Superconducting Cat States

Superconducting circuits can prepare and measure superpositions of distinguishable electromagnetic states. A microwave cavity cat state is a superposition of two coherent states of one oscillator mode; a superposition of circuit currents is a different physical construction. Both require evidence for coherence between alternatives, not only evidence that the alternatives can be populated.

Cats in an oscillator mode

For dimensionless coherent-state amplitude $\alpha$, define the even and odd cats

$$\lvert C_\pm\rangle= \frac{\lvert\alpha\rangle\pm\lvert-\alpha\rangle} {\sqrt{2(1\pm e^{-2|\alpha|^2})}}.$$

Their branch overlap is $\langle-\alpha|\alpha\rangle=e^{-2|\alpha|^2}$. Increasing $|\alpha|$ makes the two packets distinguishable in phase space. The odd state is defined for $\alpha\ne0$; its normalized small-amplitude limit is the one-photon state.

Expanding coherent states in number states shows that $\lvert C_+\rangle$ contains only even photon numbers and $\lvert C_-\rangle$ only odd ones. Hence photon parity $\Pi=(-1)^{a^\dagger a}$ gives $\langle\Pi\rangle=\pm1$ in the ideal cats. An equal incoherent mixture of the two coherent branches instead has $\langle\Pi\rangle=e^{-2|\alpha|^2}$.

At $\alpha=2$, that mixture has parity about $3.35\times10^{-4}$, whereas the ideal even cat has parity one. This is a direct example of an observable sensitive to coherence between the branches.

How the circuit measures coherence

Dispersive coupling to an ancillary superconducting qubit can map cavity parity onto qubit readout. Repeating the measurement after phase-space displacements samples the Wigner function, conventionally

$$W(\beta)=\frac{2}{\pi} \operatorname{tr}\!\left[D^\dagger(\beta)\rho D(\beta)\Pi\right].$$

Interference oscillations between phase-space lobes distinguish coherent cats from a mixture. Readout errors, finite sampling, displacement calibration, and cavity loss must enter any reconstructed state or witness.

Vlastakis and collaborators demonstrated deterministic encoding of quantum information in cavity cat states in 2013. The result establishes controlled oscillator coherence in that device and protocol. The phrase “100-photon cat” should not be interpreted as the entire refrigerator occupying two spatially separated positions.

Loss and the choice of macroscopic variable

Photon loss carries information about phase-space branches and changes cat parity. In a zero-temperature amplitude-damping channel with energy transmission $\eta$, the branch interference is multiplied by

$$\exp[-2(1-\eta)|\alpha|^2],$$

while the coherent amplitudes shrink to $\sqrt\eta\,\alpha$. Bigger separation therefore improves distinguishability but increases susceptibility to loss. This is the physical tradeoff exploited and managed in bosonic encodings.

A collapse model coupled primarily to mass density need not predict the same sensitivity for a photonic cat as for two separated massive objects. Translating circuit coherence into an exclusion therefore requires the model’s coupling to the actual circuit degrees of freedom. Cat-state preparation alone does not resolve the outcome problem or uniquely support an interpretation.

Exercise

For $\eta=0.99$, compare the loss-induced interference factors of cats with $|\alpha|^2=4$ and $25$.

Solution

They are $e^{-0.08}\simeq0.923$ and $e^{-0.5}\simeq0.607$. Equal fractional energy loss can have substantially different effects on branch coherence. This calculation isolates the loss channel; it omits preparation and readout imperfections.

References

• Bassi, A., et al., “Models of wave-function collapse, underlying theories, and experimental tests,” Reviews of Modern Physics 85, 471–527 (2013), doi:10.1103/RevModPhys.85.471 — why model coupling determines foundational sensitivity.
• Vlastakis, B., et al., “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013), doi:10.1126/science.1243289 — circuit preparation and phase-space characterization.