Filling a cavity with photons, and watching them leave

Figure 1: By coupling a nonlinear system, such as an atom, to the electromagnetic field, it is possible to create Fock states (eigenstates of the harmonic oscillator). (Top) Brune et al. send atoms (left) into a cavity (center). The atoms are prepared with pulse P1 to be in a superposition of states |e〉 and |g〉 before they enter the cavity. The relative phase between these states, which is converted to probability amplitudes for |e〉 and |g〉 with pulse P2 when the atoms exit the cavity, depends on the number of photons in the cavity. (Bottom) In place of a cavity, Wang et al. create an electromagnetic field in a microwave resonator (blue). A superconducting qubit, acting as an artificial atom, couples to the center conductor.

The harmonic oscillator is one of the most fundamental systems in quantum mechanics. Equipped with its solution—one of the first that every physics student learns to calculate exactly—it is possible to describe realistic problems, from phonons in a crystal to the interaction of light with an atom. It is perhaps ironic, then, how challenging it is to actually prepare a pure harmonic oscillator state with a well-defined excitation number n, also known as a Fock state. Now, using different methods, two groups—Michel Brune and colleagues at the Laboratoire Kastler Brossel of the CRNS and Collège de France, both in Paris, and Haohua Wang and colleagues at the University of California in Santa Barbara (UCSB)—have created these nonclassical states of the harmonic oscillator and performed a detailed study of how they decay in time. The experiments, reported in Physical Review Letters [1, 2] demonstrate that the lifetime of a Fock state with excitation number n scales as 1/n, as predicted by theory.

Since it is of relevance to both experiments, consider one of the simplest realizations of the harmonic oscillator: the electromagnetic field. Its excited states are photons and a Fock state corresponds to the creation of n photons with the same energy, ħω. However, when using a classical source with a well-defined frequency (such as a laser) to generate an electromagnetic field, the result is a coherent state: a superposition of Fock states that is nearly indistinguishable from a classical state. The reason is that the energy spectrum of the harmonic oscillator is linear, such that the energy provided by the source will spread over a wide distribution of Fock states. Instead, to directly prepare a purely nonclassical photon state and observe quantum effects, we need to make a sufficiently strong interaction between the electromagnetic field and an additional, nonlinear, component. This is the heart of the experiments from Brune et al. and Wang et al.

In the work from the CNRS group, the nonlinear components are atoms with excited states that are not evenly spaced in energy. In particular, they use circular Rydberg atoms (atoms in highly excited states and with maximum angular momentum: l=|m|=n-1 ). These atoms have large dipole moments that couple strongly to the microwave photons that are used in the experiments. To enhance this coupling, the photons are confined to a cavity made out of two high-quality mirrors that face each other. The CNRS group has studied the interaction between light and matter in a cavity, also known as cavity quantum electrodynamics (QED) [3, 4, 5], with circular Rydberg atoms over the last 20 years. Thus, they have been able to carefully design the experiment so that the coupling strength between the circular Rydberg atoms and the photons in the cavity overwhelms all the decay rates of the combined system: the single-photon decay rate κ out of the cavity, the atomic decay rate Γ1, and the atomic dephasing rate Γϕ. It is worth noting that cavities can now be fabricated with extremely large quality factors such that the single photon lifetime 1/κ can be as large as 0.1 seconds [6]. This is long enough for photons to travel 39,000 km back and forth between the two mirrors separated by 2.7 cm!

The experiment by the CNRS group relies on the conditional preparation of Fock states. They first create a coherent state of the cavity field with a microwave source. Then, to prepare a pure but randomly chosen Fock state, they rely upon the magic of quantum measurements: they perform a measurement of photon number with result n, which projects the classical field to the quantum state |n〉. The CNRS team’s measurement is special in the sense that it involves no energy exchange. In these so-called quantum nondemolition measurements, atoms that are nonresonant with the photons in the cavity are sent one by one across the cavity. Before entering the cavity, each atom is prepared in a superposition of two of its internal states, labeled |g〉 and |e〉 for ground and excited states (see Fig.1, top ). During the time that the atom is in the cavity, this superposition acquires a relative phase proportional to the number of photons in the cavity. After leaving the cavity, a second pulse converts this phase information to probability amplitudes for |g〉 and |e〉, which are then measured by state-selective ionization of the atom. By repeating this process with sufficiently many atoms (roughly 110 in the experiment), the Fock state, which was randomly selected from the initial field distribution, is prepared with high accuracy.

