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Physical Review A

Coherent population trapping under periodic polarization modulation: Appearance of the CPT doublet

M. Huang, James C. Camparo

Published Jan 12, 2012• 20 citations
Publisher

Abstract

In order to understand how stochastic processes might enter and influence coherent atomic dynamics, we have studied the behavior of a \ensuremath{\Lambda} system under periodic polarization modulation. Polarization, in addition to amplitude and phase, is a defining feature of a classical vector field. However, to date, there has been little study concerning the response of quantum systems to temporal variations in polarization, even though some lasers are known to exhibit stochastic polarization fluctuations. In our work, we square-wave modulate the polarization of a laser that induces coherent-population trapping (CPT) in ${}^{87}$Rb. At low-modulation frequencies, we find that the amplitude of the CPT resonance increases with modulation frequency because the polarization variations limit the number of atoms confined to the system's trapping state. Surprisingly, at higher-modulation frequencies, we find that the CPT resonance splits into a doublet. We have developed an analytical theory of CPT in the presence of polarization modulation that captures the primary features of our experimental findings and shows that the doublet is a consequence of ground-state coherence modulation in the \ensuremath{\Lambda} system. The present results lay a foundation for understanding how more complicated (i.e., stochastic) temporal variations in laser polarization could influence \ensuremath{\Lambda}-system dynamics.

DOI
10.1103/physreva.85.012509

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We have developed and tested an efficient mathematical model for calculating a coherent population trapping (CPT) resonance in a periodically modulated laser field when the modulation frequency $f$ is close to the fractional part of the hyperfine splitting frequency (HFS) in the ground state ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}/N$ (where $N=1,2,3,\dots{}$). In such a polychromatic light, only two frequency components that are most resonant with the working optical transitions in the atom are taken into account exactly, while all other off-resonant frequency components are taken into account by using the second-order perturbation theory in the field magnitude. Within the framework of our concept, an equation for the atomic density matrix is obtained, in which all off-resonant frequency components are described by two operators (which are nondiagonal in the general case): a shift operator and a relaxation operator. This model has been verified by numerical calculations for various dependencies, in which we have not found visual differences from the results of exact calculations. Thus, in our bichromatic model, the infinite variety of periodically modulated fields in CPT spectroscopy can be uniformly described by only five parameters. In addition to a significant mathematical simplification, our model provides a clear physical picture of the various features of CPT spectroscopy in a periodically modulated laser field, including some effects that have not yet been discussed in the scientific literature. In particular, we show a widespread viewpoint that the CPT resonance shift is determined by standard ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}$ between different off-resonant frequency components can be comparable to the value of standard ac Stark shift (or even dominate it). Therefore, even if we have detailed information on the spectral composition of the modulated field (e.g., using a spectrum analyzer), this is, in general, absolutely insufficient for determining the light shift of the CPT resonance.

