Abstract
Anharmonic oscillators, such as the pendulum, are widely used for precision measurement^{1} and to model nonlinear phenomena^{2}. Fluctuations—such as thermal or quantum mechanical noise—can excite random motion in the oscillator, ultimately imposing a bound on measurement sensitivity. In systems where equilibrium is established with the environment, noiseinduced broadening scales with the intensity of fluctuations. But how does noise affect an outofequilibrium oscillator where the motion is varied faster than energy is exchanged with the environment? We create such a scenario by applying fast, frequencychirped voltage pulses to a nonlinear superconducting resonator where the ringdown time is longer than the pulse duration. Under these conditions, the circuit oscillates with either small or large amplitude depending on whether the drive voltage is below or above a critical value^{3}. This phenomenon, known as autoresonance, is significant in planetary dynamics^{4} and plasmas^{5}, enables the excitation of particles in cyclotron accelerators^{6} and may even be used to detect the state of a quantum twolevel system^{7}. Our results show that the amplitude of fluctuations determines the initial conditions of such a nonequilibrium oscillator and does not affect its time evolution.
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Our oscillator is a superconducting electrical circuit based on a Josephson tunnel junction formed by two thin aluminium films separated by an oxide barrier. In response to a periodic current drive I(t), the junction voltage V (t) oscillates according to the Josephson relations, which describe the motion of a particle with coordinate δ in a sinusoidal potential U(δ)=[ℏI_{0}/(2e)]cos(δ), where I_{0} is the junction critical current, δ is the phase difference across the junction and e is the electron charge. This system is analogous to a mechanical pendulum with angular coordinate θ=δ (Fig. 1a,b). The response of such an anharmonic oscillator is illustrated in Fig. 1c; as the drive strength is increased the resonant response shifts to lower frequency and ultimately exhibits bistability. In this regime, if the oscillator is excited with a constantfrequency drive of increasing amplitude, as depicted by the lightgrey arrow in Fig. 1c, it abruptly switches from a low to a highamplitude oscillation state. This effect has been used to realize a threshold amplifier^{8} to read out the state of quantum bits^{9,10,11}. As the system is in equilibrium, fluctuationinduced switching from one oscillation state to the other is modelled as a stationary process involving escape over an effective energy barrier^{12,13}. The sensitivity of this amplifier improves with decreasing intensity of noise until the quantum regime is reached^{14}.
The bistable regime may also be accessed by driving the anharmonic oscillator with a fixedamplitude, chirped drive as depicted by the black arrows in Fig. 1c. For a given chirp rate α and oscillator quality factor Q, if the drive amplitude is above a critical voltage V_{c}∝α^{3/4}, the oscillator phase locks with the drive signal and climbs to the highamplitude oscillation state^{3}. Although known in plasma and accelerator physics since the 1950s, autoresonance has only recently been observed in an electrical circuit, where it was noted that this process could be used for sensitive amplification^{7}. Unlike the amplitudedriven case, autoresonant excitation is by construction a nonequilibrium process, and in this letter we analyse the finite width of the threshold and the role of quantum fluctuations in determining the ultimate sensitivity of a measurement device that uses this effect.
The oscillator circuit consists of a tunnel junction embedded in the centre of a linear cavity (see Fig. 1d), realized using a capacitively isolated section of coplanar waveguide transmission line that forms a 6 GHz Fabry–Perot resonator. This architecture allows us to engineer the electrical impedance shunting the junction and hence tune the oscillator quality factor (Q), and provides a simple means to couple microwave frequency signals into and out of the oscillator. The equation of motion for the embedded system at large excitation differs from that of a physical pendulum; however, the autoresonant dynamics are unchanged^{15}. We excite the circuit with a frequency chirp, created using a voltagecontrolled oscillator (VCO) modulated by an arbitrary waveform generator. This signal is sent through heavily filtered coaxial lines to the chip, which is cooled to temperatures as low as 15 mK in a dilution refrigerator. The transmitted voltage is amplified at 2.3 K and demodulated using quadrature mixers at room temperature to obtain the signal magnitude relative to the drive.
