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We show experimentally that a dc-biased Josephson junction in series with two microwave resonators emits entangled beams of microwaves leaking out of the resonators. In the absence of a stationary phase reference for characterizing the entanglement of the outgoing beams, we measure second-order coherence functions for proving entanglement up to an emission rate of 2.5 billion photon pairs per second. The experimental results are found in quantitative agreement with theory, proving that the low frequency noise of the dc bias is the main limitation for the coherence time of the entangled beams. This agreement allows us to evaluate the entropy of entanglement of the resonators, and to identify the improvements that could bring this device closer to a useful bright source of entangled microwaves for quantum-technological applications.

In their Comment [P. Hakonen and E. B. Sonin, preceding comment, Phys. Rev. X 11, 018001 (2021)PRXHAE2160-3308], Hakonen and Sonin (HS) object to our conclusion on the absence of a dissipation-induced superconducting-to-insulating quantum phase transition (DQPT) in resistively shunted Josephson junctions (RSJJs) originally predicted by Schmid and Bulgadaev (SB). Their objections are based on the account they make of a theory explaining the DQPT in terms of Bloch bands which was developed in the 1980s and early 1990s. In this Reply, we point to several issues in the Comment which undermine the objections HS formulate against our work.

We show experimentally that a dc biased Josephson junction in series with a high-enough-impedance microwave resonator emits antibunched photons. Our resonator is made of a simple microfabricated spiral coil that resonates at 4.4 GHz and reaches a $1.97\text{ }\text{ }\mathrm{k}\mathrm{\ensuremath{\Omega}}$ characteristic impedance. The second order correlation function of the power leaking out of the resonator drops down to 0.3 at zero delay, which demonstrates the antibunching of the photons emitted by the circuit at a rate of $6\ifmmode\times\else\texttimes\fi{}{10}^{7}$ photons per second. Results are found in quantitative agreement with our theoretical predictions. This simple scheme could offer an efficient and bright single-photon source in the microwave domain.

International audience; Single-photon generation is an important proof of the underlying quantum nature of a physical process and a ubiquitous tool for scientific exploration, with applications ranging from spectroscopy and metrology to quantum computing. In the microwave regime, emission of antibunched radiation has so far relied on coherent control of Josephson qubits requiring precisely calibrated microwave pulses. In this work, we experimentally demonstrate the operation of a bright on-demand source of quantum microwave radiation driven by a simple dc voltage bias across a Josephson junction. Our source is based on photon emission into a microwave resonator through inelastic Cooper pair tunneling regulated by a high-impedance RC circuit preventing simultaneous tunnel events. It is characterized by its normalized second-order correlation function of g(2)(0)≈0.43 corresponding to antibunching in the single-photon regime. Our device can be triggered, and its in situ tunable emission rate exceeds those obtained with current microwave single-photon sources by more than 1 order of magnitude.

We theoretically investigate a near-quantum-limited parametric amplifier based on the nonlinear dynamics of quasiparticles flowing through a superconducting-insulator-superconducting junction. Photon-assisted tunneling, resulting from the combination of dc- and ac-voltage bias, gives rise to a strong parametric interaction for the electromagnetic modes reflected by the junction coupled to a transmission line. We show phase-sensitive and phase-preserving amplification, together with single- and two-mode squeezing. For an aluminum junction pumped at twice the center frequency, $\omega_0/2\pi=6$~GHz, we predict narrow-band phase-sensitive amplification of microwaves signals to more than 20 dB, and broadband phase-preserving amplification of 20 dB over a 1.2 GHz 3-dB bandwidth. We also predict single- and two-mode squeezing reaching more than -12 dB over 5.3 GHz 3-dB bandwidth. Moreover, with a simple impedance matching circuit, we demonstrate 3 dB bandwidth reaching 4.3 GHz for 20 dB of gain. A key feature of the device is that its performance can be controlled in-situ with the applied dc- and ac-voltage biases., Comment: Accepted for publication at the Physical Review Applied. 12 pages and 9 figures

