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We resolve phonon number states in the spectrum of a superconducting qubit coupled to a multimode acoustic cavity. Crucial to this resolution is the sharp frequency dependence in the qubit-phonon interaction engineered by coupling the qubit to surface acoustic waves in two locations separated by $\sim40$ acoustic wavelengths. In analogy to double-slit diffraction, the resulting self-interference generates high-contrast frequency structure in the qubit-phonon interaction. We observe this frequency structure both in the coupling rate to multiple cavity modes and in the qubit spontaneous emission rate into unconfined modes. We use this sharp frequency structure to resolve single phonons by tuning the qubit to a frequency of destructive interference where all acoustic interactions are dispersive. By exciting several detuned yet strongly-coupled phononic modes and measuring the resulting qubit spectrum, we observe that, for two modes, the device enters the strong dispersive regime where single phonons are spectrally resolved., 9 pages, 8 figures; revised arguments in paragraphs 3 and 8, added Hamiltonian description, and corrected typos

International audience; We use the strong intrinsic nonlinearity of a microwave superconducting qubit with a 4 GHz transition frequency to directly detect and control the energy of a micromechanical oscillator vibrating at 25 MHz. The qubit and the oscillator are coupled electrostatically at a rate of approximately 2π × 22 MHz. In this far off-resonant regime, the qubit frequency is shifted by 0.52 MHz per oscillator phonon, or about 14% of the 3.7 MHz qubit linewidth. The qubit behaves as a vibrational energy detector and from its line shape we extract the phonon number distribution of the oscillator. We manipulate this distribution by driving number state sensitive sideband transitions and creating profoundly nonthermal states. Finally, by driving the lower frequency sideband transition, we cool the oscillator and increase its ground state population up to 0.48 AE 0.13, close to a factor of 8 above its value at thermal equilibrium. These results demonstrate a new class of electromechanics experiments that are a promising strategy for quantum nondemolition measurements and nonclassical state preparation. The ability to bring manmade acoustical or mechanical structures into the quantum regime has been demonstrated in a variety of devices, from micromechanical oscillators in opto-and electromechanics experiments [1,2], to acoustic resonators in circuit quantum electrodynamics (cQED) experiments [3]. Mechanical oscillators are generally very linear harmonic oscillators at the quantum scale, and to achieve arbitrary quantum control, one needs an extrinsic nonlinearity [4]. Performing nonlinear detection is also a way to enable quantum nondemolition measurement by measuring energy instead of position or momentum [5-7]. One strategy is to use the Josephson junction used in superconducting microwave circuits. It provides a dissipa-tionless strong nonlinearity and has enabled the demonstration of landmark results in quantum science from the preparation of arbitrary quantum states of microwave light [8,9] to the demonstration of early-stage quantum computers [10,11]. By using piezoelectric materials, resonant coupling between superconducting qubits and high frequency (GHz) acoustic wave resonators has been demonstrated [3,12]. This resonant approach is, however, restricted to a small class of acoustic oscillators and loses many of the advantages of the micromechanical oscillators used in opto-and electromechanics experiments [13]. In these experiments, a wide variety of techniques have been developed and have made these mass-on-a-spring-like oscillators very versatile. They can be used to interface otherwise incompatible quantum systems such as superconducting circuits and optical light [14], they are extraordinarily sensitive detectors of force and strain [15,16], and they can be engineered to have extremely long lifetimes [17]. However, these low frequency mechanical oscillators have proven to be more challenging to couple to superconducting qubits. One strategy is to use a linear cavity to transfer nonclassical microwave fields created by a qubit to a mechanical oscillator by using the radiation pressure interaction [18,19]. This approach has to battle the incompatibility of large microwave pump powers with qubits as well as the loss during the state propagation or transfer. Low frequency mechanical oscillators have also been directly coupled to qubits [20,21], but so far the interaction strengths have been too weak to achieve control or detection of motion at the scale of few phonons. In this Letter, we directly couple a superconducting qubit to a mechanical oscillator, achieving an ultrastrong interaction of g m ≈ 2π × 22 MHz, comparable to the oscillator's resonance frequency ω m ≈ 2π × 25 MHz. Similar to quad-ratic optomechanics proposals [6], we detect the energy of the oscillator instead of its position. More precisely, a mechanical ac-Stark effect shifts the qubit frequency by 0.52 MHz per oscillator phonon, or about 14% of the 3.7 MHz qubit linewidth. The qubit line shape therefore encodes the phonon number statistics, which we extract using a Bayesian-based algorithm. The qubit-oscillator system also exhibits blue and red sideband transitions, analogous to those found in optomechanics and trapped ions systems [13,22], at the sum (blue) and difference (red) of frequencies. In contrast to optomechanics, the qubit nonlinearity makes these sideband transitions number state dependent. Using this property, we demonstrate control of populations in the Fock space with a resolution of about 7 quanta. By driving the lower frequency sideband transition, we cool the oscillator and increase its ground state PHYSICAL

