Bistability in a mesoscopic Josephson junction array resonator

We present an experimental investigation of stochastic switching of a bistable Josephson junction array resonator with a resonance frequency in the GHz range. As the device is in the regime where the anharmonicity is on the order of the linewidth, the bistability appears for a pump strength of only a few photons. We measure the dynamics of the bistability by continuously observing the jumps between the two metastable states, which occur with a rate ranging from a few Hz down to a few mHz. The switching rate strongly depends on the pump strength, readout strength and the temperature, following Kramers' theory. The interplay between nonlinearity and coupling, in this little explored regime, could provide a new resource for nondemolition measurements, single photon switches, or even elements for autonomous quantum error correction.

Figure 1

(a) Photograph of one half of a rectangular copper waveguide with a 6 GHz cutoff. A JJA with a microwave antenna is fabricated on a piece of sapphire and placed in the center of the waveguide. (b) Schematic representation of an array of Josephson junctions inside a waveguide. $C_0$ is the capacitance of the islands to ground, $C_S$ is the shunt capacitance for the array, $C_J$ is the junction capacitance, and $C_g$ denotes the coupling capacitance of the antenna to the waveguide. $L_J$ is the Josephson junction inductance. The input and output couplers to the waveguide are shown on the top left and top right of the schematic. (c) Optical image of the JJA coupled to a 6-mm-long antenna and a shunt capacitance. The inset shows a zoom-in on the junction array. (d) Electron beam image (blue box in the inset) of ten of the ${10}^3$ Josephson junctions.

Figure 2

Two-tone measurement. (a) Shift ${\mathrm{\Delta }}_R$ of the resonance frequency of the readout mode ${\omega }_R/2\pi =7.105$ GHz upon application of a pump tone. The pump tone is detuned from the resonance ${\omega }_P/2\pi =4.8156$ GHz of the fifth mode of the JJA by a detuning ${\mathrm{\Delta }}_P$ and has a pump strength of ${\overline{n}}_P=115$ photons. Due to the nonlinearity of the JJA, the resonance frequency of the readout mode is shifted when the pump tone matches mode five. (b) Zoom in on the bistable region of mode five. For a detuning of about ${\mathrm{\Delta }}_P=-2.58\left(4\right)$ MHz two different resonance frequencies of the readout mode can be observed. The shifted and unshifted resonances correspond to 115 and 1 circulating photons in mode five, respectively.

Figure 3

(a) Transitions from the low amplitude state to the high amplitude state for ${\overline{n}}_P=9$ photons at a detuning of ${\mathrm{\Delta }}_P\approx {\mathrm{\Delta }}_P^{\mathrm{Max}}=-0.54\left(1\right)$ MHz from mode five. The readout mode seven was driven with ${\overline{n}}_R=0.5$ photons on resonance. The trace displayed here is a 15 s segment out of a total recorded time of 20 000 s. (b) Histogram of the amplitude distribution $\rho \left(A\right)$ for the data displayed in (a). (c) Histogram of $\rho \left(A\right)$ for a detuning of ${\mathrm{\Delta }}_P=-0.59\left(1\right)$ MHz. (d) Histogram of the amplitude distribution $\rho \left(A\right)$ for a detuning of ${\mathrm{\Delta }}_P=-0.49\left(1\right)$ MHz. (e). Dependence of the normalized state population in the high and low photon state on ${\mathrm{\Delta }}_P$ relative to the bistable point. The solid line is a fit to a sigmoid function.

Figure 4

(a) Dependence of switching rate $\mathrm{\Gamma }$ on ${\mathrm{\Delta }}_P$ for three different pump strengths: $•- {\overline{n}}_P=2.6$ photons, $•- {\overline{n}}_P=4$ photons, $•- {\overline{n}}_P=6$ photons. The solid lines are Lorentzian fits to the data. (b) Extracted ${\mathrm{\Gamma }}_{\mathrm{Max}}$ and corresponding ${\mathrm{\Delta }}_P^{\mathrm{Max}}$ as a function of ${\overline{n}}_P$. The black and red lines are from a fit to our theoretical model (see text). (c) $\mathrm{\Gamma }$ measured for constant ${\overline{n}}_P=6$ photons while varying the readout strength: $•- {\overline{n}}_R=0.5$ photons, $•- {\overline{n}}_R=1.5$ photons, $•- {\overline{n}}_R=2.5$ photons. (d) $\mathrm{\Gamma }$ measured for different cryostat base temperatures with ${\overline{n}}_P=6$. In (a), (b), and (d) ${\overline{n}}_R=0.5$.

