Unraveling the Mystery of Coherent States: A Guide for Graduate Students

[Sunnyocean]

Hello,

I would like to understand as well as possible what (quantum) coherent states are. Can anyone advise on what books (or other materials) I should read?
Please assume I have an introductory level in Quantum Physics (where by "introductory" I mean the material introduced in books like, for example, "Introduction to Quantum Mechanics" by David Griffiths.
So considering these, what books / materials I should read/ study to understand (and even master) coherent states?
I would like to say that I am not interested in "easy" materials i.e. materials that do away with the maths (unless by some miracle they happen to be particularly enlightening, which I doubt a material that does away with the maths or summarises the maths can be); on the contrary, I am interested in gaining an understanding of coherent states as thorough and as rigorous as possible.

Also since I have a BSc in physics I chose "graduate" for the prefix; however I am not sure so please if what I need is lower than graduate-level textbooks / materials, do not hesitate to recommend them to me.

Thank you very much in advance!
:)

 

[Cthugha]

Well, coherent states arise mostly in quantum optics (where they are the most classical states possible), but also in other branches. The book you should read, depends on your math background. Some relevant literature in order of increasing difficulty is:

Quantum Optics: an introduction by Mark Fox - good introduction on the level of a Bachelor student
Introductory Quantum Optics by Gerry Knight - more difficult, but still easy going
Quantum Optics by Scully and Zubairy - good medium difficulty textbook
Quantum Optics by Vogel and Welsch - Difficult, but elegant. If you think learning classical mechanics from Landau/Lifshitz is a good idea, this is your book to start with. Otherwise, you better start with a different book

Other "special interest" books:
Statistical Methods in Quantum Optics by Carmichael - Specializes on master equations and open systems and is a good read if you actually want to implement some calculations
Coherent states in Quantum Physics by Gazeau - Not really a great book, but it is solely about all kinds of coherent states. If you really are interested only in coherent states, this might be of interest to you.

 

[strangerep]

Since Cthugha didn't mention it, I'll also recommend Mandel & Wolf, called by some the "bible" of its field. (It's a graduate-level textbook.)

Also, google for "generalized coherent states", and scholarly articles by Perelomov. That whole subject is thoroughly fascinating, illuminating the group theoretic foundations of quantum physics.

@A. Neumaier will no doubt suggest further reading...

 

[vanhees71]

Before you go into the more complicated realm of QED and quantum optics, just look at the simple harmonic oscillator:

https://en.wikipedia.org/wiki/Coherent_state#Quantum_mechanical_definition

Another very good book on quantum optics is

J. Garrison, R. Chiao, Quantum optics, Oxford University
Press, New York (2008).
https://dx.doi.org/10.1093/acprof:oso/9780198508861.001.0001

 

[A. Neumaier]

I would like to understand as well as possible what (quantum) coherent states are.

It depends on what you want to use your knowledge for. There are lots of different kinds of coherent states, depending on the application. There are also different levels of depth and exposition, the most powerful being based on Lie groups.

The simplest are Schrödinger's coherent states for a harmonic oscillator, corresponding to the 1 DOF Heisenberg-Weyl group. Next come Glauber's coherent states for laser light, corresponding to an infinite-dimensional Heisenberg-Weyl group. Then squeezed coherent states for nonclassical light, corresponding to symplectic groups, and spin coherent states for Fermions, corrresponding to the group SO(3) of spatial rotations.

All these are special cases of the Perelomov coherent states for arbitrary Lie groups with a triangular decomposition, whose understanding requires a solid background on Lie algebra and Lie group theory, for example that obtained from reading my online book

• A. Neumaier and D. Westra, Classical and quantum mechanics via Lie algebras, 2nd ed. 2011.

A very informative and thorough physics-oriented exposition of Perelomov coherent states is given in

• W.M. Zhang, D.H. Feng and R. Gilmore, Coherent states: theory and some applications, Rev. Mod. Phys. 62 (1990), 867--927.

At the most general level, coherent states are associated to coherent spaces, which do not need to have the symmetries of the group case. These provide an efficient tool for the quantization of classical theories and the semiclassical approximation of quantum theories, see Chapters 5 and 6 of my recent book

• A. Neumaier, Coherent Quantum Physics: A Reinterpretation of the Tradition, de Gruyter, Berlin 2019.

Earlier preprint related to the material in these chapters are here and here.

I would like to understand as well as possible what (quantum) coherent states are.

For the best possible understanding you'll need to digest all of the above.

 

[Sunnyocean]

Thank you very much to everyone who answered my question.

Currently I am in doubt as to which book to start with. It would be helpful if someone could say exactly what kind of mathematics I need, rather than "graduate-level math" or "undergraduate-level math" etc.

My experience is that what people - even in physics - mean by "undergraduate-level maths", "post-graduate level maths" (and so on) varies greatly from one person to another.