We first demonstrate threshold behaviour when the oscillator is driven with a phasecontinuous voltage pulse that starts at 6.075 GHz and decreases linearly in time to 5.775 GHz. The smallsignal resonance frequency of the oscillator is ω/(2π)=5.987 GHz with Q=8,200. Thus, both the start and stop frequencies are many linewidths away from resonance. The transmitted voltage () developed in response to the drive is plotted in Fig. 2a as a function of drive frequency and amplitude. The data are acquired by averaging 5,000 frequency sweeps at a fixed amplitude and then stepping its value over a 10 nV interval. As the drive approaches the resonance condition, energy is transferred and oscillation amplitude begins to build. When the drive amplitude is weak, we observe that after a slight excitation the oscillator relaxes back to very small oscillations. On the other hand, for stronger drive, the oscillator builds up energy, behaviour that is indicative of autoresonant phaselocking of the oscillations to the drive.
A constantfrequency slice of the data in Fig. 2a, shown in the left panel, shows the threshold for autoresonance at V_{c}≃27 nV. Here, the average junction voltage varies between a low level (V_{l}) corresponding to the unlocked events and a high level (V_{h}) corresponding to locked events. Within the threshold, in a given frequency sweep the oscillator either locks into autoresonance or does not. We define the locking probability such that P=1 when and P=0 when . The locking probability versus drive power is shown for three different chirp rates in Fig. 2b. The threshold location, V_{c}, where P=1/2, is plotted versus chirp rate in Fig. 2c, and agrees with theory^{7}.
We now turn to the width of the autoresonant threshold. In the absence of fluctuations, the capture probability would be an infinitely sharp step at V_{c} as indicated in Fig. 2d,e. At the start of a chirp sequence, the resonator has an initial excitation that is the result of interactions with the fluctuating bath up to that time. The threshold for a given initial amplitude A_{0} is thus either augmented or diminished depending on the initial phase mismatch, Δ, between the drive signal and the oscillator motion, , where κ=0.245 is a prefactor determined by numerical simulations. A finite threshold width is obtained by integrating the locking probability (see Fig. 2d) over all values of Δ (assumed to be uniformly distributed between 0 and 2π) and over a thermal distribution of initial amplitudes at temperature T_{eff}, where k_{B}T_{eff}=(ℏω/2)coth(ℏω/(2k_{B}T)). We note that the quantum ground state is equivalent to a thermal distribution with T_{eff}=ℏω/2. The threshold width, defined as the inverse slope of the locking probability at P=1/2, is then , where L is the total inductance of the resonator and k_{B} is Boltzmann’s constant^{16}.
The quantum counterpart to classical autoresonance is quantum mechanical ladder climbing. The transition between these two regimes depends on the degree of anharmonicity. When the anharmonicity is sufficiently large, high excitation is achieved by climbing the ladder of accessible states, as opposed to a continuous classical evolution^{17}. To confirm the validity of our classical description when k_{B}T≪ℏω, we compute the dynamics in the presence of quantum fluctuations. The state of an oscillator can be described by its position and momentum, which for our electrical circuit correspond to the dimensionless charge (q) and current (j), respectively. In Fig. 3, we plot the Wigner quasiprobability distribution that results after the ground state—a Gaussian centred about zero—is evolved using a voltage excitation 20% greater than V_{c}. a–c correspond to decreasing anharmonicity with undriven energylevel shifts 100, 49 and 25 times larger than our experimental parameters. d is the classical result for the same parameters and initial conditions as in c. We observe the majority of the population in the phaselocked state, as indicated by dashed line. The data in Fig. 3c show nearly the same distribution as those of the classical calculation. For higher anharmonicity, the phaselocked state remains qualitatively unchanged, but the population in the lowamplitude state exhibits a characteristic quantum interference pattern indicating the participation of a relatively small number of discrete levels. Our sample parameters, however, correspond to an oscillator with 25timesweaker anharmonicity than Fig. 3c, and we thus expect the classical model for the transition width to be valid at all T_{eff}.
By varying the physical temperature of the dilution refrigerator we measured the transition width as a function of temperature. To compare with theory we plot the square of the transition width, scaled to temperature units as ΔV^{2}/(8κπL α k_{B}), versus temperature in Fig. 4. For temperatures above 200 mK, we observe a clear T^{1/2} dependence of the width. At the lowest temperatures, saturation is observed corresponding to T_{eff}=144 mK, which is precisely ℏω/(2k_{B}). The solid line indicates our theoretical prediction for the width with no adjustable parameters. To verify the quantum origin of this saturation, we plot the thresholdwidth data for a 1.6 GHz resonator (Fig. 4 inset). Here, we observe a T^{1/2} scaling over the entire temperature range, with a suggestive flattening of the transition width at the lowest temperatures, where k_{B}T≃ℏω/2.