Coupling double-quantum-dot circuits to microwave cavities provides a powerful means to control, couple, and manipulate qubits based on the charge or spin of individual electrons. Here, we revisit this standard configuration by adding superconductivity to the circuit. We combine theory and experiment to study a superconductor-double-quantum-dot circuit coupled to microwave cavity photons. First, we use the cavity as a spectroscopic probe. This allows us to determine the low-energy spectrum of the device and to reveal directly Cooper-pair-assisted tunneling between the two dots. Second, we observe a vacuum Rabi splitting which is a signature of strong charge photon coupling and a premiere with carbon-nanotube-based quantum-dot circuits. We show that our circuit design intrinsically combines a set of key features to achieve the strong coupling regime to the cavity. A low charging energy reduces the device sensitivity to charge noise, while sufficient coupling is provided by the shaping of the spectrum of the double quantum dot by the superconducting reservoir. Our findings could be adapted to many other circuit designs and shed light on the coupling of superconducting nanoscale devices to microwave fields.

Driving a quantum system at finite frequency allows one to explore its dynamics. This has become a well-mastered resource for controlling the quantum state of two-level systems in the context of quantum information processing. However, this can also be of fundamental interest, especially with many-body systems which display an intricate finite-frequency behavior. In condensed matter, the Kondo effect epitomizes strong electronic correlations, but the study of its dynamics and the related scaling laws has remained elusive so far. Here, we fill this gap by studying a carbon-nanotube-based Kondo quantum dot at half filling driven by a microwave signal. Our findings not only confirm long-standing theoretical predictions but also allow us to establish a simple ansatz for the scaling laws on the Kondo problem at finite frequency. More generally, our technique opens a path for understanding the dynamics of complex quantum dot circuits in the context of quantum simulation of strongly correlated electron fluids.

We investigate an acoustical analog of circuit quantum electrodynamics that facilitates compact high-Q (${>}20,000$) microwave-frequency cavities with dense spectra. We fabricate and characterize a device that comprises a flux tunable transmon coupled to a $300\,\mathrm{��m}$ long surface acoustic wave resonator. For some modes, the qubit-cavity coupling reaches $6.5\,\mathrm{MHz}$, exceeding the cavity loss rate ($200\,\mathrm{kHz}$), qubit linewidth ($1.1\,\mathrm{MHz}$), and the cavity free spectral range ($4.8\,\mathrm{MHz}$), placing the device in both the strong coupling and strong multimode regimes. With the qubit detuned from the cavity, we show that the dispersive shift behaves according to predictions from a generalized Jaynes-Cummings Hamiltonian. Finally, we observe that the qubit linewidth strongly depends on its frequency, as expected for spontaneous emission of phonons, and we identify operating frequencies where this emission rate is suppressed.

