List of talks

Cavity quantum electrodynamics, the field that deals with the interaction of atoms and light quanta in optical cavities, has paved the route towards a deeper understanding of fundamental quantum phenomena. Its more recent realization is circuit-QED, where atoms are replaced by `articial atoms', particularly designed (often superconducting) two-level systems, in microwave cavities. The theory of quantum electrodynamics, however, has a much broader range and implies the interaction of fermionic with bosonic matter in general. In solid state physics, one of the most interesting processes is the transfer of charges due to external voltage sources. In this field of quantum electronics fascinating progress has been achieved as well in the last decades with accurate control down to the level of individual charge carriers. Activities to combine these two previously basically distinct fields, circuit-QED and quantum electronics, have appeared only very recently. Particularly, superconducting circuits open a new playground to study a wealth of phenomena close and far from thermal equilibrium including quantum-classical crossovers, Coulomb blockade, nonlinear resonances, and non-classical microwave sources with squeezed or entangled photons. In this talk I will discuss recent theoretical developments in the context of Josephson photonics.

Embedding a voltage-biased Josephson junction within a high-Q superconducting microwave cavity provides a novel way of exploring strongly non-linear quantum dynamics. At resonances where the energy given to a tunnelling Cooper pair by the voltage bias is equal to a multiple of the cavity photon frequency the cavity can be pumped to far-from-equilibrium states containing many microwave photons. Intriguingly, the cavity states produced by the flow of Cooper-pairs have distinct non-classical features. In this talk I will describe a simple theoretical model for this type of system and outline how the coupled dynamics of the cavity photons and Cooper-pairs can be uncovered using a combination of analytical approximations and numerical methods.

We consider a Josephson junction formed by a quantum dot connected to two bulk superconductors in the presence of Coulomb interaction and coupling to both an electromagnetic environment and a finite density of electronic quasiparticles. In the limit of a large superconducting gap we obtain a Born-Markov description of the relevant Andreev bound-states dynamics [1]. We calculate the current-phase relation and we find that the experimentally unavoidable presence of quasiparticles can dramatically modify the 0-pi standard transition picture [1]. We show that photon-assisted quasiparticle absorption allows the dynamic switching from the 0 to the pi state and vice versa, washing out the 0-pi transition predicted by purely thermodynamic arguments. Spectroscopic signatures of Andreev bound-states broadening are investigated by considering microwave irradiation.

Reference :
[1] R. Avriller and F. Pistolesi, Phys. Rev. Lett. 114, 037003 (2015).

Using low-temperature scanning tunneling microscopy the electron transport through atomic-scale contacts is investigated from the tunnelling range to ballistic transport.  The talk will address the light emission from single-atom and single-molecule contacts as well as electrical measurements of the shot noise of the current.

We study the interplay of geometric frustration and interactions in a non-equilibrium photonic lattice system exhibiting a polariton flat band as described by a variant of the Jaynes-Cummings-Hubbard model. We develop a semi-analytic projective method and employ an open system version of the time-evolving block decimation algorithm (TEBD) in order to calculate the non-equilibrium steady state of the array subject to drive and dissipation. We find that frustration strongly enhances photon repulsion in a flat band leading to an incompressible state of light. The latter manifests itself in strong spatial correlations, i.e., on-site and nearest-neighbor anti-bunching combined with extended density-wave oscillations at larger distances. We propose a state-of-the-art circuit QED realization of our system, which is tunable in-situ.

A mesoscopic hybrid Normal/ Superconducting (NS) ring is characterized by an Andreev spectrum with a phase dependent minigap. We have implemented a high frequency phase bias to an NS ring coupled to a multimode superconducting resonator. These experiments reveal the dynamics of a Josephson SNS close to equilibrium in a wide frequency range which cannot be accessed in current or voltage biased experiments. We find that the current response contains, beside the well known dissipation-less Josephson contribution, a large dissipative component. At high frequency compared to the minigap and low temperature we find that the dissipation is due to transitions above the minigap. In contrast, at lower frequency there is range of temperature for which dissipation is caused predominantly by the relaxation of the Andreev states' population. This dissipative response, related via the fluctuation dissipation theorem to a non intuitive zero frequency thermal noise of supercurrent, is characterized by a phase dependence dominated by its second harmonic. It can be interpreted in terms of an effective dissipative conductance much larger than the classical Drude one.

Understanding the flow of information is crucial in the analysis of feedback control schemes [1], e.g. in Maxwell demon type models with  bi-partite structure [2] (controller and the process to be controlled). Other recently discussed models use information tapes (somewhat similar to Turing machines) [3], or are defined on networks [4]. 

Here, I propose a model where the feedback is achieved via the mutual interaction among many individual stochastic processes that each describe, e.g., transport of charges through nanostructures such as quantum dots.   In its simplest version, N transport channels are coupled by interactions  that arise through charge interactions like in a network of capacitors. These interactions make  the individual transitions rates dependent upon the channel (reservoir) state of the entire system. 

I start from an infinite system of rate equations for the Full Counting Statistics (FCS) that can be derived in a Nakajima-Zwanzig-type fashion [5] and by adiabatically eliminating the  connections (quantum dots) between source and sink reservoirs.  This being still too complicated for a transparent analysis, I proceed by deriving a Fokker--Planck equation that corresponds to a Gaussian approximation of the FCS but still captures the essential physics.
 
