Dmitri A. Uzdensky

University of Colorado at Boulder , Boulder, Colorado, United States

Are you Dmitri A. Uzdensky?

Claim your profile

Publications (81)232.49 Total impact

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Using two-dimensional particle-in-cell simulations, we characterize the energy spectra of particles accelerated by relativistic magnetic reconnection (without guide field) in collisionless electron-positron plasmas, for a wide range of upstream magnetizations $\sigma$ and system sizes $L$. The particle spectra are well-represented by a power law $\gamma^{-\alpha}$, with a combination of exponential and super-exponential high-energy cutoffs, proportional to $\sigma$ and $L$, respectively. For large $L$ and $\sigma$, the power-law index $\alpha$ approaches about 1.2.
    09/2014;
  • Source
    Dmitri A Uzdensky, Shane Rightley
    [Show abstract] [Hide abstract]
    ABSTRACT: Among the incredibly diverse variety of astrophysical objects, there are some that are characterized by very extreme physical conditions not encountered anywhere else in the Universe. Of special interest are ultra-magnetized systems that possess magnetic fields exceeding the critical quantum field of about 44 TG. There are basically only two classes of such objects: magnetars, whose magnetic activity is manifested, e.g., via their very short but intense gamma-ray flares, and central engines of supernovae (SNe) and gamma-ray bursts (GRBs)-the most powerful explosions in the modern Universe. Figuring out how these complex systems work necessarily requires understanding various plasma processes, both small-scale kinetic and large-scale magnetohydrodynamic (MHD), that govern their behavior. However, the presence of an ultra-strong magnetic field modifies the underlying basic physics to such a great extent that relying on conventional, classical plasma physics is often not justified. Instead, plasma-physical problems relevant to these extreme astrophysical environments call for constructing relativistic quantum plasma (RQP) physics based on quantum electrodynamics (QED). In this review, after briefly describing the astrophysical systems of interest and identifying some of the key plasma-physical problems important to them, we survey the recent progress in the development of such a theory. We first discuss the ways in which the presence of a super-critical field modifies the properties of vacuum and matter and then outline the basic theoretical framework for describing both non-relativistic and RQPs. We then turn to some specific astrophysical applications of relativistic QED plasma physics relevant to magnetar magnetospheres and to central engines of core-collapse SNe and long GRBs. Specifically, we discuss the propagation of light through a magnetar magnetosphere; large-scale MHD processes driving magnetar activity and responsible for jet launching and propagation in GRBs; energy-transport processes governing the thermodynamics of extreme plasma environments; micro-scale kinetic plasma processes important in the interaction of intense electric currents flowing through a magnetar magnetosphere with the neutron star surface; and magnetic reconnection of ultra-strong magnetic fields. Finally, we point out that future progress in applying RQP physics to real astrophysical problems will require the development of suitable numerical modeling capabilities.
    Reports on Progress in Physics 03/2014; 77(3):036902. · 13.23 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The Crab Nebula was formed after the collapse of a massive star about a thousand years ago, leaving behind a pulsar that inflates a bubble of ultra-relativistic electron-positron pairs permeated with magnetic field. The observation of brief but bright flares of energetic gamma rays suggests that pairs are accelerated to PeV energies within a few days; such rapid acceleration cannot be driven by shocks. Here, it is argued that the flares may be the smoking gun of magnetic dissipation in the Nebula. Using 2D and 3D particle-in-cell simulations, it is shown that the observations are consistent with relativistic magnetic reconnection, where pairs are subject to strong radiative cooling. The Crab flares may highlight the importance of relativistic magnetic reconnection in astrophysical sources.
    01/2014; 21(5).
