Physical Review A

Coherent population trapping in open systems: A coupled/noncoupled-state analysis

Ferruccio Renzoni, Albrecht Lindner, E. Arimondo

Published Jul 1, 1999• 50 citations

Abstract

Coherent population trapping (CPT) with losses towards off-resonant levels is analyzed on the basis of the coupled/noncoupled states. We derive that at a long interaction time the noncoupled state is depopulated at a rate that is decreasing for decreasing Raman detuning. This behavior produces a narrowing of the dark resonance with the interaction time. The asymptotic evolution of the atomic system is completely characterized by the effective width ${\ensuremath{\Gamma}}_{\mathrm{NC}}$ of the noncoupled state. The analytic expression for this width evidences a quadratic dependence on the Raman detuning. This dependence implies that the width of the dark resonance decreases asymptotically with the inverse of the square root of the interaction time. This narrowing law, independent of the interaction scheme, has a general validity for CPT in open systems. Our analysis points out the differences between CPT with losses and CPT in closed systems.

DOI: 10.1103/physreva.60.450

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Population-loss-induced narrowing of dark resonances

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Depopulation and repopulation pumping in coherent population trapping

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Physical Review A

Coherent population trapping in open systems: A coupled/noncoupled-state analysis

Ferruccio Renzoni, Albrecht Lindner, E. Arimondo

Published Jul 1, 1999• 50 citations

Publisher

Abstract

{\ensuremath{\Gamma}}_{\mathrm{NC}}

of the noncoupled state. The analytic expression for this width evidences a quadratic dependence on the Raman detuning. This dependence implies that the width of the dark resonance decreases asymptotically with the inverse of the square root of the interaction time. This narrowing law, independent of the interaction scheme, has a general validity for CPT in open systems. Our analysis points out the differences between CPT with losses and CPT in closed systems.

DOI: 10.1103/physreva.60.450

Related papers

Population-loss-induced narrowing of dark resonances

Ferruccio Renzoni, E. Arimondo

The dependence of the width of the coherent population trapping (CPT) resonance on the atomic interaction time is studied theoretically. The optical Bloch equations are solved for the Hanle/CPT schemes corresponding to the open ${F}_{g}\ensuremath{\rightarrow}{F}_{e}$ transitions of the sodium ${D}_{1}$ line interacting with linearly polarized laser light. The mechanism producing the narrowing of the Hanle/CPT resonance is examined, the role of the population losses is pointed out, and the law describing the narrowing of the width of the resonance with the interaction time is determined. Our results are compared with the Hanle/CPT linewidth dependence on the interaction time occurring for a closed optical transition.The dependence of the width of the coherent population trapping (CPT) resonance on the atomic interaction time is studied theoretically. The optical Bloch equations are solved for the Hanle/CPT schemes corresponding to the open ${F}_{g}\ensuremath{\rightarrow}{F}_{e}$ transitions of the sodium ${D}_{1}$ line interacting with linearly polarized laser light. The mechanism producing the narrowing of the Hanle/CPT resonance is examined, the role of the population losses is pointed out, and the law describing the narrowing of the width of the resonance with the interaction time is determined. Our results are compared with the Hanle/CPT linewidth dependence on the interaction time occurring for a closed optical transition.

Physical Review ADec 1, 199840 citations

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Ferruccio Renzoni, E. Arimondo

Europhysics Letters (EPL)Jun 15, 199914 citations

Analytical expressions for parameters of the dark resonance in a vacuum vapour cell

Denis V. Brazhnikov, Moustafa Abdel Hafiz, Grégoire Coget +1 more

In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)Jul 1, 20170 citations

Transient coherent population trapping in a closed loop interaction scheme

Wolfgang Maichen, Ferruccio Renzoni, I. E. Mazets +2 more

We investigated the existence of coherent population trapping (CPT) as a transient effect dependent on the rate of change of interaction parameters between an atom and light waves. For that purpose we exploited the phase dependence of CPT in a closed double-\ensuremath{\Lambda} interaction scheme, varying the total phase of the atomic transition loop linearly in time by detuning the light waves from exact four-photon resonance. For a slowly changing phase (that is, for small detunings) we observed the subsequent creation and destruction of CPT depending on the varying phase, while for higher speed of phase change CPT disappears. An analytical solution for a special case for a double-\ensuremath{\Lambda} interaction scheme that can be used to understand the principles of this phenomenon is presented. A qualitative comparison with our experiment shows good agreement within the limits of our theoretical considerations, confirming the validity of the basic idea. \textcopyright{} 1996 The American Physical Society.We investigated the existence of coherent population trapping (CPT) as a transient effect dependent on the rate of change of interaction parameters between an atom and light waves. For that purpose we exploited the phase dependence of CPT in a closed double-\ensuremath{\Lambda} interaction scheme, varying the total phase of the atomic transition loop linearly in time by detuning the light waves from exact four-photon resonance. For a slowly changing phase (that is, for small detunings) we observed the subsequent creation and destruction of CPT depending on the varying phase, while for higher speed of phase change CPT disappears. An analytical solution for a special case for a double-\ensuremath{\Lambda} interaction scheme that can be used to understand the principles of this phenomenon is presented. A qualitative comparison with our experiment shows good agreement within the limits of our theoretical considerations, confirming the validity of the basic idea. \textcopyright{} 1996 The American Physical Society.

