One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator

Bistable dynamical systems are widely employed to robustly encode classical bits of information. However, they owe their robustness to inherent losses, making them unsuitable to encode quantum information. Surprisingly, there exists a loss mechanism, known as two-photon dissipation, that provides stability without inducing decoherence. An oscillator exchanging pairs of photons with its environment is expected to reach macroscopic bit-flip times between dynamical states containing only a handful of photons. However, previous implementations have observed bit-flip times saturating in the millisecond range. In this experiment, we design a superconducting resonator endowed with two-photon dissipation, and free of all suspected sources of instabilities and inessential ancillary systems. We attain bit-flip times exceeding 100 s in between states containing about 40 photons. Although a full quantum model is necessary to explain our data, the preparation of coherent superposition states remains inaccessible. This experiment demonstrates that macroscopic bit-flip times are attainable with mesoscopic photon numbers in a two-photon dissipative oscillator.

Popular Summary

Classical bits are notoriously stable. In fact, in modern computer chips, a bit will only randomly flip every 10,000 years. This stability is due to regular friction that dampens erroneous diffusion between states. However, friction (or dissipation) is also known to cause decoherence, making these bits unsuitable for processing quantum information. Remarkably, there exists a dissipative mechanism, known as two-photon dissipation, that provides stability without causing decoherence. Previous implementations have observed bit-flip times saturate in the millisecond range, way below the targeted macroscopic values. In this experiment, we have identified and removed the sources responsible for this saturation and observed 100-second bit-flip times.

Our two-photon dissipative system is implemented in a superconducting nonlinear circuit. Our implementation has two main technical innovations: first, to operate this circuit in a parameter regime expected to evade dynamical instabilities; second, to remove the transmon qubit previously employed for tomography and suspected of inducing bit flips.

Future work will need to recover the ability—absent in this experiment—to control and measure quantum mechanical superpositions of these macroscopically stable dynamical states. Such qubits could then be assembled into one-dimensional chains to form a fully protected logical qubit.

Figure 1

(Top) Principle of the experiment. A cavity is endowed with a special mechanism (dashed left mirror) that exchanges pairs of photons (blue double waves) at variable intensity (control knob) with a cold bath. Two metastable pointer states emerge, represented by the blue distributions centered around amplitudes $\pm \alpha$. The resemblance of the steady states to these circular Gaussian distributions is valid in the limit of large $|\alpha |$. A fraction of the cavity field (blue waves) escapes through the weakly transmissive mirror and is collected by our heterodyne detector. By monitoring the signal over time, we track individual trajectories undergoing bit flips (blue time trace). (Bottom) False-color optical micrograph of the experimental superconducting circuit in a coplanar waveguide geometry. The cavity is a $\lambda /2$ resonator (blue) that radiates a field $a_{\mathrm{out}}$ through a weakly coupled 50-$\mathrm{\Omega }$ port. It also couples to a two-photon exchange device composed of a buffer mode (red) shunted to ground through an asymmetrically threaded SQUID (ATS) as emphasized in the insets. dc currents enter through on-chip bias tees (green) and impose phase biases ${\phi }_{L,R}$. A differential pump (purple arrows) and a common buffer drive (red arrows) are channeled through filtered transmission lines (orange).

Figure 2

Emergence of two metastable pointer states from a nonlinear dissipative phase transition. (a1)–(a3) Radiated energy from the memory mode in units of circulating photon number (color) as a function of the detuning from the frequency-matching condition ${\mathrm{\Delta }}_a=\frac{1}{2}({\omega }_p-(2{\omega }_a-{\omega }_d))$ ($x$ axis), and the detuning from the buffer resonance ${\mathrm{\Delta }}_b={\omega }_d-{\omega }_b$ ($y$ axis). We denote ${\omega }_{a,b,p,d}$ the memory, buffer, pump, and drive angular frequencies, respectively. Both data and semiclassical numerical simulations are shown in different regions of each panel corresponding to the specified drive amplitude ${ϵ}_d$. (b) Radiated energy from the memory in units of circulating photon number ($y$ axis) at ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_b=0$ as a function of the drive amplitude ($x$ axis). The data correspond to an integration time of $10\text{μ}\mathrm{s}$ with single averaging (circles) and 10 000 averages (crosses). A semiclassical model (green solid line) captures the appearance of a critical point around ${ϵ}_d/2\pi \approx 3\mathrm{MHz}$ above which the vacuum state becomes unstable (green dashed line). A full quantum model (red solid line) is necessary to capture the curvature at the critical point, as emphasized by the enlargement in the inset panel. (c) Histogram (color) of the $I$ quadrature integrated over 1 ms of the field radiated by the memory ($y$ axis) in units of the square root of circulating photon number as a function of the drive amplitude ($x$ axis). Passed the critical point, the memory field transits from the vacuum into two metastable pointer states.

