Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation

Manipulating the state of a logical quantum bit (qubit) usually comes at the expense of exposing it to decoherence. Fault-tolerant quantum computing tackles this problem by manipulating quantum information within a stable manifold of a larger Hilbert space, whose symmetries restrict the number of independent errors. The remaining errors do not affect the quantum computation and are correctable after the fact. Here we implement the autonomous stabilization of an encoding manifold spanned by Schrödinger cat states in a superconducting cavity. We show Zeno-driven coherent oscillations between these states analogous to the Rabi rotation of a qubit protected against phase flips. Such gates are compatible with quantum error correction and hence are crucial for fault-tolerant logical qubits.

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Friction dissipates precious energy from a system, but it can also provide stability. Shock absorbers, for example, employ a damping piston to smooth a car ride over a bumpy road. However, this approach to stability comes with a price. If the system has two equivalent (or degenerate) lowest-energy states, then motion from one to the other, which might be desirable in some cases, will also be damped. In quantum-mechanical systems, however, this is not necessarily the case. This possibility of moving between multiple degenerate states is called the dynamical quantum Zeno effect. Here, friction transforms a force that would normally push the system out of the collection of degenerate states into a force that rotates the relative phases of the superposition of these states. In this paper, we present experimental evidence for this effect.

We engineer a superconducting microwave cavity to exchange pairs of photons with its frictional environment. Under these circumstances, the electromagnetic field inside the cavity evolves irreversibly to one of two coherent states with fields pointing in opposite directions. These states constitute a manifold of two stable degenerate steady states of the cavity. The stabilization that forces the field to settle into these states does not distinguish between one or the other—the cavity therefore hosts any quantum superposition of the two states. By properly adjusting the drive of the cavity, we observe a hallmark of quantum Zeno dynamics: a continuous change in the relative phase of the superposed states.

Our setup could be employed as an error-corrected qubit, the essential element of information in a quantum computer. With some reasonable improvements, our system would avoid the limitations observed in other methods for realizing a qubit such as phase flips.

Figure 1

Quantum Zeno dynamics and its implementation. (a) Conceptual representation of the experiment. The quantum state of a harmonic oscillator is represented here by a point in a 2D plane (not to be confused with phase space). The dark blue circle represents a cross section of the Bloch sphere of a two-state manifold in the larger Hilbert space of the oscillator. The quantum Zeno effect observed in our experiment corresponds to motion along the circle. A weak excitation drive is applied to the oscillator and the resulting trajectory has a component both along the circle and out of it. The nonlinear dissipation and drive (orange) cancels the movement outside the circle while being blind to the position of the quantum state on the circle. (b) Schematics of the experimental device. The quantum manifold is stabilized within the Hilbert space of the fundamental mode of an aluminium post cavity (cyan, storage in the text). This resonator is coupled to two Josephson junctions on sapphire (yellow for the reservoir and crimson for the Wigner transmon, see text), which are read out by stripline resonators (gray). Three couplers (brown) bring microwave drives into the system and carry signals out of it.

Figure 2

Experimental protocol and Wigner tomography result. (a) Sequence of different drives outlined in Fig. 1. The storage is initialized either in a cat state $\mathcal{N}(|{\alpha }_{\infty }⟩\pm |-{\alpha }_{\infty }⟩)$ with optimal control pulses or in a coherent state $|\pm {\alpha }_{\infty }⟩$ with a displacement $D(\pm {\alpha }_{\infty })$. The drives stabilizing the manifold are turned on in 24 ns (purple and yellow). They are on for a duration $\mathrm{\Delta }t$ during which the storage drive (cyan) can be turned on to induce the parity oscillations. The drives are left on for another 500 ns and then turned off in 24 ns, after which the Wigner function is measured. (b) In the left-hand column is the Wigner functions of the storage cavity after initialization in an even or odd cat state $({|{\alpha }_{\infty }|}^2=3)$. The right-hand column shows the corresponding Wigner functions after a quarter of an oscillation. The color maps are averaged raw data of the Wigner function measurement (see text) and the orange circles are cuts along $\mathrm{Re}(\alpha )=0$. The gray solid lines are theoretical curves corresponding to even or odd cats (left-hand column, lines 1 and 2, respectively) and parityless cats (right-hand column, lines 1 and 2). The only fit parameter in the theory is the renormalization of the amplitude by a factor 0.87 on the left, and 0.65 on the right. These factors account for the fidelity of the parity measurement and the decay of the fringes of the cat states during the stabilization.

Figure 3

Characterization of the oscillations. (a) Evolution of the measured parity as a function of time. The initial cat states are even, with $\overline{n}={|{\alpha }_{\infty }|}^2=2$, 3, 5 (circles, squares, diamonds). The storage drive is either off (black markers) or on (colored markers) with various strengths given in units of a chosen base strength ${\epsilon }_0$. Simulations are shown as solid lines. The minimum of each experimental curve is emphasized for each drive strength (full marker with black contour). (b) A fit of the data gives the frequency $\mathrm{\Omega }$ and the time constant $\tau$ of the decaying oscillations. The former is plotted as a function of the relative drive strength $\epsilon /{\epsilon }_0$ (top panel). The case $\overline{n}=2$ is fitted with a linear function (dashed line). Based on this, we make predictions for $\overline{n}=3$, 5 (solid lines). The bottom panel shows the characteristic decay time of the oscillations $\tau$, normalized by the decay time of the nondriven case ${\tau }_0$.

Figure 4

Gate on cardinal points of the Bloch sphere of the protected manifold. An arbitrary cat state $\mathcal{N}[\cos (\theta /2)|{\mathcal{C}}_{{\alpha }_{\infty }}^{+}⟩+\sin (\theta /2)e^{i\varphi }|{\mathcal{C}}_{{\alpha }_{\infty }}^{-}⟩]$ is represented by a point on a sphere. Six initial states are chosen, corresponding to the cardinal points $(\theta ,\varphi )=(0,0)$, $(\pi ,0)$, $(\pm \pi /2,0)$, $(\pi /2,\pm \pi /2)$, with ${|{\alpha }_{\infty }|}^2=3$, and their equivalent Pauli operators are measured. The markers corresponding to each initial state are, respectively, red and blue circles, gray up or down triangles, and black up or down triangles. The initial octahedron formed by those points (left) is either transformed under the action of the identity (upper right) or a rotation of $\pi /2$ around the $X$ axis (lower right).