After preparing the Fock state |n〉, every atom that is then sent through the cavity reveals information about the subsequent evolution of the cavity field. In this way, Brune et al. follow the time evolution of Fock states n=0 through n=7, and can track how these states decay, something known as quantum process tomography. As expected from theory, they find that the decay rate of a Fock state with n photons is nκ, which is n times faster than for a Fock state with n=1. This enhanced rate simply reflects the fact that each additional photon has its own probability to decay, speeding up the relaxation. Since in this experiment preparing a Fock state is a random process, completely characterizing the state is a costly enterprise, requiring up to a million single atom measurements.

In parallel, researchers have been developing an on-chip version of cavity QED, also known as circuit QED. In this system, the many Rydberg atoms are replaced by a single superconducting qubit and the cavity is a transmission-line resonator, essentially a one-dimensional superconducting coaxial cable (see Fig. 1, bottom). Gaps in the center conductor of the resonator play the role of the mirrors in cavity QED. Moreover, similarly to the cavity used by the CNRS group, excitations of the resonator are microwave photons. These essentially one-dimensional cavities have a small mode volume, resulting in a large electric field per photon. Superconducting qubits are electrical circuits based on Josephson junctions. With their well-defined energy levels, they behave as artificial atoms, providing the essential nonlinearity. In addition, superconducting qubits have a large effective dipole moment. As a result, this system can easily reach the strong coupling regime of cavity QED [7]. Groups at Yale [8] and Delft (in this last case using a different type of on-chip cavity) [9] first demonstrated strong coupling between a superconducting qubit and a microwave resonator in 2004. Because of the very strong coupling, it was predicted [10] and soon confirmed [11] that in circuit QED, Fock states could be resolved by measuring the qubit absorption spectrum. Earlier this year, the UCSB group showed they could prepare Fock states with up to n=6 photons [12].

In the new experiments from the UCSB group [2], the superconducting qubit, playing the role of the atom, is capacitively coupled to the center conductor of the resonator (Fig. 1, bottom). An advantage of this artificial atom is that the energy difference between its |g〉 and |e〉 states can be tuned into and out of resonance with the resonator frequency. Starting with the qubit out of resonance and in its ground state |g〉, a classical source is used to pump it to |e〉. This energy quantum is then transferred to the resonator by tuning the qubit so it is in resonance with the microwave resonator for an appropriate amount of time. By repeating this process, Fock states with n up to 15 have been created. The microwave resonator’s state can in turn be determined by tuning the qubit into resonance with the resonator and measuring the undriven Rabi oscillations of the qubit between |g〉 and |e〉. Since the frequency of these oscillations depends characteristically on the photon number in the microwave resonator, Wang et al. can extract information about the photon distribution of the Fock state. Similarly to the CNRS group, they find that the n-photon Fock state decays at the enhanced rate nκ.

The experiments now reported by Brune et al. and Wang et al. go beyond the groups’ earlier work in that both are able to create Fock states with large n and reach a level of precision with which to probe the decay of these states. By combining two prototypical systems—harmonic oscillators and two-level systems—cavity QED has established itself in the last 20 years as a unique test-bed for fundamental investigations of quantum mechanics. With the recent developments, such as cavities with high-quality factors and circuit QED, new ways to generate, control and measure non-classical states of light are now possible and more surprises are sure to be on their way.

