The electronic density of states of graphene is equivalent to that of relativistic electrons [1-3]. In the absence of disorder or external doping the Fermi energy lies at the Dirac point where the density of states vanishes. Although transport measurements at high carrier densities indicate rather high mobilities [4-6], many questions pertaining to disorder [7-14] remain unanswered. In particular, it has been argued theoretically, that when the average carrier density is zero, the inescapable presence of disorder will lead to electron and hole puddles with equal probability. In this work, we use a scanning single electron transistor to image the carrier density landscape of graphene in the vicinity of the neutrality point. Our results clearly show the electron-hole puddles expected theoretically [13]. In addition, our measurement technique enables to determine locally the density of states in graphene. In contrast to previously studied massive two dimensional electron systems, the kinetic contribution to the density of states accounts quantitatively for the measured signal. Our results suggests that exchange and correlation effects are either weak or have canceling contributions. The kinetic energy of Dirac particles in graphene increases linearly with momentum [1,15,16]. The total energy per particle however, also referred to as the chemical potential, µ , contains additional exchange and correlations contributions that arise from the Coulomb interactionHere K E , ex E and c E are the kinetic, exchange and correlation terms respectively. Therefore, the chemical potential and its derivative with respect to density, known as the inverse compressibility or density of states, provide direct insight into the properties of the Coulomb interaction in such a system. Compressibility measurements of conventional, massive, two dimensional electron systems made for example of Si or GaAs have been carried out by several groups [19,[23][24][25][26]. It has been shown that at zero magnetic field the chemical potential may be described rather accurately within the Hartree-Fock approximation. Quantitatively it has been found that Coulomb interactions add a substantial contribution to the compressibility and become dominant at low carrier densities. In a perfectly uniform and clean graphene sample, the inverse compressibility is expected to diverge at the Dirac point in view of the vanishing density of states. This divergence, however, is expected to be rounded off by disorder on length scales smaller than our tip size. Long range disorder on the other hand will cause local shifts in the Dirac point indicative of a non-zero local density. In this work we measure the spatial dependence of the local compressibility versus carrier density across the sample. Fluctuations in the Dirac point across the sample are translated into

Ultracold atom-ion mixtures are gaining increasing interest due to their potential applications in ultracold and state-controlled chemistry, quantum computing, and many-body physics. Here, we studied the dynamics of a single ground-state cooled ion during few, to many, Langevin (spiraling) collisions with ultracold atoms. We measured the ion's energy distribution and observed a clear deviation from the Maxwell-Boltzmann distribution, characterized by an exponential tail, to a power-law distribution best described by a Tsallis function. Unlike previous experiments, the energy scale of atom-ion interactions is not determined by either the atomic cloud temperature or the ion's trap residual excess-micromotion energy. Instead, it is determined by the force the atom exerts on the ion during a collision which is then amplified by the trap dynamics. This effect is intrinsic to ion Paul traps and sets the lower bound of atom-ion steady-state interaction energy in these systems. Despite the fact that our system is eventually driven out of the ultracold regime, we are capable of studying quantum effects by limiting the interaction to the first collision when the ion is initialized in the ground state of the trap.

Quantum metrology uses tools from quantum information science to improve measurement signal-to-noise ratios. The challenge is to increase sensitivity while reducing susceptibility to noise, tasks that are often in conflict. Lock-in measurement is a detection scheme designed to overcome this difficulty by spectrally separating signal from noise. Here we report on the implementation of a quantum analogue to the classical lock-in amplifier. All the lock-in operations--modulation, detection and mixing--are performed through the application of non-commuting quantum operators to the electronic spin state of a single, trapped Sr(+) ion. We significantly increase its sensitivity to external fields while extending phase coherence by three orders of magnitude, to more than one second. Using this technique, we measure frequency shifts with a sensitivity of 0.42 Hz Hz(-1/2) (corresponding to a magnetic field measurement sensitivity of 15 pT Hz(-1/2)), obtaining an uncertainty of less than 10 mHz (350 fT) after 3,720 seconds of averaging. These sensitivities are limited by quantum projection noise and improve on other single-spin probe technologies by two orders of magnitude. Our reported sensitivity is sufficient for the measurement of parity non-conservation, as well as the detection of the magnetic field of a single electronic spin one micrometre from an ion detector with nanometre resolution. As a first application, we perform light shift spectroscopy of a narrow optical quadrupole transition. Finally, we emphasize that the quantum lock-in technique is generic and can potentially enhance the sensitivity of any quantum sensor.

Electrons have an intrinsic, indivisible, magnetic dipole aligned with their internal angular momentum (spin)1 . The magnetic interaction between two electrons can therefore impose a change in their spin orientation. Similar dipolar magnetic interactions exists between other spin systems and were studied experimentally. Examples include the interaction between an electron and its nucleus or between several multi-electron spin complexes 2-8 . The process for two electrons, however, was never observed in experiment. The challenge is two-fold.At the atomic scale, where the coupling is relatively large, the magnetic interaction is often overshadowed by the much larger coulomb exchange counterpart 2 . In typical situations where exchange is negligible, magnetic interactions are also very weak and well below ambient magnetic noise. Here we report on the first measurement of the magnetic interaction between two electronic spins. To this end, we used the ground state valence electrons of two 88 Sr + ions, co-trapped in an electric Paul trap and separated by more than two micrometers.We measured the weak, millihertz scale (alternatively 10 −18 eV or 10 −14 K), magnetic interaction between their electronic spins. This, in the presence of magnetic noise that was six * Current address: Physical Measurement

Particle localization is an essential ingredient in quantum Hall physics [1,2]. In conventional high mobility two-dimensional electron systems Coulomb interactions were shown to compete with disorder and to play a central role in particle localization [3]. Here we address the nature of localization in graphene where the carrier mobility, quantifying the disorder, is two to four orders of magnitude smaller [4,5,6,7,8,9,10]. We image the electronic density of states and the localized state spectrum of a graphene flake in the quantum Hall regime with a scanning single electron transistor [11]. Our microscopic approach provides direct insight into the nature of localization. Surprisingly, despite strong disorder, our findings indicate that localization in graphene is not dominated by single particle physics, but rather by a competition between the underlying disorder potential and the repulsive Coulomb interaction responsible for screening.Comment: 18 pages, including 5 figure