The first observation is the reduction of the width of the charge fluctuations in the individual channels by a factor 1/N. For large N, this effectively leads to a freezing of the  FCS very much in the same way as in a previous model [6], where the feedback was achieved not via mutual interactions but in a non-autonomous manner by synchronization with an external clock. Next, for large N and nearest neighbour interactions among the channels, the spreading of the feedback signal becomes itself a diffusive process  akin to the discussion of scale invariance in the dynmaics of surfaces by Kardar [7]. Depending on the dimensionality of the  feedback channel 'lattice', one can then extract various temporal  regimes for the feedback spreading.
 
The last part concerns the role of information and entropy in this collective feedback process. For identical channels, I can extract analytical expressions for the transient entropy flow and the information current, which describes the dynamics of the FCS freezing at large N. For N=2, there is a close analogy with previous results for two coupled Brownian oscillators with the possibility of stationary information flows [8]. 
For infinitely long range interactions, the information current vanishes for N→∞, since the notion of 'boundary' between subsystems becomes meaningless and the difference between local and (scaled) global entropy vanishes.

[1] J. M. Horowitz, M. Esposito, Phys. Rev. X 4, 031015 (2014);  D. Hartich, A. C. Barato, U. Seifert, J. Stat. Mech. P02016 (2014). 
[2] P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Phys. Rev. Lett. 110, 040601 (2013).
[3] P. Strasberg, G. Schaller, T. Brandes, and C. Jarzynski, Phys. Rev. E 90, 062107 (2014).
[4] S. Ito, T. Sagawa,  Phys. Rev. Lett. 111, 180603 (2013). 
[5] M. Schubotz and T. Brandes, Phys. Rev. B 84, 075340 (2011).
[6] T. Brandes, Phys. Rev. Lett. 105, 060602 (2010).
[7] M. Kardar, 'Statistical Physics of Fields', Cambridge University Press (2007).  
[8] A. E. Allahverdyan, D. Janzing, G. Mahler, J. Stat. Mech. P09011 (2009).

Distant entangled electrons would be of great interest in the context of quantum information. It has been suggested to make use of natural spin entanglement in superconductors by splitting a Cooper pair into two different electronic orbitals. It is possible to implement such a Cooper pair splitter in a carbon nanotube based double quantum dot architecture [1]. 
Conductance measurements demonstrated splitting of Cooper pairs but were not able to probe their coherence. 
I will present simultaneous transport and microwave measurement of a Cooper pair splitter coupled to a cavity. We aim to access the quantum coherence of the split pair by performing the spectroscopy of the device coupled to the microwave cavity using a two tone measurement scheme [2, 3].

References
[1] L.G. Herrmann et al., PRL 104, 026801 (2010)
[2] A. Cottet, T. Kontos, and A. Levy Yeyati, PRL 108, 166803 (2012)
[3] A. Cottet, PRB 90, 125139 (2014)

to be announced

In this work we present a theoretical study of a hybrid circuit-QED system in which two non-interacting charge qubits are coupled to a common bosonic mode in a microwave resonator [1]. The qubits are defined in terms of the charge states of two spatially separated double-quantum dots (DQDs). In particular, we analyze a transport setup for the qubits and show that the inelastic current across each DQD reflects an indirect qubit-qubit interaction mediated by off-resonant photons in the resonator. Furthermore, as a result of this interaction, both charge qubits are entangled in the steady state. We demonstrate that it is possible to extract information about this nontrivial steady-state entanglement from the shot noise cross-correlations between currents across distant DQDs.
To fully characterize and understand these effects, the time evolution of the hybrid system is also analyzed and we explore the emergence of non-Markovianity in the dynamics of the qubits [2].

[1]. L.D. Contreras-Pulido et al., New J. Phys. 15, 095008 (2013)
[2]. L.D. Contreras-Pulido et al., In progress.

    A new type of experiments combining microwave cavities and mesoscopic circuits gathering nanoconductors and fermionic reservoirs has recently appeared [1,2,3]. This mesoscopic Quantum Electrodynamics (QED) offers many new possibilities like for instance quantum computing schemes based on localized electronic spins, or a powerful photonic study of electronic transport. In this talk, I will introduce a general theoretical framework to describe these experiments and predict new ones [4]. This task faces two challenges. First, one has to quantize the electromagnetic field properly by taking into account electromagnetic boundary conditions which are naturally omitted in atomic cavity QED, due to the smallness of an atom. Second, in the nanocircuits, one has to take into account collective plasmonic modes, as well as individual electronic states which are absent from circuit QED performed with superconducting quantum bits. I will discuss an effective model where these two types of modes are physically separated. This leads to a description which involves a linear coupling between tunneling electrons and a photonic pseudo-potential. I will discuss in details the case of a superconducting nanostructure enclosing Majorana quasiparticles directly coupled to a microwave cavity. The cavity provides a powerful means to test the properties of these exotic quasiparticles, and in particular their self-adjoint character [5,6].
    [1] M.R. Delbecq et al, Phys. Rev. Lett. 107, 256804 (2011).
    [2] T. Frey et al, Phys. Rev. Lett. 108 046807 (2012).
    [3] K. D. Petersson et al., Nature 490, 380 (2012) 
    [4] A. Cottet, T. Kontos and B. Douçot, arXiv:1501.00803
    [5] A. Cottet, T. Kontos and B. Douçot, Phys. Rev. B 88, 195415 (2013).
    [6] A. Cottet et al, unpublished