Figure 5

Experimental setup for performing two tone spectroscopies. For the switching rate measurements the VNA is replaced with a signal generator and mixers to down convert the transmitted signal in order to digitize it with a 1 GS analog-to-digital converter.

Figure 6

Transmission measurements, measured with a power corresponding to about one photon circulating in the resonator. The solid line in (a), (c), and (d) is a fit using Eq. (A1) to extract $Q_{\text{tot}}$ and ${\omega }_R$. (a) Resonator response measurement for mode five. (b) Relaxation time $T_1$ on mode five with a pump strength of ${\overline{n}}_P=115$ photons. The solid line is an exponential fit to the data with $T_1=3\mu \mathrm{s}$. (c) Resonator response measurement for mode seven. (d) Resonator response measurement for mode nine.

Figure 7

Two-tone measurements and dispersion curve. (a), (b), and (c) show the transmitted amplitude of the readout mode with frequency ${\omega }_R/2\pi =7.105$ GHz as a function of the pump tone frequency ${\omega }_P^D$ and detuning ${\mathrm{\Delta }}_R$. The black arrows mark the frequencies when the pump tone matches a mode of the resonator, ${\omega }_P^D$, leading to frequency shift in the read out tone. (d) Measured resonant mode frequencies of the JJA from the two-tone spectroscopy. represents the calculated dispersion relation without including corrections due to the cross coupling between segments of the array, and represents the measured resonant mode frequencies up to 20 GHz.

Figure 8

Self-Kerr and cross-Kerr measurements. The error on each data point is about point size. (a) Dependence of the self-Kerr frequency shift on the input power of mode five. The red line is a third order polynomial fit—see text. (b) Dependence of the self-Kerr frequency shift on the input power of mode seven. The red line is a linear fit—see text. (c) Cross-Kerr frequency shift of mode seven when applying input power to mode five. The red line is calculated using the theoretical prediction for the linear self-Kerr term—see text. The read-out mode seven is driven with about ${\overline{n}}_R\approx 0.5$ photons. (d) Cross-Kerr frequency shift of mode five when applying input power to mode seven. The red line is a polynomial fit to the data. The read-out mode five is driven with about ${\overline{n}}_R\approx 1$ photons.

Figure 9

Cross-Kerr measurements $K_{75}$. (a), (b), (c), (d) Two-tone spectroscopy measurements for different pump strength. Shift ${\mathrm{\Delta }}_R$ of the resonance frequency of the readout mode ${\omega }_R/2\pi =7.105$ GHz with $n_R=0.5$ photons upon application of a pump tone to mode five. The pump frequency is detuned by ${\mathrm{\Delta }}_P$ from the resonance of mode five ${\omega }_p=4.1856$ GHz. (a),(b), (c), and (d) corresponds to a pump power of $\approx 0.01$ nW, $\approx 0.7$ nW, $\approx 1$ nW, and $\approx 2$ nW, respectively; the solid black line corresponds to a fit where we extract the resonance frequency of the readout mode for each pump frequency. From the fits we then extract the maximal frequency shift of the readout mode for a given pump power in mode five.

Figure 10

(a) Linewidth ${\kappa }_5$ vs input power on mode five. For the lowest drive power we still have a good enough signal to noise ratio to fit the data. We find ${\kappa }_5({\overline{n}}_P\to 0)=181$ kHz. The black solid line corresponds to a fit of a $\sqrt{\text{Power}}$ dependency. (b) Linewidth ${\kappa }_7$ vs input power on mode seven. On the lowest drive power we still have a good enough signal to noise ratio to fit the data with a ${\kappa }_7({\overline{n}}_P\to 0)=7.5$ MHz.

Figure 11

Schematic of the potential landscape for bistable switching.

Figure 12

Width $W_{\mathrm{\Gamma }}$ of the bistable region for different pump strengths ${\overline{n}}_p$.

Figure 13

(a),(b) Probability of being either in the low-photon state or in the high-photon state in a bistable region. , represents the measured data,  — ,  —  are from fits using Eq. (B6), and  —$·$ ,  —$·$  are fits to a sigmoid function. The pump strength is ${\overline{n}}_P=9$ for (a) and ${\overline{n}}_P=6$ for (b). The inset in (a), (b) shows the histogram of the residence time $T_{\mathrm{UP}}$ with pump strength of ${\overline{n}}_P=9,{\overline{n}}_P=6$ and for a detuning of ${\mathrm{\Delta }}_P={\mathrm{\Delta }}_P^{\mathrm{Max}}$ MHz from mode five. The red line is an exponential fit to the data giving $〈T_{\mathrm{UP}}〉=14.4$ s, $〈T_{\mathrm{UP}}〉=6.5$ s.