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A glassy counterpart to supersolids

Figure 1: (a) A classical system of Brownian spheres may crystallize at low temperatures if cooled slowly. By contrast, a bi-disperse system (i.e., consisting of two different kinds of hard spheres) such as the one shown cannot crystallize and will jam into a glass at high densities and low temperatures. Similarly, a rapidly cooled system of hard spheres does not have enough “time” to crystallize and forms a glass instead. Using the mapping of [7], it is seen that the quantum analog of the classical hard-sphere Brownian system is that of spheres with sticky interactions. In the classical hard-sphere limit, only pair interactions appear in the corresponding quantum system. (b) The pair potential in the corresponding quantum system given by Biroli et al. [1] (for two values of λ, a parameter that adjusts how hard the classical potential is, where r is the interparticle distance and σ is the sphere diameter) and the corresponding phase diagrams, shown in (c). Top: The solid curve indicates the classical Brownian hard-sphere phase diagram (pressure P versus volume fraction ϕ) for uniform annealed systems wherein spheres forming a liquid at low densities pack into a face-centered-cubic (FCC) crystal structure at high densities. The dashed curve shows the phase diagram of a rapidly quenched or bi-disperse system in which crystallization is thwarted and the system becomes a glass instead [random close packing (RCP)]. Bottom: The corresponding quantum phases obtained by the map of [7]. The liquid-to-solid transition of the classical Brownian sphere system maps into a superfluid to supersolid transition. Similarly, the superfluid to superglass transition constitutes an analog of the classical liquid to glass transition. [ Panels (b) and (c) adapted from Biroli et al. [1].]

Glasses are liquids that have ceased to flow on experimentally measurable time scales. By constrast, superfluids flow without any resistance. The existence of a phase characterized by simultaneous glassiness and superfluidity may seem like a clear contradiction. However, in a paper in Physical Review B, Giulio Biroli (Institut de Physique Théorique, France), Claudio Chamon (Boston University), and Francesco Zamponi (École Normale Supérieure, France) prove that this is not so [1] and illustrate theoretically the possibility of a “superglass” phase. This phase forms an intriguing amorphous counterpart to the “supersolid” phase [2, 3] that has seen a surge of interest in recent years [4]. Within a “supersolid” phase, superfluidity can occur without disrupting crystalline order.

Interacting quantum particles can indeed form such a superglass phase at very low temperatures and high density, and the work of Biroli et al. confirms the earlier suggestion by Boninsegni, Prokof ’ev, and Svistunov [5] and an investigation by Wu and Philips [6] of such a phase. The superglass phase is characterized by an amorphous density profile, yet at the same time a finite fraction of the particles flow without any resistance as if they were superfluid. Thus the superglass constitutes a glassy counterpart to the supersolid phase.

The approach invoked by Biroli et al. to prove the existence of a superglass is particularly elegant. It relies on mapping [7] viscous classical systems, whose properties are well known, to new many-body quantum systems. In realizing the link between classical and quantum systems to gain insight into the quantum many-body phases, Biroli et al. nicely add an important new result to earlier investigations that built on such similar insights elsewhere. Chester [3] suggested the existence of a supersolid by relying on such a connection. In a similar fashion, Laughlin invoked a highly inspirational analogy [8] between variational (the so-called Jastrow type) wave functions describing fractional quantum Hall systems and a previously studied system of classical charged particles interacting via a logarithmic potential. By using the classical plasma analogy and using known results on it, Laughlin was able to make headway on the challenging many-body quantum problem and construct his highly successful wave functions.

The mapping used by Biroli et al. similarly enables exact results on the quantum problem of superglasses and a detailed correspondence of spatial and temporal correlations between the classical and quantum systems. The authors apply this mapping to a classical system well known to exhibit glassy dynamics—the Brownian hard sphere problem. The quantum counterpart of the classical hard sphere problem is a natural system containing hard sphere interactions [Fig. 1(a),1(b)]. On the classical side of the correspondence, the hard sphere system has been heavily investigated [9, 10, 11]. When the sphere packing density is slowly varied, the classical Brownian hard sphere system undergoes a transition from a liquid at low density to an ordered crystal at high density [9]. When crystallization is thwarted by a rapid increase of the packing density or by, for example, a change of the particle geometry, the system cannot order nicely into a crystal and instead jams into a dense amorphous glass [10, 11].

