Multi-Photon Dissipation: Replacing the Pump with a Battery

Almost every cat qubit built so far, ours, and everyone else’s, is held together by a microwave pump. This signal, delivered down a cable into the fridge, forces a Josephson circuit to trade photons between two resonators in exactly the right way: two photons of the memory resonator are swapped for a single photon of the buffer resonator. That pair-by-pair removal is what holds the cat’s two states in place. The pump is what makes the cat a cat.

But it is also what limits it. A pumped coupler never does only the thing you asked for. Alongside the useful process, the Josephson nonlinearity produces unwanted parasitic terms (self-Kerr, cross-Kerr, pump-induced frequency shifts). They distort the qubit and get worse as you push the pump harder.

In our recent experiment (1), carried out at ENS Lyon with Alice & Bob, we removed the pump entirely and replaced it with something much less glamorous: a battery. A dc voltage bias of about one microvolt across a SQUID. With it, we activated one-, two-, and four-photon exchange between a high-Q memory resonator and a lossy buffer, at a two-photon exchange rate of up to 18.5 MHz, and we did it in a regime where the parasitic terms are not carefully cancelled, but averaged away by the mechanism itself.

This is not a cat qubit. It is a new kind of coupler that future cat qubits could be built around, and one that does things that are prohibitively hard to do with pumped couplers.

Before we dive in, the key results:

One battery, three ways to trade photons. One-, two-, and four-photon swaps, all on the same device, selected purely by the bias voltage

The parasitic effects stay small and don’t get worse as we push harder. Memory self-Kerr bounded at < 10 kHz with the two-photon process running, and with no visible growth as we turn the coupling up

*Figure 1 – Circuit schematic of the device.*

The engine room of a cat qubit

Quick refresher, for readers who don’t spend their days in the lab trying to swap photons between modes.

A cat qubit stores information not in a two-level system but in a resonator, which is a harmonic oscillator (like an oscillating swing or a mass hanging by a spring) that can hold many photons. The two logical states are so called coherent states of the resonators with opposite phase, and are usually written $|\alpha\rangle$ and $|-\alpha\rangle$ . Flipping between them means dragging the whole oscillation through phase space, which is exponentially unlikely to happen spontaneously as you add photons. That is where the cat qubit’s bit-flip protection comes from: the more photons we put in the resonator, the further apart the two logical states are in phase space, and the harder it will be to accidentally go from one state to the other (bit-flip) when something perturbs the system.

But a resonator left alone just leaks. Something has to actively hold those two states in place. That something is two-photon dissipation: an engineered process that removes photons from the memory strictly in pairs.

The trick to building it is to use a second mode, the buffer, that is deliberately made very lossy, and to couple the two modes so that two memory photons are converted into one buffer photon. The buffer photon escapes into the measurement line almost instantly and is gone. Net effect: the memory loses photons two at a time, irreversibly. That is exactly the dissipation that pins the cat.

The problem is energy bookkeeping. Two memory photons carry energy $2\hbar\omega_m$ ; one buffer photon carries $\hbar\omega_b$ . Those are not equal, so the conversion cannot just happen. Something has to supply the difference, $2\hbar\omega_a - \hbar\omega_b$ .

In every cat qubit device to date, that something is a microwave pump at precisely that frequency difference.

What the pump costs you

Pumping works. But it comes with an unwanted passenger.

A Josephson junction is a cosine. Expand that cosine and you get an infinite series of terms: the useful conversion you wanted, and a self-Kerr that makes the memory’s frequency depend on how many photons it holds, and a cross-Kerr between memory and buffer, and pump-power-dependent frequency shifts. These extra terms distort the cat states, degrade gates, and shift your operating point as you turn the pump up.

Clever circuit design fights back. The ATS, on which the current leading cat architectures are based, is built so that the unwanted terms cancel out interferometrically. But cancellation is only ever as good as your fabrication: two junctions that should be identical never are, and unavoidable stray inductances in the ATS limit its effectiveness. The residual nonlinearity is set by fabrication tolerances.

Overcoming parasitic terms, rather than mitigating them, needs a mechanism where the parasitic terms are absent by construction.

A battery instead of a pump

Here is the idea:

Bias a Josephson junction with a constant dc voltage $V$ , and the superconducting phase across it winds at a constant rate. The junction starts oscillating all by itself, at the Josephson frequency $\omega_J = 2eV/\hbar$ . This is the ac Josephson effect: a dc voltage in, a microwave tone out.

Equivalently, in the language of particles: every Cooper pair that tunnels across the junction must hand over (or take up) an energy $2eV$ from the rest of the circuit. So if you arrange

$2eV = 2\hbar\omega_a - \hbar\omega_b$

then the energy released by a single tunneling Cooper pair is exactly the amount needed to turn two memory photons into one buffer photon. The process becomes energy-allowed, and it switches on.