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The discovery of rapid synchrotron gamma-ray flares above 100 MeV from the Crab Nebula has attracted new interest in alternative particle acceleration mechanisms in pulsar wind nebulae. Diffuse shock-acceleration fails to explain the flares because particle acceleration and emission occur during a single or even sub-Larmor timescale. In this regime, the synchrotron energy losses induce a drag force on the particle motion that balances the electric acceleration and prevents the emission of synchrotron radiation above 160 MeV. Previous analytical studies and 2D particle-in-cell (PIC) simulations indicate that relativistic reconnection is a viable mechanism to circumvent the above difficulties. The reconnection electric field localized at X-points linearly accelerates particles with little radiative energy losses. In this paper, we check whether this mechanism survives in 3D, using a set of large PIC simulations with radiation reaction force and with a guide field. In agreement with earlier works, we find that the relativistic drift kink instability deforms and then disrupts the layer, resulting in significant plasma heating but few non-thermal particles. A moderate guide field stabilizes the layer and enables particle acceleration. We report that 3D magnetic reconnection can accelerate particles above the standard radiation reaction limit, although the effect is less pronounced than in 2D with no guide field. We confirm that the highest energy particles form compact bunches within magnetic flux ropes, and a beam tightly confined within the reconnection layer, which could result in the observed Crab flares when, by chance, the beam crosses our line of sight.
    11/2013; 782(2).
  • Dmitri A. Uzdensky
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we consider two outstanding intertwined problems in modern high-energy astrophysics: (1) the vertical-thermal structure of an optically thick accretion disk heated by the dissipation of magnetohydrodynamic turbulence driven by the magnetorotational instability (MRI), and (2) determining the fraction of the accretion power released in the corona above the disk. For simplicity, we consider a gas-pressure-dominated disk and assume a constant opacity. We argue that the local turbulent dissipation rate due to the disruption of the MRI channel flows by secondary parasitic instabilities should be uniform across most of the disk, almost up to the disk photosphere. We then obtain a self-consistent analytical solution for the vertical thermal structure of the disk, governed by the balance between the heating by MRI turbulence and the cooling by radiative diffusion. Next, we argue that the coronal power fraction is determined by the competition between the Parker instability, viewed as a parasitic instability feeding off of MRI channel flows, and other parasitic instabilities. We show that the Parker instability inevitably becomes important near the disk surface, leading to a certain lower limit on the coronal power. While most of the analysis in this paper focuses on the case of a disk threaded by an externally imposed vertical magnetic field, we also discuss the zero net flux case, in which the magnetic field is produced by the MRI dynamo itself, and show that most of our arguments and conclusions should be valid in this case as well.
    The Astrophysical Journal 09/2013; 775(2):103. · 6.73 Impact Factor
  • Jianhua Cheng, Scott E. Parker, Yang Chen, Dmitri A. Uzdensky
    [Show abstract] [Hide abstract]
    ABSTRACT: A second-order accurate semi-implicit Lorentz force ions, fluid electrons δfδf hybrid model has been developed using a “current closure” scheme. The model assumes quasi-neutrality and is fully electromagnetic. The implicit field solver improves numerical accuracy by separating the equilibrium terms in the presence of small perturbations. The equilibrium part of the generalized Ohm’s law is solved by direct matrix inversion along the direction of gradients for every Fourier mode in the other two directions, while the nonlinear part is solved iteratively. The simulation has been benchmarked on Alfvén waves, ion sound waves and whistler waves against analytical dispersion relation in a slab. In particular, the first-order and second-order schemes are compared by studying the numerical damping of whistler waves. The full evolution of the resistive tearing mode using the Harris sheet equilibrium is also investigated. The linear growth rate and mode structure are compared with the resistive MHD theory. Important tearing mode nonlinear phenomena such as the Rutherford regime and saturation are demonstrated. We also presented systematic study of Rutherford growth rates and saturation island width, which is consistent with previous MHD studies.
    Journal of Computational Physics 07/2013; 245:364–375. · 2.14 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: Magnetic reconnection converts magnetic field energy into particle kinetic energy, accelerating particles to sufficient energies to emit gamma-ray synchrotron radiation in astrophysical contexts, possibly including pulsar wind nebulae, Gamma-Ray Bursts, and blazar jets. A balance between acceleration (by the electric field E) and synchrotron braking (while orbiting a B-field line) limits particle energy so that synchrotron processes cannot emit photons above about 160 MeV, unless E > B. However, short, intense gamma-ray flares of much higher energies have recently been observed in the Crab nebula. This work demonstrates, using 2D simulations, that reconnection in relativistic electron-positron pair plasmas can accelerate particles in Speiser orbits around the magnetic null (where E > B) such that the particles can emit synchrotron photons above the 160 MeV limit. Furthermore, reconnection bunches particles and focuses them into beams; high-energy synchrotron radiation is also strongly beamed, and the sweeping of the beam across the observer's line of sight can explain the fast time variability of the flares.