Biroli et al. noticed that by using the mapping between quantum systems with classical glass-forming systems such as the Brownian spheres, they can obtain nontrivial results. In its simplest form, the mapping of [7] casts the first-order differential equation in time for the dynamics of viscous classical particles as a Schrödinger equation with an effective Hamiltonian. Biroli et al. find that under this mapping, the glassy phase of the classical system translates into a quantum glass of a Bose system. Similarly, the classical solid maps onto a quantum bosonic crystal, resulting in an interesting phase diagram [Fig.1(c)]. The spatiotemporal correlations of the (bosonic) quantum counterpart may be computed by mapping to the classical system. Both the glassy and solid phases harbor a finite Bose-Einstein condensate fraction. Putting all of the pieces together, Biroli et al. provide an important proof of concept of the superglass phase in a simple and precise way. This route may be replicated for classical systems other than the Brownian hard sphere that also display solid and glass phases.

What physical systems might exhibit the new superglass phase? Recent experiments [4, 12] on solid helium-4 exhibit supersolid-type features and have led to a flurry of activity. In the simplest explanation of observations, a fraction of the medium becomes, at low temperatures, a superfluid that decouples from the measurement apparatus. However, the condensate fraction that is required for such an explanation to account for the data does not simply conform with thermodynamic measurement [12]. Rittner and Reppy [13] further found that the measured putative supersolid-type feature is acutely sensitive to the quench rate for solidifying the liquid, while Aoki, Keiderling, and Kojima observed rich hysteresis and memory effects [14]. All of these features can arise from glassy characteristics alone [15, 16]—precisely as in the superglass phase discussed by Biroli et al. It may be that a confluence of both superfluid and glassy features (and their effects on elastic properties of a medium) [17] is at work.

These effects should be observable as experimental consequences of (super-) glassy dynamics, such as disparate relaxation times that could be measured [15]. Typical glass formers indeed typically exhibit relaxations on two different time scales. Cold atom systems may provide another realization of a superglass state. Indeed, a supersolid state of cold atoms in a confining optical lattice was very recently achieved [18]. It is natural to expect a superglass analog of these cold atomic systems.

Superglasses may also have realizations in other areas such as superconductivity and I speculate on these below. For example, consider a lattice version of the continuum system investigated by Biroli et al.: a “lattice superglass.” For charged bosons (e.g., Cooper pairs) on a lattice, such a superglass would correspond to a superconductor with glassy dynamics. In a similar vein, a “lattice supersolid” of Cooper pairs would correspond to a superconductor concomitant with well-defined crystalline (i.e., charge-density wave) order. Indeed, in some heavy fermion compounds as well as in the cuprate and the newly discovered iron arsenide family of high-temperature superconductors [19] there are some indications of nonuniform mesoscale spatial electronic structures and glassy dynamics. Classical glass formers are known to exhibit “dynamical heterogeneities”—a nonuniform distribution of local velocities [20]. I also speculate that “quantum dynamical heterogeneities” may be derived by applying the mapping used by Biroli et al. to classical glass forming systems that exhibit dynamical heterogeneities..

“Spin superglasses” are another possibility. Quantum spin systems in a magnetic field [21] can exhibit a delicate interplay between the formation of singlet states and the tendency of spins to align with the field direction. These systems can be mapped onto a system of bosons with hard-core interactions—just as in the system investigated by Biroli et al. In some spin S=1/2 antiferromagnets in an external magnetic field, triplet states with spins aligned along the field direction can be regarded as hard-core bosons. In many other systems, interactions between quantum spins may also be mapped onto hard-core-type bosonic systems [22, 23]. Invoking these bosonic representations, if a solid or glassy phase appears in a classical Brownian system, then a mapping similar to that of Biroli et al. suggests supersolidity/superglassiness in the corresponding quantum spin system. Recently, there has been much work examining supersolidity in such spin systems, e.g., [23]. It is highly natural to expect new lattice spin superglass counterparts

