Electronic phase-locking for three-color, two-pathway coherent control

TL;DR

This work tackles the measurement of very weak optical transitions by enabling coherent interference between multiple optical pathways. It introduces a three-color, electronically phase-locked cw-laser scheme to realize two-pathway coherent control beyond prior two-color methods, and derives explicit two-photon transition amplitudes for parallel and perpendicular polarizations, including angular-momentum considerations. The authors demonstrate the technique experimentally in cesium by phase-locking three lasers to produce a coherent three-color interaction that interferes with a Stark-induced one-photon path, achieving phase-sensitive detection with a measurable modulation at $150$ Hz and observing interference for both $\\Delta F = 0$ and $\\Delta F = +1$. This approach expands the applicability of coherent control to weak transitions, with potential implications for precision parity-violation studies, Rydberg excitation, and microwave-to-optical signal conversion or electric-field sensing.

Abstract

We report a new method of two-pathway coherent control using three narrow-band cw laser sources, phase locked in an optical phase-lock loop, to maintain the high degree of optical coherence required for the coherent control process. In addition, we derive expressions for two-photon transition amplitudes and demonstrate their dependence on the polarization of the field components. This phase-locking technique expands the set of interactions to which coherent control techniques may be applied. It also allows for a constant low-frequency offset between the optical interactions, producing a continuous and constant phase ramp between the interactions, facilitating phase-sensitive detection of the modulating atomic signal. We illustrate this technique with two-photon vs.~one-photon excitation of a $ΔF = 1$ component, and alternatively a $ΔF = 0$ component, of the $6s \: ^2S_{1/2} \rightarrow 7s \: ^2S_{1/2}$ transition of atomic cesium.

Figures (5)

• Figure 1: Energy level diagram of the relevant energy levels of cesium in the coherent control interaction. The $6s \rightarrow 7s$ transition is driven via a single-photon ($\lambda = 539.5$ nm) interaction, represented by the green arrow, and concurrently by the two-photon interaction, represented by the red arrows (one photon at 852 nm, which is close to, but not resonant with, the $6s \rightarrow p_{3/2}$ transition; the second at 1470 nm). The transition shown in this plot is an example of a $\Delta F = 0$ transition.
• Figure 2: A high-level diagram depicting the technique that generates laser fields necessary to drive coherent one- and two-photon transitions. ECDL - external cavity diode laser, with the wavelength of the output (in nm) indicated for each, AOM - acousto-optic modulator, SHG - second harmonic generation crystal, SFG - sum-frequency generation crystal, AWG - arbitrary waveform generator, PD - photodiode, $\bigotimes$ - frequency mixer. The FALC110 is a fast servo and the ADF4002 chip is used to generate an error signal from the beatnote between the SHG beam and the SFG beam. We list here these specific instruments, not necessarily as an endorsement/requirement, but what we chose to use. A sufficiently fast servo or a comparable digital phase lock loop (PLL) chip would also work.
• Figure 3: Phase-locked beatnote spectrum between the SHG and the SFG beams used to generate coherence between the two-photon and single-photon transitions. The resolution bandwidth is 1 kHz.
• Figure 4: Representative example of the 150 Hz modulation of the atomic excitation signal produced when the interfering excitations (single-photon and two-photon) differ by a constant frequency offset of 150 Hz. Figure a) shows the interference directly in a single trace observed on a digital oscilloscope. Figure b) shows the same interference after averaging eight traces. Here the $\Delta \nu = 150$ Hz reference signal is used as the trigger input to the oscilloscope.
• Figure 5: Representative examples of two-pathway coherent control signal collected while using the electronic phase-locking technique described in this report. a) A $\Delta F = 0$ transition ($F = 3 \rightarrow F^{\prime} = 3$), and b) a $\Delta F = +1$ transition ($F = 3 \rightarrow F^{\prime} = 4$), In each case, we show the output of the lock-in amplifier as we ramp the phase difference between the single-photon and two-photon transitions at 1 Hz.