At the hearth of superconducting circuits, the Josephson junction provides the required non-linearity to enable quantum mechanical effects. Beyond its use for quantum applications, Josephson junctions also allow for the study of light matter interactions in a more general setting. Motivated by the growing demand for devices operating at the quantum limit, important efforts have been made to utilize Josephson junctions to control large fields much like electronic transistors, or to provide quantum limited amplifiers. In this context, Josephson Junction Arrays (JJAs) exhibit large non-linearities that enables pronounced bistability at the single photon level. Concretely, the cross and self-Kerr interactions of the JJA allow for the strong dependence of a given mode of the array to the photon number in other modes.

Fig. 1: Photograph of a Josephson Junction Array (JJA) consisting of 1,000 cascaded junctions on a sapphire substrate. The large external metallic pads form a microwave antenna to enable the coupling of the JJA to a microwave copper waveguide. The response of the JJA is obtained by measuring the transmission through the waveguide. The sample, microwave waveguide and JJA, is maintained at 10 mK in a cryogen-free dilution refrigerator.

## Single photon microwave switch

By harnessing the strong non-linearity of Josephson Junctions, JJAs can be engineered to realise single-photon microwave switches: the large cross-Kerr non-linearity between two given modes is used to control the switching of one of the modes by adding a few photons in the second mode.

In [1], we experimentally studied the bistability of the 1,000 cascaded JJA resonator shown in Fig. 1. This device was tuned to provide a self and cross-Kerr non-linearity comparable with the coupling to the waveguide. This regime is particularly interesting as it ensures pronounced bistability down to the single to few photon level. In the experiment we pump the 5th mode of the JJA and measure its photon number occupation by monitoring the shift it induces through the cross-Kerr non-linearity on the readout mode (mode 7). As the detuning of the pump is changed, the JJA can be brought to the bistable region where mode 5 switches between a high and a low amplitude state, see Fig. 2. Due to the large Kerr non-linearity, this effect also allows for the efficient control of mode 5 state. Indeed, when operating close to the bistability region of mode 5, a change of a few photon in another mode (e.g. mode 7) is sufficient to shift across this region. As a result, mode 5 can be controllably changed between a large photon state and an empty state with adding only 2 photons in mode 7. This proof of principle experiment demonstrates the possibility to use JJA as a microwave switch with large amplitude modulation (two orders of magnitude in the photon occupation number).

Fig. 2: Bistable switching of the JJA. (a) Time trace of the transmission through the waveguide showing the low and high amplitude state using 9 photons to pump the mode. (b,c, and d) Histogram of the amplitude distributions as a function of the detuning of the pump showing the bistable switching close to the symmetrical, favouring the low state and favouring the high state respectively. (e) Normalised state population as a function of the detuning of the pump relative to the symmetrical switching point.

We are currently designing a dedicated JJA resonator to achieve the switching of large microwave fields (several hundreds of photons) by coupling it with a superconducting qubit. This approach would provide a direct readout of the qubit state in what we expect would provide minimal backaction.

## Bistability and Kramers model

Beside technological applications, JJAs also offer the opportunity to study bistability on its own. In this context, stochastic switching is well described by the Kramers model. In this model, the bistable system can be idealised as a doubled potential well separated by an energy barrier. The switching across the barrier is stochastically enabled by fluctuations (e.g. thermal fluctuations). The switching rate, as well as the switching speed, can then be related to the exact potential shape and the dissipation in the system.

In [1], we also demonstrated that the bistability in the JJA is accurately described by the Kramers model. In collaboration with the Quantum Nanophysics, Optics and Information group at IQOQI Innsbruck, we developed a fictitious potential approach to model the JJA and could reproduce the dependence of the switching rate as a function of the pump photon number. Remarkably, both the theoretical results and the experimental results suggest that in the range of parameters we have studied the system exhibits a Kramers turnover for which the switching rate is maximal. Such turnover has only been observed experimentally recently in levitated optomechanics by varying the dissipation in the system. In this case, by varying the photon number in the pump we effectively change the fictitious potential, keeping the dissipation constant.

We are currently planning additional measurements to study the Kramers model in JJAs in more detail, and confirm the presence of a turnover. In addition, this system offers the opportunity to investigate more recent refinements of the Kramers model including quantum effects.

[1] Bistability in a Mesoscopic Josephson Junction Array Resonator

P. R. Muppalla, O. Gargiulo, S. Mirzaei, B. Venkatesh, M. L. Juan, L. Grünhaupt, I. M. Pop, G. Kirchmair

arXiv:1706.04172 (2017)