Finally, even more intriguing superglasses might be possible. In transition-metal compounds, the fractional filling of the 3d atomic shells allows for cooperative orbital ordering [24]. Perhaps low-temperature Bose-condensed glasses of orbitals could appear, forming an orbital superglass. The orbital states may be described by pseudospins [24] that may be mapped to hard-core bosons [22]. The work of Biroli et al. allows us to investigate the possibility of an orbital superglass by knowing the dynamics of hard-core Bose model derived from a classical counterpart. In addition, the classical-to-quantum map of [7] may also suggest a new quantum critical point in related systems. The classical zero temperature jamming transition [25] of hard spheres or disks is a continuous transition with known (dynamical) critical exponents, e.g., [26, 27]. Replicating the mapping used by Biroli et al., we may derive an analog quantum system harboring a zero temperature transition with similar critical exponents. Thus the classical critical point [25, 26, 27] may rear its head anew in the form of “quantum critical jamming.” All of the systems discussed above were free of quenched disorder. Applying the same mapping to classical viscous systems with quenched disorder, we may further arrive at quenched super spin glass analogs of classical spin glass systems [28]. The tantalizing superglass phase may have numerous ramifications.

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The many shapes of spinning drops

Figure 1: (Top) The bifurcation and stability diagram for rotating liquid drops that are held together by surface tension. The plot depicts dimensionless angular velocity as a function of dimensionless angular momentum. The depicted shapes are stable along the solid lines and unstable along the dashed lines. All shapes with more than 2 lobes are unstable. (Bottom) The experimental setup used by Hill and Eaves. The water droplet is suspended by a magnetic field. The field also exerts a torque on a current that runs between two thin wires embedded in the droplet, thus setting the droplet into rotation. A camera monitors the droplet shape from below.

Liquids, such as water or oil, can form droplets that are held together by surface tension—a cohesive force that causes the surface of many liquids to behave as an elastic membrane. Because liquid droplets are fairly simple to study and control under different conditions, they have fascinated experimentalists and theoreticians—including Laplace, Gauss, Poincarè, Chandrasekhar, Bohr, and Wheeler—for more than two centuries. A droplet on a surface, for example, spreads out or balls up depending on its interaction with the surface. But what happens to the shape of droplets in motion? As a freely suspended droplet rotates at higher and higher velocity, the droplet will tend to get pulled apart and deform. Droplets distort according to a minimum energy principle, always seeking the lowest energy state for a given rotational frequency, and theorists predicted that they would form the series of equilibrium shapes shown in the top of Fig. 1 [1].

To see the full range of shapes shown in Fig. 1 requires a carefully designed experiment. First, the effects of gravity must be removed, since the theoretical predictions assume a freely suspended droplet. Second, the viscous drag on the droplet must be as small as possible. Finally, the droplets must spin at a controllable rate. Writing in Physical Review Letters, Richard Hill and Laurence Eaves of the University of Nottingham in the UK have been able to achieve this by ingeniously combining diamagnetic levitation of water droplets in air with a liquid electric motor technique to spin up the droplets [2]. They are able to observe several different families of rotating water droplets, compare with theoretical expectations, and even record the centimeter-sized droplets in several movies. Because liquid drop models are used to describe systems like atomic nuclei and black holes that cannot be controlled in the laboratory, their work opens the way to a whole new series of measurements of direct interest to several branches of physics.

The first experimental investigations of rotating fluid droplets were conducted by a blind Belgian physicist named Joseph-Antoine-Ferdinand Plateau [3] (Plateau, who also performed experiments with light, once kept his naked eyes fixed for too long on the sun, which brought on a choroid inflammation and total blindness in 1843). Plateau suspended olive oil in a mixture of water and alcohol that had the same density as the oil, thereby balancing gravity with the buoyant force and leaving the droplet effectively weightless. Starting with a nonrotating drop of spherical shape, Plateau found that as he rotated the droplet at higher and higher speeds, the droplet transformed from a spherical shape (at zero rotation), to axisymmetric shapes (at slow speeds), to ellipsoidal and 2-lobed “peanut” shapes, and finally a toroidal shape at very large rotational speeds. These are the same shapes shown in Fig. 1, which were fully understood theoretically about 100 years after Plateau’s experiments.