For our device, $2\omega_a - \omega_b$ corresponds to 0.53 GHz, so the required bias is 1.1 microvolts. That is the entire control signal for the coupler. No pump tone, no high-power microwave line, no pump harmonics.

Figure 2 – (a) Energy levels of memory mode a (blue) and buffer mode b (magenta) coupled to a Josephson junction biased with a voltage V. For 2eV = 2ℏωa − ℏωb, Cooper pair tunneling enables transitions converting a pair of photons of mode a into a single photon of mode b (solid arrows) and back (dashed arrows).

The parasitic effects average themselves away

The junction’s self-generated oscillation at $\omega_J$ does not only switch on the useful process. It multiplies everything the junction produces. Go into the frame rotating at $\omega_J$ and the picture separates cleanly into two families:

the term you wanted, which sits still and acts;
every other term, self-Kerr, cross-Kerr, bias-induced detunings, which rotates at $\omega_J$ and averages to zero.

In a pumped coupler, those parasitic terms are static: they sit there and accumulate, and you have to cancel them, imperfectly, with circuit symmetry. Here they oscillate very fast and simply wash out. The suppression is not a cancellation between two nominally identical junctions; it is a dynamical averaging set by a frequency ratio you choose at design time: the parasitic terms vanish by construction.

One, two… four

A remarkable aspect of this design is its versatility: we show that, by just turning a simple knob – the dc voltage bias – we can choose to activate a swap of one, two or four photons of the memory for one photon of the buffer. This is another thing that microwave-driven devices struggle with, but comes very easy to this device.

Setting $2eV = \hbar(\omega_b - \omega_a)$ turns on the simplest process — a one-photon swap between memory and buffer. As we raise the SQUID’s Josephson energy with flux, the single buffer resonance splits into two hybridized peaks, a textbook avoided crossing. The splitting reaches $2g_1/2\pi \approx 85$ MHz.

Then we moved the voltage. At $2eV \simeq 2\hbar\omega_a - \hbar\omega_b$ , the probe photons start converting into pairs of memory photons. At $2eV \simeq 4\hbar\omega_a - \hbar\omega_b$ , into quartets. In each case the absorption feature follows its own energy-matching line as we sweep the voltage, which is what confirms the order of the process.

That four-photon swap deserves a pause. Four-photon processes are notoriously hard to isolate with pumped couplers — they are high order, they are weak, and getting them requires pump powers where all the parasitic effects you were trying to avoid come back. Here, it is just a different voltage on the same chip. (We can also activate the three-photon process; its spectroscopic signature happens to be masked by a bright one-photon swap involving the memory’s third harmonic).

Figure 3 – In (a) and (b), as we turn up the coupling, the buffer’s single resonance splits in two, which is the signature of memory and buffer swapping photons one for one. In c, d, e we also see buffer spectroscopies: here we sweep the bias and show that the absorption feature, which happens because the buffer swap photons with the memory, follows the energy matching condition for respectively one-, two- and four- to one conversion, proving that we are activating these processes.

Watching pairs of photons leave

Spectroscopy shows the exchange exists. To show it really produces dissipation, we did what we call a “deflate experiment”.

We prepare the memory in a coherent state with about 4 photons, switch on the two-to-one-photon swap process with a fast flux pulse for a variable time, then read out the memory’s full quantum state by Wigner tomography using a transmon qubit coupled to it.

The signature is unmistakable. With the process off, the photon number decays as a plain exponential at the memory’s intrinsic loss rate. With it on, the decay is not exponential at all: there is a fast collapse in the first ~50 ns as pairs are stripped out at a rate proportional to the photon number, followed by a slow tail once only zero or one photon is left and there are no pairs to remove. Two-photon loss and one-photon loss have visibly different shapes, and the data show both to highlight this qualitative difference.

Repeating this across the flux range and fitting everything jointly to a two-mode memory–buffer model gives our headline number:

$g_2^{\max}/2\pi = 18.5 \pm 0.5$ MHz

For context, cat-qubit couplers reported to date operate roughly an order of magnitude below this (we put the numbers in a table in the expert section). We also verified that in the regime where the buffer can be adiabatically eliminated, the extracted two-photon loss rate follows the expected $\kappa_2 = 4g_2^2/\kappa_b$ and is independent of the initial photon number — and that at the strongest couplings it stops doing so, exactly where the simple dissipator picture is supposed to break down.

*Figure 4: In (b), snapshots of the* *memory’s* quantum state show a coherent state of about 16 photons collapsing within 50 ns as photons leave in pairs, then slowly relaxing to empty. In (c), the photon count drops as a plain exponential with the swap off (blue), but *falls fast then flattens with the swap on (orange),* *the signature of photons leaving* *two at a time.*

How nonlinear is the memory now?

