Interfacing superconducting quantum processors, working in the GHz frequency range, with optical quantum networks and atomic qubits is a challenging task for the implementation of distributed quantum information processing as well as for quantum communication. Using spin ensembles of rare-earth ions provides an excellent opportunity to bridge microwave and optical domains at the quantum level. In this Rapid Communication, we demonstrate the ultralow-power, on-chip, electron-spin-resonance spectroscopy of Er 3+ spins doped in a Y 2 SiO 5 crystal using a high-Q, coplanar, superconducting resonator.Quantum communication is a rapidly developing field of science and technology, which allows the transmission of information in an intrinsically secure way. 1 As well as its classical counterpart, a quantum communication network can combine various types of systems which transmit, receive, and process information using quantum algorithms. 2 For example, the nodes of such a network can be implemented by superconducting (SC) quantum circuits operated in the GHz frequency range, 3 whereas fiber optics operated at near infrared can be used to link them over long distances. For the reversible transfer of quantum states between systems operating at GHz and optical frequency ranges, one must use a hybrid system. 4 Spin ensembles coupled to a microwave resonator or to a SC qubit represent one of the possible implementations of such a system. 5-8 The collective coupling strength of a spin ensemble is increased with respect to a single spin by the square root of the number of spins. Transparent crystals doped with paramagnetic ions often possess long coherence times, 9,10 and the collective coupling has been recently demonstrated with nitrogen-vacancy centers in diamond, 11-13 organic molecules, 14 and (Cr 3+ ) ions in ruby. 12 In this Rapid Communication, we report on the ultralowpower electron-spin-resonance (ESR) spectroscopy of an erbium-ion spin ensemble at sub-Kelvin temperatures using a high-Q, coplanar, SC resonator. The Er 3+ ions are distinct from other spin ensembles due to their optical transition at the telecom C band, i.e., inside the so-called erbium window at 1.54 μm wavelength, and their long measured optical coherence time. 15 The energy-level diagram of erbium ions embedded inside a crystal is shown in Fig. 1(a). The electronic configuration of a free Er 3+ ion is 4f 11 , with a 4 I term. The spin-orbit coupling splits it into several fine structure levels. An optical transition at the telecom wavelength occurs between the ground state 2S+1 L J = 4 I 15/2 and the first excited state 4 I 13/2 , where S, L, and J are the respective spin, orbital, and total magnetic momenta of the ion. The weak crystal field splits the ground state into eight (J + 1/2) Kramers doublets. 16 At cryogenic temperature, only the lowest doublet Z 1 is populated, therefore the system can be described as an effective electronic spin with S = 1/2. However, erbium has five even isotopes, 162 Er, 164 Er, 166 Er, 168 Er, and 170 Er, and one odd ...

The interplay between superconductivity and Coulomb interactions has been studied for more than twenty years now 1-13 . In low-dimensional systems, superconductivity degrades in the presence of Coulomb repulsion: interactions tend to suppress fluctuations of charge, thereby increasing fluctuations of phase. This can lead to the occurrence of a superconducting-insulator transition, as has been observed in thin superconducting films 5,6 , wires 7 and also in Josephson junction arrays 9,11-13 . The latter are very attractive systems as they enable a relatively easy control of the relevant energies involved in the competition between superconductivity and Coulomb interactions. Josephson junction chains have been successfully used to create particular electromagnetic environments for the reduction of charge fluctuations [14][15][16] . Recently, they have attracted interest as they could provide the basis for the realization of a new type of topologically protected qubit 17,18 or for the implementation of a new current standard 19 . Here we present measurements that show clearly the effect of quantum phase slips on the ground state of a Josephson junction chain. We tune in situ the strength of quantum phase fluctuations and obtain for the first time an excellent agreement with the tight-binding model initially proposed by Matveev et al. 8 .The Hamiltonian for the theoretical description of superconducting circuits can be conveniently obtained by applying Devoret's circuit theory 20 . Here, each electrical element such as an inductance, a capacitor or the Josephson element can add a degree of freedom. In the case of circuits with a small number of electrical elements, a complete analytical description that takes into account all degrees of freedom can be obtained. However, when the circuits contain an increasing number of elements, as for example Josephson junction chains, even numerical solutions of the problem become difficult to obtain when taking into account all degrees of freedom. Nevertheless our measurements demonstrate that the ground state of a phase-biased Josephson junction chain (see Fig. 1(a)) can be described by a single degree of freedom. Although the chain is a multi-dimensional object, the effect of quantum phase-slips can be described by a single variable m, that counts the number of phase-slips in the chain.We start by giving a short introduction on the lowenergy properties of a Josephson junction chain which have been studied in terms of quantum phase slips by Matveev et al. 8 . Let us consider the Josephson junction chain depicted in Fig. 1(a). The chain contains N junctions and is biased with a phase γ. We denote E J the Josephson energy of a single junction and E C = e 2 2C its charging energy. Here we consider E J E C . Let Q i be the charge on each junction and θ i the phase difference. In the nearest-neighbor-capacitance limit the Hamiltonian can be written as:Ignoring the charging energy for the moment, we find the classical ground state, that satisfies the constraint on the phase N i=1 θ i ...