Unfortunately, in Plateau’s setup the surrounding fluid exerted a large viscous drag on the rotating droplets, which induced unwanted shape deformations. The droplets were also hard to control and to spin, therefore comparisons with theoretical results were mainly qualitative. More recently, precise experiments with silicone oil droplets have been performed in orbiting spacecraft [4], and it has been possible to see the 2-lobed droplet shape. Unfortunately, special care is needed to control and spin up the droplet using acoustic pressure waves, not to mention that it requires going into space!

Hill and Eaves found an innovative way to compensate for the effects of gravity and to make the droplets spin. They solved the gravity problem by using diamagnetic levitation (Fig. 1, bottom). Many substances, including water, are diamagnetic. When acted upon by external magnetic fields, the electrons in diamagnetic materials rearrange themselves creating small currents, which oppose the external field. In a magnetic field that is not constant in space, a diamagnet will always seek the lower field strength. Thus, diamagnetic substances, despite being weakly magnetic, can easily be levitated if the exterior magnetic field has high enough intensity and a well-defined gradient (in fact, even animals, such as frogs, can be levitated diamagnetically [5]).

To set the droplet into rotation, the Nottingham group designed a “liquid electric motor,” also shown in the bottom part of Fig. 1. They inserted two thin wire electrodes in the droplet, with one wire aligned with the droplet center and with the magnetic field. The other wire was set parallel to the first a distance d away. A current I was set to flow through these wires. It follows that a Lorentz force IdB acts on the droplet, causing a torque of IBd2/2 on the droplet that accelerates it to rotate around the vertical axis with a frequency that can be easily controlled by adjusting the current.

There is one final catch to the experiment: the off-axis electrode can excite small amplitude waves on the surface of the droplet, like a boat moving on a water surface. These excited waves have an important effect, in that they can help stabilize otherwise unstable configurations [6]. For instance, 3-lobed configurations cannot usually be observed since they are unstable and rapidly decay into another configuration (typically the 2-lobed shapes). However, if the small waves excited by the off-axis electrode are in resonance with this configuration, they are able to sustain 3-lobed configurations far longer than expected. As the rotation of the droplet increases, it progresses through an elliptical (not shown in Fig. 1) and then a 2-lobed shape. As the rotation increases even further, the droplet assumes a 3-lobed configuration, which was measured accurately for the first time in this experiment. It turns out that the critical rotational frequency at which 2 and 3-lobed shapes start to develop agrees very well with theoretical predictions (see Video 1 and 2). A sequence of beautiful movies of this experiment can be seen at http://netserver.aip.org/cgi-bin/epaps?ID=E-PRLTAO-101-065848).

Attempts to understand rotating fluid droplets have been more than an academic exercise. Since surface tension is a cohesive force, it can be used to model other cohesive forces, such as gravity, that hold together stars and planets, and the nuclear interactions that hold together nuclei [7, 8]. Higher dimensional objects (that is, objects which only exist in a large number of space-time dimensions) with an event horizon seem to behave in general as fluid membranes [7, 9]. Recently, the breakup of extended black-hole-like objects that are called “black strings” was explained in terms of the same classical fluid mechanics instability that is responsible for the breakup of fluid jets in a water tap, or for the formation of water droplets in rain [7, 10]. Simultaneously, it has been understood that many field theories have a hydrodynamic description in the limit of high energy density [11]. If we combine this description with the now famous gravity/quantum field duality [12], we are left with a gravity/hydrodynamic duality, where fluids are literally dual to black holes [13]. Therefore droplets, fluid cylinders, and other fluid configurations could teach us how gravity behaves, with the experimental solution of Hill and Eaves providing a simple way to investigate these exciting issues in table-top experiments.

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High-energy physics in a new guise

Figure 1: A magnetoelectric effect in a topological insulator. (Left) A quantum Hall effect occurs without strong magnetic field when an electric field applied in the plane of the interface between a topological (red region) and an ordinary (blue region) insulator (or vacuum) induces a precisely quantized current perpendicular to the field. (Right) A magnetic field applied perpendicular to the same interface introduces (n+1/2) electrons for each flux quantum of applied field. The shaded region corresponds to the charge density, ρ, of the electrons, which mainly concentrates around the boundary between the two insulators and is largest where the magnetic field is strongest.

