## A way to induce quantum stability

Dynamical systems, whether classical or quantum, usually require a method to stabilize performance and maintain the required state. For instance, communication between computers requires error correction codes to ensure that information is transferred correctly. In a quantum system, however, the very act of measuring it can perturb it. Leghtas *et al.* show that engineering the interaction between a quantum system and its environment can induce stability for the delicate quantum states, a process that could simplify quantum information processing.

*Science*, this issue p. 853

## Abstract

Physical systems usually exhibit quantum behavior, such as superpositions and entanglement, only when they are sufficiently decoupled from a lossy environment. Paradoxically, a specially engineered interaction with the environment can become a resource for the generation and protection of quantum states. This notion can be generalized to the confinement of a system into a manifold of quantum states, consisting of all coherent superpositions of multiple stable steady states. We have confined the state of a superconducting resonator to the quantum manifold spanned by two coherent states of opposite phases and have observed a Schrödinger cat state spontaneously squeeze out of vacuum before decaying into a classical mixture. This experiment points toward robustly encoding quantum information in multidimensional steady-state manifolds.

Stabilizing the state of a system in the vicinity of a predefined state despite the presence of external perturbations plays a central role in science and engineering. On a quantum system, stabilization is a fundamentally more subtle process than on a classical system, as it requires an interaction that, quantum mechanically, is always invasive. The mere act of learning something about a system perturbs it. Carefully designed nondestructive quantum measurements have recently been incorporated in feedback loops to stabilize a single quantum state (*1*–*4*). Alternatively, adequately engineering an interaction with an auxiliary dissipative system, termed “engineered dissipation,” can also stabilize a single quantum state (*5*–*8*).

Can engineered dissipation protect all unknown superpositions of two states, thus protecting quantum information? In fact, the static random access memory of a computer chip dynamically stabilizes the states representing 0 and 1 by combining the energy supply and dissipation, providing fast access time and robustness against noise. For a quantum memory, however, the system must not only have one or two stable steady states (SSSs), but rather a whole quantum manifold composed of all coherent superpositions of two SSSs (Fig. 1A). All quantum states are attracted to this manifold and are thus protected against diffusion out of it. However, by construction, such a system does not distinguish between all of its SSSs and hence cannot correct against random transitions within the quantum manifold.

An oscillator that exchanges only pairs of photons with a dissipative auxiliary system (*9*) is a practical example that displays a manifold of SSSs. This two-photon loss will force and confine the state of the oscillator into the quantum manifold spanned by two oscillation states with opposite phases. Uncontrolled energy decay, termed “single-photon loss,” causes decoherence within the SSS manifold, and hence quantum superpositions will eventually decay into classical mixtures. Nevertheless, in the regime where pairs of photons are extracted at a rate at least as large as the single-photon decay rate, transient quantum coherence can be observed. This regime requires combining strong nonlinear interactions between modes and low single-photon decay rates.

Our experiment enters this regime and demonstrates transient quantum states through a circuit quantum electrodynamics architecture (*10*), benefiting from the strong nonlinearity and low loss of a Josephson junction. Our setup (Fig. 1B), based on a recent proposal (*11*), consists of two superconducting microwave oscillators coupled through a Josephson junction in a bridge transmon configuration (*12*). These oscillators are the fundamental modes of two superconducting cavities. One cavity, termed “the storage,” holds the manifold of SSSs and is designed to have minimal single-photon dissipation. The other, termed “the readout,” is overcoupled to a transmission line, and its role is to evacuate entropy from the storage. In a variety of nonlinear systems, the interaction of a pump tone with relevant degrees of freedom provides cooling (*13*), squeezing (*14*), and amplification (*15*, *16*). Similarly, we use the four-wave mixing capability of the Josephson junction to generate a coupling that exchanges pairs of photons in the storage with single photons in the readout.

