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What's Quantum Optics?

What is quantum optics? This is a rather personal question. A well-known scientist in this field once gave the following authoritative answer: "Whatever I do defines quantum optics!" On a more objective basis one is tempted to define this branch of physics by the pun: "Quantum optics is that branch of optics where the quantum features of light matter."

Which discovery in physics marks the birthday of quantum optics? Many phenomena come to our mind. Is it the discovery of the quantum, the development of QED, or the maser/laser? Or is it none of the above?

In this chapter we answer this question in a back handed way by summarizing some path breaking experiments that define quantum optics. Admittedly this list is not complete and chosen in a rather subjective way. The rapidly moving field of quantum optics demonstrates most clearly that even after 100 years of quantum physics there is still a lot to be learned from Planck's original discovery.

1.1 On the Road to Quantum Optics

More than hundred years ago M. Planck was struggling with the experimental data of black body radiation obtained at the Physikalisch-Technische Reichsanstalt in Berlin by H. Rubens and F. Kurlbaum. From todays point of view these experiments look rather academic. However, they were motivated by industrial applications. Indeed, standards had to be developed in order to describe light bulbs. This need triggered one of the most important problems in the physics of the 20th century: Classical electromagnetic theory cannot explain the measured black body spectrum. In a desperate but courageous attempt Planck postulated that the oscillators in the walls of the cavity can only absorb and emit radiation in discrete units. This revolutionary idea of discreteness rather than a continuum provided the celebrated radiation formula and was the starting point of quantum mechanics.

Nowadays we associate the quantization with the field rather than with the mechanical oscillators in the wall. However, wave and matrix mechanics were first developed for massive particles and then, later, transferred to the electromagnetic field leading to quantized electrodynamics.

The field of quantum electrodynamics, QED, which deals with the interaction of quantized matter with quantized electromagnetic fields started with P.A.M. Dirac. He was the first to derive the Einstein A and B coefficients of spontaneous and induced emission. The field of QED culminated on the one hand with the experimental discovery of the level shift in the hydrogen atom by W.E. Lamb and R.C. Retherford and the measurement of the anomalous moment of the electron by H.M. Foley and P. Kusch. On the other hand the theoretical works of S. Tomonaga, J. Schwinger and R. Feynman showed how to avoid the infinities that had plagued the theory since the thirties. The incredible agreement between theory and experiment established nowadays in many QED systems confirms beyond any doubt the quantized nature of light.

The development of the ammonium maser by C.H. Townes, J. Gordon and H. Zeiger, and the laser by T. Maiman following the paper Optical Masers by A. Schawlow and Townes has opened the new field of quantum electronics. Motivated by the experiments on the maser and building on his own theoretical work on water-vapor absorption W.E. Lamb developed a theory of the maser during the years 1954-1956. Later he worked out a complete semi-classical theory of laser action. Independently the group of H. Haken in Stuttgart developed their own approach. In the semiclassical treatment of Lamb and Haken the electromagnetic field was described classically and the atom quantum mechanically.

Since then laser theory has come a long way from the early approaches using birth and death equations via the semiclassical theory of the laser to the fully quantized version. The three approaches to the quantum theory of the laser are the Fokker- Planck method, pursued by H. Haken and H. Risken, the noise operator method by M. Lax and W.H. Louisell and the density matrix techniques by M.O. Scully and W.E. Lamb. Earlier the quantum theory of photon counting has been developed by R. Glauber.

Unfortunately, the quantum effects of the laser were scarce. The photon statistics of the laser and the phase diffusion were the only quantum effects that could be measured.

1.2 Resonance Fluorescence

Quantum optics has received an enormous push from the phenomenon of resonance fluorescence. The light emitted from an atom which is driven by a classical monochromatic electromagnetic field shows interesting quantum effects in its spectrum and in the statistics. We now briefly review this corner stone of quantum optics.