The ability to control electronic states at the nanoscale has contributed to our modern understanding of condensed matter. In particular, quantum dot circuits represent model systems for the study of strong electronic correlations, epitomized by the Kondo effect. Here, we show that circuit Quantum Electrodynamics architectures can be used to study the internal degrees of freedom of such a many-body phenomenon. We couple a quantum dot to a high finesse microwave cavity to measure with an unprecedented sensitivity the dot electronic compressibility i.e. the ability of the dot to accommodate charges. Because it corresponds solely to the charge response of the electronic system, this quantity is not equivalent to the conductance which involves in general other degrees of freedom such as spin. By performing dual conductance/compressibility measurements in the Kondo regime, we uncover directly the charge dynamics of this peculiar mechanism of electron transfer. Strikingly, the Kondo resonance, visible in transport measurements, is transparent to microwave photons trapped in the high finesse cavity. This reveals that, in such a many body resonance, finite conduction is achieved from a charge frozen by Coulomb interaction. This previously elusive freezing of charge dynamics is in stark contrast with the physics of a free electron gas. Our setup highlights the power of circuit quantum electrodynamics architectures to study condensed matter problems. The tools of cavity quantum electrodynamics could be used in other types of mesoscopic circuits with many-body correlations and bring a promising platform to perform quantum simulation of fermion-boson problems., Comment: minor differences with published version

Electron spins and photons are complementary quantum-mechanical objects that can be used to carry, manipulate and transform quantum information. To combine these resources, it is desirable to achieve the coherent coupling of a single spin to photons stored in a superconducting resonator. 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 achieve spin-photon coupling up to the MHz range at the single spin level. The cooperativity is found to reach 2.3, and the spin coherence time is about 60ns. We thereby demonstrate a mesoscopic device suitable for non-destructive spin read-out and distant spin coupling., Comment: minor differences with published version

Circuit QED techniques have been instrumental to manipulate and probe with exquisite sensitivity the quantum state of superconducting quantum bits coupled to microwave cavities. Recently, it has become possible to fabricate new devices where the superconducting quantum bits are replaced by hybrid mesoscopic circuits combining nanoconductors and metallic reservoirs. This mesoscopic QED provides a new experimental playground to study the light-matter interaction in electronic circuits. Here, we present the experimental state of the art of Mesoscopic QED and its theoretical description. A first class of experiments focuses on the artificial atom limit, where some quasiparticles are trapped in nanocircuit bound states. In this limit, the Circuit QED techniques can be used to manipulate and probe electronic degrees of freedom such as confined charges, spins, or Andreev pairs. A second class of experiments consists in using cavity photons to reveal the dynamics of electron tunneling between a nanoconductor and fermionic reservoirs. For instance, the Kondo effect, the charge relaxation caused by grounded metallic contacts, and the photo-emission caused by voltage-biased reservoirs have been studied. The tunnel coupling between nanoconductors and fermionic reservoirs also enable one to obtain split Cooper pairs, or Majorana bound states. Cavity photons represent a qualitatively new tool to study these exotic condensed matter states., 34 pages, 18 figures, 1 table, minor differences with the published version to appear in Journal of Physics: Condensed Matter as a topical review

Cavity quantum electrodynamics allows one to study the interaction between light and matter at the most elementary level. The methods developed in this field have taught us how to probe and manipulate individual quantum systems like atoms and superconducting quantum bits with an exquisite accuracy. There is now a strong effort to extend further these methods to other quantum systems, and in particular hybrid quantum dot circuits. This could turn out to be instrumental for a noninvasive study of quantum dot circuits and a realization of scalable spin quantum bit architectures. It could also provide an interesting platform for quantum simulation of simple fermion-boson condensed matter systems. In this short review, we discuss the experimental state of the art for hybrid circuit quantum electrodynamics with quantum dots, and we present a simple theoretical modeling of experiments., Minor differences with published version

Microwave cavities have been widely used to investigate the behavior of closed few-level systems. Here, we show that they also represent a powerful probe for the dynamics of charge transfer between a discrete electronic level and fermionic continua. We have combined experiment and theory for a carbon nanotube quantum dot coupled to normal metal and superconducting contacts. In equilibrium conditions, where our device behaves as an effective quantum dot-normal metal junction, we approach a universal photon dissipation regime governed by a quantum charge relaxation effect. We observe how photon dissipation is modified when the dot admittance turns from capacitive to inductive. When the fermionic reservoirs are voltage biased, the dot can even cause photon emission due to inelastic tunneling to/from a Bardeen-Cooper-Schrieffer peak in the density of states of the superconducting contact. We can model these numerous effects quantitatively in terms of the charge susceptibility of the quantum dot circuit. This validates an approach that could be used to study a wide class of mesoscopic QED devices., Comment: 15 pages, 8 figures, minor differences with published version