Electric and thermal transport properties of a ν = 2/3 fractional quantum Hall junction are analyzed. We investigate the evolution of the electric and thermal two-terminal conductances, G and G Q , with system size L and temperature T . This is done both for the case of strong interaction between the 1 and 1/ 3 modes (when the low-temperature physics of the interacting segment of the device is controlled by the vicinity of the strong-disorder Kane-Fisher-Polchinski fixed point) and for relatively weak interaction, for which the disorder is irrelevant at T = 0 in the renormalization-group sense. The transport properties in both cases are similar in several respects. In particular, G(L) is close to 4/3 (in units of e 2 /h) and G Q to 2 (in units of πT/6 ) for small L, independently of the interaction strength. For large L the system is in an incoherent regime, with G given by 2/3 and G Q showing the Ohmic scaling, G Q ∝ 1/L, again for any interaction strength. The hallmark of the strong-disorder fixed point is the emergence of an intermediate range of L, in which the electric conductance shows strong mesoscopic fluctuations and the thermal conductance is G Q = 1. The analysis is extended also to a device with floating 1/3 mode, as studied in a recent experiment [A. Grivnin et al, Phys. Rev. Lett. 113, 266803 (2014)].

We study the heat conductivity in Anderson insulators in the presence of power-law interaction. Particle-hole excitations built on localized electron states are viewed as two-level systems randomly distributed in space and energy and coupled due to electron-electron interaction. A small fraction of these states form resonant pairs that in turn build a complex network allowing for energy propagation. We identify the character of energy transport through this network and evaluate the thermal conductivity. For physically relevant cases of 2D and 3D spin systems with 1/r 3 dipole-dipole interaction (originating from the conventional 1/r Coulomb interaction between electrons), the found thermal conductivity κ scales with temperature as κ ∝ T 3 and κ ∝ T 4/3 , respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.

Field-theoretical approach to Anderson localization in 2D disordered fermionic systems of chiral symmetry classes (BDI, AIII, CII) is developed. Important representatives of these symmetry classes are random hopping models on bipartite lattices at the band center. As was found by Gade and Wegner two decades ago within the sigma-model formalism, quantum interference effects in these classes are absent to all orders of perturbation theory. We demonstrate that the quantum localization effects emerge when the theory is treated nonperturbatively. Specifically, they are controlled by topological vortexlike excitations of the sigma models. We derive renormalization-group equations including these nonperturbative contributions. Analyzing them, we find that the 2D disordered systems of chiral classes undergo a metal-insulator transition driven by topologically induced Anderson localization. We also show that the Wess-Zumino and Z 2 θ terms on surfaces of 3D topological insulators (in classes AIII and CII, respectively) overpower the vortex-induced localization.