    04/2013;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: It is generally accepted that astrophysical sources cannot emit synchrotron radiation above 160 MeV in their rest frame. This limit is given by the balance between the accelerating electric force and the radiation reaction force acting on the electrons. The discovery of synchrotron gamma-ray flares in the Crab Nebula, well above this limit, challenges this classical picture of particle acceleration. To overcome this limit, particles must accelerate in a region of high electric field and low magnetic field. This is possible only with a non-ideal magnetohydrodynamic process, like magnetic reconnection. We present the first numerical evidence of particle acceleration beyond the synchrotron burnoff limit, using a set of 2D particle-in-cell simulations of ultra-relativistic pair plasma reconnection. We use a new code, Zeltron, that includes self-consistently the radiation reaction force in the equation of motion of the particles. We demonstrate that the most energetic particles move back and forth across the reconnection layer, following relativistic Speiser orbits. These particles then radiate >160 MeV synchrotron radiation rapidly, within a fraction of a full gyration, after they exit the layer. Our analysis shows that the high-energy synchrotron flux is highly variable in time because of the strong anisotropy and inhomogeneity of the energetic particles. We discover a robust positive correlation between the flux and the cut-off energy of the emitted radiation, mimicking the effect of relativistic Doppler amplification. A strong guide field quenches the emission of >160 MeV synchrotron radiation. Our results are consistent with the observed properties of the Crab flares, supporting the reconnection scenario.
    The Astrophysical Journal 02/2013; 770(2). · 6.73 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We develop a framework for studying the statistical properties of current sheets in numerical simulations of 3D magnetohydrodynamic (MHD) turbulence. We describe an algorithm that identifies current sheets in a simulation snapshot and then determines their geometrical properties (including length, width, and thickness) and intensities (peak current density and total energy dissipation rate). We then apply this procedure to simulations of reduced MHD turbulence and perform a statistical analysis on the obtained population of current sheets. We evaluate the role of reconnection by separately studying the populations of current sheets which contain magnetic X-points and those which do not. We find that the statistical properties of the two populations are different in general. We compare the scaling of these properties to phenomenological predictions obtained for the inertial range of MHD turbulence. Finally, we test whether the reconnecting current sheets are consistent with the Sweet-Parker model.
    The Astrophysical Journal 02/2013; 771(2). · 6.73 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A two-dimensional (2D) linear theory of the instability of Sweet-Parker (SP) current sheets is developed in the framework of reduced magnetohydrodynamics. A local analysis is performed taking into account the dependence of a generic equilibrium profile on the outflow coordinate. The plasmoid instability [Loureiro et al., Phys. Plasmas 14, 100703 (2007)] is recovered, i.e., current sheets are unstable to the formation of a large-wave-number chain of plasmoids (k_{max}L_{CS}∼S^{3/8}, where k_{max} is the wave number of fastest growing mode, S=L_{CS}V_{A}/η is the Lundquist number, L_{CS} is the length of the sheet, V_{A} is the Alfvén speed, and η is the plasma resistivity), which grows super Alfvénically fast (γ_{max}τ_{A}∼S^{1/4}, where γ_{max} is the maximum growth rate, and τ_{A}=L_{CS}/V_{A}). For typical background profiles, the growth rate and the wave number are found to increase in the outflow direction. This is due to the presence of another mode, the Kelvin-Helmholtz (KH) instability, which is triggered at the periphery of the layer, where the outflow velocity exceeds the Alfvén speed associated with the upstream magnetic field. The KH instability grows even faster than the plasmoid instability γ_{max}τ_{A}∼k_{max}L_{CS}∼S^{1/2}. The effect of viscosity (ν) on the plasmoid instability is also addressed. In the limit of large magnetic Prandtl numbers Pm=ν/η, it is found that γ_{max}∼S^{1/4}Pm^{-5/8} and k_{max}L_{CS}∼S^{3/8}Pm^{-3/16}, leading to the prediction that the critical Lundquist number for plasmoid instability in the Pm≫1 regime is S_{crit}∼10^{4}Pm^{1/2}. These results are verified via direct numerical simulation of the linearized equations, using an analytical 2D SP equilibrium solution.
