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In the post What is coherence in quantum mechanics? and the answer by udrv in this post it seems to imply that a pure quantum state and coherent quantum state are the same thing since any pure state can be written as a projector onto the pure state when written as a density operator.

Are they equivalent? If these two concepts are not equivalent, what is a simply counterexample to illustrate the difference?

Then there is also the definition of a coherent state which defined it as a quantum state of the harmonic oscillator, quite confusing as to how these concepts are related and distinct, could someone provide some clarity on these distinctions?

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  • $\begingroup$ I think they are the same. The term "coherent" was originally used just for the harmonic oscillator but the terminology has been generalized over time. $\endgroup$ – Virgo Apr 27 '18 at 20:03
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The confusion arises because the word “coherent” evolved to have different meanings in different contexts where it is not fully qualified.

Going back to the 2-slit experiment, one shows that the intensity of the signal at a particular point $$ I_{tot}(x)\ne I_{1}(x)+I_{2}(x)\tag{1} $$ is not the simple sums of intensities of the signals from the two source slit. This is because the light from the slits is “coherent” in the sense that the signals can interfere at point. This website gives some details but basically the intensity at one point is of the form $$ I_{tot}(x)= (A(x)+B(x))^2\tag{2} $$ with cross-terms of the type $A(x)B(x)$ typical of interference between terms. (Despite the efforts of generations of students, $(A+B)^2\ne A^2+B^2$ so (2) CANNOT be the same as (1) in general).

This opposes incoherent light, where the intensity at a point is just the sum of individual intensities of the different sources: $I_{tot}=I_1+I_2$. This is what happens if you shine two flashlights at a wall: the intensity of the light is just the sum of the intensity from the two flashlights: there are no dark and bright fringes of interference.

Now, in a linear combination of wave functions, say $$ \psi(x) = \alpha \psi_1(x) +\beta \psi_2(x). \tag{3} $$ the various parts can, in general, interfere in the sense that the probability density \begin{align} \vert \psi(x)\vert^2 &= \vert \alpha\vert^2 \vert\psi_1(x)\vert^2+ \vert\beta\vert^2\vert\psi_2(x)\vert^2 \\ &\quad + \alpha^*\beta\psi_1(x)^*\psi_2(x) +\alpha\beta^*\psi_1(x)\psi_2(x)^* \end{align} is not just the sum of the probability densities of the individual components, i.e. it is contains cross terms of the type $$ \alpha^*\beta\psi_1(x)^*\psi_2(x)+\hbox{c.c.} $$ and is therefore reminiscent of (2). Thus we speak here of “coherent superposition”. The state of (3) is actually a pure state.

In a mixed state (which cannot be described by a wavefunction), the probability density is a sum of individual probability densities, i.e. something like $\vert\psi(x)\vert^2 =\vert\alpha \psi_1(x)\vert^2+\vert \beta\psi_2(x)\vert^2$ without the interference term. Note that $\psi_1(x)$ could itself be a sum, i.e. $\psi_1(x)=a \phi(x)+b \chi(x)$ so that $\vert\psi_1(x)\vert^2 = \vert a\phi(x)+b\chi(x)\vert^2$ can have cross-terms, but there would be no cross-terms between the pieces in $\psi_1(x)$ and $\psi_2(x)$.

Now as to coherent states. Glauber investigated the question of coherence in quantum optics, i.e. the coherence properties of the quantized electromagnetic field. The tool of choice here is the correlation function, and Glauber was able to find a linear combination of harmonic oscillator states that was “coherent to all order” in the sense of the correlation function. These states Glauber naturally called “coherent states”. Coherent states are pure states so the various parts can inteference and they are thus coherent in the sense that cross-terms appear in the probability density. However, whereas all pure states are coherent superposition of basis states, not all of them are “coherent” in the sense that their correlation functions do not satisfy the condition set out by Glauber.

