# Phase of coherent states and single photons

1. Nov 16, 2015

### kaje

Hello

I am confused and I cant figure out what is meant by phase when it comes to quantum mechanics for single photons and coherent states as I am a new to this field.

Best regards

2. Nov 16, 2015

### blue_leaf77

In the context of non-photon related quantum mechanics, the term phase is most of the time associated with the phase of a (complex) wavefunction. Iin quantum optics, where the main object of study is photon, the term phase can be referred to the phase of the expectation value of electric field as often as it is to the phase of the wavefunction of the photon itself. You need to figure out in which context your resource is talking about.

3. Nov 16, 2015

### kaje

Thanks, I am trying to cmpare the phase of the classical wave and in QM. When we say that a coherent state has a phase shift of π or 3π/2 by employing MZI, what that means?

4. Nov 16, 2015

### blue_leaf77

What kind of MZI arrangement are you discussing about? MZI is usually described as a 2-input 2-output device. For instance, what states are prepared at the inputs?

5. Nov 16, 2015

### kaje

Hi,
Thanks, the system that i'm intereted in is phase encoding between two time-bins. At Alice site, she prepares coherent pulses which undergo interference using interferometer before reaching to Bob. My question here in case of using single photons or coherent states, what is the physical meaning of phase between the two consecutive coherent pulses in terms of QM.
OK, for instnce for the following state, what is the pysical meaning of the phase between |1⟩ and |0
|1⟩+exp(ix) |0

Regards

6. Nov 16, 2015

### blue_leaf77

Phase difference in a superposition state like $|1\rangle + e^{ix}|0\rangle$ can be interpreted as the "interference fringe" of an expectation value of an observable to which the superposing states $|1\rangle$ and $|0\rangle$ are not the eigenstates.

7. Nov 16, 2015

### kaje

Thank you so much, it seems that I need to read more about interefernce, but what that means when we say that the relative phase is 0 , π, π/2 or 3π/2.
could you kindly suggest any refernce (books, papers) besides your informative explanation.

Best regards

8. Nov 16, 2015

### blue_leaf77

If this relative phase is related to the phase difference of states involved in a superposition, not that of some complex quantity, taking the above as an example, it will be $x = 0,\hspace{1mm}\pi, \hspace{1mm}3\pi/2,$ and so on. Just for an additional information, the phase difference (or more precisely, the coherence) of two or more states cannot be measured unless we prepare the system in a superposition between the states we are interested in, and measure an observable which is sensitive to the phase differences. In atomic physics, the phase difference between two energy eigenstates, which is cycling with frequency corresponding to the energy difference between these two states, can be measured by, for example, measuring the dipole moment in a superposition state. Those states can be made superposing (also known as coupling of states) by sending a light beam with resonant frequency.
It's not clear to me which topics in QM do you want to study? If it's about quantum mechanical MZI, I would suggest "Introductory Quantum Optics" by Gerry and Knight.

9. Nov 17, 2015

### kaje

Hi,
Many thanks..

I've uploaded here two figures that I don't understand well in this regard. In figure 1, what is the function of α and Φ for the phase modulator( I mean the physical meaning in quanum optics). What does it mean when a coherent pulse or a single photn pass through the phase modulator shown in the figure. What happens at t2, when do we have constructive and destructive. In case of destructive, does any of the detectors clicks?.
In the second figures, what is meant by 0 and Π here, why the pulse at '''0' phase looks different .

#### Attached Files:

• ###### phase modulation.pdf
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10. Nov 17, 2015

### Elemental

Kaje,

For what it's worth, single photons have no definite phase because of the number-phase uncertainty relation. Coherent states of light always have an indefinite number of photons. However, it looks from the figure that pulses from a coherent light source are prepared in states which are quantum superpositions of two states, one of which is phase shifted by φ relative to the other (by adding in an adjustable path delay between beam splitters). These states are then detected with an interferometer which adds another phase shift α between its beam splitters. The interferometer has two detectors, and if the phase delays are such that the final superposition seen at one of the two detectors has a phase difference of a multiple of 2π, that detector will fire and the other will not, and vice versa if the phase difference is an odd multiple of π. Label one detector as logic "1" and the other as logic "0" and the system is a binary demodulator. Note that these phase delays may be ascertained by taking the desired fraction of the coherent light's wavelength.

11. Nov 17, 2015

### kaje

Thanks Elemental,

For the single photons I got the idea, but does the relative phase here for coherent states mean the same concept as in the classic presentation of EM.

12. Nov 17, 2015

### Elemental

Yes, in this case you can interpret phase in the electromagnetic sense.

13. Nov 17, 2015

### kaje

Hi, see the following paragraph please, when it says that Alice modulate the succseesive pulse by 0 and pi, what that physically means in terms of quantum mechanics.

Alice sends a sequence of coherent states with the same intensity, and modulates the phase between successive pulses between 0 to code for bit “0” and π to code for bit “1”. On Bob’s side, he can unambiguously discriminate the encoded bit by interfering successive pulses with an unbalanced interferometer. More specifically, Bob can calibrate his interferometer such that the path length difference makes up for the delay between the pulses, and whenever their relative phase is 0 then detector D0 will click (likewise for relative phase and detector D1).

14. Nov 17, 2015

### Elemental

Kaje,

Quantum states are generally superpositions of other basis states. These basis states are usually chosen so that they represent the different possible outcomes of a measurement. Each basis state has a phase factor which evolves as a function of its energy and momentum, causing constructive or destructive interference with the other basis states just as light waves do, which affects the probable outcome of measurements. More will have to come from textbooks. In this case, the interference pattern is established during modulation.

Elemental