    Physical Review E 01/2013; 87(1-1):013102. · 2.31 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: We report on the first study of energetic particles and radiation angular distributions generated in relativistic collisionless pair plasma reconnection, using 2.5-dimensional particle-incell simulations. We have discovered that the energetic particles are focused within a small solid angle, and bunched into compact regions inside magnetic islands. In addition, we find that the synchrotron radiation emitted by these particles, as seen by an external observer, is tightly beamed and variable on time scales much shorter than the light-crossing time of the system. This energy dependent "kinetic beaming" differs fundamentally from the achromatic Doppler beaming usually ascribed to relativistic jets. Our findings can account for the puzzling discoveries of bright, short flares seen in high-energy gamma rays, especially from the Crab Nebula and from blazars.
    12/2012;
  • Source
    Dmitri A. Uzdensky, Anatoly Spitkovsky
    [Show abstract] [Hide abstract]
    ABSTRACT: The magnetosphere of a rotating pulsar naturally develops a current sheet beyond the light cylinder (LC). Magnetic reconnection in this current sheet inevitably dissipates a nontrivial fraction of the pulsar spin-down power within a few LC radii. We develop a basic physical picture of reconnection in this environment and discuss its implications for the observed pulsed gamma-ray emission. We argue that reconnection proceeds in the plasmoid-dominated regime, via an hierarchical chain of multiple secondary islands/flux ropes. The inter-plasmoid reconnection layers are subject to strong synchrotron cooling, leading to significant plasma compression. Using the conditions of pressure balance across these current layers, the balance between the heating by magnetic energy dissipation and synchrotron cooling, and Ampere's law, we obtain simple estimates for key parameters of the layers --- temperature, density, and layer thickness. In the comoving frame of the relativistic pulsar wind just outside of the equatorial current sheet, these basic parameters are uniquely determined by the strength of the reconnecting upstream magnetic field. For the case of the Crab pulsar, we find them to be of order 10 GeV, $10^{13} cm^{-3}$, and 10 cm, respectively. After accounting for the bulk Doppler boosting due to the pulsar wind, the synchrotron and inverse-Compton emission from the reconnecting current sheet can explain the observed pulsed high-energy (GeV) and VHE (~100 GeV) radiation, respectively. Also, we suggest that the rapid relative motions of the secondary plasmoids in the hierarchical chain may contribute to the production of the pulsar radio emission.
    The Astrophysical Journal 10/2012; · 6.73 Impact Factor
  • Jianhua Cheng, Scott Parker, Dmitri Uzdensky, Yang Chen
    [Show abstract] [Hide abstract]
    ABSTRACT: Secondary islands have been intensively studied due to their role in the energy dissipation process of reconnection. Recently, we have studied magnetic reconnection initiated by the tearing instability. The simulation uses a hybrid model with Lorentz force ions and fluid electrons. For large &'circ; tearing mode, we have observed secondary islands forming and coalescing in the nonlinear regime. The competition between the two processes strongly influences the reconnection rate and eventually leads the reconnection to a steady state. To better understand these phenomena, detailed diagnostics are performed. The kinetic treatment of ions allows us to record the abnormally heated ions and the ion flow pattern around the secondary islands. These ion diagnostics help to explain how the hot ions are heated and hence how the magnetic energy is dissipated to the ion kinetic energy. Another interesting problem is the large in-plane electric fields inside the secondary islands, which has been observed in magnetosphere. Our simulation will help understand the origin of these in-plane electric fields.
    10/2012;
  • Shane Rightley, Dmitri Uzdensky
    [Show abstract] [Hide abstract]
    ABSTRACT: Plasmas in which the inter-particle spacing approaches the thermal de Broglie wavelength are subject to quantum statistical effects due to Pauli exclusion, and many familiar plasma phenomena are modified on such length scales because of the Heisenberg uncertainty principle. The question of how to model these quantum plasmas is a naturally interesting one, as it pushes the envelope of our knowledge of plasma physics and applies the well-established principles of quantum mechanics in a novel context. Such models are important for microelectronic systems, dense laser-produced plasmas, and some extreme astrophysical environments. For completely degenerate plasmas, both kinetic and fluid theories have already been developed. In this presentation, unmagnetized Fermi-Dirac equilibrium plasmas with finite temperature and arbitrary degree of degeneracy are considered. Linear dispersion relations for electrostatic waves and oscillations, including Landau damping, are derived and analyzed. The analysis is carried out using a self-consistent mean-field quantum kinetic model (the Wigner-Poisson system). Growth of waves due to kinetic instabilities, such as the Buneman and bump-on-tail instabilities, is also considered.