Physical Review AMay 1, 199654 citations

Coherent population trapping in quantized light field

Taichenachev, A V, Tumaikin, A M, Yudin, V I

A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions $F_g\\to F_e$ , including bright transitions $F\\to F+1$ and $F''\\to F''$ with $F''$ a half-integer, for which the CPT effect is absent in a classical field.A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions $F_g\\to F_e$, including bright transitions $F\\to F+1$ and $F''\\to F''$ with $F''$ a half-integer, for which the CPT effect is absent in a classical field.

Apr 19, 20057 citations

Power dependence of the coherent-population-trapping resonances registered in fluorescence and transmission: Resonance-width narrowing effects

S. Gateva, Emilia A. Alipieva, Elena T. Taskova

The investigation of the coherent-population-trapping (CPT) resonance on the degenerate two-level system of the $({F}_{g}=2\ensuremath{\rightarrow}{F}_{e}=1)$ transition of the $^{87}\mathrm{Rb}\phantom{\rule{0.3em}{0ex}}{D}_{1}$ line by means of a Hanle effect configuration in an uncoated vacuum cell [Opt. Lett. 28, 1817 (2003)] has shown that the measured in fluorescence resonance has a complex shape---a very narrow (about 1 mG) structure superimposed on a broader one (about few tens of mG). In this work, the dependence of the width and amplitude of the CPT resonance structures are measured at different laser power densities when the registration is in fluorescence and transmission. While the narrow resonance width does not change within the limits of the accuracy of our measurements, the wide resonance width dependence is complex and is different in fluorescence and transmission. The origins of the observed resonance width narrowing of the CPT resonance structures registered in fluorescence and transmission are discussed.The investigation of the coherent-population-trapping (CPT) resonance on the degenerate two-level system of the $({F}_{g}=2\ensuremath{\rightarrow}{F}_{e}=1)$ transition of the $^{87}\mathrm{Rb}\phantom{\rule{0.3em}{0ex}}{D}_{1}$ line by means of a Hanle effect configuration in an uncoated vacuum cell [Opt. Lett. 28, 1817 (2003)] has shown that the measured in fluorescence resonance has a complex shape---a very narrow (about 1 mG) structure superimposed on a broader one (about few tens of mG). In this work, the dependence of the width and amplitude of the CPT resonance structures are measured at different laser power densities when the registration is in fluorescence and transmission. While the narrow resonance width does not change within the limits of the accuracy of our measurements, the wide resonance width dependence is complex and is different in fluorescence and transmission. The origins of the observed resonance width narrowing of the CPT resonance structures registered in fluorescence and transmission are discussed.

Physical Review AAug 17, 200512 citations

{\ensuremath{\Gamma}}_{\mathrm{NC}}

{D}_{1}

line interacting with linearly polarized laser light. The mechanism producing the narrowing of the Hanle/CPT resonance is examined, the role of the population losses is pointed out, and the law describing the narrowing of the width of the resonance with the interaction time is determined. Our results are compared with the Hanle/CPT linewidth dependence on the interaction time occurring for a closed optical transition.

F_g\\to F_e

, including bright transitions

F\\to F+1

and

F''\\to F''

with

F''

a half-integer, for which the CPT effect is absent in a classical field.

The investigation of the coherent-population-trapping (CPT) resonance on the degenerate two-level system of the $({F}_{g}=2\ensuremath{\rightarrow}{F}_{e}=1)$ transition of the

^{87}\mathrm{Rb}\phantom{\rule{0.3em}{0ex}}{D}_{1}

line by means of a Hanle effect configuration in an uncoated vacuum cell [Opt. Lett. 28, 1817 (2003)] has shown that the measured in fluorescence resonance has a complex shape---a very narrow (about 1 mG) structure superimposed on a broader one (about few tens of mG). In this work, the dependence of the width and amplitude of the CPT resonance structures are measured at different laser power densities when the registration is in fluorescence and transmission. While the narrow resonance width does not change within the limits of the accuracy of our measurements, the wide resonance width dependence is complex and is different in fluorescence and transmission. The origins of the observed resonance width narrowing of the CPT resonance structures registered in fluorescence and transmission are discussed.