Understanding photon sideband statistics and correlation for determining phonon coherence

Generating and detecting coherent high-frequency heat-carrying phonons have been topics of great interest in recent years. Although there have been successful attempts in generating and observing coherent phonons, rigorous techniques to characterize and detect phonon coherence in a crystalline material have been lagging compared to what has been achieved for photons. One main challenge is a lack of detailed understanding of how detection signals for phonons can be related to coherence. The quantum theory of photoelectric detection has greatly advanced the ability to characterize photon coherence in the past century, and a similar theory for phonon detection is necessary. Here, we reexamine the optical sideband fluorescence technique that has been used to detect high-frequency phonons in materials with optically active defects. We propose a quantum theory of phonon detection using the sideband technique and found that there are distinct differences in sideband counting statistics between thermal and coherent phonons. We further propose a second-order correlation function unique to sideband signals that allows for a rigorous distinction between thermal and coherent phonons. Our theory is relevant to a correlation measurement with nontrivial response functions at the quantum level and can potentially bridge the gap of experimentally determining phonon coherence to be on par with that of photons.

Figure 1

(a) Schematic of the sideband detection scheme. An incident optical beam gets absorbed and interacts with the local phonon population. The measured spectral properties of the transmitted photons with a grating spectrometer give information of the local phonon population. (b) Energy-level diagram of a defect electron with vibrational transitions for the ground state and excited state. Optical absorption couples the ground to the excited state where the zero-phonon line (ZPL) couples the same vibrational levels ($\alpha =\beta$) whereas any other case $\alpha \ne \beta$ implies that phonons are absorbed or emitted. The $S$ parameter determines the average phonon energy absorbed. (c) $\langle F\left(\alpha \right)\rangle$ versus the $S$ parameter and average phonon number $\langle \alpha \rangle$ for a thermal-state ensemble. (d) $\langle F\left(\alpha \right)\rangle$ versus the $S$ parameter and average phonon number $\langle \alpha \rangle$ for a coherent-state ensemble. Both (c) and (d) have similar behaviors at larger average phonon numbers, but there are noticeable differences between them at lower $S$ values.

Figure 2

(a)–(c) Plots of Eqs. (5) and (4) labeled as short (blue) and long (red), respectively. The coherent-state ensemble is labeled with a circle, and the thermal-state ensemble is labeled with a cross. The probability of sideband detection for coherent phonons is orders of magnitude lower than thermal phonons just at a few sideband counts. The short- and long-time limits do not show distinct variation in probability except for the intermediate value of the $S$ parameter. The average phonon number used here is $\langle \alpha \rangle =10$.

Figure 3

(a) Schematic of the proposed measurement of the second-order phonon correlation using sideband detection. An optical excitation generates a phonon beam which gets split by a clean interface into two paths. Detection on each path with the sideband technique allows us to measure the second-order correlation defined in Eq. (7). (b)–(d) Second-order sideband correlation for different values of the $S$ parameter and average phonon number $\langle \alpha \rangle$ at $\tau =0$. The red line indicates the coherent ensemble, and the blue line indicates the thermal ensemble. Note that the thermal state is always above the coherent state for all values of average phonon number $\langle \alpha \rangle$ and $S$ parameters.