Über die Autorin bzw. den Autor

Paul R. Berman is professor of physics at the University of Michigan. Vladimir S. Malinovsky is a visiting professor in the Physics Department at Stevens Institute of Technology.

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"This book is special in that it covers certain topics from several viewpoints. Many are presented, compared, discussed, and described in terms of their similarities and differences. I think this is beautifully done! The writing is clear, precise, and concise, and the well-done citations to other parts of the text lead the reader along logical paths to a significant conclusion."--Harold Metcalf, State University of New York, Stony Brook

"This book gives a very detailed and comprehensive treatment of theoretical quantum optics. It provides a consistent and thorough look at the whole field and will be a valuable reference."--Richard Thompson, Imperial College, London

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"This book is special in that it covers certain topics from several viewpoints. Many are presented, compared, discussed, and described in terms of their similarities and differences. I think this is beautifully done! The writing is clear, precise, and concise, and the well-done citations to other parts of the text lead the reader along logical paths to a significant conclusion."--Harold Metcalf, State University of New York, Stony Brook

"This book gives a very detailed and comprehensive treatment of theoretical quantum optics. It provides a consistent and thorough look at the whole field and will be a valuable reference."--Richard Thompson, Imperial College, London

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Principles of Laser Spectroscopy and Quantum Optics

PRINCETON UNIVERSITY PRESS

Contents

Preface.................................................................................................................xv1 Preliminaries.........................................................................................................12 Two-Level Quantum Systems.............................................................................................173 Density Matrix for a Single Atom......................................................................................564 Applications of the Density Matrix Formalism..........................................................................835 Density Matrix Equations: Atomic Center-of-Mass Motion, Elementary Atom Optics, and Laser Cooling.....................996 Maxwell-Bloch Equations...............................................................................................1207 Two-Level Atoms in Two or More Fields: Introduction to Saturation Spectroscopy........................................1368 Three-Level Atoms: Applications to Nonlinear Spectroscopy-Open Quantum Systems........................................1599 Three-Level Atoms: Dark States, Adiabatic Following, and Slow Light...................................................18410 Coherent Transients..................................................................................................20611 Atom Optics and Atom Interferometry..................................................................................24212 The Quantized, Free Radiation Field..................................................................................28013 Coherence Properties of the Electric Field...........................................................................31214 Photon Counting and Interferometry...................................................................................33915 Atom–Quantized Field Interactions..............................................................................35816 Spontaneous Decay....................................................................................................37517 Optical Pumping and Optical Lattices.................................................................................40218 Sub-Doppler Laser Cooling............................................................................................42219 Operator Approach to Atom–Field Interactions: Source-Field Equation............................................45320 Light Scattering.....................................................................................................47421 Entanglement and Spin Squeezing......................................................................................492Problems................................................................................................................505References..............................................................................................................506Bibliography............................................................................................................507Index...................................................................................................................509

Chapter One

1.1 Atoms and Fields

As any worker knows, when you come to a job, you have to have the proper tools to get the job done right. More than that, you must come to the job with the proper attitude and a high set of standards. The idea is not simply to get the job done but to achieve an end result of which you can be proud. You must be content with knowing that you are putting out your best possible effort. Physics is an extraordinarily difficult "job." To understand the underlying physical origin of many seemingly simple processes is sometimes all but impossible. Yet the satisfaction that one gets in arriving at that understanding can be exhilarating. In this book, we hope to provide a foundation on which you can build a working knowledge of atom–field interactions, with specific applications to linear and nonlinear spectroscopy. Among the topics to be discussed are absorption, emission and scattering of light, the mechanical effects of light, and quantum properties of the radiation field.

