# My review of "Wstęp do optyki kwantowej" by C.C. Gerry and P.L. Knight

- Szczegóły
- Category: Review
- Published: thursday, 31, december 2015 15:32
- Author : Janusz Szcząchor

This is one of few quantum optics handbooks which have been printed in Polish. It is the translation of the authors' *Introductory Quantum Optics* which was originally written in English. I am not going to put forward the content of the book since it has already been done. I recommend you the review by Ryszard Tanaś, who is from UAM in Poznań, placed in *Postępy Fizyki*, vol. 59, p. 227 ( 2008) . I only aim to present my comments about the book.

The authors, C.C. Gerry and P.L. Knight, did not avoid making some faults. The content of the book has some gaps in the lines of thought. They can cause some problems to the students who are not fluent in English and find difficulty in reading scientific papers. It is necessary to admit that in the introduction warned the authors themself the reader against having problems with understanding of the book.

Now I go to the heart of the matter. The presentation of the subject of quantum optics has been begun in classical way - i.e. with quantization of the electromagnetic field. The issue of quantization is depicted at length except for the problem of the fluctuation of the vacuum and energy of zero-point oscillations.

They are crucial issues concerning quantum electrodynamics but the authors provided only two examples to illustrate them. That examples are Lamb lineshift and Casimir force. The illustration is too skimpy, in my opinion.

I understand that the authors only wanted to prove that the zero-point fluctuations may give rise to a slight shift of energy levels. However, they have admitted that those are serious flaws in quantum electrodynamics. Therefore, I think that the lacks of quantization of the electromagnetic field should be discussed in wider context. After all, widely available textbooks are not intended to students from only two colleges!

In any case, the content of chapter 2.6. has not satisfied me, especially as Bethe's work dates from 1947 year and quoted Welton's paper was published only one year later.

As we know, the Universe consisting only of fields of interaction and no matter would be of little interest. However, simultaneous quantization of fundamental forces and matter leads to very complicated mathematical problems. Therefore, scientists developed many semiclassical models, in which the matter is handled quantum-mechanically, but with the external field treated as a time-dependent classical variable.

When I was a student, I was still being taught about theory of emission and absorption of radiation which had been developed by Dirac, Slater and Fermi at the turn of the 1920s and 1930s years. Certainly, their theory was semiclassical. What is more, I was not informed at all that had already existed a soluble fully quantum mechanical model of an atom in a field. It was, originally defined in 1963, the Jaynes-Cummings model.

It is thus praiseworthy that the authors devoted that model whole chapter. Nevertheless, the discussion of the model suffers from an essential defficiency.

The simplest version of the Jaynes-Cummings model comprises a single two-state atom interacting with a single near resonant quantized cavity mode of the electromagnetic field.

Chapter 4 of that book also describes another version of that model in which the initial field is expressible as a coherent superposition of photon-number states and the atom is initially unexcited. Then the initial statevector of the atom and the field is also expressible as a superposition of statevectors which evolve independently and each of them describe the evolution of the atom and one of photon-number states.

The above conclusion arises from the fact that the JCM Hamiltonian results in an infinite set of uncoupled two-state Schrödinger equations, each pair identified by the number of photons that are present when the atom is in the lowest-energy state. However, one can imagine that interaction between the atom and a one of photon-number states is not independent of interaction between the atom and another photon-number state. Consequently, the two-state Schrödinger equations would be coupled.

The book gives detailed expression, marked with number (4.120), which describes temporal evolution of previously mentioned initial statevector. However, it is impossible to obtain the expression without above mentioned information on the JCM Hamiltonian but the book tells about that nothing.

The authors presented the reader the final result of the calculations and did not account for how they had obtained it. One can find the clarification of that in [2]. Fortunately, Gerry and Knight pointed out that article in references to that chapter.

Special attention deserves the chapter devoted to nonclassical light. To tell you the truth, there are no classical lights since all are quantum ones.

On the other hand , it is widely assumed that the most classical of single-mode quantum states is a coherent state. It-in a segment of the beam and for a time interval short compared with the coherence time-comes close to what is classicaly described as a wave with fixed phase and amplitude.

The other name of the state is Glauber state. The state is found from the requirement that the fluctuation of the electric field strength, averaged over some lights periods, should be minimal for a given mean photon number. A single-mode coherent state is a superposition of the Fock states |n >, where n=0,1,2,..., whose photon probabilities follow a Poisson distribution.

What is a nonclassical light?

The definition of non-classical states is very elaborated. A state is called classical when its density operator ρ allows for a so-called P representation and the function P(α) is non-negative or not more singular than the Dirac delta. Otherwise, the state is nonclassical. There are many kinds of them. For instance, squeezed states. Mainly to them is devoted that chapter.

Previously mentioned the semiclassical and the Jaynes-Cummings models do not take account of dissipation of energy to a vacuum field from the system consisting of an atom and a few photons in a cavity.