In this report we review the present state of the art of the control of propagating quantum states at the single-electron level and its potential application to quantum information processing. We give an overview of the different approaches which have been developed over the last ten years in order to gain full control over a propagating single electron in a solid state system. After a brief introduction of the basic concepts, we present experiments on flying qubit circuits for ensemble of electrons measured in the low frequency (DC) limit. We then present the basic ingredients necessary to realise such experiments at the single-electron level. This includes a review of the various single electron sources which are compatible with integrated single electron circuits. This is followed by a review of recent key experiments on electron quantum optics with single electrons. Finally we will present recent developments about the new physics that emerges using ultrashort voltage pulses. We conclude our review with an outlook and future challenges in the field., Comment: 35 pages, 24 figures, This is an author-created, un-copyedited version of an article accepted for publication in Rep. Prog. Phys. (references added)

Nature sets fundamental limits regarding how accurate the amplification of analog signals may be. For instance, a linear amplifier unavoidably adds some noise which amounts to half a photon at best. While for most applications much higher noise levels are acceptable, the readout of microwave quantum systems, such as spin or superconducting qubits, requires noise as close as possible to this ultimate limit. To date, it is approached only by parametric amplifiers exploiting non-linearities in superconducting circuits and driven by a strong microwave pump tone. However, this microwave drive makes them much more difficult to implement and operate than conventional DC powered amplifiers, which so far suffer from much higher noise. Here we present the first experimental proof that a simple DC-powered setup allows for amplification close to the quantum limit. Our amplification scheme is based on the stimulated microwave photon emission accompanying inelastic Cooper pair tunneling through a DC-biased Josephson junction, with the key to low noise lying in a well defined auxiliary idler mode, in analogy to parametric amplifiers., Comment: 6 pages, 4 figures

This corrects the article DOI: 10.1103/PhysRevLett.114.126801.

International audience; We show that a properly dc-biased Josephson junction in series with two microwave resonators of different frequencies emits photon pairs in the resonators. By measuring auto- and intercorrelations of the power leaking out of the resonators, we demonstrate two-mode amplitude squeezing below the classical limit. This nonclassical microwave light emission is found to be in quantitative agreement with our theoretical predictions, up to an emission rate of 2 billion photon pairs per second.

International audience; When combining lumped mesoscopic electronic components to form a circuit, quantum fluctuations of electrical quantities lead to a non-linear electromagnetic interaction between the components that is not generally understood. The Landauer-B\"uttiker formalism that is frequently used to describe non-interacting coherent mesoscopic components is not directly suited to describe such circuits since it assumes perfect voltage bias, i.e. the absence of fluctuations. Here, we show that for short coherent conductors of arbitrary transmission, the Landauer-B\"uttiker formalism can be extended to take into account quantum voltage fluctuations similarly to what is done for tunnel junctions. The electrodynamics of the whole circuit is then formally worked out disregarding the non-Gaussianity of fluctuations. This reveals how the aforementioned non-linear interaction operates in short coherent conductors: voltage fluctuations induce a reduction of conductance through the phenomenon of dynamical Coulomb blockade but they also modify their internal density of states leading to an additional electrostatic modification of the transmission. Using this approach we can account quantitatively for conductance measurements performed on Quantum Point Contacts in series with impedances of the order of $R_K = h / e^2$. Our work should enable a better engineering of quantum circuits with targeted properties.

In this paper we present an experiment where we measured the quantum coherence of a quasiparticle injected at a well-defined energy above the Fermi sea into the edge states of the integer quantum Hall regime. Electrons are introduced in an electronic Mach-Zehnder interferometer after passing through a quantum dot that plays the role of an energy filter. Measurements show that above a threshold injection energy, the visibility of the quantum interferences is almost independent of the energy. This is true even for high energies, up to 130~$\mu$eV, well above the thermal energy of the measured sample. This result is in strong contradiction with our theoretical predictions, which instead predict a continuous decrease of the interference visibility with increasing energy. This experiment raises serious questions concerning the understanding of excitations in the integer quantum Hall regime.