This book is divided roughly into two parts. In the first part, we examine the interaction of classical electromagnetic fields with quantum-mechanical atoms. The external fields, such as laser fields, can be monochromatic, quasi-monochromatic, or pulsed in nature, and can even contain noise, but any quantum noise effects associated with the fields are neglected. Theories in which the fields are treated classically and the atoms quantum-mechanically are often referred to as semiclassical theories. For virtually all problems in laser spectroscopy, the semiclassical approach is all that is needed. Processes such as the photoelectric effect and Compton scattering, which are often offered as evidence for photons and the quantum nature of the radiation field can, in fact, be explained rather simply with the use of classical external fields. The price one pays in the semiclassical approach is the use of a time-dependent Hamiltonian for which the energy is no longer a constant of the motion.

Although the semiclassical approach is sufficient for a wide range of problems, it is not always possible to consider optical fields as classical in nature. One might ask when such quantum optics effects begin to play a role. Atoms are remarkable devices. If you place an atom in an excited state, it radiates a uniquely quantum-type field, the one-photon state. One of the authors (PRB) is a former student of Willis Lamb, who claimed that it should be necessary for people to apply for a license before they can use the word photon. Lamb was not opposed to the idea of a quantized field mode, but he felt that the word photon was misused on a regular basis. We will try to explain the distinction between a one-photon field and a photon when we begin our discussion of the quantized radiation field.

The field radiated by an atom in an excited state has a uniquely quantum character. In fact, any field in which the average value of the number operator for the field (average number of photons in the field) is less than or on the order of the number of atoms with which the field interacts must usually be treated using a quantized field approach. Thus, the second, or quantum optics, part of this book incorporates a fully quantized approach, one in which both the atoms and the fields are treated as quantum-mechanical entities. The advantage of using quantized fields is that one recovers a Hamiltonian that is perfectly Hermitian and independent of time. The most common quantum optics effects are those associated with spontaneous emission, scattering of external fields by atoms, quantum noise, and cavity quantum electrodynamics. There is another class of problems related to quantized field effects involving van der Waals forces and Casimir effects, but we do not discuss these in any detail.

1.2 Important Parameters

Why did the invention of the laser cause such a revolution in physics? Laser fields differ from conventional optical sources in their coherence properties and intensity. In this book, we look at applications that exploit the coherence properties of lasers, although complementary textbooks could be written in which the emphasis is on strong field–matter interactions. Moreover, we touch only briefly on the current advances in atto-second science that have been enabled using nonlinear atom–field interactions. Even if we deal mainly with the coherence properties of the fields, our plate is quite full. Historically, the coherence properties of optical fields have been one of the limiting factors in determining the ultimate resolution one can achieve in characterizing the transition frequencies of atomic, molecular, and condensed phase systems. It will prove useful to list some of the relevant frequencies that one encounters in considering such problems.

First and foremost are the transition frequencies themselves. We focus mainly on optical transitions in this text, for which the transition frequencies are of order ω₀/2π [equivalent] 5 × 10¹⁴ Hz. The laser fields needed to probe such transitions must have comparable frequencies. The first gas and solid-state lasers had a very limited range of tunability, but the invention of the dye laser allowed for an expanded range of tunability in the visible part of the spectrum. One might even go so far as to say that it was the dye laser that really launched the field of laser spectroscopy. Since that time, the development of tunable semiconductor-based and titanium-sapphire lasers operating at infrared frequencies, combined with frequency doublers (nonlinear optical crystals) and frequency dividers (optical parametric amplifiers and oscillators), has enabled the creation of tunable coherent sources over a wide range of frequencies from the ultraviolet to the far-infrared.