We present a new type of mechanical quantum device, where propagating surface acoustic wave (SAW) phonons serve as carriers for quantum information. At the core of our device is a superconducting qubit, designed to couple to SAW waves in the underlying substrate through the piezoelectric effect. This type of coupling can be very strong, and in our case exceeds the coupling to any external electromagnetic mode. The SAW waves propagate freely on the surface of the substrate, and we use a remote electro-acoustic transducer to address the qubit acoustically. Three different experiments are presented:
i)	Exciting the qubit with an electromagnetic signal we can “listen” to the SAW phonons emitted by the qubit. The low speed of sound also allows us to observe the emission of the qubit in the time domain, which gives clear proof that the dominant coupling is acoustic.
ii)	Reflecting a SAW wave off the qubit, we observe a nonlinear reflection with strong reflection at low power and low reflection at high power.
iii)	Exciting the qubit with both an electromagnetic signal and with a SAW signal, we can do two tone spectroscopy on the qubit
In all of these experiments we find a good agreement between experiment and theory.
In comparison to photons, SAW phonons have several striking features. Their speed of propagation is around 105 times lower, and the wavelength at a given frequency correspondingly shorter, indeed very similar to the wavelength of visible light. The slow speed means that qubits can be tuned much faster than SAWs traverse inter-qubit distances on a chip, which can enable new dynamic schemes for trapping and processing quanta. SAW phonons furthermore give access to a regime where the size of the qubit substantially exceeds the wavelength of the quanta it interacts with, opposite to the point-like interaction in photonic systems.

A superconducting qubit coupled to an open transmission line represents an implementation of the spin-boson model with a broadband environment. We show that this environment can be engineered by introducing partial reflectors into the transmission line, allowing one to shape the spectral function, 𝐽(ω), of the spin-boson model [1]. The spectral function can be accessed by measuring the resonance fluorescence of the qubit, which provides information on both the engineered environment and the coupling between qubit and transmission line. The spectral function of a transmission line without partial reflectors is found to be Ohmic over a wide frequency range, whereas a peaked spectral density is found for the shaped environment. Our work lays the ground for future quantum simulations of other, more involved, impurity models with superconducting circuits.
We acknowledge support by the German Research Foundation through SFB 631, the EU project PROMISCE, and the Elite Network of Bavaria through the program ExQM.
[1] M. Haeberlein et al., manuscript in preparation (2015).

We predict a dynamical bistability in the photon emission from a microwave cavity coupled to the electronic transport of a nearby double quantum dot. The switching rates of the bistability can be extracted from the average electrical current and the shot noise in the quantum dots. Using large-deviation techniques we demonstrate that the full counting statistics of emitted photons is captured by the universal shape of a tilted ellipse whose form can be controlled by modulating the electronic transport in the quantum dots. Our prediction is robust against moderate electronic decoherence and dephasing due to phonons and may be tested using existing experimental setups.

I will discuss a spectroscopy based on embedding a superconducting constriction, formed by a single-level molecule junction, in a microwave QED cavity environment. In the electron-dressed cavity spectrum we find a polariton excitation at twice the Andreev bound state energy, and a superconducting-phase-dependent ac Stark shift of the cavity frequency. Dispersive measurement of this frequency shift can be used for Andreev bound state spectroscopy.

Weak links between superconductors host localized quasiparticle states, the “Andreev bound states”. In contrast with quasiparticle states in bulk superconductors, the Andreev states are discrete and are expected to be long-lived. As a consequence, the minimal energy photonic excitation at a weak link should not be seen as a “broken Copper pair”, but as an excited pair which remains coherent. 
In order to illustrate these concepts, I will present experiments performed on the simplest weak link: a single-atom contact between superconductors, which contains only a few Andreev states. This contact is coupled to a coplanar resonator, and time-domain manipulation of Andreev states is performed.

Carbon nanotubes (CNTs) are extremely versatile quantum conductors: depending on the coupling strength between leads and contacts various transport regimes can be reached. For strong coupling multiple reflections at the interface give rise to Fabry-Perot interference patterns in the current voltage characteristics, whose fine structure is determined by spin-orbit and nonlinear effects.  For intermediate coupling interaction effects give rise to Kondo assisted transport. Again the spin-orbit coupling and chirality strongly affect the fine-structure of the Kondo resonance. Novel selection rules for Kondo-enahnced transport in 
the non-perturbative regime are discussed.

