Cavity Quantum ElectrodynamicsSerge Haroche, and Daniel Kleppner

Citation: Physics Today 42, 1, 24 (1989); doi: 10.1063/1.881201View online: https://doi.org/10.1063/1.881201View Table of Contents: https://physicstoday.scitation.org/toc/pto/42/1Published by the American Institute of Physics

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CAVITY QUANTUMELECTRODYNAMICS

A new generation of experiments shows that spontaneousradiation from excited atoms can be greatly suppressed orenhanced by placing the atoms between mirrors or in cavities.

Serge Horoche and Daniel Kleppner

Ever since Einstein demonstrated that spontaneous emis-sion must occur if matter and radiation are to achievethermal equilibrium, physicists have generally believedthat excited atoms inevitably radiate.1 Spontaneousemission is so fundamental that it is usually regarded asan inherent property of matter. This view, however,overlooks the fact that spontaneous emission is not aproperty of an isolated atom but of an atom-vacuumsystem. The most distinctive feature of such emission,irreversibility, comes about because an infinity of vacuumstates is available to the radiated photon. If these statesare modified—for instance, by placing the excited atombetween mirrors or in a cavity—spontaneous emission canbe greatly inhibited or enhanced.

Recently developed atomic and optical techniqueshave made it possible to control and manipulate spontane-ous emission (figure 1). Experiments have demonstratedthat spontaneous emission can be virtually eliminated orelse made to display features of reversibility: Instead ofradiatively decaying to a lower energy state, an atom canexchange energy periodically with a cavity.

Serge Haroche is a professor of physics at the University ofParis VI and at the Ecole Normale Superieure, in Paris, and atYale University, in New Haven, Connecticut. DanielKleppner is Lester Wolfe Professor of Physics at theMassachusetts Institute of Technology, in Cambridge,Massachusetts.

The recent research on atom-vacuum interactionsbelongs to a new field of atomic physics and quantumoptics called cavity quantum electrodynamics. In additionto demonstrating dramatic changes in spontaneous emis-sion, cavity QED has led to the creation of new kinds of mi-croscopic masers that operate with a single atom and a fewphotons or with photons emitted in pairs in a two-photontransition.

Emission in free spaceWe can introduce cavity QED with a brief review ofspontaneous emission in free space. Consider a one-electron atom with two electronic levels e and /"separatedby an energy interval Ee — Ef = fuo. Spontaneous emis-sion appears as a jump of the electron from level e to level faccompanied by the emission of a photon. This process canbe understood as resulting from the coupling of the atomicelectron to the electromagnetic field in its "vacuum" state.

A radiation field in space is usually described in termsof an infinite set of harmonic oscillators, one for each modeof radiation. The levels of this oscillator correspond tostates with 0, 1, 2, . . . , n photons of energy fuo. In itsground state each oscillator has a "zero-point" energyfio/2 associated with its quantum fluctuations.

The rms vacuum electric-field amplitude £vac in amode of frequency a> is [fc/(2e0V)]1/2, where e0 is thepermittivity of free space, V is the size of an arbitraryquantization volume and the units are SI. The coupling ofthe atom to each field mode is described by the elementary

24 PHYSICS TODAY JANUARY 1989 © 1989 American Insriture of Physic

Resonant superconducting cavity.The cylindrical niobium cavity, situatedbetween the two shiny, rectangularblocks at the top of the apparatus, isused for cavity QED experiments onRydberg atoms at the Ecole NormaleSuperieure in Paris. It is cooled by ahelium cryostat, the large cylinder at thebottom of the photograph. The atomicbeam passes along the cavity axis,entering and leaving through smallholes. Figure 1

frequencyJfl

Here Def is the matrix element of the electric dipole of theatom between the two levels, and flef, which is oftenreferred to as the Rabi frequency of the vacuum, is the fre-quency at which the atom and the field would exchangeenergy if there were only a single mode of the field. An es-sential feature of spontaneous emission in free space,however, is that the atom can radiate into any mode thatsatisfies the conservation of energy and momentum. Thetime of emission and the particular mode in which thephoton is observed are random variables.