By off-resonantly pumping the readout at an angular frequency ω_{p} = 2ω_{s} – ω_{r}
(1)where ω_{r} and ω_{s} are the readout and storage angular frequencies, respectively, the pump stimulates the conversion of two storage photons into one readout photon and one pump photon. The readout photon then rapidly dissipates through the transmission line, resulting in a loss in photon pairs for the storage (Fig. 1D). This engineered dissipation is the key ingredient in our experiment. The input power that balances this dissipation is provided by the readout drive, a weak resonant irradiation of the readout. Due to the nonlinear mixing with the pump, these input readout photons are converted into pairs of storage photons (Fig. 1E). Unlike the usual linearly driven dissipative oscillator, which adopts only one oscillation state, our nonlinearly driven dissipative system displays a quantum manifold of SSSs corresponding to all superpositions of two oscillation states with opposite phases.

We used a third mode besides the storage and readout: the excitation of the bridge transmon qubit, restricted to its ground and first excited state. It served as a calibration tool for all of the experimental parameters and as a means to directly measure the Wigner function of the storage. Our system is well described by the effective Hamiltonian for the storage and the readout [(*17*), section 2]

The readout and storage annihilation operators are denoted **a**_{r} and **a**_{s}, respectively, and *ħ* is Planck’s constant *h* divided by 2π. The first four terms constitute a microscopic Hamiltonian of the degenerate parametric oscillator [(*18*), section 9.2.4] with

where χ_{rs}/2π = 206 kHz is the dispersive coupling between the readout and the storage, κ_{r} is the decay rate of the readout, and and are the pump and drive amplitudes, respectively. The terms in *g*_{2} correspond to the conversion of pairs of photons in the storage into single photons in the readout (Fig. 1, D and E). The readout and storage have a Kerr nonlinearity: χ_{rr}/2π = 2.14 MHz and χ_{ss}/2π = 4 kHz, respectively. The Kerr interactions can be considered as perturbations that do not substantially disturb the two-photon conversion effects [(*17*), section 2.3]. The storage and readout single-photon lifetimes are, respectively, 1/κ_{s} = 20 μs and 1/κ_{r} = 25 ns.

The two-photon processes (Fig. 1, D and E) are activated only when the frequency-matching condition (Eq. 1) is met. We satisfy this condition by performing a calibration experiment (Fig. 2). We excited the readout with a weak continuous wave (CW) probe tone (≈one photon) and measured its transmitted power in the presence of the pump tone while sweeping the frequency of both tones. The pump power is kept fixed during this measurement, and its value was chosen as the largest that did not degrade the coherence times of our system [(*17*), fig. S5]. When the frequency-matching condition is met, the probe photons are converted back and forth into pairs of storage photons (Fig. 1, D and E). When equilibrium is reached for this process, the input probe photons interfere destructively with the back-converted storage photons and are now reflected back into the probe input port [(*18*), section 12.1.1]: the readout is in an induced dark state. The dip (Fig. 2, A and B) is a signature of this interference. Its depth indicates that we have achieved a large nonlinear coupling [(*17*), section 2.2]. For the subsequent experiments, we fixed the pump frequency to ω_{p}/2π = 8.011 GHz, which makes the dip coincide with the readout resonance frequency.

We demonstrate that photons are inserted in the storage by measuring the probability of having *n* > 0 photons in the storage while sweeping the readout drive frequency (Fig. 2C). A 10-μs square pulse is applied from the pump and drive tones simultaneously, and then the qubit is excited from its ground to its excited state, conditioned on there being *n* = 0 photons in the storage (*19*). Reading out the qubit state then answers the question of whether there are 0 photons in the storage. The peak at zero detuning shows that the readout drive and the pump combine nonlinearly to insert photons into the storage. The drive tone frequency is chosen such that it maximizes the number of photons in the storage, and the drive power is fixed to ensure an equilibrium average photon number in the storage of about four [(*17*), section 1.4.6].