The zero-point energy stored in the modes of an electromagnetic cavity has experimentally detectable effects, giving rise even to an attractive interaction between the opposite walls, the static Casimir effect. A dynamic version of this effect, called the dynamical Casimir effect, occurs when the vacuum energy is modulated either by moving the walls of the cavity or by changing the index of refraction, resulting in the conversion of vacuum fluctuations into real photons. 
We have established the dynamical Casimir effect using a Josephson metamaterial embedded in a microwave cavity resonant at 5.4 GHz.  In our experiment, we modulate the effective length of the cavity by flux-biasing the SQUID-based metamaterial, which results in a few percent variation in the speed of light. We extract the full 4 x 4 covariance matrix of the emitted microwave radiation, demonstrating that photons at frequencies symmetric with respect to half the modulation frequency are generated in pairs. At large detunings of the cavity from half the modulation frequency, we find bimodal power spectra that show clearly the theoretically-predicted hallmarks of the Casimir effect.  
Recently, we have explored dynamical Casimir effect under the influence of two external pumps (ω- and ω+). In such a “double dynamical Casimir effect”, specific correlations emerge in the cavity even when the generation of Casimir photons is small and the average power in the cavity is at the single quantum level. In particular, in the lowest order one can describe this state as a tripartite state with squeezing correlations between neighboring frequency points (ω1 + ω2 = ω- or ω+) and beam-splitter correlations between the extremal frequency points (ω1 + ω2 = ω-/2 + ω+/2). This, in fact, results in the quenching of the amplified vacuum fluctuations in the orthogonal superposition of the extremal modes, which can be regarded as a parametric dark state. These phase-sensitive effects can be fully controlled by the relative phase of the pumps. Using pulsing of the pumps, we can access the analog of spatial which-way-information in the frequency space.  

A crucial prerequisite for realizing quantum information processing are qubits which are protected against the occurrence of errors to the best possible degree. Here, protecting qubits via topological properties of the employed physical systems has become a very actively pursued route in recent years. A breakthrough that significantly triggered this research direction was the discovery of the Toric Code by Kitaev. His approach considers a two-dimensional spin lattice with quasi local four-body interactions that, on a torus, features 4 degenerate topologically protected states which can be employed as  logical qubits. Yet despite the enormous interest in this model, a physical implementation is still absent.

In this talk I will present a scheme for realising the Toric Code Hamiltonain by  generating four-body interactions between superconducting qubits via a coupling circuits containing a driven SQUID. Since superconducting circuit technology permits manipulations and measurements of individual qubits as well as measurements of correlations between several qubits, the scheme allows for experimental verifications of its topological properties. A minimal realisation requires a lattice of 8 qubits and is thus in reach with existing technology.

We have recently shown that a simple dc voltage-bias on a small Josephson junction leads to microwave emission via inelastic Cooper-pair tunneling [2]. In this process a tunneling Cooper pair emits one or several microwave photons with a total energy of 2eV. The observed average photon emission rate is well explained within the so-called P(E) theory, but this theory does not make any predictions about the statistics of the emitted photons.

I will show experiments showing that the statistics of the emitted photons can be highly nontrivial, in agreement with recent theory [2-5]: Depending on the bias conditions and the impedance of the circuit in which the junction is embedded, we observe statistics ranging from super-bunched to strongly anti-bunched.

This type of devices might therefore offer a new way of generating useful photon states for circuit quantum optics experiments, without the need of carefully calibrated control pulses.

[1] M. Hofheinz et al., PRL {bf 106}, 217005 (2011)
[2] C. Padurariu et al., PRB {bf 86}, 054514 (2012)
[3] V. Gramich et al., PRL {bf 111}, 247002 (2013)
[4] J. Leppäkangas et al., NJP {bf 16}, 015015 (2014)
[5] J. Leppäkangas, submitted

Understanding tunneling from an atomically sharp tip to a metallic surface requires us to account for interactions on a nanoscopic scale. Inelastic tunneling of electrons generates emission of photons, whose energies intuitively should be limited by the applied bias voltage. However, experiments [1] indicate that more complex processes involving the interaction of electrons with plasmon polaritons lead to photon emission characterized by overbias energies. We propose a model of this observation in analogy to the dynamical Coulomb blockade [2], originally developed for treating the electronic environment in mesoscopic circuits. We explain the experimental finding quantitatively by the correlated tunneling of two electrons interacting with a LRC circuit modeling the local plasmon-polariton mode. To explain the overbias emission, the non-Gaussian statistics of the tunneling dynamics of the electrons is essential.

[1] G. Schull et al., Phys. Rev. Lett. 102, 057401 (2009).
[2] F. Xu, C. Holmqvist, and W. Belzig, Phys. Rev. Lett. 113, 066801 (2014).

The mechanical quality factor of a carbon nanotube nano-electromechanical resonator can rise above 105 in vacuum at cryogenic temperatures. At the same time, the 
electronically nonlinear behaviour of the quantum dot forming inside the carbon nanotube enables straightforward detection of the mechanical motion.

This presents a rich system where single-electron tunneling directly couples to and influences mechanical motion. A dc voltage alone is sufficient to excite vibration
via feedback effects. In turn, vibrations can be suppressed with a magnetic field. Tuning of the mechanical frequency takes place both via mechanical tension and strong electrostatic softening of the vibration mode.

In addition, the embedded quantum dot provides a clean quantum-mechanical few-carrier system. Mechanical effects can be used to probe its properties without a separate detector circuit- all the way from strong Coulomb blockade via the Kondo regime to the loss of charge quantization in a Fabry-Perot like system.

I'll briefly discuss the use of scattering theory for microwave photons in connection with mesoscopic charge transport.