We derive fluctuation-dissipation relations for a tunnel junction driven by a high impedance microwave resonator, displaying strong quantum fluctuations. We find that the fluctuation-dissipation relations derived for classical forces hold, provided the effect of the circuit's quantum fluctuations is incorporated into a modified non-linear $I(V)$ curve. We also demonstrate that all quantities measured under a coherent time dependent bias can be reconstructed from their dc counterpart with a photo-assisted tunneling relation. We confirm these predictions by implementing the circuit and measuring the dc current through the junction, its high frequency admittance and its current noise at the frequency of the resonator., Comment: Publisehd as Physical Review Letters, 114, 126801

We present an input-output formalism describing a tunnel junction strongly coupled to its electromagnetic environment. We exploit it in order to investigate the dynamics of the radiation being emitted and scattered by the junction. We find that the non-linearity imprinted in the electronic transport by a properly designed environment generates strongly squeezed radiation. Our results show that the interaction between a quantum conductor and electromagnetic fields can be exploited as a resource to design simple sources of non-classical radiation., Comment: 14 pages, 4 figures, includes Supplementary

We report the efficient coupling of a $50\,\Omega$ microwave circuit to a high impedance conductor. We use an impedance transformer consisting of a $\lambda/4$ co-planar resonator whose inner conductor contains an array of superconducting quantum interference devices (SQUIDs), providing the resonator with a large and tunable lineic inductance $\mathcal{L}\sim 80 \mu_0$, resulting in a large characteristic impedance $Z_C\sim 1\,\mathrm{k}\Omega$. The impedance matching efficiency is characterized by measuring the shot noise power emitted by a dc biased high resistance tunnel junction connected to the resonator. We demonstrate matching to impedances in the $15$ to $35\,\mathrm{k}\Omega$ range with bandwidths above $100\,\mathrm{MHz}$ around a resonant frequency tunable in the $4$ to $6\,\mathrm{GHz}$ range., Comment: Published in Applied Physics Letters

Nonlinear transport in the low-temperature vortex glass state of single crystal ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ has been investigated with fast current pulses driven along the $\mathrm{ab}$ plane. Field cooled preparation shows a higher threshold current (marking a jump or break in slope in the voltage response) than the zero field cooled (ZFC) one, but it is found to be metastable and convertible to the ZFC response by a small field excursion. The metastability appears on the low temperature side of the peak in the temperature dependence of the ZFC threshold current. It is suggested that the onset of metastability signals a different thermodynamic ground state.

We present an original method to measure the visibility of interferences in an electronic Mach–Zehnder interferometer in the presence of low frequency phase fluctuations. We studied the visibility of the interferences as a function of the energy and showed a gaussian variation of the decoherence and/or phase averaging at finite energy. This gaussian variation leads to a visibility modulation with a single side lobe.

We report on direct measurements of the electronic shot noise of a quantum point contact (QPC) at frequencies v in the range 4-8 GHz. Both the thermal energy and the QPC drain-source voltage V are comparable to the photon energy leading to observation of the shot noise suppression when V

International audience; We observe the suppression of the finite frequency shot-noise produced by a voltage biased tunnel junction due to its interaction with a single electromagnetic mode of high impedance. The tunnel junction is embedded in a quarter wavelength resonator containing a dense SQUID array providing it with a characteristic impedance in the kOhms range and a resonant frequency tunable in the 4-6 GHz range. Such high impedance gives rise to a sizeable Coulomb blockade on the tunnel junction (roughly 30% reduction in the differential conductance) and allows an efficient measurement of the spectral density of the current fluctuations at the resonator frequency. The observed blockade of shot-noise is found in agreement with an extension of the dynamical Coulomb blockade theory.

