Optical studies of broken time reversal symmetry in solids

Polar Kerr effect is a magneto-optic phenomenon in which normally incident plane-polarized light, upon reflection from a ferromagnetic medium, has its major axis of polarization rotated relative to that of the incident beam. The angle between polarization axes of the incident and reflected beams, called the Kerr angle, is then proportional to the component of the sample′s magnetic moment parallel to the incident light.

Although Kerr rotation in materials commonly arises from an out-of-plane ferromagnetic moment, it is well worth noting that any effect that breaks time reversal symmetry with an out-of-plane moment can give rise to a Kerr signal. The deviation of Kerr angle from zero, in particular, is of interest for detection of such time reversal symmetry-breaking states, regardless of their origin.

The Sagnac interferometry project grew out of a collaboration between Aharon Kapitulnik and Prof. Marty Fejer to design an instrument that could detect possible signatures of anyon superconductivity in high-Tc superconductors. The basic principle - the Sagnac effect- has been applied to many problems in measurement since its first use in 1913. Here we use it to measure polar Kerr effect (or Faraday effect, in transmission).

The current generation of this project uses a modified interferometer design developed by Jing Xia (now a faculty at UC Irvine - webpage) and Peter Beyersdorf (now at San José State University) [ref: Modified Sagnac interferometer for high-sensitivity magneto-optic measurements at cryogenic temperatures. Jing Xia, Peter T. Beyersdorf, M. M. Fejer, and Aharon Kapitulnik, Appl. Phys. Lett. 89, 062508 (2006) ]. By routing light along the fast and slow axes of polarization-maintaining (PM) fiber, we are able to effectively create a zero-area Sagnac loop that is sensitive (to one part in 10^6) to nonreciprocal effects arising only from interaction with the sample. As built, this instrument is shot noise limited above 10 uW incident optical power, with very low drifts with temperature (~50 nrad from 4K to room temperature) and in time (~20 nrad per day). Further improvement of the Sagnac apparatus was achieved by Alex Fried who constructed an integrated zero-area loop Sagnac system using strictly fiber-optic coupled optical components, while demonstrating operation at a new wavelength, 820 nm. A schematic of this latest interferometer is shown below.


The improved sensitivity allows us to search for signatures of broken time reversal symmetry, and to make careful studies of these transitions, in many systems of physical interest which were previously inaccessible by this technique. Some of our notable achievements include:

  • Demonstration of time-reversal symmetry breaking in Sr2RuO4 in the superconducting state [Phys. Rev. Lett 97, 167002 (2006)]. See data in zero field (left) and with training field (right):





  • Unambiguous verification that only the "B-Phase" in superconducting UPt3 breaks time reversal symmetry:  


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Quantum phase transitions & superconductor-insulator transitions

Quantum phase transitions (QPT) are at the heart of modern condensed matter physics, particularly the study of  "quantum matter."  Such transitions – where changing an external parameter in the Hamiltonian induces a transition from one quantum ground state to another, fundamentally different one – have been invoked to explain data from a variety of experiments performed on novel materials and artificial structures.  Where the transition is continuous, quantum critical phenomena are expected to give rise to interesting, universal physics which it is common practice to analyze using a straightforward scaling theory, inherited from the classical theory of finite temperature phase transitions. Working at finite temperatures, in general it is difficult to disentangle effects associated with QPT from other effects of the measurements and/or the idiosyncrasy of the system under investigation. Thus, the introduction of SIT in two-dimensions as a  "clean" model system for the study of QPT opened the door for more controlled experiments in this important field.

The physics behind the superconductor-insulator quantum phase transition in disordered films has been the subject of numerous studies. By tuning an external parameter such as the disorder or the magnetic field, the ground state of the system changes continuously from a superconductor, characterized by a resistance that tends to zero as temperature is lowered, to an insulator characterized by a diverging resistance. Experimentally, the magnetic field tuned transition is simplest to study as each experiment requires only one sample realizing a fixed amount of disorder, and the transition occurs at a particular field.

Two main scenarios have been proposed for this transition. In the “dirty boson” scenario Cooper-pairs exhibit enough integrity for the transition to be dominated by boson localization. Alternatively, pairs are broken at the transition and the system above the transition is dominated by fermion physics. In either scenario it is expected that above the transition a large enough system will exhibit insulating behavior as the temperature approaches zero.



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Topological Insulators

Topological insulators are new states of matter that are not adiabatically connected to conventional insulators and semiconductors.  A topological insulator (TI) has a full energy gap in the bulk, and contains gapless surface states that are protected by time reversal symmetry (TRS).  In two dimensions (2D), TIs are also known as quantum spin Hall (QSH) insulators. The QSH insulator state has topologically protected 1D gapless edge states that lie inside the bulk gap. The edge states have a distinct helical property: two states with opposite spin polarization counterpropagate at a given edge.  Thus, the spin is correlated with the direction of motion. The edge states come in Kramers doublets, and TRS ensures the crossing of their energy levels at special points in the Brillouin zone (BZ). Because of this level crossing, the spectrum of a QSH insulator cannot be adiabatically deformed into that of a topologically trivial insulator.

