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Physical Review A

Bistability in a quantum nonlinear oscillator

Dafna Bortman, Amiram Ron

Published Oct 1, 1995• 8 citations
Publisher

Abstract

The response of a nonlinear oscillator to an external electromagnetic field is studied in the quantum regime. It is demonstrated that bistability occurs when the shift of the resonance frequency, induced by the external field and modified by fluctuations (quantum and thermal), surpasses the natural linewidth. This theory can be applied to the study of the nonlinear response, near resonance, of a single electron, trapped in a magnetic field, to an external electromagnetic field. It is shown that, even at low quantum levels of the oscillator, bistability can be observed due to the interplay between the small relativistic nonlinearity, the external electromagnetic field, and the heat bath.

DOI
10.1103/physreva.52.3316

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The response of a nonlinear oscillator to an external electromagnetic field is studied in the quantum regime. It is demonstrated that bistability occurs when the shift of the resonance frequency, induced by the external field and modified by fluctuations (quantum and thermal), surpasses the natural linewidth. This theory can be applied to the study of the nonlinear response, near resonance, of a single electron, trapped in a magnetic field, to an external electromagnetic field. It is shown that, even at low quantum levels of the oscillator, bistability can be observed due to the interplay between the small relativistic nonlinearity, the external electromagnetic field, and the heat bath.
A nonlinear oscillator driven by both a coherent external field and a (weak) fluctuation-dissipation process of standard type is studied. It is shown that in the extremely-low-friction regime an additional random process comes into play which makes the linear-response theory work in spite of the scarce amount of standard fluctuation. This additional random process depends on the interplay of the nonlinear character of the oscillator under study and the weak fluctuation-dissipation process of standard type, but it is of transverse type, i.e., it does not involve any energy-dissipation process. On the theoretical side, this result rests on the joint use of a procedure of elimination of ``irrelevant'' variables and a rotating-wave-approximation technique which makes the role played by the transverse relaxation-process transparent. These predictions are checked by means of an analog experiment. Furthermore, light is shed onto an incorrect conclusion on the range of validity of the linear-response theory arrived at in an earlier report. This incorrect conclusion is now traced back to an ambiguous feature of the experimental results. The frequency position of the absorption peak does not change significantly upon increase of the excitation field up to a certain value of the excitation field. After this initial plateau the frequency of the absorption peak exhibits a rapid increase with increasing the intensity of the radiation field. Then the increase of the linewidth of the absorption peak becomes slower again with a parallel slow increase of the linewidth of the absorption spectrum, the shape of which, furthermore, does not show any sign of saturation, thereby generating the false impression, when the experimental investigation is carried out in this region, of a complete breakdown of the linear-response theory.
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.
Summary form only given. Optical bistability is basically a nonlinear response of the medium (here confined electrons in a QW) to an external oscillating field, in which there are three possible responses (two stable and one unstable). Which one the system chooses depends on the history of the adiabatically varying control parameters (usually electric field amplitude). The OB we consider is an intrinsic feature of the electron gas in the QW, and does not rely on the use of a Fabry-Perot resonator. Another feature of OB is that it is not a ubiquitous nonlinear response such as superharmonic generation; it can be observed only at some restricted values of the parameters. Furthermore the nonlinear response changes abruptly from one value of the output to the other. This effect can be measured by detecting the different transmitted intensities of the FIR field to the same input field. These features of OB make it a relevant choice for applications to optical switching and the realization of optical logic gates.
A novel mechanism is proposed for optical bistability by utilizing the effect of interference which occurs when two levels are coupled by a one-photon resonance and by a two-photon resonance. The resulting bistability does not require atomic saturation, so the bistability can be observed at relatively low incident intensities, and the mechanism is essentially Doppler free, since it occurs by a two-photon process involving absorption of counterpropagating beams of the same frequency. A significant feature is that two-photon bistability can be controlled not only by the magnitude of the field of the one-photon resonance but also by its phase, and an oscillation or amplification of the field of the two-photon resonance can be observed under certain conditions by the mechanism of the parametric interaction of the field of the one-photon resonance and the field of the two-photon resonance through a nonlinear medium.
One of the most studied model systems in quantum optics is a two-level atom strongly coupled to a single mode of the electromagnetic field stored in a cavity, a research field named cavity quantum electrodynamics or CQED. CQED has recently received renewed attention due to its implementation with superconducting artificial atoms and coplanar resonators in the so-called circuit quantum electrodynamics (cQED) architecture. In cQED, the couplings can be much stronger than in CQED due to the design flexibility of superconducting circuits and to the enhanced field confinement in one-dimensional cavities. This enabled the realization of fundamental quantum physics and quantum information processing experiments with a degree of control comparable to that obtained in CQED. The purpose of this chapter is to investigate the situation where the resonator to which the atom is coupled is made nonlinear with a Kerr-type nonlinearity, causing its energy levels to be nonequidistant. The system is then described by a nonlinear Jaynes-Cummings Hamiltonian. This considerably enriches the physics since a pumped nonlinear resonator displays bistability, parametric amplification, and squeezing. The interplay of strong coupling and these nonlinear effects constitutes a novel model system for quantum optics that can be implemented experimentally with superconducting circuits. This chapter is organized as follows. In a first section we present the system consisting of a superconducting Kerr nonlinear resonator strongly coupled to a transmon qubit. In the second section, we describe the response of the sole nonlinear resonator to an external drive. In the third section, we show how the resonator bistability can be used to perform a high-fidelity readout of the transmon qubit. In the last section, we investigate the quantum backaction exerted by the intracavity field on the qubit.