Laser Phase Spectroscopy in Closed-Loop Multilevel Schemes

Abstract

Atomic/molecular systems with closed-loop configurations have singular features associated with the creation of multiple interconnected coherences. The resonant condition is determined by the full-loop detuning and the full-loop phase of the laser excitation. The main theoretical features and experimental tests published so far are reviewed. Perspectives of laser spectroscopy based on closed loops are discussed.

This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.

Appendix: Triangle Density Matrix Equations and Solution

We write the $\rho$ density matrix equations including the decay rates depending on the level configuration. For the case of triangle-$\varLambda$ configuration the only nonzero decay elements are ${\gamma}_{\mathrm{g}}$ the ground state rate equilibrating the $\mid a\Big\rangle$ and $\mid c\Big\rangle$ populations, and ${\gamma}_b$ the $\mid b\Big\rangle$ state decay rate into $\mid a\Big\rangle$ and $\mid c\Big\rangle$ ground states, for simplicity with equal branching ratios. Applying the rotating-wave approximation, the Hamiltonian of Eq. 1 leads to the following equations for populations and coherences:

$$ {\displaystyle \begin{array}{cc}\hfill {\dot{\rho}}_{aa}=& {\gamma}_{\mathrm{g}}\left({\rho}_{cc}-{\rho}_{aa}\right)+\frac{\gamma_b}{2}{\rho}_{bb}\hfill \\ {}\hfill & -\left[i\frac{\varOmega_3}{2}{\mathrm{e}}^{-i{\omega}_3t}{\rho}_{ab}+i\frac{\varOmega_1}{2}{\mathrm{e}}^{-i\left({\omega}_1t-\varPhi \right)}{\rho}_{ac}+\mathrm{H}.\mathrm{c}.\right],\hfill \\ {}\hfill {\dot{\rho}}_{bb}=& -{\gamma}_b{\rho}_{bb}+\left[i\frac{\varOmega_3}{2}{\mathrm{e}}^{-i{\omega}_3t}{\rho}_{ab}-i\frac{\varOmega_2}{2}{\mathrm{e}}^{-i{\omega}_2t}{\rho}_{bc}+\mathrm{H}.\mathrm{c}.\right],\hfill \\ {}\hfill {\dot{\rho}}_{cc}=& {\gamma}_{\mathrm{g}}\left({\rho}_{aa}-{\rho}_{cc}\right)+\frac{\gamma_b}{2}{\rho}_{bb}\hfill \\ {}\hfill & +\left[i\frac{\varOmega_2}{2}{\mathrm{e}}^{-i{\omega}_2t}{\rho}_{bc}+i\frac{\varOmega_1}{2}{\mathrm{e}}^{-i\left({\omega}_1t-\varPhi \right)}{\rho}_{ac}+\mathrm{H}.\mathrm{c}.\right],\hfill \\ {}\hfill {\dot{\rho}}_{ab}=& -\left(\frac{\gamma_b+{\gamma}_{\mathrm{g}}}{2}-i{\omega}_b\right){\rho}_{ab}+i\frac{\varOmega_1}{2}{\mathrm{e}}^{i{\omega}_1t}\left({\rho}_{bb}-{\rho}_{aa}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_3}{2}{\mathrm{e}}^{i\left({\omega}_3t-{\varPhi}_T\right)}{\rho}_{cb}-i\frac{\varOmega_2}{2}{\mathrm{e}}^{-i{\omega}_2t}{\rho}_{ac},\hfill \\ {}\hfill \dot{\rho_{bc}}=& -\left[\frac{\gamma_b+{\gamma}_{\mathrm{g}}}{2}-i\left({\omega}_c-{\omega}_b\right)\right]{\rho}_{bc}+i\frac{\varOmega_2}{2}{\mathrm{e}}^{i{\omega}_2t}\left({\rho}_{cc}-{\rho}_{bb}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_3}{2}{\mathrm{e}}^{-i\left({\omega}_3t-{\varPhi}_T\right)}{\rho}_{ac}-i\frac{\varOmega_1}{2}{\mathrm{e}}^{-i{\omega}_1t}{\rho}_{ba},\hfill \\ {}\hfill {\dot{\rho}}_{ac}=& -\left({\gamma}_{\mathrm{g}}-i{\omega}_c\right){\rho}_{ca}+i\frac{\varOmega_3}{2}{\mathrm{e}}^{i\left({\omega}_3t-{\varPhi}_T\right)}\left({\rho}_{cc}-{\rho}_{aa}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_1}{2}{\mathrm{e}}^{i{\omega}_1t}{\rho}_{bc}-i\frac{\varOmega_3}{2}{\mathrm{e}}^{i\left({\omega}_3t-{\varPhi}_T\right)}{\rho}_{ab}.\hfill \end{array}} $$

(10)

