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Oxford University Press eBooks

Coherent Interactions of Fields with Atoms

Stephen Colby Rand

Published Jun 9, 2016• 0 citations
Publisher

Abstract

Chapter 5 begins the systematic application of the density matrix to describe optical interactions. Stationary two-and three-level atoms interacting with a traveling wave are compared to show that the presence of additional levels changes the saturation behavior of atoms. Then motion of the atoms is introduced and the phenomena of spatial and frequency-domain hole-burning are analyzed. Special effects due to standing waves, such as the light shift, are covered in this section. Next, tri-level coherence is considered, to extend the concept of coherences to two-photon transitions or Zeeman interactions in folded 3- or Λ-systems. Higher order interactions are also analyzed. This includes four-wave mixing and pump–probe techniques which form the basis of common nonlinear laser spectroscopies. Quantum interference is covered as a prelude to the topic of electromagnetically induced transparency. Higher order coherent interactions are also treated using a diagrammatic calculation technique based on Feynman diagrams.

DOI
10.1093/acprof:oso/9780198757450.003.0005

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Coherent Interactions of Fields and Atoms

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Reduced density matrix descriptions are developed for linear and non-linear electromagnetic interactions of moving atomic systems, taking into account applied magnetic fields. Atomic collision processes are treated as environmental interactions. Applications of interest include electromagnetically induced transparency and related pump-probe optical phenomena in atomic vapors. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general nonperturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is introduced. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear electromagnetic-response tensors occurring in a perturbation-theory treatment of the electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Intense-field electromagnetic interactions can be treated by means of an alternative method, which is based on a Liouville-space Floquet representation of the reduced density operator. Collisional interactions between atoms in a vapor can be treated in various approximations for the self-energy operator and the influence of Zeeman coherences on the electromagnetic response can be incorporated.

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Mechanical effects of light are considered in the first section of Chapter 7, together with their application to “optical tweezers.” This leads naturally to a discussion of laser cooling by a variety of techniques. Doppler cooling, magneto‐optic traps, sub‐Doppler cooling, and velocity‐selective coherent population trapping (VSCPT) are all covered. Then coherent population transfer is analyzed as a counterintuitive way of altering the occupation of energy levels in an exceptionally efficient manner. Coherent transverse optical magnetism, covered next, is the only example in the book of an optical effect mediated by the magnetic field of light. This phenomenon is characterized by intense emission of magnetic rather than electric dipole radiation at only moderate intensities and theoretically enables negative indices of refraction and spin control in unstructured, natural (nonmagnetic) materials. Electromagnetically induced transparency (EIT) is also described. This phenomenon is based on tri‐level coherence and can turn opaque media transparent over a limited bandwidth. Additionally, it can intensify high‐order nonlinear optical processes via intermediate step enhancement that is fully resonant. It has been applied to slow or store light as well. In this final chapter, squeezed light is given as an example of a coherent state with altered noise properties. Because the noise in squeezed quadratures of such a state can be reduced below the shot noise limit, it has applications to precision measurements and secure communication. The final section is on cavity quantum electrodynamics (QED). This topic treats the interaction of atoms with weak fields that are enhanced through the use of external cavities. In cavity QED, single photons can interact strongly enough with atoms to exhibit Rabi‐splitting behavior, and the strong coupling limit is considered to be a viable approach to quantum computation.

