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Edited by: Tamar Seideman, Northwestern University, United States

Reviewed by: Qing Ai, Beijing Normal University, China; Andre Bandrauk, Université de Sherbrooke, Canada

This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We propose and demonstrate a method for measuring the time evolution of the off-diagonal elements ρ_{n,n+k}(_{n,n+k}(_{n,n+1}(

Quantum coherence effects in molecular physics are largely based on the existence of the laser [

Fifty years ago the Scully-Lamb (SL) quantum theory of the laser (QTL) was developed using a density matrix formalism [

where α is the linear gain, β is the non-linear saturation coefficient, γ is the cavity loss rate, and 𝔑 is the normalization constant:

Equation (1) is plotted in

Steady state photon distribution function for coherent (orange dashed line) and laser radiation (blue solid line). The laser is taken to be 20 percent above threshold, 〈

The formalism developed in the QTL density matrix analysis has since been successfully applied to many other physical systems such as the single-atom maser(aka the micromaser) [

Parameters in laser and BEC systems.

α | Linear stimulated emission gain | Rate of cooling due to interaction with walls times the number of atom N |

β | Non-linear saturation due to the reabsorption of photons generated by stimulated emission | Non-linearity parameter due to the constraint that there are N atoms in the BEC: numerically equal to α/ |

γ | Loss rate due to photons absorbed in cavity mirrors etc. | Loss rate due to photon absorption from the thermal bath (walls) equal to |

The off-diagonal elements vanish at steady state, regressing to zero as [

where

where ν is the center frequency of the laser field and the electric field per photon is given by _{0} is the permittivity of free space and V is the laser cavity volume.

As is discussed in the following, the second order off-diagonal elements are given by the field operator averages

and the third order off-diagonal elements are given by

Equation (4) gives the time evolution associated with the first order off-diagonal elements

Many experiments have been carried out to determine the linewidth [

Experimental setup used in measuring the spectrum of the beat note between lasers 1 and 2. The beat note signal is measured by the detector (

For the first set of experiments, the first order coherence function [

where ρ = ρ_{1} ⊗ ρ_{2} is the density operator of the system, ρ_{1} and ρ_{2} represent the density operators of laser 1 and 2, ν_{1} and ν_{2} represent the center frequencies of the lasers 1 and 2, respectively.

From the above equation, we can see the only terms that carry the beat note frequency are

with its complex conjugate which contributes to the −ν_{0} frequency component, where ν_{0} ≡ ν_{2} − ν_{1}. Under the condition that the two lasers are independent, we can rewrite Equation (8) as

Taking the Fourier transform, we have a Lorentzian spectrum centered at the beat frequency ν_{0} with a width

The second and third experiments measure the spectral profile of the second and third order correlation of beat notes, the setup is shown in _{0} from the second order coherence function is

The correlated heterodyne signal is

Taking the Fourier transform, we get a Lorentzian spectral profile centered at 2ν_{0} with a width of 4

similarly, the signal of interest at frequency 3ν_{0} from the third order coherence function is

The correlated heterodyne signal is

We therefore get a Lorentzian spectral profile centered at 3ν_{0} with a width of 9

The main experimental results are shown in

Schematic setup for measuring higher order spectral line distribution up to 3rd order. Laser 1 and 2 : He-Ne lasers; P, polarizer; F, filter; A, analyzer; BS, non-polarizing beamsplitter; Mixer, frequency mixer;

Experimental results from the two sets of measurements. The bandwidths of the detectors are 50 MHz, the resolution bandwidth of the SA is 10 kHZ. The black dots are experimental data and the red curves are theory. ^{−D′t}) associated with frequency ν_{0}, as shown in Equation (10); ^{−4D′t}) associated with frequency 2ν_{0}, as shown in Equation (13). ^{−9D′t}) associated with frequency 3ν_{0}, as shown in Equation (16).

In conclusion, we have studied the time evolution of the higher degrees of off-diagonality obtained SL theory of the laser. We particularly measured the bandwidth of the laser beat note and the bandwidth of the correlated laser beat note, which reveal the evolution of the first, second, and third order off-diagonal elements of the laser density operator. The higher order spectra reveal the influence of the randomness in the phase of the laser field due to quantum fluctuation. Experimental results agreed with the SL QTL showing that the bandwidth of the third order and second order spectral profile are nine times and four times wider than that of the first order spectral profile, respectively.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

TP, YS, and MS discussed the design principle. TP and XZ performed the experiment and data analysis. TP and MS wrote the paper. All the authors polished the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Z. H. Yi, R. Nessler, H. Cai, and J. Sprigg for helpful discussion.