How Does the Elitzur-Vaidman Bomb Tester Utilize Quantum Mechanics?

[LukeD]

Hey, I'm trying to understand the Elitzur-Vaidman bomb tester since it is an essentially quantum mechanic effect, but I haven't been able to find a technical quantum mechanical treatment of the subject though I've found plenty of non-technical treatments and a few pseudo-classical treatments.

The way I understand the setup is that we start with a photon prepared in an entangled state between going on the top and bottom arms of the bomb tester. (Let's call this state |T> + |B>) The bottom photon then encounters a bomb that is live or dead. I suppose I could treat this bomb as an operator, either being a projection operator (live bomb, projects onto |T>) or identity operator (dead bomb), but this would operate on the entire state at once, and that doesn't seem quite right. Then I really don't understand what's going on with the half-silvered mirror at the end. It seems to make the top and bottom arms of the bomb tester recombine and interfere, but I can't figure out what that means in terms of operators.

Can someone help me out?

 

[Demystifier]

The bomb is not an operator of the kind you suggest. It is better to think of bomb as a specific classical potential V(x) playing a role in the Schrödinger equation. More precisely, the bomb is nothing but a mirror that reflects the wave function.

 

[LukeD]

Ok, then what about 2nd half silvered mirror? Should I treat the mirror as an operator?

I want to know the proper quantum mechanical treatment of this setup because I want to design an extension to it that I think should be possible where the bottom arm, instead of testing one bomb, tests and reports information about a set of boolean values.

 

[Mentz114]

Just so we know we're talking about the same thing ...
The bomb-tester works on the principle that if there is any chance of getting 'which-path' information, the interference at the last mirror is destroyed. A live bomb acts like a photon-detector, but a dud bomb is just a mirror. However if the bomb is live, there's a probability of 1/2 that the photon will not trigger a live bomb, but there will be no interference, telling us the bomb is live. Of course, half of the live bombs will explode, making this a rather expensive practice.

There are two set-ups here, one with a live bomb and one with a dud. You can't combine the two unless you have a probability for selecting a dud. In that case you'd have 3 possible outcomes. The state space would be a sort of tensor product of the two-state vector a|dud>+b|live> and c|boom> + d|T>. Shouldn't you write the state space out first then the operators ?

 

[Edgardo]

These papers may be helpful:

"[URL Photon quantum mechanics and beam splitters
C. H. Holbrow,a) E. Galvez, and M. E. Parks[/URL]

"[URL Interaction-Free Measurements
Lev Vaidman[/URL]

http://www.iop.org/EJ/abstract/1464-4266/3/3/311

http://www.iop.org/EJ/abstract/1355-5111/7/2/005

 

[Mentz114]

Thank you, Edgardo.

What I've described is the Penrose machine, not the EV bomb-tester. Whoops.