Adiabatically eliminating the readout from Eq. 2 [(*17*), section 2.3.1; (*18*), section 12.1], we obtain a dynamics for the storage governed by the Hamiltonianand loss operators and , where

The ϵ_{2} nonlinear drive inserts pairs of photons in the storage (Fig. 1E) and is analogous to the usual squeezing drive of a nonlinear oscillator [(*14*), equation 1.66]. The novel element in this experiment is the nonlinear decay of rate κ_{2}, which extracts only photons in pairs from the storage (Fig. 1D). In the absence of unavoidable loss κ_{s} and neglecting the effect of χ_{ss} [(*17*), section 2.3.2], the storage converges into the two-dimensional (2D) quantum manifold spanned by coherent states , where

In a classical model where quantum noise is just ordinary noise (*20*), our system behaves as a bistable oscillator with two oscillation states of amplitudes ±α_{∞}. The storage then evolves to +α_{∞} or –α_{∞}. However, in the full quantum model, the storage must evolve to +α_{∞} and –α_{∞} when initialized in the vacuum state, thus forming an even Schrödinger cat state: , where is a normalization constant (*21*–*25*).

We visualize these dynamics by measuring the state of the storage by direct Wigner tomography (*22*, *26*). The Wigner function [(*27*), section 6.5] is a representation of a quantum state defined over the complex plane as , the normalized expectation value of the parity operator for the state displaced by the operator . This quasi-probability distribution vividly displays the quantum features of a coherent superposition.

The bistable property of our system is demonstrated by initializing the storage in coherent states with a mean photon number of 6.8 with various phases and observing their convergence to the closest equilibrium state (Fig. 3) (displacement angle = {0, ±π/4, ±3π/4,π}). The upper and lower middle panels (Fig. 3) (displacement angle = ±π/2) correspond to states initialized at almost equal distance from ±α_{∞}, which randomly evolve to one equilibrium state or the other, thus converging to the statistical mixture of ±α_{∞}.

The coherent splitting of the vacuum into the quantum superposition of is demonstrated in Fig. 4. In the absence of loss in the storage, the pairwise exchange of photons between the storage and the environment conserves parity. Because the vacuum state is an even-parity state, it must transform into the even cat state: the unique even state contained in the manifold of equilibrium states. Similarly, because Fock state is an odd-parity state, it must transform into the odd cat state [(*17*), fig. S8]. In presence of κ_{s}, all coherences will ultimately disappear. However, for large enough κ_{2}, a quantum superposition transient state is observed.

In this experiment, we achieve , which implies *g*_{2}/2π = 111 kHz and κ_{2}/κ_{s} = 1.0. The quantum nature of the transient storage state is visible in the negative fringes of the Wigner function (see Fig. 4, A and B; 7 μs) and the non-Poissonian photon number statistics (Fig. 4D; 7 μs). After 7 μs of pumping, we obtain a state with an average photon number and a parity of 42%, which is larger than the parity of a thermal state (17%) or a coherent state (0.8%) with equal . After 19 μs of pumping, although the negative fringes vanish, the phase and amplitude of the SSSs are conserved. Our data are in good agreement with numerical simulations (Fig. 4C), indicating that our dominant source of imperfection is single-photon loss. These results illustrate the confinement of the storage state into the manifold of SSSs and how it transits through a quantum superposition of .

We have realized a nonlinearly driven dissipative oscillator that spontaneously evolves toward the quantum manifold spanned by two coherent states. This was achieved by attaining the regime in which the photon-pair exchange rate is of the same order as the single-photon decay rate. The ratio between these two rates can be further improved within the present technology by using an oscillator with a higher quality factor and increasing the oscillator’s nonlinear coupling to the bath. Our experiment is an essential step toward a new paradigm for universal quantum computation (*11*). By combining higher-order forms of our nonlinear dissipation with efficient error syndrome measurements (*28*), quantum information can be encoded and manipulated in a protected manifold of quantum states.

## Supplementary Materials

www.sciencemag.org/content/347/6224/853/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S11

Tables S1 and S2

## References and Notes

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**Acknowledgments:**We thank L. Jiang and V. V. Albert for helpful discussions. Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering and NSF Materials Research Science and Engineering Center Division of Material Research award 1119826. This research was supported by the Army Research Office under grant W911NF-14-1-0011. M.M. acknowledges support from the French Agence Nationale de la Recherche under the project EPOQ2 number ANR-09-JCJC-0070. S.T. acknowledges support from ENS Cachan.