The probability Fo of photon emission per unit time,more familiarly called the Einstein ^4-coefficient, isproportional to the square of the frequency fle/- and to themode density pn{(o), the number of modes available per unitfrequency interval. The mode density is given by theexpressionpa(a>) — aPV/v2^, where it is assumed that thequantization volume Vis large compared to A:i, or (2irc/cof.The probability Fo is given by the Fermi "golden rule":

The probability P,Xt) of finding an atom still excited attime t after its preparation in state e is exp( — rot). Suchan exponential decay law describes an irreversible processthat leads the atom irrevocably to its ground state. The

source of irreversibility is the continuum of field modesresonantly coupled to the atom. The vacuum field acts asa gigantic "reservoir" in which the atomic excitationdecays away.

Emission between mirrorsThe mode structure of the vacuum field is dramaticallyaltered in a cavity whose size is comparable to thewavelength. The schematic diagram in figure 2 shows anatom confined between two plane, parallel mirrors withseparation d. For an electric field polarized parallel to themirrors no mode exists unless A < 2d: As A is increased,the lowest-order mode cuts off abruptly when A exceeds 2d.As a result, an excited atom whose radiation arises froman electric dipole moment oscillating parallel to themirrors becomes infinitely long-lived when A > 2d. Classi-cally, the electron orbit of such an atom is essentiallyparallel to the mirrors; we call this a polarization. Bycontrast, for polarization normal to the mirrors—npolarization—there is no such cutoff and the excited statedecays rapidly. Such simple cavity structures are easy tobuild in the microwave domain and can now be realizeddown to the micron size for optical fields, as thephotograph in figure 2 shows.

The influence of boundary conditions on atomicradiation was pointed out long ago, and a considerablebody of literature exists on the subject.2 Until recently,however, experimental evidence for these effects had beenscarce because of the difficulties of preparing, controlling

PHYSICS TODAY JANUARY 19S9 2 5

Parallel mirrors 1.1-microns apart used forcavity QED experiments at Yale University.An excited atom placed in the cavity betweenthe mirrors will tend to remain excited. Thegap was built for optical frequencies byplacing a thin metallic spacer between two flatmirrors. The light areas in the electronmicrograph (a) are the mirror substrates,which are silica blocks coated on their innersurfaces with gold. The schematic diagram (b)shows an excited atom between two parallelmirrors. The atom remains permanentlyexcited if the mirror spacing d is smaller thanhalf the atomic transition wavelength,provided the atomic dipole is parallel to themirror surfaces. Figure 2

and observing the emission of isolated atoms or moleculesnear surfaces. In the early 1970s Karl H. Drexhage at theUniversity of Marburg, Germany, carried out pioneeringwork on the fluorescence of organic dyes deposited ondielectric films over a metallic mirror.3 He observedalterations in the rate and pattern of the emission, but theopen geometry of his experimental setup prevented anymajor reduction of the spontaneous emission rate. Morerecently several groups have studied isolated atomicradiators in cavities, and have demonstrated the inhibi-tion of spontaneous emission at wavelengths from themicrowave to the optical. A group at the University ofWashington performed the first in this generation ofexperiments, observing the radiation damping of anisolated electron undergoing cyclotron motion in anelectromagnetic trap whose walls formed a high-modecavity. They discovered large changes in the cyclotrondamping rate that depended on the position of the electronwith respect to the trap's nodal pattern.4

Physicists at MIT, Yale and the University of Romehave carried out experiments using the parallel-mirrorgeometry of figure 2. The MIT experiment demonstratedthe inhibition of spontaneous emission from Rydbergstates of cesium atoms in a beam.5 The atoms radiated at awavelength of about 0.4 mm as they passed between two20-cm-long aluminum mirrors separated by approximate-ly 0.2 mm. Just before entering the "cavity," the atomswere prepared in such a way that their radiating dipoleswere strictly parallel to the mirrors, much like the"circular" state sketched in figure 2. The atoms survivingin the initial quantum state were detected at the cavityexit by ionizing them in a small electric field. Atoms thatunderwent spontaneous emission were transferred to amore tightly bound level that remained un-ionized.