For example some people seem to think that matrix multiplication (and other operations such as calculating the determinant etc.) or the epsilon-delta characterisation of continuity (where by continuity I mean "continuity relative to the standard topology applied do the set of real numbers", as there is no such thing as absolute continuity and "continuous" is always relative to the topology you are using) are undergraduate-level maths - however, I learned them in high school. And, conversely, it was not until university that I learned other mathematics (for example statistical mathematics) while some other people seem to have learned them in high school.

For starters (*but not limited to this*), I would like to understand better the concept of coherence as it is used in Dycke's paper on superradiance ( Dicke, Robert H. (1954). "Coherence in Spontaneous Radiation Processes". Physical Review. 93 (1): 99–110. Bibcode:1954PhRv...93...99D. doi:10.1103/PhysRev.93.99. ) and also the coherence as it is used in QASER (Quantum Amplification of Superradiant Emission of Radiation - please see, for example, https://journals.aps.org/prx/abstract/10.1103/PhysRevX.3.041001 )

Of course, I am not asking for a comprehensive list of all the mathematical theorems and formulae I need - this would probably be the book itself - but some more precise pointers would be great, so that I know where (i.e. which book) to start from. (Or if I need to buy books on mathematics as well).

For example, if they could say "for this book you need vector calculus" (followed by a few names of the more significant theorems I would need to use in order to understand that book, as vector calculus is of course very vast) or "you need one-variable integrals, two-variable integrals and three-variable integrals" etc.

 

[Sunnyocean]

I started reading the paper at https://arxiv.org/pdf/1804.01402.pdf

I definitely need to study more mathematics in order to understand concepts such as "antilinear functionals"and so on - and this is just by starting reading the first chapter.

I also previewed the book by Arnold Neumeier, "Coherent Quantum Physics: A Reinterpretation of the Tradition " and from reading what was available for previewing it looks like an excellent book, but currently it is prohibitively expensive for me.

So it seems I would need to read other books before actually starting reading about coherent spaces. Could anyone please recommend to me such books?
Also, would I need them to understand coherence as it is used in quantum optics? (I am really just trying to use my time efficiently, not trying to "get away" with studying less; so if the answer is yes or even "partially yes" please do not hesitate to say so.)

 

[vanhees71]

I'd recommend to start with the RMP article mentioned in #5

https://doi.org/10.1103/RevModPhys.62.867

 

[Cthugha]

My experience is that what people - even in physics - mean by "undergraduate-level maths", "post-graduate level maths" (and so on) varies greatly from one person to another.

For example some people seem to think that matrix multiplication (and other operations such as calculating the determinant etc.) or the epsilon-delta characterisation of continuity (where by continuity I mean "continuity relative to the standard topology applied do the set of real numbers", as there is no such thing as absolute continuity and "continuous" is always relative to the topology you are using) are undergraduate-level maths - however, I learned them in high school. And, conversely, it was not until university that I learned other mathematics (for example statistical mathematics) while some other people seem to have learned them in high school.

To be honest, for serious physics books nobody makes the distinction between "high school" and undergrad math. Anything that could potentially arise in a high school context is certainly below grad level math. What you will certainyl need, is a good grasp of basic quantum mechanics and some idea about second quantization (although using the term for light fields is somewhat questionable).

For starters (*but not limited to this*), I would like to understand better the concept of coherence as it is used in Dycke's paper on superradiance ( Dicke, Robert H. (1954). "Coherence in Spontaneous Radiation Processes". Physical Review. 93 (1): 99–110. Bibcode:1954PhRv...93...99D. doi:10.1103/PhysRev.93.99. ) and also the coherence as it is used in QASER (Quantum Amplification of Superradiant Emission of Radiation - please see, for example, https://journals.aps.org/prx/abstract/10.1103/PhysRevX.3.041001 )

Dicke's paper comes from a time before the advent of quantum optics. His usage of the term "coherent" is completely unrelated to coherent states. In fact, superradiance is pretty noisy and therefore incoherent in terms of what defines coherent states. Still, you have a collective coherent interaction between an ensemble of emitters and a joint light field, which is a different beast. If you have access to journals via your research institution, a good overview on superradiance is given in
"Superradiance: An essay on the theory of collective spontaneous emission", Physics Reports 93, 301 (1982). It is authored by M. Gross and the Nobel-prize winner Serge Haroche.
However, most textbooks on quantum optics will also discuss superradiance at some point.

 

[A. Neumaier]

I started reading the paper at https://arxiv.org/pdf/1804.01402.pdf

I definitely need to study more mathematics in order to understand concepts such as "antilinear functionals"and so on - and this is just by starting reading the first chapter.

Antilinear functionals are introduced here; they are linear up to complex conjugation.

Most of my online book linked to in my previous post is less demanding and free of charge.