We explore the life time of excitations in a dispersive Luttinger liquid. We perform a bosonization supplemented by a sequence of unitary transformations that allows us to treat the problem in terms of weakly interacting quasiparticles. The relaxation described by the resulting Hamiltonian is analyzed by bosonic and (after a refermionization) by fermionic perturbation theory. We show that the the fermionic and bosonic formulations of the problem exhibit a remarkable strong-weakcoupling duality. Specifically, the fermionic theory is characterized by a dimensionless coupling constant λ = m * l 2 T and the bosonic theory by λ −1 , where 1/m * and l characterize the curvature of the fermionic and bosonic spectra, respectively, and T is the temperature.

We explore dynamics of a density pulse induced by a local quench in a one-dimensional electron system. The spectral curvature leads to an "overturn" (population inversion) of the wave. We show that beyond this time the density profile develops strong oscillations with a period much larger than the Fermi wave length. The effect is studied first for the case of free fermions by means of direct quantum simulations and via semiclassical analysis of the evolution of Wigner function. We demonstrate then that the period of oscillations is correctly reproduced by a hydrodynamic theory with an appropriate dispersive term. Finally, we explore the effect of different types of electronelectron interaction on the phenomenon. We show that sufficiently strong interaction [U (r) ≫ 1/mr 2 where m is the fermionic mass and r the relevant spatial scale] determines the dominant dispersive term in the hydrodynamic equations. Hydrodynamic theory reveals crucial dependence of the density evolution on the relative sign of the interaction and the density perturbation.

The many-body localized (MBL) phase is characterized by a complete set of quasi-local integrals of motion and area-law entanglement of excited eigenstates. We study the effect of non-Abelian continuous symmetries on MBL, considering the case of SU (2) symmetric disordered spin chains. The SU (2) symmetry imposes strong constraints on the entanglement structure of the eigenstates, precluding conventional MBL. We construct a fixed-point Hamiltonian, which realizes a non-ergodic (but non-MBL) phase characterized by eigenstates having logarithmic scaling of entanglement with the system size, as well as an incomplete set of quasi-local integrals of motion. We study the response of such a phase to local symmetric perturbations, finding that even weak perturbations induce multi-spin resonances. We conclude that the non-ergodic phase is generally unstable and that SU (2) symmetry implies thermalization. The approach introduced in this work can be used to study dynamics in disordered systems with non-Abelian symmetries, and provides a starting point for searching non-ergodic phases beyond conventional MBL. Introduction. Over the past several years, the phenomenon of many-body localization (MBL) has been attracting significant interest, both theoretically [1-17] and experimentally [18][19][20]. Many-body localization occurs in strongly disordered systems and is driven by a mechanism similar to the (single-particle) Anderson localization in the many-body Hilbert space. Isolated many-body localized systems exhibit zero conductivity and avoid thermalization, and therefore provide the only known, generic example of ergodicity breaking in many-body systems.MBL eigenstates have low, area-law entanglement entropy [8, 21], in contrast to the excited eigenstates of ergodic systems, which have thermal, volume-law entanglement. The systems in which all states are manybody localized exhibit a new kind of robust integrability: a complete set of quasi-local integrals of motion (LIOMs) emerges [8, 9] (see also [22][23][24]). Apart from providing a simple physical intuition for the ergodicity breaking in MBL phase, LIOM theory has been used to explain dynamical properties of MBL eigenstates, including logarithmic entanglement growth in a quantum quench setup [7][8][9], as well as power-law decay [25] and revivals [26] of local observables, which can be tested in cold atoms experiments.A natural question concerns the role of various symmetries on MBL and thermalization. Previous works focused mostly on MBL in the presence of discrete symmetries, such as Z 2 symmetry. It was shown [13,27,28] that in this case two distinct MBL phases are possible, one of which locally preserves Z 2 symmetry, while the other phase locally breaks that symmetry. It was also argued that MBL can protect topological [21,27] and symmetry-protected topological [29] order at finite temperatures. Ref. [30] considered the effect of a particular non-Abelian discrete symmetry on MBL.