Figure 3 shows how the signal of the atoms changes as

the ratio A/2d is varied around the critical value of 1. Thelarge increase in transmission when A exceeds 2d is clearevidence of the inhibition of spontaneous emission. Thelifetime is at least 20 times longer than it is in free space.The decrease for A/2d> 1.015 is an artifact caused by theionization of the atoms by the electric field.

The experimenters at Yale6 suppressed spontaneousemission in the near infrared using a geometry similar tothe MIT experiment but with the much smaller waveguidestructure shown in figure 2a. The Yale experimentemployed a cesium atomic beam, excited into the low-lying5d level. The inhibited transition was 5ci->6/> at awavelength of 3.5 microns, far beyond the cutoff wave-length of 2.2 microns. Excited atoms propagated throughthe tunnel for about 13 natural lifetimes without apprecia-ble decay. Application of a small magnetic field to changethe orientation of the atomic dipoles demonstrated theanisotropy of spontaneous emission between mirrors. Ifthe magnetic field has a component parallel to the mirrorsurface, the dipole precesses so that it acquires a perpen-dicular component. After rotating through only a smallfraction of a turn the dipole can spontaneously radiate a ir-polarized photon. Figure 4 shows the transmission ofexcited atoms through the microtunnel as a function of theangle between the magnetic field and the normal to themirrors: The large change in the lifetime induced by themagnetic precession is evident. In another experiment,carried out at the University of Rome, the radiativelifetime of dye molecules at optical frequencies waslengthened using a cavity formed by plane, parallelmirrors.7

Resonant cavity emissionJust as a cavity below cutoff suppresses vacuum fluctu-ations, a resonant cavity enhances them. How an atom

2 6 PHYSICS TODAY JANUARY 1989

behaves in a resonant cavity depends on the ratio of thevacuum Rabi frequency to the cavity bandwidth, ftc/-/A<yc.The cavity bandwidth is most conveniently described bythe quality factor Q, which is given by cj/A<yc. Thereciprocal of A<uc is effectively the density of modes "seen"by the atom in the cavity; alternatively, it is the lifetime ofa photon in the cavity. In open structures such as the oneshown in figure 2a, edge diffraction limits the qualityfactor Q. By employing spherical mirrors to create aFabry-Perot resonator, Q can be enhanced substantially.In the microwave regime, closed cavities are practical(figure 1). A superconducting microwave cavity canachieve values of Q up to 10H and photon storage times of afraction of a second.

In a low-Q cavity the emitted photon is dampedrapidly and an atom undergoes radiative decay much likeit does in free space, though at an enhanced rate. Theradiation rate in a cavity of volume V is

Compared with the rate in free space, the emission rate isincreased by the ratio QA.3/V, which can be large.Physicists at Ecole Normale Superieure in Paris haveobserved this regime of enhanced spontaneous emission inthe millimeter wave regime for Rydberg atoms of sodiumcoupled to a Fabry-Perot cavity.8 The spontaneousemission rate was enhanced by a factor of approximately500.

In the optical regime, physicists in the Rome group7

and at the MIT Spectroscopy Laboratory9 have observedeffects of both enhanced and inhibited spontaneousemission. The MIT experimenters employed a sphericalFabry-Perot resonator. A laser within the resonatorexcited an atomic beam of ytterbium, which fluoresces at awavelength of 556 nm. As the experimenters tuned theresonator through successive resonances, the radiationrate into the solid angle subtended by the resonator was al-ternately inhibited by a factor of 42 and enhanced by a fac-tor of 19.

The regime of very high Q, where Cler/Ao}c > 1,manifests totally new behavior. The radiation remains inthe cavity so long that there is a high probability it will bereabsorbed by the atom before it dissipates. Spontaneousemission becomes reversible as the atom and the fieldexchange excitation at the rate ilef. Such behavior is awell-known feature of the interaction of an atom with aclassical monochromatic field. These so called "Rabioscillations" are familiar in nuclear magnetic resonanceand optical transient experiments. In cavity QED, how-ever, the atom couples to its own one-photon field withoutany externally applied radiation.

The link between classical Rabi oscillations andspontaneous Rabi oscillations is described by the equationbelow for the probability Pe (t) of finding the atom in its ex-

cited state at time t, assuming it was prepared in this stateat time 0. The probability that there are n photons in thecavity is described by the distribution function p(n), andthe cavity is taken to have an infinite quality factor Q.

