blue dotted line is a dual-pump FWM interaction. The orange rings are shown for scale but are not visible simultaneously with the four modes.

Figure 2

Camera images 150 mm after the cell of the single- and double-pump configurations with pump areas subtracted. The scale of each image is 6 mm $\times$ 12 mm. Top: pump A only; middle: both pumps; bottom: pump B only. In the case of either single pump, the resultant output mode is a spontaneous FWM ring, whereas the two-pump case results in a collapse to the distinct four-mode stability.

Figure 3

Frequency shift of intensity curves for the ring and four modes (single and dual pumps, respectively). The addition of a second pump changes the optimal pump detuning $\mathrm{\Delta }$, and this shift increases linearly with power. Top: Normalized total intensity for the ring around one pump (blue) and the four modes around two pumps (orange) as a function of overall pump detuning $\mathrm{\Delta }$. Bottom: The frequency at which maximum intensity occurs for the ring (blue diamonds) and for the four modes (orange triangles) as a function of the power of pump A.

Figure 4

Top: Camera images as the power ratio of $P_A$ to $P_B$ is varied. Blue: $P_A=143,P_B=23\mathrm{mW}$; orange: $P_A=78,P_B=86\mathrm{mW}$; green: $P_A=23,P_B=149\mathrm{mW}$. Here the pumps are not subtracted from the images for visualization purposes, but when calculating intensity the pump areas are always subtracted from each image as they saturate the camera. The scale of each image shown is approximately 5 mm $\times$ 5 mm. Bottom: Total intensity of generated light versus ratio of powers $P_A$ to $P_B$. The pumps are subtracted from each camera image, i.e., this measurement only takes generated light into account.

Figure 5

Total intensity of the generated light as $P_B$ is varied with different areas excluded. The pumps are subtracted from every image, and $P_A$ is fixed at 210 mW. The two curves are normalized individually, and the error bars represent one standard deviation and are based off of ten images taken for each $P_B$. Red: Intensity of generated light in mode 2 only. Interestingly, there is a local intensity maximum before $P_B=P_A=210\mathrm{mW}$ at $P_B\approx 0.3P_A$. Blue: Intensity of generated light in the area excluding the four modes, i.e., ring light only.

Figure 6

Effects of changing cell temperature on the total intensity of the generated modes for pump A only (blue), pump B only (red), and both pumps on (yellow). Pump powers $P_A$ and $P_B$ are equal, i.e., when imaging with both pumps on, the total pump power in the cell is $2P_A$.

Figure 7

Level schemes and phase-matching diagrams for both processes involved in the four-mode FWM configuration. Top: Energy-level schemes for FWM on the ${}^{85}\mathrm{Rb}$ $D_1$ line for the single- (left) and dual- (right) pump processes. Bottom: Corresponding $k$-vector diagrams displaying the phase matching in each FWM process. Changing arrow thickness in the dual-pump case is used to illustrate the vectors entering and exiting the plane of the page.