Number of thermal photons in a real (finite Q) cavity

[f95toli]

This is probably a common questions, but I can't find it answered anywhere...

It is of course well known that the average number of thermal photons in a mode can be calculate from from the B-E distribution:

$$<n>=\frac{1}{e^{hf/k_BT}-1}$$

The usual understanding of this is that what we are considering is a cavity supporting a single mode with a frequency f. At microwave frequencies (~10 GHz) this works out to be about 1000 photons.

My question is how we calculate the number of thermal photons if we consider a real cavity with a finite qualify factor f/Δf? I.e. when the cavity BW is (possibly) quite broad? In a way this is still a single mode, but now the mode has some width.

I suspect the answer is that we will need to consider a mode density instead; but I am not sure how to do this.

It could be that there is a "language barriers" which is preventing me from finding the right reference, must references deal with optical frequencies (and people talk about finesse rather than Q); whereas I am mainly interested in microwave frequencies.

 

[Paul Colby]

In very general terms once an ideal lossless cavity is subject to any loss, more than one mode is supported inside it. For example, a finite rectangular PEC cavity with a perfectly absorbing wall could be viewed as a semi-infinite rectangular wave guild which supports an infinite number of modes. As far as the thermodynamics of the field you might look at what's called the fluctuation-dissipation theorem.

 

[Cthugha]

My question is how we calculate the number of thermal photons if we consider a real cavity with a finite qualify factor f/Δf? I.e. when the cavity BW is (possibly) quite broad? In a way this is still a single mode, but now the mode has some width.

First caveat: I am an optics guy working in the visible and not that well versed in microwave frequencies.

The good thing is: It is not just a single mode in a way. It is a single mode. Raymer and Walmsley (among others) did a lot of work on temporal modes in quantum optics and you can define a mode as anything uncertainty/Fourier-limited. Essentially this just reflects the fact that the power spectral density of your light field directly is the Fourier transform of the first-order coherence function of the same light field, which defines its coherence volume (at least most of it) and all photons within a coherence volume are indistinguishable anyway and form a mode.

It was just Dirac's take to prefer monochromatic modes - which are without any doubt very convenient in most situations - but for systems which show some time dependence, you are often better off by defining modes of finite spectral width and finite temporal duration matching the characteristic spectral/temporal features of your system. As long as the time-energy uncertainty of this mode is minimal, it qualifies as a single mode and should show the standard single mode photon number if it is in thermal equilibrium with the mode outside of the cavity. If not, things become more complicated.

From my experience the bigger problem with cavities is that they easily stop to be single mode/uncertainty-limited at finite spectral width and then it is not easy to track how many modes you actually really have. However, I have no idea how much of a problem this is in the microwave regime.

 

[f95toli]

The good thing is: It is not just a single mode in a way. It is a single mode. Raymer and Walmsley (among others) did a lot of work on temporal modes in quantum optics and you can define a mode as anything uncertainty/Fourier-limited.

Thanks. I did not know that.
I am not sure how/if this applied to MW cavities. Typically, the modes in a well-designed MW cavity are quite far apart. Most on-chip resonators/cavities used for quantum computing are essentially 1D meaning the next mode is twice the fundamental. On the other hand they tend to be over-coupled resulting in a Q of ~10 000 or; i.e. the width is about 0.1-1 MHz.

It is easy to calculate the mean photon number in a driven cavity; but the mean thermal photon number seems trickier...

 

[Cthugha]

I see. It seems to me that MW cavities indeed tend to be single mode.
In the following paper from the Wallraff group ( https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.105.163601 ), they tried to investigate the behavior of a qubit in a MW cavity at different temperatures by injecting a thermal light field to introduce an effective temperature. To me it seems that a treatment in terms of the thermal occupation of a single mode cavity seems to work well in that case. However, there may be several peculiarities of MW cavities that I simply missed and that make things more complicated.