Pe{t) =

Spontaneous emission corresponds to an initially emptycavity: p(0) = 1 . A classical field corresponds to a narrowphoton distribution peaked around a large average photonnumber n. (The Rabi frequency Cler^n + 1 is roughlyproportional to the square root of n, that is, to the electricfield amplitude in the mode.) For ordinary atoms ormolecules, Clef is intrinsically small, and observing theRabi oscillation requires an enormous average photonnumber h. For Rydberg states, on the other hand, ilef canbe relatively large—103 to 106 sec"' for principal quantumnumbers around 40—and the Rabi oscillations becomeobservable even with h = 0.

Physicists at the Max Planck Institute for QuantumOptics near Munich have observed Rabi oscillationsinduced by a small thermal field in the cavity.10 The

0.975 0.98 0.99 1.00ATOMIC WAVELENGTH A/2i

1.01 1.02 1.03

Survival of excited Rydberg atoms moving ina gap between mirrors, plotted as a functionof the wavelength for spontaneous emission inthe vicinity of the cavity cutoff.5 The signalcomes from excited atoms detected at thecavity exit. The atomic wavelength X is variedby applying a small electric field to the atoms.The sharp increase in survival when the ratioA/2d is 1 is caused by the inhibition ofspontaneous emission. Figure 3

PHYSICS TODAY JANUARY 1989 2 7

ORIENTATION OF MAGNETIC FIELD (degrees)

Transmission of excited atoms througha mirror gap causing a cutoff ofspontaneous emission in the infraredregime around 3.5 microns.6 The gapstructure is shown in figure 2a. Thesignal is plotted against the anglebetween a small applied magnetic fieldand the normal to the mirrors. Thediagrams at the bottom of the figureillustrate the magnetic fieldorientations defined as 0°, 90° and180°. At zero angle the atoms arepolarized parallel to the mirrors and thelarge transmission of excited atoms isevidence of inhibited spontaneousemission. When the angle is nonzero,Larmor precession reorients the atomicdipoles so that they can emit ir-polarized radiation, which canpropagate in the gap. As a result, theatoms radiate spontaneously and fail toreach the detector in their excitedstate. Figure 4

photon number distribution is then given by the Planckformula: p(n) = [1 — exp( — fico/kB Tj\ exp( — nfuolk^ T),where kB is the Boltzmann constant. The pattern ofoscillations is a complicated function of time resultingfrom the beating between the elementary Rabi frequen-cies weighted by the corresponding values oipin). Figure 5shows the probability Pe(t) as a function of the time ofinteraction with a cavity field containing an average of 2thermal photons. The cavity operated at 21.6 GHz at atemperature T of 2.5 K. The experimenters used Rydbergstates of rubidium—the 63P3/2^61Z)5/2 transition. Theflux of atoms crossing the cavity was low, and so the cavityfield relaxed to thermal equilibrium between successiveatoms. In this way, each atom probed the thermal field at2.5 K.

Rabi oscillations induced by small thermal fields in acavity can display "quantum collapse and revival."11

After a few periods the various terms corresponding todifferent values of n in the expression for the probabilityPe(t) interfere destructively, and there is no further traceof a beat structure. Some time later, however, the termsinterfere constructively and the beating revives. TheMunich group has obtained evidence for this novelphenomenon.10

The one-atom maserIf the rate of atoms crossing a cavity exceeds the cavitydamping rate co/Q, the photon released by each atom isstored long enough to interact with the next atom. Theatom-field coupling becomes stronger and stronger as thefield builds up, eventually evolving into a steady state.The system is a new kind of maser, which operates with ex-ceedingly small numbers of atoms and photons.12 Atomicfluxes as small as 100 atoms per second have generatedmaser action. For such a low flux there is never more thana single atom in the resonator—in fact, most of the timethe cavity is empty. These "micromaser" devices wereoperated for the first time at the Max Planck Institute forQuantum Optics.13