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In order to understand how stochastic processes might enter and influence coherent atomic dynamics, we have studied the behavior of a \ensuremath{\Lambda} system under periodic polarization modulation. Polarization, in addition to amplitude and phase, is a defining feature of a classical vector field. However, to date, there has been little study concerning the response of quantum systems to temporal variations in polarization, even though some lasers are known to exhibit stochastic polarization fluctuations. In our work, we square-wave modulate the polarization of a laser that induces coherent-population trapping (CPT) in 87{}^{87}87Rb. At low-modulation frequencies, we find that the amplitude of the CPT resonance increases with modulation frequency because the polarization variations limit the number of atoms confined to the system's trapping state. Surprisingly, at higher-modulation frequencies, we find that the CPT resonance splits into a doublet. We have developed an analytical theory of CPT in the presence of polarization modulation that captures the primary features of our experimental findings and shows that the doublet is a consequence of ground-state coherence modulation in the \ensuremath{\Lambda} system. The present results lay a foundation for understanding how more complicated (i.e., stochastic) temporal variations in laser polarization could influence \ensuremath{\Lambda}-system dynamics.
In order to understand how stochastic processes enter and influence coherent atomic dynamics, we have studied the behavior of a \ensuremath{\Lambda} system under random polarization variations. Polarization, in addition to amplitude and phase, is a defining feature of a classical vector field. However, to date there has been little study of how quantum systems respond to temporal variations of polarization, even though this problem has practical implications. In our work, we generate a Bernoulli sequence of random polarization changes, and we then examine the average 87{}^{87}87Rb coherent-population-trapping (CPT) line shape induced by this stochastic field. To quantitatively conceptualize our results, we have developed an independent-modulator approximation (IMA) theory for CPT line shapes induced by stochastic-polarization fields. The IMA theory is based on the idea that a power spectrum can be understood as a probability distribution of Fourier modulation frequencies. We compare the IMA theory with our experimental results, finding quite good agreement when the polarization correlation time is less than or equal to the CPT dephasing time, which is the regime of primary experimental and technological interest. The utility of the IMA theory lies in its intuitive nature, which we believe has merit for guiding experimentalists' general understanding of stochastic fields and quantum systems.
The Coherent-Population-Trapping (CPT) vapor-cell atomic clock represents a clear break from previous vapor-cell clock technology. As such it has advantages that may result in significant frequency stability improvements. However, along with the technology's benefits come new effects that may limit the CPT clock's capabilities, and one such effect is laser polarization noise. Here, we present our first results looking at the response of a CPT signal to modulation of a laser's polarization. Our results show that the CPT clock signal undergoes a splitting when the polarization modulation is rapid, suggesting that various Fourier components of a field's stochastic polarization fluctuations will affect the CPT clock signal in qualitatively different ways.
In ultraminiature atomic physics (UAP), where the goal is to perform low-power, 'chip-scale' precision spectroscopy, stochastic-field/atom interactions can play a primary role in defining the limits of sensitivity. Unfortunately, the transient responses of a quantum system to random changes in the amplitude, phase or polarization of a resonant field are not well understood, forming the basis of what has come to be known as the stochastic-field/atom interaction problem. As the first step in understanding the more complicated stochastic problem, the present work considers the response of a coherent-population-trapping (CPT) signal to a sudden, step-function change in laser polarization. We find that the transient behaviour depends on both the redistribution of atomic population among the atom's Zeeman sublevels and the regeneration of atomic coherence between these sublevels. Despite the complicated nature of the dynamics, we develop and experimentally validate a semi-empirical, reasonably intuitive model of the CPT transient and demonstrate that in the 'typical' CPT signal the polarization-induced transients are dominated by redistribution of atomic population among the Zeeman sublevels. Further, the amplitudes of the polarization-induced transients are relatively large and could potentially 'swamp' the CPT signals of interest for UAP.
We have developed an effective mathematical model to calculate the coherent population trapping (CPT) resonance in periodically modulated light, when the modulation frequency fff varies near the fractional part of hyperfine splitting in the ground state Δhfs/NΔ_{\rm hfs}/NΔhfs​/N (where N=1,2,...N=1,2,...N=1,2,...). In such polychromatic field, only two frequency components that are most resonant with atomic optical transitions are taken into account accurately, while all other off-resonant components are taken into account using the second-order perturbation theory in the field. Within the presented concept, equation for atomic density matrix is obtained, in which the contribution of all off-resonant components is reduced to the appearance of two new operators (non-diagonal, in general case): the shift operator and relaxation operators. In the case of three-level ΛΛΛ-system, the adequacy of presented effective model was verified by numerical calculations of various dependencies, in which we did not find visual differences from the exact calculations. In addition to a significant mathematical simplification, our model provides a clear physical picture of various features of CPT spectroscopy in a periodically modulated laser field, including effects that have not been discussed in the scientific literature before. In particular, we show that the widespread viewpoint that the CPT resonance shift is determined by usual ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency ΔhfsΔ_{\rm hfs}Δhfs​ between different off-resonant frequency components can be comparable (or even dominate) with the standard ac Stark shift. Therefore, even if we have detailed information on the modulated field spectrum (e.g. using a spectrum analyzer), this is, in general, absolutely insufficient to determine the light shift of CPT resonance.
Coherent population trapping (CPT) resonance formation is modeled numerically in a three-level Λ -system with one of the near-resonance fields being frequency-modulated. The model is based on density matrix equations in RW approximation with atomic relaxation properly taken into account. Slow modulation is shown to be equivalent to CW excitation with the frequency changed point by point. As the modulation period approaches the relaxation times, the delayed response of the system is shown to cause the CPT resonance shift and reshaping, as well as the appearance of transient oscillations of the energy level populations.
We have developed and tested an efficient mathematical model for calculating a coherent population trapping (CPT) resonance in a periodically modulated laser field when the modulation frequency fff is close to the fractional part of the hyperfine splitting frequency (HFS) in the ground state ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}/N$ (where N=1,2,3,…N=1,2,3,\dots{}N=1,2,3,…). In such a polychromatic light, only two frequency components that are most resonant with the working optical transitions in the atom are taken into account exactly, while all other off-resonant frequency components are taken into account by using the second-order perturbation theory in the field magnitude. Within the framework of our concept, an equation for the atomic density matrix is obtained, in which all off-resonant frequency components are described by two operators (which are nondiagonal in the general case): a shift operator and a relaxation operator. This model has been verified by numerical calculations for various dependencies, in which we have not found visual differences from the results of exact calculations. Thus, in our bichromatic model, the infinite variety of periodically modulated fields in CPT spectroscopy can be uniformly described by only five parameters. In addition to a significant mathematical simplification, our model provides a clear physical picture of the various features of CPT spectroscopy in a periodically modulated laser field, including some effects that have not yet been discussed in the scientific literature. In particular, we show a widespread viewpoint that the CPT resonance shift is determined by standard ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}$ between different off-resonant frequency components can be comparable to the value of standard ac Stark shift (or even dominate it). Therefore, even if we have detailed information on the spectral composition of the modulated field (e.g., using a spectrum analyzer), this is, in general, absolutely insufficient for determining the light shift of the CPT resonance.