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(a) (color plot) Sideband spectroscopy of sample B on both sides of the bistability threshold, i.e. for the resonator in the L state (P p < −105 dBm) and in the H state (P p > −105 dBm). The expected frequency of the AC Stark-shifted main qubit peak as well as the Stokes and anti-Stokes sidebands are shown as dashed white and black lines. (b) (red dots) Measured r/(1 − r), with r being the ratio of the anti-Stokes to Stokes sideband peak height. Data points are shown only above the threshold, because no measurable anti-Stokes peak is found in our data below threshold.
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We measure the quantum fluctuations of a pumped nonlinear resonator, using a
superconducting artificial atom as an in-situ probe. The qubit excitation
spectrum gives access to the frequency and temperature of the intracavity field
fluctuations. These are found to be in agreement with theoretical predictions;
in particular we experimentally observe...
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... was not the case in the sample discussed in the main text because the pump frequency was much closer to the KNR frequency in these data, resulting in a smaller separation between the Stokes sideband below trheshold and the main qubit peak (since well below the threshold˜∆threshold˜ threshold˜∆ p ≈ ω c − ω p ), which made the two lines indistinguishable. Quite remarkably, even though the Stokes sideband is clearly visible below the threshold, the anti- Stokes sideband is invisible within our detection limit (see Fig.5). Since the ratio between Stokes and anti-Stokes peaks directly yields the mean photon numberñnumberñ, this sets an upper bound on the mean photon numberñnumberñ present in the mode at˜ωat˜ at˜ω c which is shown in Fig.5 for various pump powers (the dependence on pump power is simply due to the fact that the height of the Stokes peak also diminishes at small pump power, whereas the noise in the data is constant). ...
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... remarkably, even though the Stokes sideband is clearly visible below the threshold, the anti- Stokes sideband is invisible within our detection limit (see Fig.5). Since the ratio between Stokes and anti-Stokes peaks directly yields the mean photon numberñnumberñ, this sets an upper bound on the mean photon numberñnumberñ present in the mode at˜ωat˜ at˜ω c which is shown in Fig.5 for various pump powers (the dependence on pump power is simply due to the fact that the height of the Stokes peak also diminishes at small pump power, whereas the noise in the data is constant). ...
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... the ratio between Stokes and anti-Stokes peaks directly yields the mean photon numberñnumberñ, this sets an upper bound on the mean photon numberñnumberñ present in the mode at˜ωat˜ at˜ω c which is shown in Fig.5 for various pump powers (the dependence on pump power is simply due to the fact that the height of the Stokes peak also diminishes at small pump power, whereas the noise in the data is constant). In particular, this allows us to establish that any residual photon numbers that might be present at˜ωat˜ at˜ω c due to improper thermalization or filtering is below the lower point of the dashed blue line in Fig.5, namely < 0.04. ...
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... the sign change of˜∆of˜ of˜∆ p (see Fig.1 in the main text). Finally, the effective temperature extracted from Fig.5a is shown in Fig.5b and is again in fair agreement with the prediction |ν| 2 calculated without adjustable parameter, both below and above the threshold. ...
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... On resonance δ q = a [r], and provided |2 a [r]| g sinh r, the system reduces to an enhanced resonant Jaynes-Cummings interaction, that could be unambiguously revealed through an increased vacuum-Rabi splitting [42]. However, this simple picture is blurred by the squeezed bath that populates higher BO energy levels [44][45][46], which in turn broadens the qubit spectral line [32] [ Fig. 1(a)]. References [11,12] propose to circumvent this problem by injecting orthogonally squeezed vacuum, so that the bath viewed by the BO remains in vacuum. ...
The interaction strength of an oscillator to a qubit grows with the oscillator’s vacuum field fluctuations. The well-known degenerate parametric oscillator has revived interest in the regime of strongly detuned squeezing, where its eigenstates are squeezed Fock states. Owing to the amplified field fluctuations in the antisqueezed quadrature, it was recently proposed that squeezing this oscillator would dynamically boost qubit-photon interactions. In a superconducting circuit experiment, we observe a twofold increase in the dispersive interaction between a qubit and an oscillator at 5.5 dB of squeezing, demonstrating in situ dynamical control of qubit-photon interactions. This work initiates the experimental coupling of oscillators of squeezed photons to qubits, and cautiously motivates their dissemination in experimental platforms seeking enhanced interactions.
