© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2607.23630

Effect of classical noises on the coherent population trapping based on the Green's function approach to the multiplicative stochastic processes

M. Mirzaee, B. Askari, Ali Motazedifard, A. Dalafi

quant-phSubmitted Jul 26, 2026• Announced Jul 28, 2026• 0 citations

Abstract

Inspired by the Green's function (GF) approach in quantum field theory (QFT) and many body physics, we have developed a mathematical formalism to investigate classical multiplicative stochastic processes. Based on this approach, the interacting GF of any dynamical system subjected to classical stochastic noises, which enter into the system equations of motion in a multiplicative way, can be obtained from the noninteracting (free of noise) GF through an infinite perturbative series which may converge to an exact closed form under special conditions. Using this formalism, we have studied the effects of classical noises of the driving laser on the coherent population trapping (CPT) which have a crucial role in the performance of CPT-based atomic clocks. We have shown that if the bandwidth of the colored noise is sufficiently larger than the system damping rate, the infinite series corresponding to the interacting GF can be approximated by the closed form. The presented formalism enables us to investigate all kinds of homogeneous and inhomogeneous broadening mechanisms on the CPT transmission resonance lineshape, including dephasing due to atomic collisions, power/Doppler broadenings, as well as the broadening mechanisms due to phase and amplitude fluctuations of driving laser and compared their destructive effect with each other. It should be emphasized that the presented formalism is applicable to any dynamical system with multiplicative stochastic noises and the CPT phenomenon is just a prototype for the application of the presented formalism in practice.

Related papers

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

M. Huang, James C. Camparo

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.

Physical Review A•Jan 12, 2012•20 citations

Coherent population trapping and polarization fluctuations: The independent-modulator approximation for coherent-population-trapping line shapes

M. Huang, Travis U. Driskell, James C. Camparo

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 ${}^{87}$Rb 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.

Physical Review A•May 31, 2013•1 citation

Inhomogeneous light shifts of coherent population trapping resonances

J. W. Pollock, Valeriy I. Yudin, А. В. Тайченачев +5 more

Coherent population trapping (CPT) in atomic vapors using all-optical interrogation has enabled the miniaturization of microwave atomic clocks. Light shifts induced by the CPT driving fields can impact the spectral profile of CPT resonances and are a common limit to the long-term stability of CPT clocks. Nonlinear light shifts have been observed in several CPT systems and have not been explored in detail. In this Letter, we demonstrate that nonlinear light shifts in CPT clocks can arise from spatially inhomogeneous CPT driving fields. We measure this effect using Gaussian laser beams in a buffer gas cell and show strong agreement with a four-level model describing the CPT Λ-system with a noninteracting “trap” state. We estimate the effect of this nonlinearity on recently developed light shift mitigation techniques and suggest improvements to existing techniques.

Applied Physics Letters•Apr 11, 2022•11 citations

Ultra-Compact Frequency Standards Using CPT Phenomenon

Kyoya Fukuda

In recent years, the coherent population trapping (CPT) phenomenon using a very small cell has attracted considerable attention for its potential application to ultra-compact atomic clock. For the miniaturization of CPT clock, it is important to clarify influence by the collision with atoms and cell walls. We study the effect of cell geometry on the CPT spectrum using a thin Cs vapor cell without buffer gas. When the cell length is below 1 mm and the laser intensity is sufficiently weak, the transit-time broadening is largely suppressed because only slow atoms have enough time to settle down on the dark state and contribute to the CPT signals. Furthermore, we show that introduction of buffer gas may broaden the CPT resonance in extremely thin cells. We also show that sub-Doppler signal which can be used for stabilization of laser frequency is obtained by using the CPT signal.

