QM: Formalism of 2-state systems

In summary: I think the only way to solve this is to find out what the problem is really about. Okay, thanks for your input! I'll try to look for more information about the problem.In summary, the conversation discusses a problem involving a Mach-Zehnder interferometer with an additional phase shift element, with the question being to calculate the probability of detector A clicking as a function of the phase shift. The conversation also touches on the effect of random phase shifts and the interpretation of a uniformly distributed random variable.
  • #1
WWCY
479
12

Homework Statement



Hi all, I'm working on the following problem and would like some help. Many thanks in advance!

The Figure below presents the Mach-Zehnder interferometer with an additional phase shift element in the upper path.
$$\left( \begin{array}{cc}
e^{i\phi} & 0 \\
0 & 1
\end{array} \right)$$
with ##\phi## as the phase.
a) Calculate the probability that detector A clicks as a function of ##\phi##

b) In practice there is always some random phase shift even in the absence of the additional element in the upper path. It comes from misalignment of beam splitters and mirrors. Assume ##\phi## is a uniformly distributed random variable. What is the probability now that detector A clicks as a function of ##\phi##?

Screen Shot 2018-02-09 at 9.10.36 PM.png


Homework Equations

The Attempt at a Solution



a) The following vectors define the "right-moving" and "upwards-moving" photon states
$$|r> = (1,0)$$
$$|u> = (0,-1)$$
The initial right mover hits the beam splitter (defined by the matrix below) and the resulting mixed state is
$$

\left( \begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{array} \right)
%
\left( \begin{array}{cc}
1 \\
0
\end{array} \right)
= \frac{1}{\sqrt{2}} \big ( (1,0) + (0,-1) \big)
$$
Both beams then hit mirrors, so I apply the mirror (matrix below) to the mixed state
$$
\left( \begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array} \right)
%
\left( \begin{array}{cc}
1/\sqrt{2} \\
-1/\sqrt{2}
\end{array} \right)
=\frac{1}{\sqrt{2}} \big ( (-1,0) + (0,-1) \big)
$$
Only the top beam hits the phase shifter
$$
\frac{-1}{\sqrt{2}}
\left( \begin{array}{cc}
e^{i\phi} & 0 \\
0 & 1
\end{array} \right)
%
\left( \begin{array}{cc}
1 \\
0
\end{array} \right)
= \frac{-1}{\sqrt{2}} \big ( e^{i\phi},0 \big)
$$
Applying the beam-splitter to both states again yields
$$
\frac{-1}{\sqrt{2}}
\left( \begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{array} \right)
%
\left( \begin{array}{cc}
e^{i\phi} \\
1
\end{array} \right)
= \frac{-1}{2} (e^{i\phi} + 1 \ , 1 - e^{i\phi} )$$

The probability of the detector "a" clicking is thus given by
$$| \frac{1 - e^{i\phi}}{2} |^2 = \frac{1}{2}(1 - \cos\phi )$$

b) I have not formally studied probability theory, but I'm assuming that the phrase "assume ##\phi## is a uniformly distributed random variable" just means that all ##\phi##'s are equally likely to occur. I'm also assuming that this phenomenon affects both paths, and to calculate it I'd apply the phase shifter to both "u" and "r" states after they hit the first mirror.

May I know if I have applied concepts correctly? Thanks in advance.

(Spotted some errors, hence the edits)
 

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  • #2
Yes, assume a uniform random variable means that any phase between 0 and 360 degrees is equally likely.
 
  • #3
Maybe it would be sufficient to only apply the random phase shift to one of the two photon states.
 
  • #4
Thank you for your response.

Gene Naden said:
Maybe it would be sufficient to only apply the random phase shift to one of the two photon states.

May I know why this would be sufficient? Would that not mean that the photons taking one path is more "vulnerable" to random phase changes than the other?

Thank you!
 
  • #5
Oh, sure. I was thinking that what counts is the relative phase between the two photons (causing interference), so that adding a random phase to both of them would not introduce more physics than adding phase to just one of them. Certainly, both paths would be vulnerable to random phase shift.
 
  • #6
Gene Naden said:
Oh, sure. I was thinking that what counts is the relative phase between the two photons (causing interference), so that adding a random phase to both of them would not introduce more physics than adding phase to just one of them. Certainly, both paths would be vulnerable to random phase shift.

Thank you for the clarification!

With regards to part c), say I have correctly found that the probability of detector A clicking in part b) to be ##P(det A) = \frac{1}{2}(1 - \cos\phi)## which is bounded between 1 and 0. Does the fact that all ##\phi##'s being equally probable mean that ##P(det A)## in part c) is now 0.5, independent of ##\phi##?
 
  • #7
I hope one of the mentors or advisers will weigh in on this. I am having some difficulty interpreting the problem. But your math seems OK. You have indicated that the probability of detection is ##\frac{1}{2}(1 - \cos\phi )##. The average of that, assuming ##\phi## is a random variable would be just ##\frac{1}{2}##.
 
  • #8
Gene Naden said:
I hope one of the mentors or advisers will weigh in on this. I am having some difficulty interpreting the problem. But your math seems OK. You have indicated that the probability of detection is ##\frac{1}{2}(1 - \cos\phi )##. The average of that, assuming ##\phi## is a random variable would be just ##\frac{1}{2}##.

No worries, thank you very much!
 
  • #9
I see nobody has picked up your thread, so I will go on with it. Basically, what is bothering me about this problem is that I expect they are trying to get at some kind of interference phenomena, which would produce visible interference fringes. That is what I saw with the interferometer in my optics class. But I don't see how to get that out of your equations. There has to be some kind of distance variable. So these doubts about the problem are why I said I hoped a mentor would come in.
 

1. What is the formalism of 2-state systems in quantum mechanics?

The formalism of 2-state systems in quantum mechanics refers to the mathematical framework used to describe the behavior and properties of a system with two possible states. This formalism involves using vectors and matrices to represent the states of the system and operators to describe the evolution of the system over time.

2. How does the formalism of 2-state systems differ from classical mechanics?

The formalism of 2-state systems in quantum mechanics differs from classical mechanics in several ways. In classical mechanics, the state of a system can be precisely determined and predicted, while in quantum mechanics, the state of a system is described by a wave function that represents the probability of finding the system in a particular state. Additionally, classical mechanics follows deterministic laws, while quantum mechanics involves inherent uncertainty and probability.

3. What are some common examples of 2-state systems in quantum mechanics?

Some common examples of 2-state systems in quantum mechanics include spin-1/2 particles (such as electrons), qubits used in quantum computing, and the polarization of photons. These systems have two possible states that can be represented by a binary value (0 or 1) or a two-dimensional vector.

4. How are measurements of 2-state systems represented in the formalism of quantum mechanics?

In the formalism of 2-state systems in quantum mechanics, measurements are represented by operators that act on the state of the system. These operators correspond to physical observables, such as position or energy, and the result of the measurement is one of the eigenvalues of the operator. The state of the system then "collapses" to the corresponding eigenstate.

5. What are the implications of the formalism of 2-state systems in understanding the behavior of particles in the quantum world?

The formalism of 2-state systems in quantum mechanics has significant implications for understanding the behavior of particles in the quantum world. It allows for the description of phenomena such as superposition, entanglement, and the uncertainty principle. It also plays a crucial role in the development of technologies such as quantum computing and quantum cryptography.

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