Assuming for the moment that such sources are nearly monochromatic (typical line widths range from kHz to GHz), there are still underlying processes that limit the resolution one can achieve using laser sources to probe atoms. In other words, suppose that two transition frequencies in an atom differ by an amount Δf. What is the minimum value of Δf for which the transitions can be resolved? The ultimate limiting factor for any transition is the natural width associated with that transition. The natural width arises from interactions of atoms with the vacuum radiation field, leading to spontaneous emission. Typical natural widths for allowed optical transitions are in the range γ₂/2π [equivalent] 10⁷ - 10⁸ Hz, where γ₂ is a spontaneous emission decay rate. For "forbidden" transitions, such as those envisioned as the basis for optical frequency standards, natural line widths can be as small as a Hz or so. The fact that an allowed transition has a natural width equal to 10⁸ Hz does not imply that the transition frequency can be determined only to this accuracy. By fitting experimental line shapes to theory, one can hope to reduce this resolution by a factor of 100 or more.

The natural width is referred to as a homogeneous width since it is the same for all atoms in a sample and cannot be circumvented. Another example of a homogeneous width in a vapor is the collision line width that arises as a result of energy shifts of atomic levels that occur during collisions. If the collision duration (typically of order 5 ps) is much less than all relevant timescales in the problem, except the optical period, then collisions add a homogeneous width of order 10 MHz per Torr of perturber gas pressure [1 Torr = (1/760) atm ≈ 133 Pa ≈ 1 mmHg]. This width is often referred to as a pressure broadening width.

Even if there are no collisions in a vapor, linear absorption or emission line shapes can be broadened by an inhomogeneous line broadening mechanism, as was first appreciated by Maxwell. In a vapor, the moving atoms are characterized by a velocity distribution. As viewed in the laboratory frame, any radiation emitted by an atom is Doppler shifted by an amount (ω₀/2π)(v/c) (Hz), where v is the atom's speed and c is the speed of light. For a typical vapor at room temperature, the velocity width is of order 5 × 10² m/s, leading to a Doppler width of order 1.0 GHz or so. In a solid, crystal strain and fluctuating fields can give rise to inhomogeneous widths that can be factors of 10 to 100 times larger than Doppler widths in vapors. As you will see, it is possible to eliminate inhomogeneous contributions to line widths using methods of nonlinear laser spectroscopy.

Another contribution to absorption or stimulated emission line widths is so-called power broadening. The atom–field interaction strength in frequency units is Ω₀/2π µ₁₂ E/h, where µ12 is a dipole moment matrix element, E is the amplitude of the applied field that is driving the transition, h = 2π[??] = 6.63 × 10^-34 J × s is Planck's constant, and Ω₀ is referred to as the Rabi frequency. For a 1-mW laser focused to a 1-mm² spot size, Ω₀/2π is of the order of several MHz and grows as the square root of the intensity. Of course, power broadening can be reduced by using weaker fields.

For vapors, there is an additional cause of line broadening. Owing to their motion, atoms may stay in the atom–field interaction region for a finite time τ, which gives rise to a broadening in Hz of order 1/(2πτ). For laser-cooled atoms, such transit-time broadening is usually negligible (on the order of a Hz or so), but in a thermal vapor it can be as large as a hundred KHz for laser beam diameters equal to 1 mm.

The broadening limits the resolution that one can achieve in probing atomic transitions with optical fields. One must also contend with shifts of the optical transition frequency resulting from atom–field interactions. If the optical fields are sufficiently strong, they can give rise to light shifts of the transition frequency that are of order Ω²₀/(2πδ) (Hz), where δ/2π is the frequency mismatch between the the atomic transition and the applied field frequencies in Hz (assumed here to be larger than the natural or Doppler widths). Light shifts range from 1 Hz to 1 MHz for typical powers of continuous-wave laser fields.

Magnetic fields also result in a shift and splitting of energy levels, commonly referred to as a Zeeman splitting. The magnetic interaction strength in frequency units is of order µ_B/h [equivalent] 14 GHz/T, where µ_B = 9.27 × 10^-24 JT^-1 is the Bohr magneton. As a consequence, typical level splittings in the Earth's magnetic field are on the order of a MHz.