The Review of Laser Engineering•Jan 1, 2007•1 citation

Theoretical study of quantum dissipation and laser-noise effects on the atomic response

Ouafae El Akramine, A. Makhoute, Mohamed Zitane +1 more

The nonlinear dynamics of dissipative quantum systems in incoherent laser fields is studied in the framework of a master equation with the random telegraph model describing the laser noise and the Markovian approximation dealing with the system-bath couplings. Floquet theory and time-dependent perturbation methods are used to facilitate both analytical and numerical solutions. We develop a theoretical formalism that provides a powerful tool for the detailed analysis of the dissipative quantum dynamics of multilevel systems driven by intense stochastic laser fields. It is found that the system relaxes to a steady state from the effect of the laser phase and frequency noise and the kinetics of this relaxation increases with the addition of dissipative terms, introduced by the coupling to the reservoir. Amplitude fluctuations show a different behavior. Other results concerning the destruction of quantum coherence and the dynamical localization will be established and further relaxation mechanisms such as spontaneous emission and the ionization process will also be considered.

arXiv:physics/9911034•Physical Review A•Dec 1, 1998•7 citations

The influence of laser polarization variations on CPT atomic clock signals

Michael C. Y. Huang, James C. Camparo

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.

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)•May 1, 2011•2 citations

Inspired by the Green's function (GF) approach in quantum field theory (QFT) and many body physics, we have developed a mathematical formalism to investigate classical multiplicative stochastic processes. Based on this approach, the interacting GF of any dynamical system subjected to classical stochastic noises, which enter into the system equations of motion in a multiplicative way, can be obtained from the noninteracting (free of noise) GF through an infinite perturbative series which may converge to an exact closed form under special conditions. Using this formalism, we have studied the effects of classical noises of the driving laser on the coherent population trapping (CPT) which have a crucial role in the performance of CPT-based atomic clocks. We have shown that if the bandwidth of the colored noise is sufficiently larger than the system damping rate, the infinite series corresponding to the interacting GF can be approximated by the closed form. The presented formalism enables us to investigate all kinds of homogeneous and inhomogeneous broadening mechanisms on the CPT transmission resonance lineshape, including dephasing due to atomic collisions, power/Doppler broadenings, as well as the broadening mechanisms due to phase and amplitude fluctuations of driving laser and compared their destructive effect with each other. It should be emphasized that the presented formalism is applicable to any dynamical system with multiplicative stochastic noises and the CPT phenomenon is just a prototype for the application of the presented formalism in practice.

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.

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}

Rb 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.

Coherent population trapping (CPT) in atomic vapors using all-optical interrogation has enabled the miniaturization of microwave atomic clocks. Light shifts induced by the CPT driving fields can impact the spectral profile of CPT resonances and are a common limit to the long-term stability of CPT clocks. Nonlinear light shifts have been observed in several CPT systems and have not been explored in detail. In this Letter, we demonstrate that nonlinear light shifts in CPT clocks can arise from spatially inhomogeneous CPT driving fields. We measure this effect using Gaussian laser beams in a buffer gas cell and show strong agreement with a four-level model describing the CPT Λ-system with a noninteracting “trap” state. We estimate the effect of this nonlinearity on recently developed light shift mitigation techniques and suggest improvements to existing techniques.

In recent years, the coherent population trapping (CPT) phenomenon using a very small cell has attracted considerable attention for its potential application to ultra-compact atomic clock. For the miniaturization of CPT clock, it is important to clarify influence by the collision with atoms and cell walls. We study the effect of cell geometry on the CPT spectrum using a thin Cs vapor cell without buffer gas. When the cell length is below 1 mm and the laser intensity is sufficiently weak, the transit-time broadening is largely suppressed because only slow atoms have enough time to settle down on the dark state and contribute to the CPT signals. Furthermore, we show that introduction of buffer gas may broaden the CPT resonance in extremely thin cells. We also show that sub-Doppler signal which can be used for stabilization of laser frequency is obtained by using the CPT signal.

The nonlinear dynamics of dissipative quantum systems in incoherent laser fields is studied in the framework of a master equation with the random telegraph model describing the laser noise and the Markovian approximation dealing with the system-bath couplings. Floquet theory and time-dependent perturbation methods are used to facilitate both analytical and numerical solutions. We develop a theoretical formalism that provides a powerful tool for the detailed analysis of the dissipative quantum dynamics of multilevel systems driven by intense stochastic laser fields. It is found that the system relaxes to a steady state from the effect of the laser phase and frequency noise and the kinetics of this relaxation increases with the addition of dissipative terms, introduced by the coupling to the reservoir. Amplitude fluctuations show a different behavior. Other results concerning the destruction of quantum coherence and the dynamical localization will be established and further relaxation mechanisms such as spontaneous emission and the ionization process will also be considered.

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.