Light emission from plasmonic contacts, i.e. nanoscale contacts supporting localized surface-plasmon polaritons, is studied within the framework of the nonequilibrium Green's function formalism. A fundamental link between the finite-frequency quantum noise and ac conductance of the contact and the light emission is established. Calculating the quantum noise to higher orders in the electron-plasmon interaction, we identify a plasmon-induced electron-electron interaction as the source of experimentally observed above-threshold light emission from biased STM contacts.

Electron spins and photons are complementary quantum mechanical objects which can be used to carry, manipulate and transform quantum information. Combining them into a scalable architecture is an outstanding challenge. In this context, the coherent coupling of a single spin to photons stored in a superconducting resonator is an essential milestone. Using a circuit design based on a nanoscale spin-valve, we coherently hybridize the individual spin and charge states of a double quantum dot while preserving spin coherence. This scheme allows us to increase by five orders of magnitude the natural (magnetic) spin-photon coupling, up to the MHz range at the single spin level. Our coupling strength yields a cooperativity which reaches 2.3, with a spin coherence time of about 60ns. We thereby demonstrate a mesoscopic device suitable for non-destructive spin read-out and distant spin coupling.

Surface acoustic waves (SAWs) are mechanical modes that travel along the surface of a piezoelectric crystal, are widely used in radio and microwave frequency signal processing, and have recently received attention as components integrated into superconducting devices [1]. In this talk, I will report on measurements of SAW resonators in the frequency range 0.5-5 GHz on quartz and ZnO [2], fabricated with superconducting electrodes and measured at cryogenic temperatures in the quantum regime, at which quality factors in the range Q > 104 are consistently realised. Our results indicate that strong coupling circuit QED should be reachable using SAW phonons and superconducting qubits.

[1] MV Gustafsson et al., Science 346, 207 (2014)
[2] EB Magnusson et al., APL 106, 063509 (2015)

We demonstrate theoretically that charge transport across a Josephson junction, voltage-biased through a resistive environment,  produces anti-bunched photons. We develop a continuous-mode description of the emitted radiation field in a semi-infinite transmission line terminated by the Josephson junction. Within a perturbative treatment in powers of the tunnelling coupling across the Josephson junction, we capture effects originating in charging dynamics of consecutively tunnelling Cooper pairs. We find that within a feasible experimental setup, the Coulomb-blockade provided by high zero-frequency impedance, can be used to create anti-bunched photons at a very high rate.

If we consider a charged particle on a small island as the degree of freedom, we have to take into account the strong coupling to the electromagnetic field. This allows for strong coupling between the states of the island and the photons of a transmission-line resonator, but it also generates strong coupling to unwanted environmental degrees of freedom. We study a model which can describe a superconducting single electron transistor (SSET) or a double quantum dot coupled to transmission-line oscillator. We consider the case where a lasing condition is established and study the dependence of the average photon number in the resonator on the spectral function of the electromagnetic environment. Three important cases are considered: a strongly coupled environment with a small cut-off frequency, a structured environment peaked at a specific frequency and 1/f noise.  We find that the electromagnetic environment can have a substantial impact on the photon creation. Resonance peaks are in general broadened and additional resonances can appear. While in general strong coupling to an environment decreases the number of photons  which are created, we discuss a special case where a condition of inversionless lasing can be established. In this case the environment can effectively be an advantage. 

We study dissipative squeezing generated in a microwave resonator by coupling it to a mesoscopic device such as a tunnel junction or a quantum dot. The  conductor is biased by an AC voltage oscillating at twice the resonator frequency. Photon-assisted tunneling of electrons is accompanied by the emission of pairs of photons in the cavity, thereby engineering a driven squeezed state. For a tunnel junction, we show that squeezing is minimized by a pulse shape consisting of a periodic series of delta peaks. Squeezing is generally enhanced by non-linearities and an energy-selective electronic transmission. We indeed find that perfect vacuum squeezing can be produced in an asymmetric quantum dot in the presence of a particular periodic Leviton pulse.

Motivated by recent experiments, we study theoretically the full counting statistics of radiation emitted in a driven Josephson junction circuit. In contrast to most optical systems, a significant part of emitted radiation can be collected and converted to an output signal. This permits studying the correlations of the radiation as well as its statistics. 

I will present a theory describing the recently proposed and realized microwave amplifier based on the negative resistance of a selectively damped Josephson junction [P. Lahteenmaki et al., Sci. Rep. 2, 276 (2012)]. The standard linear theory using the Gaussian expansion around the limit cycle [A. Kamal et al. Phys. Rev. B 86, 144510 (2012)] does yield nearly perfect results for the gain characterisrtics of the device, but it completely fails for the noise. I show that an extended theory accounting for a subtle interplay between the linear response and nonlinear dynamics of the phase along the limit cycle can explain also the noise-temperature experimental data. Implications of these findings on the prospects of achieving the originally intended quantum-limited amplification will be discussed. 