We investigate electron dynamics at the graphene edge by studying the propagation of collective edge magnetoplasmon (EMP) excitations. By timing the travel of narrow wave-packets on picosecond time scales around exfoliated samples, we find chiral propagation with low attenuation at a velocity which is quantized on Hall plateaus. We extract the carrier drift contribution from the EMP propagation and find it to be slightly less than the Fermi velocity, as expected for an abrupt edge. We also extract the characteristic length for Coulomb interaction at the edge and find it to be smaller than for soft, depletion edge systems., Comment: 5 pages, 3 figures of main text and 6 pages, 6 figures of supplemental material

International audience; The on-demand generation of pure quantum excitations is important for the operation of quantum systems, but it is particularly difficult for a system of fermions. This is because any perturbation affects all states below the Fermi energy, resulting in a complex superposition of particle and hole excitations. However, it was predicted nearly 20 years ago1, 2, 3 that a Lorentzian time-dependent potential with quantized flux generates a minimal excitation with only one particle and no hole. Here we report that such quasiparticles (hereafter termed levitons) can be generated on demand in a conductor by applying voltage pulses to a contact. Partitioning the excitations with an electronic beam splitter generates a current noise that we use to measure their number. Minimal-excitation states are observed for Lorentzian pulses, whereas for other pulse shapes there are significant contributions from holes. Further identification of levitons is provided in the energy domain with shot-noise spectroscopy, and in the time domain with electronic Hong–Ou–Mandel noise correlations4, 5, 6, 7, 8. The latter, obtained by colliding synchronized levitons on a beam splitter, exemplifies the potential use of levitons for quantum information: using linear electron quantum optics9 in ballistic conductors, it is possible to imagine flying-qubit10, 11 operation in which the Fermi statistics are exploited12, 13, 14 to entangle synchronized electrons emitted by distinct sources15, 16, 17, 18. Compared with electron sources based on quantum dots19, 20, 21, the generation of levitons does not require delicate nanolithography, considerably simplifying the circuitry for scalability. Levitons are not limited to carrying a single charge, and so in a broader context n-particle levitons could find application in the study of full electron counting statistics22, 23, 24, 25. But they can also carry a fraction of charge if they are implemented in Luttinger liquids3 or in fractional quantum Hall edge channels26; this allows the study of Abelian and non-Abelian quasiparticles in the time domain. Finally, the generation technique could be applied to cold atomic gases27, 28, leading to the possibility of atomic levitons.

We present an experiment where the quantum coherence in the edge states of the integer quantum Hall regime is tuned with a decoupling gate. The coherence length is determined by measuring the visibility of quantum interferences in a Mach-Zehnder interferometer as a function of temperature, in the quantum Hall regime at filling factor two. The temperature dependence of the coherence length can be varied by a factor of two. The strengthening of the phase coherence at finite temperature is shown to arise from a reduction of the coupling between co-propagating edge states. This opens the way for a strong improvement of the phase coherence of Quantum Hall systems. The decoupling gate also allows us to investigate how inter-edge state coupling influence the quantum interferences' dependence on the injection bias. We find that the finite bias visibility can be decomposed into two contributions: a Gaussian envelop which is surprisingly insensitive to the coupling, and a beating component which, on the contrary, is strongly affected by the coupling., Comment: 4 pages, 5 figures

The dynamical Coulomb blockade (DCB) of tunneling is a quantum phenomenon in which tunneling of charge through a small tunnel junction is modified by its electromagnetic environment [1‐4]. This environment is described as an impedance in series with the tunnel element [see Fig. 1(a)]. The sudden charge transfer associated with tunneling can generate photons in the electromagnetic modes of the environment. In a normal metal tunnel junction, biased at voltage V, the energy eV of a tunneling electron can be dissipated both into quasiparticle excitations in the electrodes and into photons. At low temperature energy conservation forbids tunneling processes emitting photons with total energy higher than eV. This suppression reduces the conductance at low bias voltage [1,2,4]. In a Josephson junction, DCB effects are more prominent since at bias voltages smaller than the gap voltage 2� =e quasiparticle excitations cannot take away energy. Therefore, as sketched in Fig. 1(a), the entire energy 2eV of tunneling Cooper pairs has to be transformed into photons in the impedance for a dc current to flow through the junction [3,4]. Experiments have confirmed the predictions of DCB theory for the tunneling current, both in the normal [5‐7] and superconducting case [8,9] but the associated emission of photons into the environment has never been investigated. The aim of this work is to fill this gap by exploring the photonic side of DCB. We do so by embedding a Josephson junction into a well controlled electromagnetic environment provided by a microwave resonator. The resonator in turn leaks photons into an amplifier, allowing us to measure the rate and spectrum of photons emitted by the junction.