    10/2012;
  • [Show abstract] [Hide abstract]
    ABSTRACT: Magnetic reconnection is one of a few astrophysical mechanisms that can accelerate particles to energies sufficient to emit observable high-energy radiation. This work reports on 2D simulations of reconnection in relativistic electron-positron pair plasmas, which may power gamma-ray emission from pulsar wind nebulae (PWNe), Gamma-Ray Bursts (GRBs), and blazar jets. The most important new discovery is the strong, energy-dependent angular anisotropy and spatial inhomogeneity of accelerated particles: high-energy particles are bunched in space and focused into beams mostly confined to the reconnection layer midplane. Another important advance is the calculation of the associated radiative signatures (spectra and light curves) seen by a distant observer. The synchrotron and inverse Compton radiation from the high-energy particles is likewise focused in narrow beams. The beams sweep back and forth within the midplane, so that an observer sees intense bursts (only) when a beam crosses the line of sight. The resulting rapid variability, on timescales much shorter than the light-crossing time of the reconnection region, could explain the short, intense gamma-ray flares observed in blazar jets and PWNe, including the GeV flares recently discovered in the Crab nebula.
    10/2012;
  • Dmitri Uzdensky, Anatoly Spitkovsky
    [Show abstract] [Hide abstract]
    ABSTRACT: The magnetosphere of a rotating pulsar naturally develops a current sheet beyond the light cylinder (LC). Magnetic reconnection in this current sheet inevitably dissipates a nontrivial fraction of the pulsar spin-down power within a few LC radii. In this presentation, a basic physical picture of reconnection in this environment is developed. It is shown that reconnection proceeds in the plasmoid-dominated regime, via an hierarchical chain of multiple secondary islands/flux ropes. The inter-plasmoid reconnection layers are subject to strong synchrotron cooling, leading to significant plasma compression. The basic parameters of these current layers --- temperature, density, and layer thickness --- are estimated in terms of the upstream magnetic field. It is argued that, after accounting for the bulk Doppler boosting, the synchrotron and inverse-Compton emission mechanisms can explain the observed pulsed high-energy (GeV) and VHE (˜ 100 GeV) radiation, respectively. The motions of the secondary plasmoids may contribute to the pulsar's radio emission.
    10/2012;
  • Source
    Dmitri A. Uzdensky
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we consider two outstanding intertwined problems in modern high-energy astrophysics: (1) the vertical thermal structure of an optically thick accretion disk heated by the dissipation of magnetohydrodynamic (MHD) turbulence driven by the magneto-rotational instability (MRI), and (2) determining the fraction of the accretion power released in the corona above the disk. For simplicity, we consider a gas-pressure-dominated disk and assume a constant opacity. We argue that the local turbulent dissipation rate due to the disruption of MRI channel flows by secondary parasitic instabilities should be uniform across most of the disk, almost up to the disk photosphere. We then obtain a self-consistent analytical solution for the vertical thermal structure of the disk, governed by the balance between the heating by MRI turbulence and the cooling by radiative diffusion. Next, we argue that the coronal power fraction is determined by the competition between the Parker instability, viewed as a parasitic instability feeding off of MRI channel flows, and other parasitic instabilities. We show that the Parker instability inevitably becomes important near the disk surface, leading to a certain lower limit on the coronal power. While most of the analysis in this paper focuses on the case of a disk threaded by an externally imposed vertical magnetic field, we also discuss the zero-net-flux case, in which the magnetic field is produced by the MRI dynamo itself, and show that most of our arguments and conclusions should be valid in this case as well.