... Investigating the effect of local fields is also important in the application point of view. As pointed out in Ref. [38], the output probability distribution of the KPO network obeys a Boltzmann distribution in the presence of dissipation due to a heating process called quantum heating [40][41][42]. Verifying the Boltzmann distribution of output phases with variable local fields is an interesting topic for the future study. ...
We study simultaneous parametric oscillations in a system composed of two distributed-element-circuit Josephson parametric oscillators in the single-photon Kerr regime coupled via a static capacitance. The energy of the system is described by a two-bit Ising Hamiltonian with an effective coupling whose amplitude and sign depend on the relative phase between parametric pumps. We demonstrate that the binary phases of the parametric oscillations are correlated with each other, and that the parity and strength of the correlation can be controlled by adjusting the pump phase. The observed correlation is reproduced in our simulation taking pure dephasing into account. The present result demonstrates the tunability of the Hamiltonian parameters by the phase of external microwave, which can be used in the the Ising machine hardware composed of the KPO network.
... On resonance δ q = Ω a [r], and provided Ω a [r] sinh r, the system reduces to an enhanced resonant Jaynes-Cummings interaction, that could be unambiguously revealed through an increased vacuum-Rabi splitting. However, this simple picture is blurred by the squeezed bath that populates higher BO energy levels [25][26][27], which in turn broadens the qubit spectral line [28] (Fig. 1a). Ref. [7] proposes to circumvent this problem by injecting orthogonally squeezed vacuum, so that the bath viewed by the BO remains in vacuum. ...
The interaction strength of an oscillator to a qubit grows with the oscillator's vacuum field fluctuations. The well known degenerate parametric oscillator has revived interest in the regime of strongly detuned squeezing, where its eigenstates are squeezed Fock states. Owing to these amplified field fluctuations, it was recently proposed that squeezing this oscillator would dynamically boost its coupling to a qubit. In a superconducting circuit experiment, we observe a two-fold increase in the dispersive interaction between a qubit and an oscillator at 5.5 dB of squeezing, demonstrating in-situ dynamical control of qubit-oscillator interactions. This work initiates the experimental coupling of oscillators of squeezed photons to qubits, and cautiously motivates their dissemination in experimental platforms seeking enhanced interactions.
... Up to now, the bifurcation readout has been realized in the weak anharmonicity limit U ≪ κ in which the bistability regime is achieved when the photon number n in the nonlinear resonator exceeds the critical number N crit = κ/(3 √ 3U ) ≫ 1 [28,29]. However, this large photon number, needed to reach the bistability and thus bifurcation, exposes the qubit to excess backaction of the nonlinear cavity [28][29][30][31][32] like inducing qubit state transitions and thus rendering the readout not QND. ...
... Up to now, the bifurcation readout has been realized in the weak anharmonicity limit U ≪ κ in which the bistability regime is achieved when the photon number n in the nonlinear resonator exceeds the critical number N crit = κ/(3 √ 3U ) ≫ 1 [28,29]. However, this large photon number, needed to reach the bistability and thus bifurcation, exposes the qubit to excess backaction of the nonlinear cavity [28][29][30][31][32] like inducing qubit state transitions and thus rendering the readout not QND. Instead of the usual classical regime of weak anharmonicity and large photon number (U ≪ κ, N crit ≫ 1), the excess backaction on the qubit could be weakened in the mesoscopic regime (U ∼ κ, N crit ∼ 1) where bistability appears with photon number close to unity n ≳ N crit ∼ 1. ...
We explore the nonlinear response to a strong drive of polaritonic meters for superconducting qubit state readout. The two polaritonic meters result from the strong hybridization between a bosonic mode of a 3D microwave cavity and an anharmonic ancilla mode of the superconducting circuit. Both polaritons inherit a self-Kerr nonlinearity $U$, and decay rate $\kappa$ from the ancilla and cavity, respectively. They are coupled to a transmon qubit via a non-perturbative cross-Kerr coupling resulting in a large cavity pull $2\chi > \kappa, ~U$. By applying magnitic flux, the ancilla mode frequency varies modifying the hybridization conditions and thus the properties of the readout polariton modes. Using this, the hybridisation is tuned in the mesoscopic regime of the non-linear dissipative polariton where the self-Kerr and decay rates of one polariton are similar $U\sim \kappa$ leading to bistability and bifurcation behavior at small photon number. This bistability and bifurcation behavior depends on the qubit state and we report qubit state readout in a latching-like manner thanks to the bifurcation of the upper polariton. Without any external quantum-limited amplifier, we obtain a single-shot fidelity of $98.6\%$ in a $500$ ns integration time.