Keeping the sample at T=15 mK, we also injected white noise into the resonator to simulate a thermal bath. The results of this experiment are shown as the blue open circles in Fig. 4. The observed threshold width scales as T^{1/2} and has an offset corresponding to the half quantum of zeropoint motion of the oscillator, corroborating again a quantumnoisebroadened threshold at the lowest temperature. Thus, quantum mechanics in our system enters only through the initial fluctuations, characterized by T_{eff}, and the system otherwise can be treated classically^{18}.
Our experiment is an example of the interaction of an oscillator out of equilibrium with a noise bath. The quantum saturation of the threshold width, which we note can be tuned by adjusting α, ultimately sets the resolution of a digital detector based on autoresonance. Such a detector can be used for the readout of a quantum bit. As an example, for the parameters used in this experiment a 2% variation in I_{0} could be detected with unit signaltonoise ratio in a time of ∼200 ns. This compares to a typical readout based on switching between metastable states where ∼1% variations in I_{0} are detected in similar time^{14}. In contrast to abrupt switching between metastable states^{19}, autoresonance involves continuous evolution to either the locked or unlocked state, raising interesting questions about the measurement back action.
Methods
The oscillator was fabricated from Al deposited on a highresistivity Si wafer. At low excitation power, a total quality factor of Q=8,230 resulted from internal losses characterized by Q_{int}=17,200 and coupling to the 50 Ω environment given by Q_{ext}=15,800. The critical power for bifurcation was measured to be P_{c}=−123 dBm, corresponding to a junction critical current I_{0}=1.8 μA and a total inductance L=2.3 nH. At drive powers near P_{c} we observed a decrease in the total quality factor of the oscillator to Q=6,480.
The chirped pulse was created by driving a VCO operating near 8 GHz with an arbitrary waveform generator and mixing it with a tone at 2.312 GHz. The transmitted power from the sample was mixed with the output of the VCO to form a fixedfrequency signal at 2.312 GHz. This signal was demodulated with a local oscillator at 2.412 GHz, and the I and Q quadratures were digitized at 500 MS s^{−1}.
We modelled our oscillator by a weakly nonlinear Hamiltonian, H=j^{2}/2+q^{2}/2−β q^{4}/4+ɛ qcos_{d}, on the phase space of dimensionless charge, q, normalized to q_{0}=j_{0}/ω and current, j, normalized to . The anharmonicity is given by β=(_{0}ω^{2}q_{0}^{2})/(6L I_{0}^{3}), where _{0}=ℏ/2e is the flux quantum. The dimensionless drive voltage ɛ=V_{d}/(L q_{0}ω^{2}) and the phase of the drive . We numerically solved the dimensionless quantum Liouville equation for the Wigner function, f(q,j,τ), where τ=ω t:
The parameter γ=ℏω/(k_{B}T_{eff}) allows us to solve both the classical γ0 and quantum γ2 limits of the system (note that in the classical limit the expected temperature dependence is embedded in the reduced coordinates). We solved the Liouville equation using a standard pseudospectral method^{20} for the parameters in our experiment, but for β=n^{2}×3.55×10^{−6}, ɛ=0.0246/n and α=10^{−6}ω^{2} Hz^{2}, where n={10,7,5} correspond to nonlinear energylevel shifts that are 100, 49 and 25 times that of the experiment, and drive strengths that are 20% above the autoresonant threshold. The classical evolution was approximated by solving the Liouville equation with γ=10^{−4}.
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Acknowledgements
We thank A. G. Shagalov, who developed the pseudospectral code used for solving the Liouville equation. This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the US Government. R.V. acknowledges funding from AFOSR under Grant No FA95500810104; I.B. and L.F. acknowledge support from the Israel Science Foundation under Grant No 451/10.
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K.W.M. and R.V. carried out the experiments and analysed the data for the 6 GHz samples. R.V. and J.A. fabricated the samples at 6 GHz and 1.6 GHz, respectively. O.N. carried out the experiments for the 1.6 GHz sample. I.B. and L.F. contributed to the theoretical understanding and calculated the Wigner distribution. K.W.M. and I.S. wrote the manuscript. All work was supervised by I.S.
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Murch, K., Vijay, R., Barth, I. et al. Quantum fluctuations in the chirped pendulum. Nature Phys 7, 105–108 (2011). https://doi.org/10.1038/nphys1867
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DOI: https://doi.org/10.1038/nphys1867
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