    05/2012;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We report on the first study of the angular distribution of energetic particles and radiation generated in relativistic collisionless electron-positron pair plasma reconnection, using two-dimensional particle-in-cell simulations. We discover a strong anisotropy of the particles accelerated by reconnection and the associated strong beaming of their radiation. The focusing of particles and radiation increases with their energy; in this sense, this "kinetic beaming" effect differs fundamentally from the relativistic Doppler beaming usually invoked in high-energy astrophysics, in which all photons are focused and boosted achromatically. We also present, for the first time, the modeling of the synchrotron emission as seen by an external observer during the reconnection process. The expected lightcurves comprise several bright symmetric sub-flares emitted by the energetic beam of particles sweeping across the line of sight intermittently, and exhibit super-fast time variability as short as about one tenth of the system light-crossing time. The concentration of the energetic particles into compact regions inside magnetic islands and particle anisotropy explain the rapid variability. This radiative signature of reconnection can account for the brightness and variability of the gamma-ray flares in the Crab Nebula and in blazars.
    The Astrophysical Journal Letters 05/2012; 754(2). · 6.35 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Magnetic reconnection is a well known plasma process believed to lie at the heart of a variety of phenomena such as sub-storms in the Earth's magnetosphere, solar/stellar and accretion-disk flares, sawteeth activity in fusion devices, etc. During reconnection, the global magnetic field topology changes rapidly, leading to the violent release of magnetic energy. Over the past few years, the basic understanding of this fundamental process has undergone profound changes. The validity of the most basic, and widely accepted, reconnection paradigm - the famous Sweet-Parker (SP) model, which predicts that, in MHD, reconnection is extremely slow, its rate scaling as S-1/2, where S is the Lundquist number of the system - has been called into question as it was analytically demonstrated that, for S ≫ 1, SP-like current sheets are violently unstable to the formation of a large number of secondary islands, or plasmoids. Subsequent numerical simulations in 2D have confirmed the validity of the linear theory, and shown that plasmoids quickly grow to become wider than the thickness of the original SP current sheet, thus effectively changing the underlying reconnection geometry. Ensuing numerical work has revealed that the process of plasmoid formation, coalescence and ejection from the sheet drastically modifies the steady state picture assumed by Sweet and Parker, and leads to the unexpected result that MHD reconnection is independent of S. In this talk, we review these recent developments and present results from three-dimensional simulations of high-Lundquist number reconnection in the presence of a guide field. A parametric study varying the strength of the guide field is presented. Plasmoid flux and width distribution functions are quantified and compared with corresponding two dimensional simulations.
    04/2012;
  • Jianhua Cheng, Yang Chen, Scott Parker, Dmitri Uzdensky
    [Show abstract] [Hide abstract]
    ABSTRACT: We have developed a second-order accurate semi-implicit δ method for kinetic MHD simulation with Lorentz force ions and fluid electrons. The model has been used to study the resistive tearing mode instability, which involves multiple spatial scales. In small 'circ; cases, the linear growth rate and eigenmode structure are consistent with resistive MHD analysis. The Rutherford stage and saturation are demonstrated, but the simulation exhibits different saturation island widths compared with previous MHD simulations. In large 'circ; cases, nonlinear simulations show multiple islands forming, followed by the islands coalescing at later times. The competition between these two processes strongly influences the reconnection rates and eventually leads to a steady state reconnection. We will present various parameter studies and show that our hybrid results agree with fluid analysis in certain limits (e.g., relatively large resisitivities).
    03/2012;

Publication Stats

1k Citations
232.49 Total Impact Points

Institutions

  • 2010–2014
    • University of Colorado at Boulder
      • • Department of Physics
      • • Center for Integrated Plasma Studies
      Boulder, Colorado, United States
  • 2013
    • Technical University of Lisbon
      Lisboa, Lisbon, Portugal
  • 2011–2012
    • University of Colorado
      • Department of Physics
      Denver, Colorado, United States
  • 2009
    • University of California, Berkeley
      Berkeley, California, United States
  • 2000–2008
    • Princeton University
      • • Department of Astrophysical Sciences
      • • Princeton Plasma Physics Laboratory
      Princeton, NJ, United States
  • 2002–2005
    • University of California, Santa Barbara
      • Kavli Institute for Theoretical Physics
      Santa Barbara, California, United States
    • Massachusetts Institute of Technology
      Cambridge, Massachusetts, United States