Non-collinear spin transport is at the heart of spin or magnetization control in spintronics devices. The use of nanoscale conductors exhibiting quantum effects in transport could provide new paths for that purpose. Here we study non-collinear spin transport in a quantum dot. We use a device made out of a single-wall carbon nanotube connected to orthogonal ferromagnetic electrodes. In the spin transport signals, we observe signatures of out of equilibrium spin precession that are electrically tunable through dissipation. This could provide a new path to harness spin precession in nanoscale conductors., Control over the orientation of electronic spins forms the basis for spintronic devices in both classical and quantum systems. Here, the authors observe electrically-tunable dissipation-controlled spin precession in a carbon nanotube quantum dot bridging two non-collinearly magnetized electrodes.

Electron spins and photons are complementary quantum-mechanical objects that can be used to carry, manipulate, and transform quantum information. To combine these resources, it is desirable to achieve the coherent coupling of a single spin to photons stored in a superconducting resonator. 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 achieve spin-photon coupling up to the megahertz range at the single-spin level. The cooperativity is found to reach 2.3, and the spin coherence time is about 60 nanoseconds. We thereby demonstrate a mesoscopic device suitable for nondestructive spin readout and distant spin coupling.

The recent development of hybrid circuit quantum electrodynamics allows one to study how cavity photons interact with a system driven out of equilibrium by fermionic reservoirs. We study here one of the simplest combination: a double quantum dot coupled to a single mode of the electromagnetic field. We are able to couple resonantly the charge levels of a carbon-nanotube-based double dot to cavity photons. We perform a microwave readout of the charge states of this system, which allows us to unveil features of the out-of-equilibrium charge dynamics, otherwise invisible in the DC current. We extract the relaxation rate, dephasing rate, and photon number of the hybrid system using a theory based on a master equation technique. These findings open the path for manipulating other degrees of freedom, e.g., the spin and/or the valley in nanotube-based double dots using microwave light.

We demonstrate a hybrid architecture consisting of a quantum dot circuit coupled to a single mode of the electromagnetic field. We use single wall carbon nanotube based circuits inserted in superconducting microwave cavities. By probing the nanotube-dot using a dispersive read-out in the Coulomb blockade and the Kondo regime, we determine an electron-photon coupling strength which should enable circuit QED experiments with more complex quantum dot circuits., 4 pages, 4 figures

We report on a dry transfer technique for single wall carbon nanotube devices which allows to embed them in high finesse microwave cavity. We demonstrate the ground state charge readout and a quality factor of about 3000 down to the single photon regime. This technique allows to make devices such as double quantum dots which could be instrumental for achieving the strong spin photon coupling. It can easily be extended to generic carbon nanotube based microwave devices., Version similar to the one accepted

International audience; Optomechanics and electromechanics have made it possible to prepare macroscopic mechanical oscillators in their quantum ground states, in quadrature squeezed states, and in entangled states of motion. In addition to coaxing ever larger and more tangible objects into a regime of quantum behavior, this new capability has encouraged ideas of using mechanical oscillators in the processing and communication of quantum information and as precision force sensors operating beyond the standard quantum limit. But the effectively linear interaction between motion and light or electricity precludes access to the broader class of quantum states of motion, such as cat states or energy squeezed states. Indeed, early optomechanical proposals noted the possibility to escape this restriction by creating strong quadratic coupling of motion to light. Although there have been experimental demonstrations of quadratically coupled optomechanical systems, these have not yet accessed nonclassical states of motion. Here we create nonclassical states by quadratically coupling motion to the energy levels of a Cooper-pair box (CPB) qubit. By monitoring the qubit's transition frequency, we detect the oscillator's phonon distribution rather than its position. Through microwave frequency drives that change both the state of the oscillator and qubit, we then dissipatively stabilize the oscillator in a state with a large mean phonon number of 43 and sub-Poissonian number fluctuations of approximately 3. In this energy squeezed state we observe a striking feature of the quadratic coupling: the recoil of the mechanical oscillator caused by qubit transitions, closely analogous to the vibronic transitions in molecules.