Figure 3

Real-time oscillator dynamics revealed by individual trajectories. For photon numbers $\overline{n}=11,28,43$ (top, middle, bottom), we, respectively, set the integration time to $T_m=0.1,1,5$ ms and the total measurement duration to $T_{\mathrm{tot}}=10,1000,5000$ s. (Left) Histogram of the $(I,Q)$ quadratures of the radiated field. The narrowing of the distributions from top to bottom is due to the increasing integration time $T_m$. (Center) Trajectory of the $I$ quadrature as a function of time cropped from the full dataset. (Right) Cumulative distribution function of the stochastic time interval ${\tau }_{\mathrm{jump}}$ in between two consecutive jumps. Its average value, that defines the bit-flip time $T_{\text{bit~flip}}$, is printed on each panel. The solid lines represent the cumulative distribution for Poisson’s statistics $e^{-t/T_{\text{bit-flip}}}$.

Figure 4

Exponential suppression of bit flips. The bit-flip time ($y$ axis, log scale) is measured (open circles) as a function of the number of photons contained in the pointer states ${|0/1⟩}_{\alpha }$ ($x$ axis). The bit-flip time increases exponentially, multiplying by 1.4 per photon (tilted dashed line) before saturating at approximately 127 s (horizontal dashed line).

Figure 5

(a) Optical micrograph of the ATS made of aluminium (light gray) deposited on the niobium circuit (gray) over a silicon substrate (dark gray). (b)–(d) Scanning electron microscope images of the small single junctions (b),(d) forming the SQUID loop and the five array junctions (c) forming the inductive shunt. The small junctions are $275\times 700{\mathrm{nm}}^2$. The array junctions, which would ideally all be equal in area, are in fact composed of three $600\mathrm{nm}\times 3.9\text{μ}\mathrm{m}$ and two $270\mathrm{nm}\times 3.9\text{μ}\mathrm{m}$ junctions. This results in a critical current density of about $450\mathrm{nA}/\text{μ}{\mathrm{m}}^2$. For clarity, small arrows point to the location of each junction.

Figure 6

Lumped-element model of the circuit. The buffer (red) with bare frequency ${\omega }_{b,0}/2\pi$ is constituted with an ATS (inductive energy $E_L$, mean Josephson energy $E_J$, asymmetry $\mathrm{\Delta }{E}_J$), and a capacitor with charging energy $E_C$. The ATS loops are threaded with fluxes ${\phi }_L$, ${\phi }_R$ (green) and is connected to the memory (blue), with bare frequency ${\omega }_{a,0}/2\pi$. The phase $\phi$ is indicated with an arrow (black).

Figure 7

Measured (top) and simulated (bottom) frequencies (color) of the buffer (left) and memory (right) as a function of the differential flux ($x$ axis) and common flux ($y$ axis) in the ATS loop. The orange cross marks the flux point at which we operate the experiment.

Figure 8

Wiring of the experiment. Measurement apparatus for the memory (blue labels), buffer (red labels), and TWPA pump (black label) connect to the experiment through rf lines (black lines). dc voltage sources are used to drive flux lines (green lines). Dashed lines indicate the different temperature stages of the dilution refrigerator. Additional information is provided in the legend (gray background), annotations and in the text.

Figure 9

Each panel displays the measured relative amplitude (color) of the reflected signal on the buffer port at the frequency $f$ specified in each label box, as a function of the differential ($x$ axis) and common ($y$ axis) phase biases. The dashed gray lines are guides for the eye corresponding to ${\phi }_{\mathrm{\Delta }}=\pm \pi /2$ and ${\phi }_{\mathrm{\Sigma }}=\pi /2$.

