Two-mode squeezed states and thermal horizons

Here is a piece of physics that does double duty in this programme: it is ordinary quantum optics, and it is also the reason an accelerating universe looks warm from the inside. Two-mode squeezed states sit at the centre of both stories, and you can check every formula in this post on a napkin.

The state, and its numbers

A two-mode squeezed vacuum is what a parametric amplifier produces: vacuum in, pairs of photons out, one in each mode, correlated. Write r for the squeezing parameter. Three facts, all exact.

First, each mode alone has mean occupation n bar equal to sinh squared r. Second, the two modes together are not thermal at all; they are in a pure entangled state, with cross-correlation sinh r times cosh r, which is one half of sinh of 2 r. Third, the photon number in each single mode is geometrically distributed, P of n equals one minus lambda, times lambda to the n, with lambda equal to tanh squared r. A geometric distribution in n is precisely the Bose-Einstein distribution of a thermal state with that occupation. So each mode, taken alone, is indistinguishable from thermal, while the pair, taken together, is pure. The information about the purity is not in either mode. It is in the correlations between them.

I verified these identities numerically as part of the derivation programme, to a tolerance of one part in 10 to the 12, and I would encourage anyone to redo it: square the sinh, compare to the covariance entries, and expand the geometric distribution’s mean, which is lambda over one minus lambda, and watch tanh squared r over one minus tanh squared r equal sinh squared r. It does.

What this has to do with horizons

Now change the words. Replace the parametric amplifier with an accelerating expansion. A observer in de Sitter space, the maximally accelerating universe, is causally cut off from regions beyond the horizon, and the field modes near the horizon are related to interior modes exactly the way the two arms of a parametric amplifier are related: by a two-mode squeeze. The calculation in quantum optics and the calculation in de Sitter are the same algebra with different symbols.

So each horizon-external mode, traced out, looks thermal, with temperature H over 2 pi, the Gibbons-Hawking temperature: about 2.3 times 10 to the minus 30 kelvin for the present-day expansion rate. Absurdly cold, but not zero, and its smallness is misleading, because the thermodynamic consequences scale with horizon area. The de Sitter horizon carries an entropy of order 10 to the 122 in nats, and that enormous number is the quiet reservoir that three bounds in the next post will all land on.

Why the framework cares

The programme’s multiverse branches are modelled as superselection sectors, and the cleanest known way to get superselection sectors that are individually physical and mutually inaccessible is exactly this structure: a pure entangled global state whose sectors, viewed locally, are thermal and unable to signal each other. That is what an alpha-vacuum of de Sitter is, and the thermal single-mode reduction above is why papers on the competing avenues treat alpha-vacua as the strongest existing physics for many realities without new postulates.

The honest limit: the thermal nature of each sector is textbook physics. That nature’s sectors match the branches of this multiverse is the programme’s hypothesis, not the textbook’s. What the calculation buys is a shape: if branch isolation degrades, the degradation inherits this algebra, and patterned leakage looks like a slight mis-squeeze rather than noise. That is a prediction about structure, and it is the sort of thing a blinded experiment could, in principle, look for.