To make matters worse, Peremolov realized that the Glauber coherent states could be generalized mathematically. Perelomov observed that the Glauber coherent states could be written as $$ \vert\alpha\rangle = T(\alpha)\vert 0 \tag{4} $$ where $T(\alpha)$ is displacement in the plane: $$ T(\alpha)=e^{i(\alpha a^\dagger - \alpha^* a)}\vert 0\rangle\, . $$ Perelomov used this last property to introduced “generalized coherent states”, which are just displacements of some special state (see for instance Perelomov, A. (2012). Generalized coherent states and their applications. Springer Science & Business Media.) . Hence, “spin coherent states” are defined using rotations, i.e. displacements on the sphere, by $$ \vert\theta,\phi\rangle = R_z(\phi)R_y(\theta)\vert JJ\rangle\, . \tag{5} $$ (One can also displace the $\vert J,-J\rangle$ state.)

One can show that the Glauber coherent state $\vert\alpha\rangle$ of Eq.(4) turns out to be an eigenstate of the annihilation operator $a$, i.e. $a\vert\alpha\rangle=\alpha\vert\alpha\rangle$. Obviously this cannot happen when the Hilbert space is finite dimension so angular momentum coherent states of (5) are not eigenstates of either $J_+$ or $J_-$. However, they share with many properties of the Glauber coherent states. Both sets of states have minimum uncertainty (when the angular momentum operators are properly defined), and both states produce specific factorization properties when computing some quantities. Generalized coherent states are not limited to angular momentum but have been defined for a variety of cases, either by insisting they have minimum uncertainty or they are translate of some distinguished state.

Summary: pure states are coherent superpositions of basis states. Mixed states are incoherent superpositions of states. Glauber coherent states (or harmonic oscillator coherent states) are pure states but also satisfy additional properties as laid out by Glauber in terms of correlation functions. Generalized coherent states were introduced by Perelomov; they are pure states which share some properties of the Glauber coherent states.

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  • $\begingroup$ @JohnDoe This will definitely require some attentive reading. It's all done in the density matrix formalism and where "coherences" can refer to the off-diagonal entries of the density matrix, which is related to interference. You can check the QM textbook by Cohen-Tannoudji, Diu, Laloe, Complement EIII, or the textbook by Blum "Density matrix" for additional details on coherences as off-diagonal entries of the density matrix. $\endgroup$ – ZeroTheHero May 4 '18 at 19:29
  • $\begingroup$ Please see my other QM question and proposed solution if you have a chance. $\endgroup$ – John Doe Jun 15 '18 at 16:47
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A great way to understand the relationship between the terms is to consider the difference between a single photon, and the beam of many photons that comes out of a laser. The single photon, prepared identically every time, is as pure a pure state as you can get. The laser puts out photons that are all the same frequency and polarization, and so that also sounds like a pure state. But what is different is when you start considering how many there are.

Coming out of the laser, there are of course many, many photons. In a $1\,mW$ green laser you would have (power/(hc/wavelength)) $2.7*10^{15}$ photons per second on average. It is key to understand here that I am not just saying on average because the value is imprecise. I am saying on average because the ensemble of photons coming out of the laser is a coherent state that is defined by a Poisson distribution, which simply says that a given photon is equally likely to arrive at any time (the arrival times are independent and random). When you consider many photons, the Poisson distributions tells you how likely it is in any time interval to get say 1 photon, 2 photons, ... 100 photons, etc.

So how can you get a single photon out of a laser beam? One way to try would be to open a shutter for a very short time. You would adjust the amount of time the shutter is open for to let on average just one photon through. Most of the time you do this you will get just one photon. But because the photon arrival times are independent and random, sometimes you will get zero photons, and sometimes you will get two photons. Sometimes you might even get 3, 4, or 5, ... but it gets less and less likely.

Another way to try to get a single photon out of a laser beam would be to put filters in the path of the beam that dim the beam so much, that at the end you can observe the photons as single clicks on a sensitive photodiode. But you can't observe the photons and still use them, so you can't just wait until you see a click. You have to do the same thing as above, where you adjust how many filters you use so that you get one photon on average in a specific time interval. But now you are stuck again with a state that usually has 1 photon, but sometimes has 0, and sometimes 2,3,4,5 (with decreasing probability).