Last, there is a small shift associated with the recoil that an atom undergoes when it absorbs, emits, or scatters radiation. This recoil shift in Hz is of order ([??]k)²/(2hM), where [??]k is the momentum associated with a photon in the radiation field, and M is the atomic mass. Typical recoil shifts are in the 10 to 100 kHz range.

These frequency widths and shifts are summarized in table 1.1. The resolution achievable in a given experiment depends on the manner in which these shifts or widths affect the overall absorption, emission, or scattering line shapes.

As we go through applications, the approximations that we can use are dictated by the values of these parameters. If you keep these values stored in your memory, you will be well on your way to understanding the relative contributions of these terms and the validity of the approximations that will be employed.

1.3 Maxwell's Equations

Throughout this text, we are interested in situations where there are no free currents or free charges in the volume of interest. That is, we often look at situations where an external field is applied to an ensemble of atoms that induces a polarization in the ensemble. We set B = µ₀H (neglecting any effects arising from magnetization), but do not take D = ε₀E. Rather, we set D = ε₀E + P, where the polarization P is the electric dipole moment per unit volume. We adopt this approach since the polarization is calculated using a theory in which the atomic medium is treated quantum-mechanically.

With no free currents or charges and with B = µ₀H, Maxwell's equations can be written as

∇ × (ε₀E + P) = 0, (1.1a) ∇ × E = ∂B/∂t, (1.1b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1c) ∇ × B = 0. (1.1d)

The quantity

µ₀ = 4π × 10^-7 T × m/A (1.2)

is the permeability of free space, while

ε₀ ≈ 8.85 × 10^-7 C²/N × m² (1.3)

is the permittivity of free space. All field variables are assumed to be functions of position R and time t.

From equation (1.1), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

In free space, ∇ × E = 0 and P = 0, leading to the wave equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where the wave propagation speed in free space is equal to

v₀ = 1/√µ₀ε₀. (1.7)

Historically, by comparing the electromagnetic (i.e., that based on the force between electrical circuits) and the electrostatic units of electrical charge, Wilhelm Weber had shown by 1855 that the value of 1/(µ₀ε₀)^1/2 was equal to the speed of light within experimental error. This led Maxwell to conjecture that light is an electromagnetic phenomenon. One can only imagine the excitement Maxwell felt at this discovery.

We return to Maxwell's equations later in this text, but for now, let us consider plane-wave solutions of equations (1.5) and (1.1) for which we can take ∇ × E = 0. We still do not have enough information to solve equation (1.5) since we do not know the relationship between P(R, t) and E(R, t). In general, one can write P(R, t) = ε₀χ_e × E(R, t), where χ_e is the electric susceptibility tensor, but this does not resolve our problem, since χ_e is not yet specified. To obtain an expression for χ_e, one must model the medium–field interaction in some manner. Ultimately, we calculate χ_e using a quantum-mechanical theory to describe the atomic medium.

For the time being, however, let us the assume that the medium is linear, homogeneous, and isotropic, implying that χ_e is a constant times the unit tensor and independent of the electric field intensity. Moreover, if we neglect dispersion and assume that χ_e is independent of frequency over the range of incident field frequencies, then it is convenient to rewrite χ_e as

χ_e = n² - 1, (1.8)

where n is the index of refraction of the medium. In these limits, equation (1.5) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

where c is the speed of light in vacuum. Neglecting dispersion, the fields propagate in the medium with speed v = c/n, as expected.

For a monochromatic or nearly monochromatic field having angular frequency centered at ω, the magnetic field (or, more precisely, the magnetic induction) B is related to the electric field via

B = [k × E]/ω, (1.10)

where k is the propagation vector having magnitude k = nω/c. It then follows that the time average of the Poynting vector, S = E × H = E × B/µ₀, is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.11)

for optical fields having electric field amplitude |E| and propagation direction [??] = k/k.

(Continues...)

Excerpted from Principles of Laser Spectroscopy and Quantum Opticsby Paul R. Berman Vladimir S. Malinovsky Copyright © 2011 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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