Photon coherence time as the wavepacket length and dephasing

[benway]

Quite often one can see descriptions saying that the coherence time of single photons corresponds to the length of the single photon wavepacket (for example Jelezko et al, PRA 67 041802(R) (2003) http://pra.aps.org/abstract/PRA/v67/i4/e041802). I find it hard to come to terms with this picture. There are some threads discussing related topics but I find that none of them really helps a lot. I will try to describe my trouble and hopefully someone can join the discussion:

Consider a single photon emitted from the decay of a simple two-level system with radiative lifetime T. The probability amplitude of detecting the photon I usually picture as a sharp wavefront propagating out from the emitter, with an exponential tail decaying as exp(-t/T) (or spatially exp(-r/cT)). I can understand the direct correspondence between the length of a wavepacket such as this and the spectral distribution of the photon, but what about cases when dephasing of the excited state is present? Dephasing limits the coherence time of the emitted photons in solid state systems, and single photon interferometry is often used as a probe of the dephasing of the emitter, for example in the paper linked above. What if the dephasing is such that the energy level of the excited state is modulated during the emission of the wavepacket? This would would obviously broaden the linewidth and shorten the coherence time of the photon, and – in my world – correspond to a frequency modulated wavepacket in the time domain. But how is this compatible with the notion that shorter coherence time equals shorter wavepacket?

Looking forward to hearing your comments and ideas.

 

[Bob S]

Two things

1) The sodium D lines are broadened in high pressure gasses because the collision time is shorter than the natural lifetime:

The finite duration of the radiation process of electron transition leads to a finite width of line, in accordance with Heisenberg's uncertainty principle. For a high pressure gas, radiating times can be much greater than the interval between atomic collisions, and this perturbation by colliding atoms causes the premature transition and emission of a photon. The decreased lifetime of the state creates an increased uncertainty in photon energy, broadening the emission line. See http://www.fas.harvard.edu/~scdiroff/lds/QuantumRelativity/CollisionalBroadening/CollisionalBroadening.html

2) The Mossbauer Fe⁵⁷ nuclear 14.4 keV line is very narrow (1 part in 10¹²) due to the long lifetime (100 ns): http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/mossfe.html. I don't know how to artificially shorten the lifetime though.

 

[Cthugha]

What if the dephasing is such that the energy level of the excited state is modulated during the emission of the wavepacket? This would would obviously broaden the linewidth and shorten the coherence time of the photon, and – in my world – correspond to a frequency modulated wavepacket in the time domain. But how is this compatible with the notion that shorter coherence time equals shorter wavepacket?

That depends a bit on how fast and strong you are going to modulate the energy level. Small modulations do not really change much as long as the modulation is small compared to the intrinsic uncertainty of the transition which is already present. If you modulate it strongly, but slowly, you will get single photon emission with central energy varying over time, but roughly constant coherence time. This is of course only sensible if you have a situation which roughly resembles CW conditions. If you have a fast and strong modulation you still need to obey the uncertainty relation. You will typically just get some transition which is strongly broadened around some (maybe slightly shifted) central wavelength, but no deterministic modulation on small timescales. The coherence time will now of course shorten.

 

[benway]

Thanks folks for joining the discussion,


Cthugha:
I interpret what you are saying as if there is a weak modulation happening already early on during the "emission cycle" that won't change the coherence time much, whereas a strong one would (+ broaden the line). I can accept that intuitively. But I don't think it is right to say that the uncertainty relationship dictates that the pulse must be short. After all it is an inequality, we can have Δω.Δt >> 1/2 too. I agree that the coherence time should be shorter though, and this is what I am after here: why do people say that the photon "wavepacket" or pulse is of the length of the coherence time? I still don't see it.
Sidenote: what do you mean by "no deterministic modulation on short timescales"? Perhaps I am missing something you are saying.

Bob S:
if the pressure broadened emitter is "tickled" to a premature emission, does it mean that the photons will be bandwidth limited in this situation?

I just get the feeling that there is actually a great wealth of situations, but often an equality is made "coherence time = wavepacket length". I am still not ready to accept it =)

Looking forward to your further responses!

 

[Cthugha]

I interpret what you are saying as if there is a weak modulation happening already early on during the "emission cycle" that won't change the coherence time much, whereas a strong one would (+ broaden the line). I can accept that intuitively. But I don't think it is right to say that the uncertainty relationship dictates that the pulse must be short.

Indeed it does not dictate that. It just sets the limit. The pulse can of course be longer on average than the coherence time. This is especially obvious if you operate a single photon source using cw pumping.

After all it is an inequality, we can have Δω.Δt >> 1/2 too. I agree that the coherence time should be shorter though, and this is what I am after here: why do people say that the photon "wavepacket" or pulse is of the length of the coherence time? I still don't see it.

You have to distinguish two situations here. If you are investigating the average over many emitted single photon pulses and consider the time dependence of the averaged emission pulses (relative to your pump beam or whatever you use to get the single photon source going), the width of that averaged emission will be longer than your coherence time. This will happen because there will be stochastical jitter relative to the pump beam and the emission energy may vary and so on.

However, if you perform autocorrelation and interferometric measurements, you measure something different. Here you measure the interference at a relative delay between the pulse and a shifted copy of itself. This should not really vary much from pulse to pulse. And as you interfere the pulse with itself, it is quite straightforward to identify the duration/width of the ability to interfere with the duration/width of the single photon wavepacket. I mean: imagine you have a single shot single photon pulse which is longer than the coherence length of that single photon. In that case you would end up in a situation where a single photon does not interfere with itself at some position/time. While I imagine one could formulate a theory in which such things occur, it is not quite intuitive or easy to motivate.

Sidenote: what do you mean by "no deterministic modulation on short timescales"? Perhaps I am missing something you are saying.

I just wanted to say if you shift around the energy level really quickly, you necessarily also broaden the transition rather than shift a narrow energy level around.

 

[Bob S]

After you have solved the above coherence problem, consider this one. The mean lifetime of an electron in the 2p atomic state of hydrogen is about 1.6 nanoseconds (see equation 1138 in http://farside.ph.utexas.edu/teaching/qmech/lectures/node122.html). Suppose I have a very intense short (say 1 picosecond) pulse of UV light, and I knock a million electrons in hydrogen from the 1s up to to the 2p state (energy about 0.75 x 13.5 eV). The electrons start falling back to the 1s state with the 1.6 ns mean lifetime. But some electrons fall back in less thn 1 ns, and some take over 3 ns to fall back. Do all the emitted photons have the same coherence time?