We revisit the design and fabrication of the persistent-current flux qubit [1]. By adding a high-Q capacitor, we dramatically improve its reproducibility and coherence times while retaining 800 MHz anharmonicity in the longest lived devices [2]. We discuss in a detail a device with T1 = 55 us. We identify quasiparticles as causing temporal variability in the T1, and we demonstrate the ability to pump these quasiparticles away to stabilize and improve T1 [3]. The Hahn echo time T2E = 40 us does not reach the 2T1 limit, as is often observed with transmons coupled to resonators. We demonstrate that this is due to dephasing caused by the shot noise of residual thermal photons in the readout resonator. We use noise spectroscopy techniques to measure the lorentzian noise spectrum of the photon noise, and we then use CPMG dynamical decoupling to recover T2CPMG ~ 2T1 in a manner consistent with the noise spectrum.

[1] W.D. Oliver and P.B. Welander, MRS Bulletin, 38, 816 (2013)
[2] F. Yan et al., in preparation (2015).
[3] S. Gustavsson et al., in preparation (2015).

Recent progress in the field of optomechanics may soon allow the realization of optomechanical arrays, i.e. periodic arrangements of optical and vibrational modes whose interaction can be tuned in-situ by a laser drive. Possible implementations range from the optical to the microwave domain. The flow of light and sound in such  devices could be tailored by engineering the coherent drive, e. g. creating effective potential landscapes, tuning the phonon hopping range, or creating artificial gauge fields. Even the topology of the optomechanical band structure could be tuned in-situ. In particular, one could create for the first time a Chern insulator of phonons. The resulting chiral, topologically protected phonon transport can be probed completely optically. 

Pioneering experiments by Fujisawa et al. in the late 1990’s examined the detuning dependence of the current in semiconductor double quantum dots (DQD) and highlighted the important role of electron-phonon coupling in inelastic transport.[1] Later experiments by the same group directly measured orbital relaxations rates, which were consistent with a phonon-mediated relaxation process.[2] By placing semiconductor DQDs inside of high quality factor microwave cavities it is now possible to achieve charge-cavity coupling rates g ~ 10 to 100 MHz.[3] These rates are comparable to the phonon emission rate.
I will describe recent experiments that examine photoemission and masing in cavity-coupled semiconductor DQDs. The application of a source-drain bias results in population inversion and single electron tunneling. The interdot tunneling process generates photons and leads to gain in the cavity transmission.[4] We measure the detuning dependence of the gain and find that the gain feature is very broad compared to the cavity linewidth. Recent theory accounts for the broad gain feature by considering a second order process that involves the simultaneous emission of a cavity photon and a phonon.[5] With sufficient cavity pumping, it is feasible to achieve above-threshold maser action, which is verified by comparing the statistics of the emitted microwave field above and below the maser threshold.[6]
1. T. Fujisawa et al., Science 282, 932 (1998).
2. T. Fujisawa et al., Nature 419, 278 (2002).
3. T. Frey et al., Phys. Rev. Lett. 108, 046807 (2012).
4. Y.-Y. Liu et al., Phys. Rev. Lett. 113, 036801 (2014).
5. M. J. Gullans et al., Phys. Rev. Lett. 114, 196802 (2015).
6. Y.-Y. Liu et al., Science 347, 285 (2015).
Research was performed in collaboration with Yinyu Liu, Jiri Stehlik, Christopher Eichler, Michael Gullans, and Jacob Taylor. We acknowledge support from the Sloan and Packard Foundations, ARO, DARPA, and the NSF.

Some operations on quantum states are not restricted by the Heisenberg uncertainty principle. One example is teleportation, which allows for both the position and the momentum be transferred without added noise [1]. Perhaps even more surprisingly, a trajectory of an oscillator can be measured with an accuracy exceeding the Heisenberg uncertainty following the approach we have recently developed [2, 3]. The key feature is to monitor the oscillator trajectory in a quantized reference frame with a negative mass with which the oscillator is entangled. In the talk I will first present a oscillator with a negative mass and report the results of tracing the magnetic spin oscillator trajectory beyond the Heisenberg uncertainty.  I will then describe progress towards tracing a trajectory of a mechanical oscillator with the precision not restricted by the Heisenberg uncertainty principle. Finally, I will outline perspectives for performing similar operations with an electrical oscillator [4].

1.	H. Kraute et al. Nature Phys., 9, 400 (2013).
2.	E.S. Polzik and K. Hammerer, Ann. Phys. (Berlin) 527, No. 1–2, A15–A20 (2015);  W. Wasilewski et al. Phys. Rev. Lett., 104, 133601 (2010). 
3.	K. Hammerer, M. Aspelmeyer, E.S. Polzik, P. Zoller.  Phys. Rev. Lett. 102, 020501 (2009).
4.	T. Bagci et al. Nature 507, 81–85 (2014).

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]. The sudden charge transfer associated with tunneling generates 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, which reduces the conductance at low bias voltage [1]. In a Josephson junction, DCB effects are more prominent since at bias voltages smaller than the gap voltage 2Δ⁄e quasiparticle excitations cannot take away energy and the entire energy of tunneling Cooper pairs has to be transformed into photons in the impedance for a dc current to flow through the junction [1]. We have recently observed and characterized the radiation associated to the flow of Cooper pairs connected to a microwave resonator [2]. Our new experiment can be schematically represented as follows: a Josephson junction is placed between two microwave resonators with different resonant frequencies, ν1 and ν2, and is voltage biased at voltage V, such that 2eV=hν1+hν2. The tunneling of a Cooper pair is then associated to the emission of one photon into each of the resonators. Our setup routes the photons leaking out of each resonators into separate microwave amplifiers, allowing to detect and analyse the emitted radiation at room temperature. It has recently been shown by Leppägankas [3] and co-workers that the radiation emitted by these two photon-processes is non classical, and that it violates a classical Cauchy-Schwarz inequality for two-mode power cross-correlated fluctuations. The basic idea is that the probability of emitting a photon in each of the resonators during the observation time is higher that the geometric mean of the probability of emitting two photons in either one of them. I will present our experimental results where the classical inequality is violated.