Figure 10

(a) Relative amplitude (color) of the reflected signal on the buffer port as a function of the pump frequency ${\omega }_p$ (left $y$ axis), drive local oscillator frequency ${\omega }_d^{\mathrm{LO}}$ (right $y$ axis) and drive intermediate frequency ${\omega }_d^{\mathrm{IF}}$ ($x$ axis). The drive frequency is given by ${\omega }_d={\omega }_d^{\mathrm{IF}}+{\omega }_d^{\mathrm{LO}}$. For each pump frequency, the drive LO frequency is set to ${\omega }_d^{\mathrm{LO}}=(2{\omega }_a-{\omega }_p)+200\mathrm{MHz}$, such that vertical lines correspond to constant detuning from the frequency-matching condition (${\mathrm{\Delta }}_a=cte$). The buffer drive resonance condition ${\mathrm{\Delta }}_b=0$ is determined by fitting each horizontal cuts of the map (dashed red line). In the vicinity of ${\mathrm{\Delta }}_a=0$ and ${\mathrm{\Delta }}_b=0$, a sharp feature indicates that the two-to-one photon exchange transition is resonant (dotted green line). Spurious transitions appear near the frequency-matching condition ${\omega }_p+{\omega }_d={\omega }_{2a}$ (blue dashed line), where ${\omega }_{2a}$ is the frequency of the second harmonic of the memory $\lambda /2$-resonator measured independently. (b) Enlargement of the two-to-one photons’ exchange transition for increasing drive amplitude ${ϵ}_d$. (c) Radiated energy from the memory in units of circulating photon number (color) as a function of the pump frequency ${\omega }_p$ (left $y$ axis) and two-photon drive intermediate frequency ${\omega }_{\mathrm{dm}}^{\mathrm{IF}}$ ($x$ axis), for increasing drive amplitude. On these panels, the two-photon drive LO frequency is set to ${\omega }_{\mathrm{dm}}^{\mathrm{LO}}={\omega }_a+100\mathrm{MHz}$. When the two-to-one photon exchange transition is resonant, the engineered two-photon drive populates the memory. The average occupation of the memory is determined thanks to an undercoupled port via heterodyne detection. (d) Radiated energy from the memory in units of circulating photon number (color) as a function of the pump and drive detuning from the frequency-matching condition ${\mathrm{\Delta }}_a$ ($x$ axis), and the drive detuning from the buffer ${\mathrm{\Delta }}_b$ ($y$ axis). In these coordinates, the feature takes the shape of a regular diamond.

Figure 11

Radiated energy from the memory in units of circulating photon number (color) as a function of the detuning from the frequency matching condition ${\mathrm{\Delta }}_a$ ($x$ axis), and the detuning from the buffer resonance ${\mathrm{\Delta }}_b$ ($y$ axis). Left column displays data, right column displays semiclassical simulations for the corresponding drive amplitude. Orange cross shows the position of the maximum of the autocorrelation for the largest drive amplitude. Red cross shows the zero detuning point given by direct fit of memory and buffer spectroscopy. Red circle is the point at which well-averaged data are taken to perform the fit of $g_2$.

Figure 12

Two-photon coupling calibration. (a) Radiated energy from the memory ($y$ axis) as a function of drive amplitude ($x$ axis). There are two regimes: when the drive amplitude is small, single-photon loss overcomes the two-photon drive, and the memory stays in the vacuum. Passed the critical point, the memory gets populated by a coherent state with photon number asymptotically proportional to the drive amplitude. The axes units are chosen so that the critical point is at ${\kappa }_a{\kappa }_b/8$ and the asymptotic slope is 1. The data correspond to an integration time of $10\text{μ}\mathrm{s}$ with 10 000 averages (crosses). The semiclassical model (green solid line) captures the position of the critical point but fails to explain the curvature of the experimental data. A full numerical simulation is used to reproduce the data where the only fitting parameter is $g_2$ (red lines). (b) Enlargement of the curvature around the critical point [gray rectangle from (a)], emphasizing the agreement between simulations and experimental data.

Figure 13

Radiated energy from the memory in units of circulating photon number (color) as a function of the detunings ${\mathrm{\Delta }}_a$ ($x$ axis), and ${\mathrm{\Delta }}_b$ ($y$ axis) defined in the text, for the drive amplitude ${ϵ}_d/2\pi =12.1$ MHz. The red solid line displays the fitted slope of the top-right and bottom-left edge, yielding the ratio ${\kappa }_b/{\kappa }_a$ [see (B27)]. The red dashed-dotted lines and red dashed lines, respectively, give the upper and lower bound on this parameter (determined by graphical reading).

Figure 14

False-color optical micrograph of the detection efficiency chip in $\mathrm{Nb}$ (gray) on $\mathrm{Si}$ (dark blue). The memory resonator (blue) is capacitively coupled to a transmon (green). The inset is centered on the transmon and its Josephson junction in $\mathrm{Al}$ (light gray). We can address the memory and collect the reflected signal (blue waves) via the bottom $50\mathrm{\Omega }$ port. The left $50\mathrm{\Omega }$ port is dedicated to drive the transmon (green waves). This sample is also used to measure the memory thermal population of about 1%.

Figure 15

Calibration of the detection efficiency $\eta$. (Top) Qubit spectroscopy showing photon-number splitting: data (solid lines) and fit (dashed lines). (Bottom) Product $\overline{n}\eta$ computed using Eq. (D6) as a function of the square input signal $S_{\mathrm{in}}^2$ in units of photon number.

Figure 16

Numerical simulations (solid lines) of the bit-flip time ($y$ axis) for three values of $g_2$ (labels) as a function of the number of photons in the memory $\overline{n}$ multiplied by $(g_2/2\pi {)}^2$ ($x$ axis). The data (dots) from Fig. 4 of the main text qualitatively matches the simulations for $g_2/2\pi =39$ kHz, thus confirming our calibration of $g_2$.