It is interesting to analyze how the field grows in anideal micromaser operating at 0 K with a fixed atom-cavity interaction time tint. Assume there is an idealcavity (infinite Q) and an ideal field ionization detectorthat allows us to determine the state e or /'in which eachatom leaves the cavity. After the first atom (atom # 1) hascrossed the resonator, the state of the atom-field system isa linear superposition of states |e,0> and \f,V)- (Thenotation indicates an atom in state e correlated with 0photons in the cavity and an atom in state f correlatedwith one photon.) Using standard notation we can write

|*,> = cos(ncftint 12) |e,0> + sin(fteftint 12) \fX>If fleftint is small, 0.1 for example, there is a large

probability of finding atom # 1 in state e. If this occurs,then immediately thereafter |*!> = |e,0>, and so the fieldwill contain exactly zero photons. The next atom will theninteract with an empty cavity. At some time, however, afirst atom, say atom N, will be found in state |/>, correlatedto the instantaneous "appearance" of one photon in thecavity. Atom (N + 1) will then undergo a differentevolution: The state of the system "atom (TV + 1) + cavityfield" after atom (7V+ 1) has left the cavity will be

+ sin(nefj2tint 12)Continuous monitoring of the system would reveal a kindof random walk of the photon number: Each time an atomis detected in state |/> the field has gained one photon.The process is essentially random because the probabilityfor an atom to flip from e to /"is governed by quantum me-chanical chance. It is a random process with memory,however, because the probability law for each stepdepends on the outcome of the previous steps. This simplebuild-up process has been studied theoretically andprovides one of the simplest illustrations of the quantumtheory of measurement.14 The field can be viewed as aquantum harmonic oscillator. The two-level atoms cross-

28 PHYSICS TODAY JANUARY 1989

ing the cavity play a double role: They modify the field bythe quantum mechanical "kicks" they deliver, and theyserve as measurement devices that detect the field as itgrows. A steady state is reached if there exists a photonnumber n0 such that the quantity (Clef^ln0 + 1 tint )/2 is anintegral multiple of n. The photon number remainstrapped with the value n0 because each subsequent atomwill leave the cavity in the e state. Such a radiation stateis highly nonclassical, for the field has a precisely definedenergy but a completely random phase. It has noamplitude fluctuations, whereas an ordinary electromag-netic field has quantum-limited intensity fluctuationsproportional to -Jh. Such nonclassical states of microwaveradiation fields are now being studied at the Max PlanckInstitute for Quantum Optics.

The two-photon maser.At the Ecole Normale Superieure physicists have createda microwave maser that uses a Rydberg atom-cavitysystem but that operates on a two-photon transition.15 Intwo-photon emission an atom simultaneously radiates apair of photons with frequencies <y, and co2 while jumpingfrom an initial level e to a final level f. Energyconservation requires that Ee — Ef = #(&>, + a>2). Becausetwo-photon emission is a second-order process, it usuallyoccurs only from metastable levels that are forbidden toradiate by a single-photon process because of someselection rule. If one- and two-photon emission channelsare both open for the decay of an atomic level, the one-pho-ton process dominates and two-photon effects are negligi-ble. In the level configuration of figure 6a, which is oftenfound in the structure of Rydberg atoms, the upper level ecan decay by one-photon emission to the opposite paritylevel i or by two-photon emission to the same parity level f.In free space the former process is overwhelmingly moreprobable. In a cavity that is resonant at a frequency toequal to (Ee — Ef)/2fi, however, the spontaneous emissionof two "degenerate" photons at frequency w is enhanced,whereas the one-photon rate is suppressed.

This effect has been experimentally demonstrated ina beam of rubidium atoms prepared in the 40S state. Theatoms traversed a superconducting cavity tuned to 68.4GHz, half the frequency of the 40S-39S transition. Theresonant transfer of atoms between these two levels,shown in figure 6b, demonstrates two-photon maseraction. This new quantum oscillator displays uniquefeatures because the photons are emitted in pairs. Forinstance, startup of the system is characterized by longdelays, very different from the prompt startup of a one-photon maser.16

Other cavity QED effectsIn addition to changing the radiation rate of atoms, atom-cavity coupling also induces energy shifts. This phenome-

0.3 50 100 150TIME OF FLIGHT THROUGH CAVITY (microseconds)