So the coherent state is NOT the same as having access to many repetitions of single photons. To get single photons you would need a different sort of machine, perhaps the decay from a single excited atom, or perhaps take your laser and put it through parametric down-conversion so that you can herald the generation of a single photon and know that you have exactly 1 (and not 0 or 2,3,4,5...)

So that is the distinction between a pure state and a coherent state. The coherent state is a type pure state with particular statistics for photon number (or as you mentioned, for particular occupation level in a harmonic oscillator).

But what is the relationship between a coherent state and coherence? Well, what is of note is how the laser gave us the coherent state. In the case of the laser, coherence refers to the fact that we can measure the phase of the electric field of the beam at some point, and then predict reasonably well what that phase will be measured at either a distant point or a later time. The longer that phase relationship holds, the higher the coherence. The phase would be quite well predictable if all the photons had exactly the same frequency. In a laser this is nearly true, and the extent to which it is not true is expressed as the 'linewidth' of the laser. For white light, the frequencies of the photons are all different, and so there is no predictable phase relationship for the beam, which is why white light is 'incoherent'.

Now consider how you might get a single photon out of a white light beam. If it is truly, truly incoherent, then every photon will have a different frequency (they can be arbitrarily close, but there are also arbitrarily many possible frequencies), so I can make a filter that only lets through a certain very specific color of light. As this filter gets infinitely narrow (in terms of the frequencies or colors of light it allows through), I will have to wait longer to get one photon, but I can make the filter infinitely narrow and wait infinitely long and get one photon with certainty. We could not do this with the coherent beam.

You might ask, well what if you used this infinitely narrow filter trick on the laser beam? In so far as the laser has a finite linewidth, this will work, but that is also a measure of the fact that the laser beam is not perfectly coherent. If the ensemble of photons put out by the laser was perfectly coherent, they would all have the same frequency, and the linewidth would be infinitely narrow, and no matter how narrow you make the filter you will still get all the photons through.

So it is precisely the property of coherence that gives us a coherent state, and vice-versa.

Another way to discuss the relationship between the laser beam and the single photon is to consider what the "wavepacket" looks like. For the coherent beam, it is a sine-wave that goes on forever, and in fact there is so much amplitude that it is basically a classical wave where the quantization into individual photons matters not at all. As you make a laser pulse shorter and shorter and shorter, eventually you end up with a single photon which still has an electric field oscillating at the same frequency, but with a very short envelope. A little knowledge of Fourier transforms tells us that a shorter pulse in time must have a broader spectrum in frequency. So there's no way the single photon can be the same as a coherent state, because it inherently has some linewidth due to its finite time span. A photon is a 'minimum uncertainty wavepacket' and so it has the least possible frequency spread of any wave that has this duration, so it can have some 'coherence', but a single photon (if we know there is for certain one and only one of them) cannot be a 'coherent state'.

Again, I'll restate my conclusion from above: It is precisely the property of coherence that gives us a coherent state, and vice-versa.

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Coherent quantum states are special types of pure quantum states. The term coherent is only meaningful when there is an Fock algebra of ladder operators. Coherent states will be eigenvector of the (non-Hermitian) annihilation operator

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    $\begingroup$ I don't think this answer is very illuminating, especially given the level of understanding demonstrated by the questioner. $\endgroup$ – Virgo Apr 27 '18 at 20:00
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Coherence have a wide range of definitions from the simpler ones in the other answers to more complex ones based in resource theory and fisher information. Roughly they all try to quantify the ability of a state to display interference in various properties. This ability depends on what’s making changes(your Hamilton) and how your making a measurement. Purity and coherence don’t have to overlap and are in general different concepts . Depending on your excitement a specific mixed state might display more interference the a specific pure stare. But the cleanest interference will always be from a pure state.

A coherent state is a specific pure state used for lasers and are the most coherent for the normal interferometers.

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