Introducing the standard transformation to a rotating frame [16]

$$ {\displaystyle \begin{array}{cc}\hfill {\rho}_{ba}& ={\sigma}_{ba}{\mathrm{e}}^{-i{\omega}_1t},{\rho}_{cb}={\sigma}_{cb}{\mathrm{e}}^{-i{\omega}_2t},\hfill \\ {}\hfill {\rho}_{ca}& ={\sigma}_{ca}{\mathrm{e}}^{-i\left({\omega}_3t-{\varPhi}_T\right)},{\rho}_{ii}={\sigma}_{ii},\hfill \end{array}} $$

(11)

with $\left(i=a,b,c\right)$, we obtain a new set of equations

$$ {\displaystyle \begin{array}{cc}\hfill {\dot{\sigma}}_{aa}=& {\gamma}_{\mathrm{g}}\left({\sigma}_{cc}-{\sigma}_{aa}\right)+\frac{\gamma_b}{2}{\sigma}_{bb}-i\frac{\varOmega_1}{2}\left({\sigma}_{ab}-{\sigma}_{ba}\right)-i\frac{\varOmega_3}{2}\left({\sigma}_{ac}-{\sigma}_{ca}\right),\hfill \\ {}\hfill {\dot{\sigma}}_{bb}=& -{\gamma}_b{\sigma}_{bb}+i\frac{\varOmega_1}{2}\left({\sigma}_{ab}-{\sigma}_{ba}\right)-i\frac{\varOmega_2}{2}\left({\sigma}_{bc}-{\sigma}_{cb}\right),\hfill \\ {}\hfill {\dot{\sigma}}_{cc}=& {\gamma}_{\mathrm{g}}\left({\sigma}_{aa}-{\sigma}_{cc}\right)+\frac{\gamma_b}{2}{\sigma}_{bb}+i\frac{\varOmega_2}{2}\left({\sigma}_{bc}-{\sigma}_{cb}\right)+i\frac{\varOmega_3}{2}\left({\sigma}_{ac}-{\sigma}_{ca}\right),\hfill \\ {}\hfill {\dot{\sigma}}_{ab}=& -\left(\frac{\gamma_b+{\gamma}_{\mathrm{g}}}{2}+i{\delta}_1\right){\sigma}_{ab}+i\frac{\varOmega_1}{2}\left({\sigma}_{bb}-{\sigma}_{aa}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_3}{2}{\mathrm{e}}^{i{\Delta}_Tt}{\mathrm{e}}^{-i{\varPhi}_T}{\sigma}_{cb}-i\frac{\varOmega_2}{2}{\mathrm{e}}^{i{\Delta}_Tt}{\mathrm{e}}^{-i{\varPhi}_T}{\sigma}_{ac},\hfill \\ {}\hfill {\dot{\sigma}}_{bc}=& -\left(\frac{\gamma_b+{\gamma}_{\mathrm{g}}}{2}+i{\delta}_2\right){\sigma}_{bc}+i\frac{\varOmega_2}{2}\left({\sigma}_{cc}-{\sigma}_{bb}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_1}{2}{\mathrm{e}}^{i{\Delta}_Tt}{\mathrm{e}}^{-i{\varPhi}_T}{\sigma}_{ac}-i\frac{\varOmega_3}{2}{\mathrm{e}}^{i{\Delta}_Tt}{\mathrm{e}}^{-i{\varPhi}_T}{\sigma}_{ba},\hfill \\ {}\hfill {\dot{\sigma}}_{ac}=& -\left({\gamma}_{\mathrm{g}}+i{\delta}_3\right){\sigma}_{ac}+i\frac{\varOmega_3}{2}\left({\sigma}_{cc}-{\sigma}_{aa}\right)\hfill \\ {}\hfill & +i\frac{\varOmega_1}{2}{\mathrm{e}}^{-i{\Delta}_Tt}{\mathrm{e}}^{i{\varPhi}_T}{\sigma}_{bc}-i\frac{\varOmega_2}{2}{\mathrm{e}}^{-i{\Delta}_Tt}{\mathrm{e}}^{i{\varPhi}_T}{\sigma}_{ab}.\hfill \end{array}} $$

(12)

These equations become time-independent only in the case of ${\Delta}_T=0$, leading to time-independent populations ${\rho}_{ii}$ and to ${\rho}_{ij}$ coherences with oscillations at the corresponding laser driving frequency given by Eq. 11.

For ${\Delta}_T\ne 0$ the time-dependent density matrix equations were solved in Ref. [20] through a series expansion in terms of the Rabi frequencies, where the connection between Rabi frequency power dependence and the high-order multiphoton processes was pointed out.

Within the approach of Eq. 6, each density matrix element is expanded into a Fourier expansion with all the harmonics of the ${\Delta}_T$ angular frequency. The Fourier harmonics are proportional to the Rabi frequencies, higher-order terms containing higher powers of the Rabi frequencies, and therefore produced by high-order multiphoton processes. The Fourier components satisfy recurrent equations that can be solved through a continued fraction solution as in refs. [23, 49]. The triangle configuration with a $9\times 9$ density matrix leads to a continued fraction of matrices with the same order. Instead the four-level system is solved on the basis of $16\times 16$ continued fraction matrices.