This is the measurement that tests the central claim.

A residual self-Kerr $K$ makes the memory shear in phase space: each Fock state picks up a phase that grows with photon number, so a large coherent state gets progressively smeared into a banana. That deformation looks different from a rigid rotation (which is a detuning) and different from an isotropic blurring (which is dephasing), so we can bound $K$ even when other imperfections are present.

We took a Wigner tomography of the memory when its state is large, and thus more sensitive to Kerr. Specifically we prepared a coherent state with 9.4 photons, and compared the Wigner function with simulations at different $K$ . The data are consistent with the simulation up to about 10 kHz, and clearly inconsistent by 25 kHz.

$K/2\pi \lesssim 10$ kHz with the two-photon conversion running, and no visible increase as the Josephson energy is raised.

Before the coupler is turned on at all, the memory’s bare self-Kerr is 3 kHz. So the process we switch on does not measurably nonlinearize the mode. That is the behaviour the dc-bias mechanism predicts, and it is the property that a scalable cat architecture needs!

What doesn’t work yet

We would rather tell you this ourselves:

The memory is short-lived. $T_1 = 3.1$ µs, corresponding to $Q \approx 8 \times 10^4$ , well below the $10^6$ that tantalum resonators routinely reach. We traced this to the termination of the flux line, not to materials or fabrication (the transmon on the same chip lives 100 µs). HFSS simulations say a simple modification of the flux line fixes it.

There is a spurious mode on the bias line. A standing wave at 227 MHz, trapped between a bias tee and the chip. The biased SQUID drives it, it ac-Stark-shifts the memory, and that makes the memory frequency depend on the bias voltage. Residual voltage noise then turns into memory dephasing, up to 12 MHz at our strongest coupling. It is the reason the transient Wigner functions in this experiment stay positive instead of developing the negativities that pure two-photon loss would produce. The mode is a wiring artifact, not physics. It can be fixed with different wiring. Shortening that section of line between cooldowns moved its frequency, proving that we know the origin of the mode.

The bias is noisy at the nanovolt level. We measure a Josephson-frequency linewidth of 1.5 MHz, corresponding to 3.1 nV rms across the junction. Fine for engineering dissipation (it is 50× below the dissipation buffer linewidth, so 50x below the requirement for dissipation). Not fine for anything requiring a phase reference. Switching off the pulse-tube cryocooler drops that noise to 620 picovolts, which is state of the art. This indicates that the fridge’s own vibrations are a substantial part of the voltage noise we measure.

And this is not a stabilized a cat. Two-photon dissipation is half of a cat qubit; the other half is a two-photon drive that sets where the cat sits. Adding it is the obvious next experiment.

What this unlocks

Step back from this particular chip, and what we think we have is a design primitive with three properties that matter for cat architectures.

A coupler that doesn’t contaminate the memory. Strong engineered dissipation on a high-Q mode, with residual nonlinearity suppressed by construction rather than mitigated imperfectly.

Access to $n > 2$ dissipation. Four-photon dissipation is the ingredient for four-component cat qubits, which encode information across four coherent states instead of two and can autonomously detect and correct single-photon loss, the dominant error in bosonic qubits.

Control by voltage. Microwave pump lines are expensive at scale: they carry power, they heat, they need filtering and isolation, and their harmonics excite modes you did not design. A dc bias line is a much cheaper object to route, multiplex and filter. And because the process is selected by the bias voltage, one physical coupler can perform different conversions at different operating points, with no redesign.

Getting from this device to a cat qubit means killing the phase diffusion of the bias,using a dc-Josephson voltage standard, or driving the memory directly bypassing the dc-biased junction, and cleaning up the bias-line environment. We know how to approach both these issues.

Wrapping up

On the same device we can activate one-, two- and four- to one photon swaps between memory and buffer.

The 2-to-1 and 4-to-1 photon swap processes could be used to make respectively a 2-component or 4-component cat
Memory self-Kerr with the coupler on: < 10 kHz
Effective two-photon loss rate measured in the deflate experiment: up to $\kappa_2/2\pi = 1.3$ MHz
Bias-voltage noise: 3.1 nV rms (1.5 MHz of Josephson linewidth), dropping to 620 pV with the pulse tube off
Memory $T_1$ : 3.1 µs, limited by a fixable flux-line design flaw

The mechanism works, the rates are large, and the parasitic terms stay where they belong. The next steps are clear, and we know how to approach them.

Read the pre-print to dive deeper.

References

[1] M. Paradina et al., arXiv:2608.31154 [quant-ph] (Sep. 28, 2026); https://arxiv.org/abs/2608.31154