Abstract Synchronization is a widespread phenomenon encountered in many natural and engineered systems with nonlinear classical dynamics. How synchronization concepts and mechanisms transfer to the quantum realm and whether features are universal or platform specific are timely questions of fundamental interest. They can be studied in superconducting electrical circuits which provide a well-established platform for nonlinear quantum dynamics. Here, we consider a Josephson-photonics device, where a dc-biased Josephson junction creates (non-classical) light in a microwave cavity. The combined quantum compound constitutes a self-sustained oscillator: a system susceptible to synchronization. This is due to the inherent effect of an in-series resistance, which realizes an autonomous feedback mechanism of the charge transport on the driving voltage. Accounting for the full counting statistics of transported charge not only yields phase diffusion, but allows us to describe phase locking to an ac-signal and the mutual synchronization of two such devices. Thereby one can observe phase stabilization leading to a sharp emission spectrum as well as unique charge transport statistics revealing shot noise induced phase slips. Two-time perturbation theory is used to obtain a reduced description of the oscillators phase dynamics in form of a Fokker–Planck equation in generalization of classical synchronization theories.
Abstract Synchronization is a widespread phenomenon encountered in many natural and engineered systems with nonlinear classical dynamics. How synchronization concepts and mechanisms transfer to the quantum realm and whether features are universal or platform specific are timely questions of fundamental interest. They can be studied in superconducting electrical circuits which provide a well-established platform for nonlinear quantum dynamics. Here, we consider a Josephson-photonics device, where a dc-biased Josephson junction creates (non-classical) light in a microwave cavity. The combined quantum compound constitutes a self-sustained oscillator: a system susceptible to synchronization. This is due to the inherent effect of an in-series resistance, which realizes an autonomous feedback mechanism of the charge transport on the driving voltage. Accounting for the full counting statistics of transported charge not only yields phase diffusion, but allows us to describe phase locking to an ac-signal and the mutual synchronization of two such devices. Thereby one can observe phase stabilization leading to a sharp emission spectrum as well as unique charge transport statistics revealing shot noise induced phase slips. Two-time perturbation theory is used to obtain a reduced description of the oscillators phase dynamics in form of a Fokker–Planck equation in generalization of classical synchronization theories.
Injection locking can stabilize a source of radiation, leading to an efficient suppression of noise-induced spectral broadening and therefore, to a narrow spectrum. The technique is well established in laser physics, where a phenomenological description due to Adler is usually sufficient. Recently, locking experiments were performed in Josephson photonics devices, where microwave radiation is created by inelastic Cooper pair tunneling across a dc-biased Josephson junction connected in-series with a microwave resonator. An in-depth theory of locking for such devices, accounting for the Josephson nonlinearity and the specific engineered environments, is lacking. Here, we study injection locking in a typical Josephson photonics device where the environment consists of a single mode cavity, operated in the classical regime. We show that an in-series resistance, however small, is an important ingredient in describing self-sustained Josephson oscillations and enables the locking region. We derive a dynamical equation describing locking, similar to an Adler equation, from the specific circuit equations. The effect of noise on the locked Josephson phase is described in terms of phase slips in a modified washboard potential. For weak noise, the spectral broadening is reduced exponentially with the injection signal. When this signal is provided from a second Josephson device, the two devices synchronize. In the linearized limit, we recover the Kuramoto model of synchronized oscillators. The picture of classical phase slips established here suggests a natural extension towards a theory of locking in the quantum regime.
A resistively shunted Josephson junction with a DC-bias exhibits Josephson oscillations that link voltage and frequency through fundamental constants. These oscillations can synchronize their frequency to an external microwave signal achieving the metrological precision that underlies the modern Josephson voltage standard. Its electromagnetic dual, the phase-slip junction, exhibits Bloch oscillations that connect current to frequency in an analogous way. A current standard based on synchronized Bloch oscillations could close the quantum metrology triangle, providing a consistent test of the underlying fundamental relations between voltage, current, and resistance. The experimental realization, however, remains challenging, as external microwave lines introduce parasitic capacitances that can destroy the high-impedance environment required for Bloch oscillations. This work explores the use of on-chip synchronization to stabilize both Josephson and Bloch oscillations without the need for an external microwave drive. We review the theoretical framework of the synchronization between two coupled oscillators and the dynamics of superconducting circuits with a focus on quantum metrology. Building on this, we analyze a recent experiment on the mutual synchronization of Bloch oscillations in a pair of ultrasmall Josephson junctions. Our model in terms of correlated phase-slip junctions realized in the ground state of the device shows good agreement with the experimental results, confirming synchronized Bloch oscillations. At larger currents, nonadiabatic transitions wash out the coherence of the Bloch oscillations and therefore the synchronization features. Motivated by these findings, we propose an alternative circuit setup where the Bloch oscillations in a single phase-slip junction are stabilized by synchronization to on-chip Josephson oscillations. We study the back-action of the Bloch oscillations on the Josephson oscillations and identify a parameter regime in which it is minimized, rendering the latter an effectively rigid driving signal. Moreover, we find that the small remaining back-action further enhances the thermal stability of the synchronized Bloch oscillations compared to the externally driven case. Without a back-action mechanism, the coherence of Josephson oscillations is severely limited by thermal fluctuations. We relate this limitation to the thermodynamic uncertainty relation (TUR) in the field of stochastic thermodynamics. Recent findings show, that the TUR can be broken by a classical pendulum clock. We propose a circuit that mimics the dynamics of a pendulum clock as a flexible on-chip microwave source for high impedance environments. Due to an internal synchronization, the clock circuit emits coherent radiation beyond the limits of the TUR. Finally, we explore the use of symmetry breaking drives as a resource for nonreciprocal transport in superconducting circuits. We demonstrate that a biharmonic microwave signal applied to a conventional Josephson junction induces an ideal supercurrent diode effect with a tunable direction.