... The population of the HEL space is also suppressed by 44% at 100 µs. This population leakage has been called quantum heating-a heating process induced by quantum jumps due to dissipation (in this case, singlephoton loss) in quasienergy levels of driven quantum nonlinear systems [54,[63][64][65]. The steady state solution in Supplementary Table 1 is indeed independent of γ KPO , which is a signature of quantum heating [54,64]. ...
Autonomous quantum error correction has gained considerable attention to avoid complicated measurements and feedback. Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point -- it requires only a single continuous microwave tone. We also introduce an unconditional reset scheme that requires one more continuous microwave tone in addition to that for the error correction. The key properties underlying this simplicity are protected quasienergy states of a four-photon Kerr parametric oscillator and the degeneracy in its quasienergy level structure. These properties eliminate the need for state-by-state correction in the Fock basis. Our schemes greatly reduce the complexity of autonomous quantum error correction and thus may accelerate the use of the bosonic code for practical quantum computation.
... While the linewidth narrowing effect is induced by qubit driving, a variety of other nonlinear effects can be observed with strong cavity mode driving [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. In section V we focus on the lineshape of the cavity transmission in the nonlinear region. ...
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Period tripling in driven quantum oscillators reveals unique features absent for linear and parametric drive, but generic for all higher-order resonances. Here, we focus at zero temperature on the relaxation dynamics towards a stationary state starting initially from a domain around a classical fixed point in phase space. Beyond a certain threshold for the driving strength, the long-time dynamics is governed by a single time constant that sets the rate for switching between different states with broken time translation symmetry. By analyzing the lowest eigenvalues of the corresponding time evolution generator for the dissipative dynamics, we find that near the threshold the gap between these eigenvalues nearly closes. The closing becomes complete for a vanishing quantum parameter. We demonstrate that this behavior, reminiscent of a quantum phase transition, is associated with a transition from a stationary state which is localized in phase space to a delocalized one. We further show, that switching between domains of classical fixed points happens via quantum activation, however, with rates that differ from those obtained by a standard semiclassical treatment. As period tripling has been explored with superconducting circuits mainly in the quasi-classical regime recently, our findings may trigger new activities towards the deep quantum realm.
... The last few years has seen rapid progress in engineering and probing the properties of nonlinear oscillators in the quantum regime [1][2][3][4]. This has stimulated renewed theoretical interest in the properties of such systems, both at the level of individual oscillators and for more complex many-body realizations [5][6][7][8]. ...
We introduce a simple model system to study synchronization theoretically in quantum oscillators that are not just in limit-cycle states, but rather display a more complex bistable dynamics. Our oscillator model is purely dissipative, with a two-photon gain balanced by single- and three-photon loss processes. When the gain rate is low, loss processes dominate and the oscillator has a very low photon occupation number. In contrast, for large gain rates, the oscillator is driven into a limit-cycle state where photon numbers can become large. The bistability emerges between these limiting cases with a region of coexistence of limit-cycle and low-occupation states. Although an individual oscillator has no preferred phase, when two of them are coupled together a relative phase preference is generated which can indicate synchronization of the dynamics. We find that the form and strength of the relative phase preference varies widely depending on the dynamical states of the oscillators. In the limit-cycle regime, the phase distribution is $\pi$-periodic with peaks at $0$ and $\pi$, whilst in the low-occupation regime $\pi$-periodic phase distributions can be produced with peaks at $\pi/2$ and $3\pi/2$. Tuning the coupled system between these two regimes reveals a region where the relative phase distribution has $\pi/2$-periodicity.
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A crucial limit to measurement efficiencies of superconducting circuits comes from losses involved when coupling to an external quantum amplifier. Here, we realize a device circumventing this problem by directly embedding an artificial atom, comprised of a transmon qubit, within a flux-pumped Josephson parametric amplifier. This configuration is able to enhance dispersive measurement without exposing the qubit to appreciable excess backaction. Near-optimal backaction is obtained by engineering the circuit to permit high-power operation that reduces information loss to unmonitored channels associated with the amplification and squeezing of quantum noise. By mitigating the effects of off-chip losses downstream, the on-chip gain of this device produces end-to-end measurement efficiencies of up to 80%. Our theoretical model accurately describes the observed interplay of gain and measurement backaction and delineates the parameter space for future improvement. The device is compatible with standard fabrication and measurement techniques and, thus, provides a route for definitive investigations of fundamental quantum effects and quantum control protocols.