[1] G.-L. Ingold, and Y. Nazarov, "Charge Tunneling Rate in Ultrasmall Junctions", in Single Charge Tunneling, edited by H; Grabert and M. H. Devoret (Plenum, New York, 1992).
[2] M. Hofheinz, F. Portier, Q. Baudouin, P. Joyez, D. Vion, P. Bertet, P. Roche, and D. Esteve, "Bright Side of the Coulomb Blockade", Phys. Rev. Lett. 106, 217005-4 (2011).
[3] J. Leppäkangas, G. Jahansson, M. Marthaler, and M. Fogelstrom, "Nonclassical Photon Pair production in a Voltage-Biased Josephson Junction", Phys. Rev. Lett. 110, 267004-5 (2013).

I will discuss the spin-dependent transport in a quantum dot coupled to a mechanical resonator at low temperature. This can e.g. be realized with a suspended carbon nanotube quantum dot in contact with two ferromagnets or with a ferromagnet and a superconducting lead. Due to spin-orbit interaction and/or an external magnetic gradient, the spin on the dot couples directly to the flexural eigenmodes. Owing to this interaction, the mechanical motion induces spin-flips of the electrons passing through the dot. The inelastic vibration-assisted spin-flips give rise to a mechanical damping and for an applied bias-voltage to a steady nonequilibrium phonon occupation of the resonator. 

For two ferromagnetic leads, I will show that a spin-polarized current causes either heating or active cooling of the mechanical modes, depending on the gate voltage and on the magnetic configuration of the  leads. Optimal cooling is achieved at resonance transport when the energy splitting between two dot levels of opposite spin equals the resonator frequency. Moderate polarization can achieve ground state cooling of the resonator. On the other hand, the system can also approach a regime of mechanical instability in which the damping coefficient vanishes. Moreover, signatures of the nanomechanical motion appear in the current-voltage characteristic. 
I will discuss similar results for a hybrid system in which the leads are formed by a ferromagnet and a  superconductor. For this case, I will focus on the sub-gap transport characterised by inelastic spin-dependent Andreev reflections.

Finally, I will give some perspectives for exploiting spin-dependent transport in order to achieve quantum vibrational states in such systems (e.g. coherent amplification, entanglement). 

The quantum mechanical interaction of light with matter has long been a source of fascination for physicists.  Recent developments in superconducting electronics have deepened this fascination by extending the techniques of quantum optics to condensed matter systems, and allowing new regimes of light-matter interaction to be accessed.   As an example, in our recent work we use the ac Josephson effect of a Cooper pair transistor to pump microwave photons into a high-Q microwave cavity.  These cavity photons in turn have a strong backaction on electrical transport through the transistor, so that the cavity-embedded Cooper-pair transistor (cCPT) can only be viewed as a single quantum system with both electrical and photonic components.  A rich interplay between electrical transport and photon emission arises, exhibiting strong self-oscillations, i.e. lasing.  The cCPT and related structures also provide an excellent platform for studying the quantum dynamics of a nonlinear system.  Furthermore, the cCPT offers the promise of inducing strong light-matter interactions between microwave photons and other structures as nanomechanical resonators even at the level of single quanta.

Recent experimental progress in coupling nanoscale conductors to superconducting microwave cavities has opened up for transport investigations of the deep quantum limit of light-matter interactions, with tunneling electrons strongly coupled to individual cavity photons. We have investigated theoretically the most basic cavity-conductor system with strong, single photon induced nonlocal transport effects: two spatially separated double quantum dots (DQDs) resonantly coupled to the fundamental cavity mode. The system, described by a generalized Tavis-Cummings model, is investigated within a quantum master equation formalism, allowing us to account for both the electronic transport properties through the DQDs as well as the coherent, nonequilibrium cavity photon state. We find sizable nonlocally induced current and current cross-correlations mediated by individual photons. From a full statistical description of the electron transport we further reveal a dynamical channel blockade in one DQD lifted by photon emission due to tunneling through the other DQD. Moreover, large entanglement between the orbital states of electrons in the two DQDs is found for small DQD-lead temperatures.

to be announced

Micromechanical resonators affected by radiation pressure forces allow to address fundamental questions on quantum properties of mechanical objects. An interesting setup for the purpose is an on-chip microwave cavity coupled to a mechanical resonator. We run various experiments in the setup, such as mechanical microwave amplification, sideband cooling in a multimode system, or optomechanics with graphene membranes. One can add intriguing features to the basic optomechanics setup by including a superconducting qubit made with Josephson junctions. The nonlinearity of the two-level system allows for much richer physics than is possible with linear cavities. We present a design where the on-chip microwave cavity includes a Josephson charge qubit. This way we were able to boost the coupling in the setup by six orders of magnitude up to the MHz regime, thus allowing us to approach the threshold of strong coupling. We observe nonlinear phenomena, such as as enhanced damping due to the two-level system.