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Linear and non-linear interactions in electromagnetically induced transparency and related pump-probe optical phenomena involving moving many-electron atomic systems are investigated using a reduced-density-matrix approach. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is adopted. However, it is emphasized that a quantized-electromagnetic-field approach is essential for a self-consistent quantum-mechanical formulation. Our primary result is the derivation of compact Liouville-space operator expressions for the linear and the general (n'th order) non-linear macroscopic electromagnetic-response tensors. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Collisional interactions between atoms can be treated in various approximations for the self-energy operator, and the influence of Zeeman coherences on the electromagnetic-pulse propagation can be investigated by including an applied magnetic field on an equal footing with the electromagnetic fields.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Dec 28, 2009•0 citations
Chapter 5 begins the systematic application of the density matrix to describe optical interactions. Stationary two-and three-level atoms interacting with a traveling wave are compared to show that the presence of additional levels changes the saturation behavior of atoms. Then motion of the atoms is introduced and the phenomena of spatial and frequency-domain hole-burning are analyzed. Special effects due to standing waves, such as the light shift, are covered in this section. Next, tri-level coherence is considered, to extend the concept of coherences to two-photon transitions or Zeeman interactions in folded 3- or Λ-systems. Higher order interactions are also analyzed. This includes four-wave mixing and pump–probe techniques which form the basis of common nonlinear laser spectroscopies. Quantum interference is covered as a prelude to the topic of electromagnetically induced transparency. Higher order coherent interactions are also treated using a diagrammatic calculation technique based on Feynman diagrams.
Chapter 5 begins the systematic application of the density matrix to describe optical interactions. First, stationary two‐ and three‐level atoms interacting with a traveling wave are compared to show that the presence of additional levels changes the saturation behavior of atoms. Then motion of the atoms is introduced and the phenomena of spatial and frequency‐domain hole‐burning are analyzed. Special effects due to standing waves, such as the light shift are covered in this section. Next, tri‐level coherence is considered, to extend the concept of coherences to two‐photon transitions or Zeeman interactions in folded Λ‐ or V‐systems. Higher (third‐)order interactions are also analyzed. This includes four‐wave mixing and pump‐probe spectroscopy which form the basis of common nonlinear laser spectrocopies. The chapter closes by pointing out that higher order coherent interactions, which can sometimes be important if they are deliberately phase‐matched, are best treated using a diagrammatic calculational technique based on Feynman diagrams.
Reduced density matrix descriptions are developed for linear and non-linear electromagnetic interactions of moving atomic systems, taking into account applied magnetic fields. Atomic collision processes are treated as environmental interactions. Applications of interest include electromagnetically induced transparency and related pump-probe optical phenomena in atomic vapors. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general nonperturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is introduced. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear electromagnetic-response tensors occurring in a perturbation-theory treatment of the electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Intense-field electromagnetic interactions can be treated by means of an alternative method, which is based on a Liouville-space Floquet representation of the reduced density operator. Collisional interactions between atoms in a vapor can be treated in various approximations for the self-energy operator and the influence of Zeeman coherences on the electromagnetic response can be incorporated.
Reduced density matrix descriptions are developed for linear and non-linear electromagnetic interactions of moving atomic systems, taking into account applied magnetic fields as well as atomic collisions together with other environmental decoherence and relaxation processes. Applications of interest include electromagnetically induced transparency and related pump-probe optical phenomena in warm atomic vapors. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowestorder perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is adopted. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear macroscopic electromagnetic-response tensors occurring in a perturbation-theory treatment of the semiclassical electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Collisional interactions between atoms can be treated in various approximations for the selfenergy operator, and the influence of Zeeman coherences on the macroscopic electromagnetic response can be investigated.
In this paper we report on the study of two and multi-level atoms interacting with multiple laser beams. The semi-classical approach is used to describe the system in which the atoms are treated quantum mechanically via the density matrix operator, while the laser beams are treated classically using Maxwells equations. We present results of a two level atom interacting with single and multiple laser beams and demonstrate Rabi oscillations between the levels. The effects of laser modulation on the dynamics of the atom (atomic populations and coherences) are examined by solving the optical Bloch equations. Plots of the density matrix elements as a function of time are presented for various parameters such as laser intensity, detuning, modulation etc. In addition, phase-space plots and Fourier analysis of the density matrix elements are provided. The atomic polarization, estimated from the coherence terms of the density matrix elements, is used in the numerical solution of Maxwells equations to determine the behaviour of the laser beams as they propagate through the atomic ensemble. The effects of saturation and hole-burning are demonstrated in the case of two counter propagating beams with one being a strong beam and the other being very weak. The above work is extended to include four-wave mixing in four level atoms in a diamond configuration. Two co-propagating beams of different wavelengths drive the atoms from a ground state |1〉 to an excited state |3〉 via an intermediate state |2〉. The atoms then move back to the ground state via another intermediate state |4〉, resulting in the generation of two additional correlated photon beams. The characteristics of these additional photons are studied.
Mechanical effects of light are considered in the first section of Chapter 7, together with their application to “optical tweezers.” This leads naturally to a discussion of laser cooling by a variety of techniques. Doppler cooling, magneto‐optic traps, sub‐Doppler cooling, and velocity‐selective coherent population trapping (VSCPT) are all covered. Then coherent population transfer is analyzed as a counterintuitive way of altering the occupation of energy levels in an exceptionally efficient manner. Coherent transverse optical magnetism, covered next, is the only example in the book of an optical effect mediated by the magnetic field of light. This phenomenon is characterized by intense emission of magnetic rather than electric dipole radiation at only moderate intensities and theoretically enables negative indices of refraction and spin control in unstructured, natural (nonmagnetic) materials. Electromagnetically induced transparency (EIT) is also described. This phenomenon is based on tri‐level coherence and can turn opaque media transparent over a limited bandwidth. Additionally, it can intensify high‐order nonlinear optical processes via intermediate step enhancement that is fully resonant. It has been applied to slow or store light as well. In this final chapter, squeezed light is given as an example of a coherent state with altered noise properties. Because the noise in squeezed quadratures of such a state can be reduced below the shot noise limit, it has applications to precision measurements and secure communication. The final section is on cavity quantum electrodynamics (QED). This topic treats the interaction of atoms with weak fields that are enhanced through the use of external cavities. In cavity QED, single photons can interact strongly enough with atoms to exhibit Rabi‐splitting behavior, and the strong coupling limit is considered to be a viable approach to quantum computation.
Linear and non-linear interactions in electromagnetically induced transparency and related pump-probe optical phenomena involving moving many-electron atomic systems are investigated using a reduced-density-matrix approach. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is adopted. However, it is emphasized that a quantized-electromagnetic-field approach is essential for a self-consistent quantum-mechanical formulation. Our primary result is the derivation of compact Liouville-space operator expressions for the linear and the general (n'th order) non-linear macroscopic electromagnetic-response tensors. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Collisional interactions between atoms can be treated in various approximations for the self-energy operator, and the influence of Zeeman coherences on the electromagnetic-pulse propagation can be investigated by including an applied magnetic field on an equal footing with the electromagnetic fields.