In a prescient but until recently largely forgotten 1987 paper, Frank Wilczek [1] analyzed the effect of hypothetical elementary particles called “axions” on the laws of electricity and magnetism. Axions had been postulated in 1977 in an attempt to explain the absence of charge-parity (CP) violation in the strong interaction between quarks. Wilczek showed that the electrodynamics of axions can be described if one adds a term of the form ΔLaxion=θ(e2/2πhc)B⋅E to the ordinary Maxwell Lagrangian that governs the behavior of the electromagnetic field, where θ describes the strength of the axion field. Such a term is allowed by symmetry, but causes nontrivial modifications to Maxwell’s equations.

As far as we know today, axions do not occur in empty space, and the electrodynamics of these particles appeared to have gone down in history as an interesting curiosity, not relevant to the universe we live in. In a paper appearing in Physical Review B, however, Xiao-Liang Qi, Tayor Hughes, and Shou-Cheng Zhang of Stanford University [2] show that a term ΔLaxion, analogous to what was predicted in high-energy physics, is present in the theoretical description of a class of crystalline solids called topological insulators. The existence of topological insulators—materials characterized by a bulk energy gap and the presence of conducting surface states that are robust (or “topologically protected”) to impurities and defects—had been predicted in a series of recent theoretical works [3, 4, 5] and was confirmed experimentally this year in the semiconducting alloy Bi1-xSbx by a group at Princeton [6]. All of these developments have propelled axion electrodynamics from an idle curiosity to an experimentally observable reality. Aside from establishing the axion term, Qi et al. [2] provide a number of important insights into the physics of topological insulators and make connections with other known topological states of matter, notably the quantum Hall liquids.

Wilczek showed that the axion term has two important consequences: it modifies Gauss’ law by adding to the source term an extra charge density, so that ∇⋅E=ρ becomes ∇⋅E=ρ-(e2/2πhc)∇θ⋅B, and revises Ampère’s law by contributing an additional current density, so that ∇×B=∂tE+j becomes ∇×B=∂tE+j+(e2/2πhc)(∇θ×E+∂tθB).

The extra charge and current density only appear when the quantity θ varies in space or time. Of course, θ in crystalline solids describes something much different than the original “axions” that were hypothesized in high-energy physics. The Stanford group show that, provided the electrons’ equations of motion are time-reversal invariant, all three-dimensional insulating solids can be characterized by a quantized value of the axion field in their bulk: θ=2πn, with n integer in ordinary insulators, while θ=2π(n+1/2) in topological insulators. It follows then that θ must vary near any boundary between two insulators characterized by different bulk values of θ, and the effects of axion electrodynamics should become apparent in that region of space.

There is an important subtlety in the above classification of time-reversal invariant insulators. It turns out that while the boundary between either two ordinary or two topological insulators does not necessarily exhibit interesting behavior vis-à-vis axion electrodynamics, the boundary between a topological insulator and an ordinary insulator (or the vacuum, which presumably has θ=0) is a very special place where Maxwell laws do not hold in their conventional form.

The modifications to Maxwell’s equations give rise to some very interesting phenomena illustrated in Fig. 1. Perhaps the most remarkable of these is the finding that the boundary between a topological insulator and a normal insulator exhibits a quantum Hall effect: when an electric field E is applied in the plane of the boundary, a current flows in the direction perpendicular to E (Fig. 1, left). This is a direct consequence of the modification to Ampère’s law. This Hall current is dissipationless, a property that could be potentially useful in future electronic devices. The magnitude of the current is given by σHE with the “Hall” conductance quantized as σH=(e2/h)(n+1/2). The factor of 1/2 means that the surface exhibits a fractional quantum Hall effect. It is also notable that, unlike what is found in two-dimensional electron gasses, the Hall effect occurs here in the absence of a strong external magnetic field (although a weak magnetic field, or another time-reversal breaking perturbation is needed to determine the direction of the Hall current [1, 2]).