In 1930 Weisskopf and Wigner improved semiclassical theory through introducing exponential decay of the excited level of the atom. It was crowning achievement of that theory. Nonetheless, it is well known that a necessary condition for a pure exponential decay is that the energy spectrum of the electromagnetic field has no lower band. This is not the case of electromagnetic field neither in the free space nor in a cavity. What is more, the deviation from the exponential Weisskopf-Wigner decay of an atom in free space has been demonstrated in a number of experiments.

Fortunately, the Jaynes-Cummings model is a good point of departure for constructing new models which can help us with investigating upper mentioned problem.

The issue of dissipation of energy can be dealt with two ways. First, one can use optical Bloch equations as a master equation. However, if we consider an atomic system with N states, the master equation treatment requires the simultaneous solution of N^{2} equations. Second, you can use the wave-function treatment but it needs repeated "gedanken measurements" on the atomic system simulating the detection of the spontaneous photons. In spite of everything, in this approach we have to look for the evolution of no more than N variables. However, in order to exploit the new approach you have to understand how works the Monte Carlo method.

*Wstęp do optyki kwantowej* describes the Monte Carlo wave-function approach but the discussion is not enough in-depth, in my opinion. One can hardly understand the details of the method and is forced to study some of the references or other handbooks.

The Born’s statistical interpretation of wave-function which includes the meaning of the statevector given by the probability law and the predictivity of formalism only for the average behaviour of a group of similar events induced some scientist to claim that experiments on single quantum objects are impossible and unphysical. However, now we know that facts stand differently.

Chapter 9 puts forward several of such experiments. I would like to draw your attention to one of them, namely Hong, Ou and Mandel's experiment described in [7].

In this experiment two photons simultaneously generated from one photon in the process of parametric down-conversion pass through a beam splitter and interfere and are detected by two photodetectors. If the two photons are indistinguishable and impinge simultaneously on different entrance ports of a 50:50 beam splitter, both photons will leave the beam splitter at the same output port. In the process, it is impossible for them to leave the beam-splitter at the different output ports. In my opinion, this experiment clears up at last what the wave-particle duality is.

When I was a student I analysed many reasonings trying to clarify this duality but all of them had unclear moments in the line of thoughts. Simply, they did not convince me. Hong, Ou and Mandel's experiment enables us to understand the duality with the naked eye.

Chapter 10, the last one I would like to present, is devoted to QED in a cavity and trapped ions. I am going to focus only on the first issue.

The Jaynes-Cummings model, the first soluble fully quantum mechanical one, was realized in practice in the form of one-atom maser.

The maser has two important ingredients. The first is a highly excited rubidium atom with a very large principal quantum number n. It is called Rydberg atom. The second is the superconducting cavity which can contain photons conforming to various statistics. In order to be allowed only one transition between selected two levels, the Rydberg atoms are appropriately prepared and are used suitable cavities. The probability of finding the atom in the upper maser level shows clearly the collapse and revival predicted by the Jaynes-Cummings model. The revivals can be considered as a pure quantum effect not predicted by any of the semiclassical theories and caused by the interaction between the atom and each of photon-number states.

As far as the editorial aspect is concerned, the book contains many typographical errors. Several dozen were pointed out by prof. Ryszard Tanaś. However, the text of the book has them more. E.g. obvious miscalculations are contained in chapter 7 on page 153. That is why you cannot believe the formulae and yourself absolutely have to check them.

To sum up, I regard the quantum optics handbook as quality one on the Polish publishing market. However, in order to study it you will need at least the course book *Wykłady z optyki kwantowej *by Ryszard Tanaś available on UAM's website and some of references written in English, of course.

Finally, I would like to mention that I have learnt the subject of quantum optics and mechanics in English from the following papers.

## References

[1] Van Vleck, J. H., Huber, D. L. ,1977, Rev. Mod. Phys. **49**, 939.

[2] Shore, B. W., and Knight, P. L., 1993, J.Mod.Opt. **40**, 1195.

[3] Paul, H., 1986, Rev.Mod.Phys. **58**, 209.

[4] Buzek, V., Drobny, G., Min Gyu Kim, Havukainen, H., and Knight, P. L.,1999, Phys. Rev. A **60**, 582.

[5] Dalibard, J., Castin, Y., and Molmer, K., 1992, Phys.Rev.Lett. **68**, 580.

[6] Cramer, J. G., 1986, Rev.Mod.Phys. **58**, 647.

[7] Hong, C.K., Ou, Z. Y., and Mandel, L., 1987, Phys.Rev.Lett. **59**, 2044

[8] Legero, T., Wilk, T., Kuhn, A., and Rempe, G., 2003, Appl.Phys. B ,**77**, 797.

[9] Rempe, G., Walther, H., and Klein, N.,1987, Phys.Rev.Lett. **58**, 353.

[10] Weisskopf, V.F., 1949, Rev.Mod.Phys. **21**, 305.