On the other hand, you had stated

that I am not interested in "easy" materials i.e. materials that do away with the maths [...] I am interested in gaining an understanding of coherent states as thorough and as rigorous as possible.

To meet the last condition you need a lot of math.

Also, would I need them to understand coherence as it is used in quantum optics?

For quantum optics you just need Glauber coherent states and squeezed coherent states. At an informal level you can get the required understanding by following the Wikipedia links given. It is anyway advisable to first get a less rigorous initial understanding before dealing with the abstract concepts in full generality.

 

[Sunnyocean]

Thank you very much again for your reply.

It is anyway advisable to first get a less rigorous initial understanding before dealing with the abstract concepts in full generality.

That is exactly what I think. To me lack of a rigorous understanding means no understanding. Not because I am fussy but this is the reality when you try to actually understand and find out there are errors or some information is missing.

I looked at squeezed coherent state on Wikipedia (the link you sent) but I do not think wiki explains it well; for example even in the first few lines it says "In physics, a squeezed coherent state is a quantum state that [...]" and shortly after that "Trivial examples, which are in fact not squeezed, are [...]". In other words, what they are saying is "here are some examples of squeezed states which are in fact *not* squeezed". Or if you scroll down further you find "Depending on the phase angle at which the state's width is reduced, one can distinguish amplitude-squeezed, phase-squeezed, and general quadrature-squeezed states. " And after they say that, they don't explain what they mean by each of the three types. These are not the only errors or places where information is missing; they were just examples.

So it is as you said: it is an informal introduction (in fact I think you were too kind towards wiki when you characterised the wiki material on squeezed states as "informal") and definitely not what I need for a rigorous understanding.

For Glauber coherent states, when I clicked it said "page not found", could you please re-post?

I wrote the paragraph above (regarding the material on wiki about squeezed states) in order to give a better idea of what I would like: a continuous presentation, without missing information, without errors and also a material that is as much as possible self-contained. Even if it takes thousands of pages, I am fine with that. What I am not fine with (again, not because I am fussy but because I think it would be counterproductive) is a material in which I constantly have to "jump" to other materials where there are errors or missing information, which introduces a lot of interruptions and thus makes one use time in a very inefficient and wasteful manner - like the material on wiki.

To meet the last condition you need a lot of math.

Yes, true, and I am more than happy and even eager to learn it in a constructive, rigorous, step-by-step manner. For example the material at the link on antilinear functions that you posted ( https://en.wikipedia.org/wiki/Complex_conjugate_vector_space ) seemed to me very good and much closer to what I need (i.e. I need material with rigorous maths, no steps skipped etc.). However, I also realized I need to learn more about rings, vector spaces, isomorphism etc. (I did some algebra years ago but it was an introduction).
So, in this regard:
1) Do you think I need this algebra for a rigorous (i.e. fundamental-level) understanding of coherent states (and not only)?
2) Irrespective of your answer at point 1), could you please recommend some good, rigorous books / materials for this area of algebra?

Thank you very much again.

 

[A. Neumaier]

I don't think the desired book or reference exists. The more rigorous a tratment the more sketchy it is in details. Thus either you need to interpolate the missing details or you need to interpolate the missing rigor.

"Trivial examples, which are in fact not squeezed, are [...]".

This is like introducing complex numbers and then saying, ''Trivial examples, which are in fact not complex, are the real numbers''. This makes complete sense once one knows that real numbers are a subset of complex numbers, and that when talking of complex numbers, one may or may not exclude those with imaginary part zero. Similarly, Glauber states are amon the sueezed states (by their general definition) though they are not squeezed in the same sense as real numbers are not complex.

To me lack of a rigorous understanding means no understanding. Not because I am fussy but this is the reality when you try to actually understand and find out there are errors or some information is missing.

Well, your understanding is complete once you can spot all the errors and interpolate all the missing information to your satisfaction. This is the only test of true understanding. You may or may not need rigor for achieving this.

1) Do you think I need this algebra for a rigorous (i.e. fundamental-level) understanding of coherent states (and not only)?
2) Irrespective of your answer at point 1), could you please recommend some good, rigorous books / materials for this area of algebra?

There are no rigorous books on coherent states; the closest is wht I referred to earlier. You need basics on groups, rings and algebras, and in particular on Lie groups and Lie algebras. At some point you need to know basics about differential geometry and groups acting on manifolds. All this can be found in Wikipedia by following the links to whatever you do not yet thoroughly understand but need it to digest something on the way to your goal. You can read in parallel my free online book

• A. Neumaier and D. Westra, Classical and quantum mechanics via Lie algebras, 2nd ed. 2011.

and the survey

• W.M. Zhang, D.H. Feng and R. Gilmore, Coherent states: theory and some applications, Rev. Mod. Phys. 62 (1990), 867--927.

and look up in wikipedia all the stuff that you need for understanding these two sources.