Dynamical Casimir effect / Microwave photon pairs

Dynamical Casimir Photon Pairs

How can you tell that a modulated boundary emits photons in pairs, when each mode alone looks like thermal noise? Look at the two modes together. Either alone looks exactly like thermal light of the same brightness. The pairs show up in their shared fluctuations: with the receiver phase set right, the noise in suitable combinations of the two fields drops below the vacuum level, which no unentangled source can do. Loss and thermal input wash this out. The energy comes from the pump driving the boundary.

Modulate a superconducting boundary and examine two matched microwave modes. Rotate the receiver phases, compare the output with thermal light of equal brightness, and collect the fluctuations that the two modes share.

The model is an effective two-mode squeezer, inspired by a waveguide terminated by a SQUID. The energy comes from a prescribed pump. Both inputs have the same thermal occupation, and both modes lose photons to vacuum at the same rate. The model does not solve the broadband circuit, does not calibrate how flux converts to gain, and ignores pump depletion. The scene is schematic, and the mean field is zero.

A modulated boundary and two matched modes

JOINT NOISE / VACUUM REFERENCE

0

BRIGHTNESS OF EITHER OUTPUT MODE

0

01PUMP / SQUID

02FREQUENCY SORTER

03RECEIVERS A / B

Schematic chip. Envelope height scales with square root of photon occupation, above vacuum. Purple blocks mark loss. Curves are not instantaneous fields; selected frequencies sum to the pump frequency.

r=0.9

01 / Joint fluctuations

The shape tells you what brightness cannot.

Cyan density: model for the X setting. Gold dots: saved X records only. Drag horizontally to rotate both receivers; arrows change one degree. Square-root density colors, fixed scale.

$X_A\ \longrightarrow\quad X_B\ \uparrow$

02 / Rotate the measurement axis

Find the quiet direction.

Model joint noise versus shared receiver phase; gold marks the selected angle. The dashed reference is one. Both orthogonal measurement settings are required for the witness.

$\theta\ ({}^\circ)\ \longrightarrow\quad W$

Each trial starts from a fresh preparation

Measure both outputs together and keep every sample.

0 / 20000

Independent matched-mode preparations, alternating the X and orthogonal P settings. Each trial jointly measures one quadrature per mode; incompatible quadratures are never measured on the same copy. Ideal homodyne reception after the declared loss. Fixed seed, all samples retained.

Recorded joint noise

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Sampling standard error

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Model joint noise

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Uncertainty check

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Each output is noisy on its own; together they are quiet.

Gold: recorded combinations, X difference and P sum pooled. Cyan: the model Gaussian with integrated bin probabilities. Outer bins include every tail beyond −5 and 5.

Latest 20 records · export all

Trial	Setting	A	B

Model, conventions and sources

The real pump phase defines zero. Both receivers rotate through the same angle. Each vacuum quadrature has variance one half. Separate X and P preparations give the two variances; a witness below one is sufficient for entanglement in this trusted ideal measurement model. A value above one at a poor angle does not prove separability.

a'=a\cosh r+b^\dagger\sinh r

X=(a+a^\dagger)/\sqrt2,\quad P=(a-a^\dagger)/(i\sqrt2)

W=\tfrac12[\operatorname{Var}(X_A-X_B)+\operatorname{Var}(P_A+P_B)]

W_\theta=1-\eta+\eta(2n_{\mathrm{th}}+1)[\cosh(2r)-\sinh(2r)\cos(2\theta)]

The model keeps only two selected temporal-frequency modes. Its exponential squeeze map is not the full broadband SQUID input–output calculation. The two input modes have equal occupation; this does not mean they share one physical temperature at their different frequencies. Symmetric attenuation mixes each output with independent vacuum. Amplifier noise, phase drift, bandwidth inversion, pump depletion and circuit calibration are left out.

Start with the essentials

Focus question: How can you tell that a modulated boundary emits photons in pairs, when each mode alone looks like thermal noise?
One-sentence intuition: Each mode on its own looks like thermal light. Only the fluctuations the two modes share reveal that photons are made in pairs.

Core mathematical model

A measurable signature

W=\frac{\operatorname{Var}(X_A-X_B)+\operatorname{Var}(P_A+P_B)}{2}

If each quadrature has vacuum variance one half, every separable state gives W of at least one. The two combinations are measured on separate preparations.

Common difficulties

What really causes the effect

Typical misconceptionThe vacuum supplies free energy.

Better mental modelThe energy comes from the boundary, which is driven from outside. Squeezing describes the effective quantum transformation of the modes.

Model curves versus collected data

Typical misconceptionThe theory curve on screen already counts as measured data.

Better mental modelThe model is an effective two-mode squeezer, inspired by a waveguide terminated by a SQUID. The energy comes from a prescribed pump. Both inputs have the same thermal occupation, and both modes lose photons to vacuum at the same rate. The model does not solve the broadband circuit, does not calibrate how flux converts to gain, and ignores pump depletion. The scene is schematic, and the mean field is zero.

Run the experiment

01
Predict the difference
Compare the squeezed output with the equally bright thermal preset.
What to observe: The mean brightness is the same, but the joint cloud has a different shape.
02
Make the effect disappear
Rotate both receiver phases, or add thermal input and loss.
What to observe: At a poorly chosen measurement angle the correlations can vanish from view, even though the source is just as bright.
03
Collect and inspect
Collect trials and pin a run, then change one control and restore it. Export all the records as CSV.
What to observe: The gold data come only from saved simulated records, and the cyan curves are model references.

Dynamical Casimir Photon Pairs

The shape tells you what brightness cannot.

Find the quiet direction.

Measure both outputs together and keep every sample.

Each output is noisy on its own; together they are quiet.