Similarly, in three-dimensions (3D), TIs are fully gapped in the bulk but have topologically protected surface states consisting of an odd number of 2D massless cones of Dirac fermions. The 2D massless Dirac fermion is helical, in the sense that the electron spin points perpendicularly to the momentum, forming a lefthanded helical texture in momentum space. Similar to the 1D helical edge states, a single massless Dirac fermion state is ‘‘holographic,’’ in the sense that it cannot occur in a purely 2D system with TRS, but can exist as the boundary of a 3D insulator.

In our group we have been studying many aspects of topological insulators. Some of these include:



  • MBE growth of high quality Bi2Se3 films and the study of quantum interference effects (such as weak antilocalization) in these films. In particular the observation of weak localization effects as evidence for bulk quantization [Phys. Rev. B 88, 121103 (2013)].

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Search for gravity-like forces at sub-mm scale

Predictions based on recent theories of physics beyond the Standard Model would, if true, lead to deviations from Newtonian gravity on experimentally accessible length scales. To detect or constrain such deviations, we constructed two experiments, both with cantilever-based probes, to directly measure the force between two masses separated by tens of microns. The first experiment used a vibrating "bimorph" system exciting a cantilever-based sensing mass [Phys. Rev. Lett. 90, 151101 (2003)]. Several iterations of this design were used. Improvements over the initial design, mostly in terms of taking care of various systematics, and using more sophisticated statistical analysis were introduced later by Smullin et al. [Phys. Rev. D 72, 122001 (2005)]. The final improvement to this system was introduced by Geraci et al., mostly in terms of improved drive mass and equivalent magnetic calibration [Phys. Rev. D 78, 022002 (2008)].

Our recent experiment tests for variations from Newtonian gravity (1/r2) at length scales below 50 µm. In these recent experimets we use a rotating drive mass and a test mass-bearing cantilever at a separation of ~30 μm [Phys. Rev. D 77, 062006 (2008)]. The drive mass consists of a TeCu disc with one hundred radial trenches cut into the surface which are filled with a thermal-expansion matched epoxy. The trenches are then covered by a deposition of gold to create a smooth equipotential surface to eliminate electrostatic forces along with a gold pattern near the edge of the drive mass to allow for spin speed detection. The mass is then actuated by a helium gas quartz rotor. The assembly of the device with the cantilevers wafer, cantilevers compartment, the quartz rotor, and the full probe are shown here:

The cantilever design considers factors of spring constant, quality factor, the droop in earth′s gravity and resonance frequency. A small number of ~5 µg gold prism test masses are epoxied onto the cantilevers. The cantilever spring constant and test mass give a resonance frequency of about 350 Hz. The frequency of the AC gravitational signal is matched to the cantilever/test-mass resonance by feedback-controlling the spin of the rotor. Radiation pressure damping is used to lower the effective quality factor of the cantilever with out reducing sensitivity.

Force measurement sensitivity of the cantilever is ultimately limited by thermal noise. To minimize the cantilever thermal noise we operate the experiment at 4.2 K. Other sources of noise are mitigated by operating the cantilevers in a vacuum and vibrationally isolating the experiment from the floor and vacuum pumps. Cantilever position readout is achieved through use of an interferometer whose cavity is formed from the cleaved end of an optical fiber and the top surface of the test mass on the cantilever. Reflected optical power is then a simple a function of the cantilever position.

The apparatus provided the most stringent bounds for new forces below ~30µm. The complete phase-space for the strength (in units of Newton's gravity) vs. range (in meters) for a Yukawa-type gravity-like force is given below, with the red line denoting our projected sensitivity when the current version of the apparatus is completed.


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STM Studies of High-Tc Superconductors

The Scanning Tunneling Microscope (STM) has been an important tool in the study of high-temperature superconductors since their discovery.  Indeed, already in 1988 our group published the first atomic resolution study of the surface of Bi2Sr2CaCu2O8, showing the superstructure, and arguing that missing atoms are associated with this superstructure [see figure (a) below, and M.D. Kirk, J. Nogami, A.A. Baski, D.B. Mitzi, A. Kapitulnik, T.H. Geballe, and C.F. Quate, Science, 242, 1673 (1988)]

Fits to a d-wave gap were performed with "good fits" near zero-bias and "bad fits" near the coherence peaks (see (b) above). Initially, a variety of gap sizes and structures were found and introduced much controversy into the subject. Later, a more coherent consensus among different groups emerged regarding the surface properties of these high-Tc materials. To give a few examples, STM studies revealed the nature of the superstructure in BSCCO, the d-wave nature of the gap and its size, the effect of local impurities, the emergence of zero-bias anomalies, and the electronic structure of the core of vortices. More recent measurements suggest that superconductivity may not be homogeneous in high-Tc superconductors. In particular, STM measurements have found spatial variations of the gap size in BSCCO.

While gap inhomogeneities have been found to dominate the electronic structure at large measured bias, more ordered structures underlying the d-wave-like tunneling spectra have been found at lower energies. A current topic of great interest in high-Tc superconductors is the presence of spatial modulations of the charge and spin densities. Theoretical and experimental evidence has been mounting in support of the possibility that their ground state exhibits spin and charge density waves (SDW and CDW), which may be primarily one-dimensional (i.e., stripes).

More on the charge-density modulation and the relation to the STM studies is given in the review paper we wrote: S. A. Kivelson, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, “How to detect fluctuating order in the high-temperature superconductors,” Rev. Mod. Phys. 75 (2003), 1201-41.


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