International audience; The quantum Hall effect is universal and expected to occur in all two-dimensional electron systems in perpendicular high-magnetic field. We revisit quantum localization thanks to the high-energy scale of the quantum Hall effect in graphene, where the electron dynamics obey the Dirac equation. We solve a long debated question on the nature of electron transport in the transition region between Hall resistance plateaus. Is it of metallic or of variable-range hopping type as proposed, respectively, by Pruisken and Polyakov-Shklovskii? To the best of our knowledge, no experiment was able to discriminate between these models. Here, measurements of the conductance peak width scaling exponents with both temperature and current and determination of the localization length validates the variable-range hopping scenario. This shows that the usual assumption of a metallic behavior of the two-dimensional electron gas (2DEG) between Hall resistance plateaus is unnecessary for macroscopic samples.

We report on conductance measurements in carbon nanotube based double quantum dots connected to two normal electrodes and a central superconducting finger. By operating our devices as beam splitters, we provide evidence for Crossed Andreev Reflections \textit{tunable in situ}. This opens an avenue to more sophisticated quantum optics-like experiments with spin entangled electrons., submitted

We report on a microwave Hanbury-Brown Twiss experiment probing the statistics of GHz photons emitted by a tunnel junction in the shot noise regime at low temperature. By measuring the crosscorrelated fluctuations of the occupation numbers of the photon modes of both detection branches we show that, while the statistics of electrons is Poissonian, the photons obey chaotic statistics. This is observed even for low photon occupation number when the voltage across the junction is close to $h\nu/e$., Comment: Submitted to Phys.Rev.Lett

The decoupled layer quasi-two-dimensional glass phase of c-axis vortices in optimally doped Bi2Sr2CaCu2O8+delta (BSCCO) exhibits low-temperature metastability, the onset temperature of which is coincident with the peak in depinning current for zero-field-cooled sample preparation. We present experiments in strongly underdoped BSCCO showing similar metastability and develop a model for pinning dynamics where metastability results from progressive loss of thermal contact. The model accounts well for the field-temperature locus of the metastability and for the irreversibility and memory effects. We conclude that the metastability arises from a nonsingular dynamic crossover and does not mark a phase transition.

We present an experiment where we tune the decoherence in a quantum interferometer using one of the simplest objects available in the physics of quantum conductors: an Ohmic contact. For that purpose, we designed an electronic Mach-Zehnder interferometer which has one of its two arms connected to an Ohmic contact through a quantum point contact. At low temperature, we observe quantum interference patterns with a visibility up to 57%. Increasing the connection between one arm of the interferometer to the floating Ohmic contact, the voltage probe, reduces quantum interference as it probes the electron trajectory. This unique experimental realization of a voltage probe works as a trivial which-path detector whose efficiency can be simply tuned by a gate voltage.

An electronic Mach Zehnder interferometer is used in the integer quantum hall regime at filling factor 2, to study the dephasing of the interferences. This is found to be induced by the electrical noise existing in the edge states capacitively coupled to each others. Electrical shot noise created in one channel leads to phase randomization in the other, which destroys the interference pattern. These findings are extended to the dephasing induced by thermal noise instead of shot noise: it explains the underlying mechanism responsible for the finite temperature coherence time $\tau_\phi(T)$ of the edge states at filling factor 2, measured in a recent experiment. Finally, we present here a theory of the dephasing based on Gaussian noise, which is found in excellent agreement with our experimental results., Comment: ~4 pages, 4 figures

We report on direct measurements of the electronic shot noise of a quantum point contact at frequencies nu in the range 4-8 GHz. The very small energy scale used ensures energy independent transmissions of the few transmitted electronic modes and their accurate knowledge. Both the thermal energy and the quantum point contact drain-source voltage Vds are comparable to the photon energy hnu leading to observation of the shot noise suppression when $V_{ds}, Comment: Version Published in Phys. Rev. Lett. (Phys. Rev. Lett. 99, 236803 (2007))