It is well known from classical physics that weakly coupled self-sustained oscillators may spontaneously lock their phases. Just like classical synchronization is known to break down due to noise induced phase slips, we show here how the synchronization of continuous variable quantum systems breaks down by proliferation of quantum phase slips. Within a Keldysh path integral formulation of limit cycles, we analyze the phase dynamics and show how, in spite of strong phase correlations, quantum phase slips degrade the actual phase locking. This approach also allows us to address non-Markovian effects on the synchronization of limit cycles, which we illustrate explicitly for superconducting resonators coupled via a voltage biased double quantum dot.
At its heart, a Josephson-photonics device combines macroscopic superconducting electronics and microscopic quantum optics. It consists of a dc-voltage biased Josephson junction that is connected in series to one or a few microwave cavities. Cooper pairs can tunnel incoherently across the junction by depositing their energy 2eVdc in the microwave cavities, creating photonic excitations in them. Many parameters, such as the eigenfrequencies ∝ (LC)^(−1/2) or zero-point fluctuations ∝ (L/C)^(1/4) of the microwave cavities, can be experimentally designed; other parameters, such as the strength of the Josephson nonlinearity and the dc voltage are easily modified in situ. In this way, Josephson-photonics devices constitute a versatile source of microwave photons with a wide variety of features extending from the classical to the deep quantum regime. To make them of use for quantum technological applications that rely on a well-defined phase of the microwave light, a crucial drawback must be overcome. The dc voltage does not provide a stable reference phase to Josephson-photonics devices, as it is subject to experimental noise, such that all phase information is lost. In this thesis, we develop the field of Josephson photonics in two directions: As first main result, we resolve the issue of phase instability in Josephson photonics. The phase can be stabilized either by injection locking to a small external signal with a stable and well-defined reference phase or by mutual synchronization to a second device. An alternative approach mitigates the issue for some use cases by optimizing an ansatz to reconstruct the undiffused stationary quantum state from experiments with phase diffusion. Secondly, we show how Josephson-photonics devices can be used for the challenging task of reliably detecting single itinerant microwave photons. The “Inelastic Cooper Pair Tunneling Photon Multiplier” exploits a resonance condition, where the energy provided by the dc voltage enables a multiplication process from one incoming to n outgoing photons. This process can be staged to reach a large number of outgoing photons. Here, we model itinerant Gaussian single-photon pulses of length T with Mølmer’s approach. Further, we define a detection scheme based on heterodyne quadrature measurement of the output signal. Most importantly, we show that realistic devices can achieve a detection efficiency of 84.5% with a small dark-count rate of 10^(−3)/T, promising to outperform competing schemes.
Self-sustained oscillators play a central role in the stabilization and synchronization of complex dynamical systems. A number of different physical systems are currently being investigated to clarify the importance of such active components in the quantum realm. Here we explore the properties of a driven dissipative electron-photon hybrid system based on superconducting microwave resonators coupled resonantly to a voltage-biased double quantum dot (DQD). First, we establish a Hopf bifurcation at a critical value of the electron-photon coupling, beyond which an effective negative friction sustains steady limit-cycle oscillations of individual resonators. Second, we show that two such limit-cycle resonators coupled via the same voltage-biased DQD synchronize for small enough frequency detuning. A nonlinear photon Keldysh action is derived by perturbation theory in the effective circuit fine-structure constant, and the limit-cycle dynamics is analyzed in terms of resulting saddle-point, and Fokker-Planck equations. In the Markovian limit of infinite bias voltage, these results are shown to agree well with the solution of a corresponding Lindblad master equation for the DQD resonator system.
The interplay of the tunneling transfer of charges and the emission and absorption of light can be investigated in a set-up, where a voltage-biased Josephson junction is placed in series to a microwave cavity. Here, we concentrate on the emission process and show that due to the inherent nonlinearity of the Josephson junction tunneling Cooper-pairs can create a variety of non-classical states of light. Depending on experimental parameters and tuning the device can be described by effective Hamiltonians, indicating specific photon creation mechanisms which lead to strongly bunched or anti-bunched light emission. We derive explicit analytical expressions for the photon correlation function g(2)(τ=0) for these different processes and investigate their full crossover numerically. Photon distribution functions show squeezing and other interesting non-Poissonian behavior.