1.2.1 Elastic Peak: Light as a Wave

Resonance fluorescence is an old problem that has been discussed for the first time in great detail by W. Heitler in his classic book "The Quantum Theory of Radiation". He pointed out that the emitted radiation has the same frequency as the incident radiation. Thus, the spectrum is a delta-function. In this sense the atom is just a driven dipole and therefore radiates with the frequency of the driving field. This elastic component in the scattered light has been observed experimentally and is shown in Fig. 1.1.

Fig. 1.1: Heterodyne spectrum of the elastic fluorescence component of a single trapped 24Mg+ ion. A narrow peak emerges when the frequency difference between the heterodyne signal and the driving field vanishes. Taken from J.T. Höffges et al., Opt. Comm. 133, 170 (1997).

For this measurement a single magnesium ion was stored in a modified Paul trap shown in Fig. 1.2. A laser was driving an electronic transition of the ion and the emitted radiation was superimposed with the driving field. The resulting signal, commonly referred to as heterodyne signal, was analyzed with respect to frequency. This spectrum displays a narrow structure at the frequency of the incident radiation as shown in Fig. 1.1. Theoretically, the width of this line should be zero. However, it is determined by the spectrum of the exciting light.

In this context it is interesting to note that this experiment is also a verification of the wave nature of light. Indeed, the elastic peak is so narrow since the emitted wave has a fixed phase relation with the driving field. The emitted light is therefore a wave. This experiment clearly supports the wave rather than the particle concept. However, as we will see in the next section we can slightly rearrange the experiment such that the particle aspect of light emerges. This is one more manifestation of Bohr's principle of complementarity.

1.2.2 Mollow-Three-Peak Spectrum

However, this delta-function peak in the spectrum is only one part of the problem. In the late sixties B.R. Mollow investigated resonance fluorescence using quantum electrodynamics and found that the spectrum depends on the intensity of the incident radiation. For low intensities the Heitler result is valid. However, for larger intensities the spectrum displays a more complicated structure: In addition to this elastic delta-function peak there exist three broad, incoherent contributions centered at the incident frequency and at two side bands. They are shifted by a frequency determined by the electric field of the incident wave. These incoherent peaks have a different width determined by the natural line width G of the atom. Indeed, the central peak has a width of G/2 whereas the sidebands have the width 3G/4.

Fig. 1.2: Electrode configuration of the endcap trap. The trap consists of two co-linearly arranged cylinders corresponding to the cap electrodes of the traditional Paul trap. The ring electrode is simulated by two hollow cylinders which are concentric with each of the cylindrical endcaps. Additional electrodes allow for the compensation of stray electric field components. The open structure offers a large detection solid angle and good access for laser beams. Taken from J.T. Höffges et al., Opt. Comm. 133, 170 (1997).

Fig. 1.3: Experimental three-peak Mollow spectrum for increasing laser intensity. We note the emergence of the side peaks. The elastic peak ideally represented by a deltafunction located on top of the central peak is not shown. Taken from W. Hartig et al., Z. Physik A 278, 205 (1976).

Fig. 1.4: Measurement of second order correlation function. Light from a source passes a beam splitter and falls onto two detectors. We are interested in the distribution of consecutive clicks of the two detectors. The first photon hitting a detector starts the clock, the second photon hitting the other detector stops the clock. As light source we could use a thermal light bulb, a laser or the resonance fluorescence of a single ion driven by a laser field.

This spectrum has been measured experimentally by the groups of C.R. Stroud, S. Ezekiel and H. Walther in the mid-seventies. In Fig. 1.3 we show the emergence of the three-peak Mollow spectrum for increasing laser intensity.

1.2.3 Anti-Bunching

A new chapter in the book of resonance fluorescence was opened in the mid-seventies when H. Carmichael and D.F. Walls in New Zealand, and H.J. Kimble and L. Mandel in the US independently from each other analyzed the statistics of the light. They found a time delay between two successive photons emitted from the atom. The light is anti-bunched. This behavior is in sharp contrast...