Quantum transport is a diverse field, sometimes combining seemingly contradicting concepts - quantum and classical, conduction and insulating - within a single nanodevice. Quantum transport is an essential and challenging part of nanoscience, and understanding its concepts and methods is vital to the successful fabrication of devices at the nanoscale. This textbook is a comprehensive introduction to the rapidly developing field of quantum transport. The authors present the comprehensive theoretical background, and explore the groundbreaking experiments that laid the foundations of the field. Ideal for graduate students, each section contains control questions and exercises to check readers' understanding of the topics covered. Its broad scope and in-depth analysis of selected topics will appeal to researchers and professionals working in nanoscience.

The combination of low mass density, high frequency, and high quality-factor of mechanical resonators made of two-dimensional crystals such as graphene 1-8 make them attractive for applications in force sensing/mass sensing, and exploring the quantum regime of mechanical motion. Microwave optomechanics with superconducting cavities 9-14 offers exquisite position sensitivity 10 and enables the preparation and detection of mechanical systems in the quantum ground state 15,16 . Here, we demonstrate coupling between a multilayer graphene resonator with quality factors up to 220,000 and a high-Q superconducting cavity. Using thermo-mechanical noise as calibration, we achieve a displacement sensitivity of 17 fm/ √ Hz. Optomechanical coupling is demonstrated by optomechanically induced reflection (OMIR) and absorption (OMIA) of microwave photons [17][18][19] . We observe 17 dB of mechanical microwave amplification 13 and 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. Here, we present a multilayer graphene mechanical resonator coupled to a superconducting cavity. Using a deterministic all-dry transfer technique 22 and a novel microwave coupling design, we are able to combine these two without sacrificing the exceptional intrinsic properties of either. Although multilayer graphene has a higher mass than a monolayer, it could be advantageous for coupling to a superconducting cavity due to its lower electrical resistance. In Figure 2, we characterize the mechanical properties of the multilayer graphene resonator using a homodyne measurement scheme 9 . Here, the cavity is used as an interferometer to detect motion while injecting a microwave signal near ω c and exciting the mechanical resonator with an AC voltage applied to the gate. The mechanical resonance frequency is much larger than the cavity linewidth (ω m /κ ∼ 150), placing us in the sideband resolved limit, a prerequisite for ground state cooling. The cavity can also be used to detect the undriven motion, such as thermomechanical noise of the drum shown in the inset of Figure 2(a) corresponding to a mechanical mode temperature of 96 mK (see SI for additional details). The thermal motion peak serves as a calibration for the displacement sensitivity. While driving the cavity at its resonance and utilizing its full dynamic range before the electrical nonlinearity set in (-41 dBm injected power) we estimate a displacement sensitivity for mechanical motion of 17 fm/ √ Hz. Using a DC voltage applied to the gate electrode, we can also tune the frequency of the multilayer graphene resonator shown in Figure 2(b). The decrease in resonance frequency ω m for non-zero gate voltage is due to electrostatic softening of the spring constant and has been observed before 3 .In Figure 3, we demonstrate optomechanical coupling between the multilayer graphene mechanical resona...

We study the model of a molecular switch comprised of a molecule with a soft vibrational degree of freedom coupled to metallic leads. In the presence of strong electron-ion interaction, different charge states of the molecule correspond to substantially different ionic configurations, which can lead to very slow switching between energetically close configurations ͑Franck-Condon blockade͒. Application of transport voltage, however, can drive the molecule far out of thermal equilibrium and thus dramatically accelerate the switching. The tunneling electrons play the role of a heat bath with an effective temperature dependent on the applied transport voltage. Including the transport-induced "heating" self-consistently, we determine the stationary currentvoltage characteristics of the device and the switching dynamics for symmetric and asymmetric devices. We also study the effects of an extra dissipative environment and demonstrate that it can lead to enhanced nonlinearities in the transport properties of the device and dramatically suppress the switching dynamics.

We report a theory for the coherent and incoherent chiral pumping of spin waves into thin magnetic films through the dipolar coupling with a local magnetic transducer, such as a nanowire. The ferromagnetic resonance of the nanowire is broadened by the injection of unidirectional spin waves that generates a non-equilibrium magnetization in only half of the film. A temperature gradient between the local magnet and film leads to a unidirectional flow of incoherent magnons, i.e., a chiral spin Seebeck effect.

Spin waves—the elementary excitations of magnetic materials—are prime candidate signal carriers for low-dissipation information processing. Being able to image coherent spin-wave transport is crucial for developing interference-based spin-wave devices. We introduce magnetic resonance imaging of the microwave magnetic stray fields that are generated by spin waves as a new approach for imaging coherent spin-wave transport. We realize this approach using a dense layer of electronic sensor spins in a diamond chip, which combines the ability to detect small magnetic fields with a sensitivity to their polarization. Focusing on a thin-film magnetic insulator, we quantify spin-wave amplitudes, visualize spin-wave dispersion and interference, and demonstrate time-domain measurements of spin-wave packets. We theoretically explain the observed anisotropic spin-wave patterns in terms of chiral spin-wave excitation and stray-field coupling to the sensor spins. Our results pave the way for probing spin waves in atomically thin magnets, even when embedded between opaque materials.

The reduced dimensionality of two-dimensional (2D) materials results in characteristic types of magnetically and electronically ordered phases. However, only few methods are available to study this order, in particular in ultrathin insulating antiferromagnets that couple weakly to magnetic and electronic probes. Here, we demonstrate that phase transitions in thin membranes of 2D antiferromagnetic FePS3, MnPS3 and NiPS3 can be probed mechanically via the temperature-dependent resonance frequency and quality factor. The observed relation between mechanical motion and antiferromagnetic order is shown to be mediated by the specific heat and reveals a strong dependence of the Néel temperature of FePS3 on electrostatically induced strain. The methodology is not restricted to magnetic order, as we demonstrate by probing an electronic charge-density-wave phase in 2H-TaS2. It thus offers the potential to characterize phase transitions in a wide variety of materials, including those that are antiferromagnetic, insulating or so thin that conventional bulk characterization methods become unsuitable.

Measuring the quantum dynamics of a mechanical system, when few phonons are involved, remains a challenge. We show that a superconducting microwave resonator linearly coupled to the mechanical mode constitutes a very powerful probe for this scope. This new coupling can be much stronger than the usual radiation pressure interaction by adjusting a gate voltage. We focus on the detection of phonon blockade, showing that it can be observed by measuring the statistics of the light in the cavity. The underlying reason is the formation of an entangled state between the two resonators. Our scheme realizes a phonotonic Josephson junction, giving rise to coherent oscillations between phonons and photons as well as a self-trapping regime for a coupling smaller than a critical value. The transition from the self-trapping to the oscillating regime is also induced dynamically by dissipation.

In cavity optomechanics, light is used to control mechanical motion. A central goal of the field is achieving single-photon strong coupling, which would enable the creation of quantum superposition states of motion. Reaching this limit requires significant improvements in optomechanical coupling and cavity coherence. Here we introduce an optomechanical architecture consisting of a silicon nitride membrane coupled to a three-dimensional superconducting microwave cavity. Exploiting their large quality factors, we achieve an optomechanical cooperativity of 146,000 and perform sideband cooling of the kilohertz-frequency membrane motion to 34±5 μK, the lowest mechanical mode temperature reported to date. The achieved cooling is limited only by classical noise of the signal generator, and should extend deep into the ground state with superconducting filters. Our results suggest that this realization of optomechanics has the potential to reach the regimes of ultra-large cooperativity and single-photon strong coupling, opening up a new generation of experiments.