We have determined the finite temperature coherence length of edge states in the Integer Quantum Hall Effect (IQHE) regime. This was realized by measuring the visibility of electronic Mach-Zehnder interferometers of different sizes, at filling factor 2. The visibility shows an exponential decay with the temperature. The characteristic temperature scale is found inversely proportional to the length of the interferometer arm, allowing to define a coherence length $\l_\phi$. The variations of $\l_\phi$ with magnetic field are the same for all samples, with a maximum located at the upper end of the quantum hall plateau. Our results provide the first accurate determination of $\l_\phi$ in the quantum Hall regime., Comment: 4 pages, 4 figures

We present an original statistical method to measure the visibility of interferences in an electronic Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents a single side lobe structure shown to result from a gaussian phase averaging whose variance is quadratic with the bias. To reinforce our approach and validate our statistical method, the same experiment is also realized with a stable sample. It exhibits the same visibility behavior as the fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples, the dilution of the impinging current reduces the variance of the gaussian distribution., Comment: 4 pages, 5 figures

Experiments on potential differences in the low-temperature vortex solid phase of monocrystalline platelets of superconducting ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ subjected to currents driven either through an $\mathrm{ab}$ surface or from one such surface to another show evidence of a resistive/nonresistive front moving progressively out from the current contacts as the current increases. The depth of the resistive region has been measured by an in-depth voltage probe contact. The position of the front associated with an injection point appears to depend only on the current magnitude and not on its withdrawal point. It is argued that enhanced nonresistive superconducting anisotropy limits current penetration to depths less than the London length and results in a flat rectangular resistive region with simultaneous $\mathrm{ab}$ and c current breakdown which moves progressively out from the injection point with increasing current. Measurements in $\mathrm{ab}$ or c configurations are seen to give the same information, involving both $\mathrm{ab}$-plane and c-axis conduction properties.

Nonlinear charge transport in the low temperature vortex glass state of single crystal Bi 2 Sr 2 CaCu 2 O 8 superconductor has been investigated with fast current pulses driven along the ab plane. Field cooled (FC) preparation shows a higher threshold current than zero field cooled (ZFC) one, but is found to be metastable and convertible to the ZFC response by a small field excursion. The metastability appears on the low temperature side of the peak in the temperature dependence of the ZFC threshold current. It is suggested that the onset of metastability signals a new thermodynamic ground state.

The probabilistic charge transfer in tunnel junctions is a source of current noise, which can be picked up by the environment of the junction where it creates electromagnetic excitations - or photons. In this thesis, we demonstrate that a superconducting tunnel junction coupled to a tailored environment can act as a bright source of quantum microwave radiation. The energy required to create photons is extracted from a DC voltage source during the inelastic tunneling of Cooper pair through the junction. We detect this emitted light and study its properties with quantum optics tools adapted to the microwave domain. We characterize the singlephoton nature of the light emitted in a single mode strongly coupled to a junction, as well as the entanglement of photon pairs emitted in two modes at different frequencies. In addition to a better understanding of the charge-light coupling in coherent conductors, this work could lead to new ways of manipulating information at the quantum level., Le passage aléatoire des charges à travers une jonction tunnel génère un bruit en courant, qui peut être capté par l'environnement de la jonction où il crée des excitations électromagnétiques appelés photons. Dans cette thèse, nous démontrons qu'une jonction tunnel supraconductrice couplée à un environnement bien choisi peut implémenter une source brillante de rayonnement micro-onde quantique. L'énergie nécessaire pour créer les photons est fournie par la source de tension continue lors du passage tunnel inélastique d’une paire de Cooper à travers la jonction. Nous détectons cette lumière émise et étudions ses propriétés avec des outils de l’optique quantique adaptés au domaine micro-onde. Nous caractérisons le dégroupement des photons émis dans un seul mode fortement couplée à une jonction, ainsi que l’intrication de paires de photons émis dans deux modes à des fréquences différentes. Outre une meilleure compréhension du couplage charge-lumière dans les conducteurs cohérents, ces travaux pourraient déboucher sur de nouvelles façons de manipuler l'information au niveau quantique.