Probability of f inding an atom in the upperlevel of a two-state system as a funct ion of theatom-cav i ty interaction t ime.1 6 The datawere taken in a cavity tuned to 21.6 GHz , thefrequency of the 6 3 P 3 / 2 ^ 6 1 D 5 / 2 transition ofRb85. The curve is theoretical. Radiation isinduced by the thermal f ield in the cavity at2.5 K. The Rabi oscil lation is clearlyobservable. Figure 5

non can be understood as a frequency-pulling effect thatoccurs when an atomic "oscillator" interacts with areactive cavity. Quantum mechanically, one can analyzeit in terms of the effect of the cavity walls on the virtual-photon exchange processes. The metallic boundaries alterthe modes of the vacuum field around the atom and affectnot only real photon emission—that is, spontaneousemission rates—but also the virtual process responsiblefor radiative energy shifts.

A superficial analysis of the experiments on theinhibition of spontaneous emission could lead one tobelieve that suppressing the natural line width might giverise to arbitrarily narrow spectral lines and ultrahighspectroscopic resolution. The fact that the atomic energylevels are at the same time shifted from their "free space"positions reveals that one is really carrying out spectrosco-py on an atom "dressed" by the cavity vacuum, which isdifferent from the atom "dressed" by the free-fieldvacuum. Precise experiments are now beginning toobserve these effects. For example, experiments onindividual trapped electrons are reaching the pointwhere cavity-induced shifts are affecting measurements ofthe electron magnetic moment.17

The group at the MIT Spectroscopy Laboratory hasobserved frequency shifts caused by cavity effects atoptical wavelengths, using barium atoms in a concentricoptical resonator.18 They saw changes in both thetransition frequency and the linewidth caused by en-hanced and inhibited spontaneous emission.

Cavity QED has potential applications to the study ofnonlinear dynamics in small systems. A tenuous beam ofatoms crossing a high-Q cavity one at a time is a highlynonlinear system with feedback—the field radiated by oneatom reacts strongly back on the next atom. These are theingredients required for chaos in a classical system. Whensolving the classical equation of motion for such a system,one finds chaotic behavior. Studying the quantum behav-ior of the system in the regime where the classical modelexhibits bifurcations and chaos may provide insight intothe connection between quantum mechanics and nonlin-ear classical mechanics.

PHYSICS TODAY JANUARY 1989 2 9

40St/> = 68.41587 GHz

.39P.

39S,/2 39 MHz

crLU

<cr111

LJJ0.2 -

0 _68 415.80 68 415.85 68 415.90 68 415.95

CAVITY RESONANT FREQUENCY (megahertz)

Two-photon Rydberg maser.15 The maseroperates between two levels, e and f, of thesame parity in rubidium, a: The cavity, tunedto exactly half the frequency of the e-finterval, enhances the two-photon processand inhibits the usually dominant one-photonemission toward the intermediate level /. b:Plot of the ratio of the lower-state populationto the upper-state population as the cavity istuned across the two-photon resonance. Thepeak indicates two-photon maseroperation. Figure 6

The resonant-cavity-QED effects described above canbe generalized to many-atom systems. Instead of couplingthe atoms to the cavity one by one, it is possible to preparesamples of N identical Rydberg atoms in the cavity andstudy their collective behavior. Because of strong intera-tomic correlations, all the radiative phenomena occurmuch faster than they do with single atoms. Experi-menters have observed19 cooperative "superradiance"taking place within a time (NT0)~l as well as collectiveRabi oscillations involving a reversible exchange of energybetween the field and the atomic sample at the enhancedrate ilef4N.

In the case of a strongly pumped two-level system,cavity effects can profoundly alter the dynamics ofradiative relaxation and the population densities atequilibrium. A group at the University of Oregon hasstudied such effects theoretically and experimentally.20

Finally, cavity-QED experiments provide a link be-tween microscopic and macroscopic physics. As thenumber of atoms and photons in the cavity is increased,the system evolves from one in which quantum fluctu-ations are dominant to one in which statistical fluctu-ations take over. This evolution involves such phenomenaas quantum tunneling and quantum diffusion. Thesurvival of purely quantum effects in a macroscopicsystem where fluctuation and dissipation are important is

interesting, and simple atom-cavity systems are goodcandidates for studying it.

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3 0 PHYSICS TODAY JANUARY 1989