I will present some recent theoretical study on a one-dimensional p-wave superconductor capacitively coupled to a microwave cavity. By probing the light exiting from the cavity, one can reveal the electronic susceptibility of the p-wave superconductor. I will show that this susceptibility allows us to determine the topological phase transition point, the emergence of the Majorana fermions, and the parity of the ground state of the topological superconductor. All these effects, which are absent in effective theories that take into account the coupling of light to Majorana fermions only, are due to the interplay between the Majorana and the bulk states in the superconductor.

The combination of low mass density, high frequency, and high quality-factor make graphene mechanical resonators very attractive for applications such as force sensing, mass sensing, and exploring the quantum regime of mechanical motion. Microwave optomechanics with superconducting cavities offers exquisite position sensitivity and enables the preparation and detection of mechanical systems in the quantum ground state. Here, I will review our recent work [1] in which we demonstrate coupling between a graphene resonator and a high-Q superconducting cavity. We achieve a displacement sensitivity of 55 fm ⁄ √ Hz and measure mechanical quality factors up to 220,000, both significantly better than shown before for graphene. Optomechanical coupling is demonstrated by optomechanically induced reflection (OMIR) and absorption (OMIA) of microwave photons. We observe 17 dB of mechanical microwave amplification and the onset of normal mode splitting, both signatures of strong optomechanical backaction. We extract the cooperativity C, a characterization of coupling strength, quantitatively from the measurement with no free parameters and find C=8, promising for the quantum regime of graphene motion.

[1] V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, G. A. Steele. Optomechanical coupling between a graphene mechanical resonator and a superconducting microwave cavity. 

We study a voltage biased  Josephson junction coupled to two resonators of incommensurate frequencies. Using a density approach to analyze the cavity fields and an input-output description to analyze the emitted photonic fluxes and their correlation functions, we have shown, both for infinite and finite bandwidth detectors,  that the emitted radiation is non-classical in the sense that the correlators violates Cauchy-Schwarz inequalities. We have  also studied the time dependence of the photonic correlations and showed that their line-width becomes narrower with the increase of the emission rate approaching from below the threshold limit.

I will discuss our recent work [1] on generation and sustenance of entangled many-body states of spatially separated superconducting qubits. 

[1] Camille Aron, Manas Kulkarni, Hakan E. Tureci, arXiv:1412.8477.

Recent experiments on nanoscale conductors coupled to microwave cavities showed the prospect of transport investigations of electron-photon interplay in the deep quantum regime. Here we propose a pump-probe scheme to investigate the transient dynamics of individual electron-photon excitations in a double-quantum-dot-cavity system. Excitations pumped into the system decay via charge tunneling at the double dot, probed in real time. We investigate theoretically the short-time charge-transfer statistics at the dot for periodic pumping and show that this gives access to vacuum Rabi oscillations as well as excitation dynamics in the presence of double-dot dephasing and relaxation.

Semiconductor quantum dots, Rydberg atoms and superconducting qubits all possess excitations in the microwave frequency domain. For this domain we have developed a wide range of novel approaches to create, store, manipulate and detect individual photons using micro-fabricated superconducting quantum electronic circuits. A key ingredient of this approach are coplanar wave guide resonators in which the field energy of an excitation is distributed over a mode volume much smaller than that of a mirror based resonator. This feature creates sizable electromagnetic fields at the level of individual microwave photons mediating strong electromagnetic interactions with a variety of different quantum systems. In an approach known as circuit quantum electrodynamics (QED) we have learned how to probe fundamental quantum optical effects and to demonstrate basic features of quantum information processing with superconducting quantum bits [1,2]. In this presentation, I will discuss a project in which we explore the physics of semiconductor quantum dots in the context of circuit QED. We investigate the coherent dipole coupling of semiconductor double quantum dots to microwave photons [3,4] and detect radiation emitted from the dots in inelastic electron tunneling processes [5]. This may allow us to explore quantum coherent interfaces between different solid state qubits in the context of quantum science and technology in future experiments.

[1] C. Lang et al., Nat. Phys. 9, 345–348 (2013)
[2] L. Steffen et al., Nature 500, 319-322 (2013)
[3] T. Frey et al., Phys. Rev. Lett. 108, 046807 (2012)
[4] A. Wallraff et al., Phys. Rev. Lett. 111, 249701 (2013)
[5] A. Stockklauser et al., arXiv:1504.05497 (2015)

We report on the generation of quantum metrological current by periodically modulating a single quantum electronic level coupled to two superconducting leads. Single quantum dot junctions are formed by inserting individual gold nanoparticles between superconducting aluminum leads. We demonstrate current quantization up to frequencies exceeding 200 MHz, conveyed by a single quantum level. Essential differences with previous experiments based on a normal metallic island rather than a quantum dot are highlighted and explained.
In particular, a striking dynamical transport feature associated to the hybridization of the single quantum level to the superconductor single particle density of states is demonstrated.