Undergrad Are two entangled photons described by the same wavefunction?

[Nickyv2423]

Are two entangled photons described by the same wave function or wave function shape? Heres an example...
Say for example, we have a laser in TEM01 mode that is shooting individual photons (this mode as two distinct maxima). Then the individual photons are going through a BBO crystal to become a pair of entangled photons. The signal photon and the idler photon have different polarization though. Are they still described by the same wave function? Or do the two photons split up in different paths/wave functions?

 

[Ben Wilson]

Are two entangled photons described by the same wave function or wave function shape? Heres an example...
Say for example, we have a laser in TEM01 mode that is shooting individual photons (this mode as two distinct maxima). Then the individual photons are going through a BBO crystal to become a pair of entangled photons. The signal photon and the idler photon have different polarization though. Are they still described by the same wave function? Or do the two photons split up in different paths/wave functions?

the photons have their own wave function in some sense but the entanglement implies that these wave functions aren't separable from the wave function for the system of two particles - i.e. Ψ₁₊₂≠φ₁^hφ₂^v. In quantum optics equations they're described by the same wavefunction which includes a function of BOTH photon's momenta (using the dipole and rotating wave approximation).

For a maximally entangled bell state we have

Ψ₁₊₂=(1/√2)(φ₁^hφ₂^v + φ₁^vφ₂^h)

h & v are polarizations, 1 & 2 are photons

 

[PeterDonis]

Are they still described by the same wave function?

Strictly speaking, the wave function describes the system, which in this case is the pair of photons; there is no such thing as the wave function of an individual photon.

In cases where the photons are not entangled, you can separate the wave function for the system into the product of two wave functions, one for each of the individual photons. But as Ben Wilson said, if the photons are entangled, you can't do that, and in that case there is no useful sense in which the individual photons have their own wave functions; the only wave function is the wave function of the system.

 

[Nickyv2423]

Strictly speaking, the wave function describes the system, which in this case is the pair of photons; there is no such thing as the wave function of an individual photon.

In cases where the photons are not entangled, you can separate the wave function for the system into the product of two wave functions, one for each of the individual photons. But as Ben Wilson said, if the photons are entangled, you can't do that, and in that case there is no useful sense in which the individual photons have their own wave functions; the only wave function is the wave function of the system.

So if a photon had a wave function shape of the TEM01 mode (shape of two lobes or maxima), when it goes through a bbo crystal to become a two photon entangled pair, would each of the photons (both idler and signal) both be in a wave functional shape of two maxima (total of 4 maxima for the two photons)?
I know this is very confusing.
Or would each lobe be split up when the photon becomes two photons?
Does this make any sense?

 

[PeterDonis]

would each of the photons (both idler and signal) both be in a wave functional shape of two maxima (total of 4 maxima for the two photons)?

Or would each lobe be split up when the photon becomes two photons?

Neither of these are the right question. The right question is, what does the wave function of the two photon system that comes out of the crystal look like? You figure that out by figuring out what unitary operator describes the action of the crystal on the incoming photon. Since the two photons coming out are entangled, there will not be any useful sense in which the wave function of the two photon system can be interpreted as individual wave functions for each of the photons that "combine" in some way.

 

[Nickyv2423]

Neither of these are the right question. The right question is, what does the wave function of the two photon system that comes out of the crystal look like? You figure that out by figuring out what unitary operator describes the action of the crystal on the incoming photon. Since the two photons coming out are entangled, there will not be any useful sense in which the wave function of the two photon system can be interpreted as individual wave functions for each of the photons that "combine" in some way.

If i were to collapse the wave function of the position of a signal photon, would the wave function of the idle photon also collapse in its position?

 

[DrChinese]

If i were to collapse the wave function of the position of a signal photon, would the wave function of the idle photon also collapse in its position?

That's a tough question to answer to everyone's satisfaction. It would be "as if" it collapsed the idler too. The problem is that the ordering of the collapse (signal first/idler second or vice versa) does not make any difference. That makes it difficult to say what "causes" what.