Similarly, according to the modified Gauss’ law, a magnetic field applied perpendicular to the plane of the surface leads to accumulation of charge (Fig. 1, right). The total accumulated charge corresponds to e(n+1/2) per flux quantum of the applied field. Thus, interestingly, one can think of a fractional charge, equal to a half-integer number of electrons, as being bound to each flux quantum. In this context, one can hypothesize that, in analogy with the fractional quantum Hall states, these flux-charge composites will exhibit fractional exchange statistics and thus be potentially useful in schemes that seek to implement fault-tolerant quantum computation [7, 8]. Finally, the surface of a topological insulator can rotate the polarization vector of reflected light (Kerr effect) and would be another experimental signature of axion electrodynamics in these materials.

What is the microscopic picture for these effects? The key ingredient is the presence of strong spin-orbit coupling, which typically occurs in crystalline solids that are made from the heavier elements, such as Pb and Bi. Under the right conditions, the spin-orbit terms can give rise to anomalous band structure that can in turn support topologically robust gapless states at the surfaces of a bulk insulator. This means that bands associated with the surface states are guaranteed to meet at a certain number of points in the surface Brillouin zone, while the bulk bands remain separated by a gap. These surface states are chiral—specifically, the electron spin and momentum are aligned—and at low energies resemble states of massless Dirac fermions, now familiar from the physics of graphene.

The precise definition of the topological insulator involves counting these surface states: an odd number corresponds to a topological insulator, whereas an even number implies ordinariness [3, 4]. That number N is also related to θ from the above discussion of axion electrodynamics. Specifically, it holds that N=θ/π. Remarkably, whether N is odd or even depends only on the bulk properties of the insulator. This situation is reminiscent of the integer quantum Hall systems where the number of chiral edge states is determined by the bulk Hall conductance. In fact, Qi et al. make this connection more transparent by deriving the topological invariant of a three-dimensional insulator from the fictitious four-dimensional quantum Hall system employing an ingenious procedure of dimensional reduction.

Having an odd number of chiral Dirac fermions at a surface is itself odd. In fact, the famous Nielsen-Ninomyia “no-go” theorem [9] states that under very general conditions such Dirac fermions must always come in pairs of opposite chirality, e.g., as they do in graphene. A topological insulator evades this theorem by spatially separating the states of opposite chirality so that they appear on its opposite surfaces. In this way the three-dimensional system as a whole satisfies the no-go theorem but its surfaces, when viewed in isolation, seemingly violate it. Qi et al. explain how the axion phenomenology discussed above arises at the microscopic level from the odd number of topologically robust chiral surface states.

The anomalous surface states that are characteristic of topological insulators have other possible applications that go beyond axion electrodynamics. An example is the proposal put forward by Fu and Kane [10] to pair the surface electrons of a topological insulator into a superconducting state by means of the proximity effect. Owing to the chiral nature of the surface electrons, this superconducting state has unusual topological excitations, with just the right properties to be potential fundamental blocks for quantum computers [7].

The unusual phenomenology of topological insulators arises from the interplay between their unique band structure and the well understood physics of spin-orbit coupling. Thus, remarkably, these systems can be thought of as weakly interacting in the sense that electron-electron interactions play no significant role. Ever since Anderson’s famous 1972 essay “More is different” [11], no condensed-matter physicist has doubted the virtually limitless potential for discovery of new phenomena in interacting quantum many-body systems. That something radically new can appear even in noninteracting systems came both as a great surprise and a promise that profound discoveries can be made where no one expects them.

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Anti-Matter Goldmine

Billions of anti-matter particles were recently let loose at Lawrence Livermore National Laboratory. Using a short-pulse laser, a team of researchers figured out how to produce anti-electrons or positrons faster and in greater density than ever before in the laboratory.

While positrons were the only form of anti-matter produced in the experiment, not all anti-matter particles are positrons. Every particle has its own corresponding, oppositely charged anti-particle (check out last month's post on anti-matter).

The researchers struck gold; literally. By shooting a laser through a gold sample the size of the head of a push pin, approximately 100 billion positron particles were generated, shooting out of the sample in a cone-shaped plasma "jet".

Accelerated and ionized or charged by the laser, electrons plough through the gold sample, hitting gold nuclei along the way. The electron-gold nuclei interactions serve as a catalyst to create positrons, kind of like how fertilizer assists in the growth of plants. The laser is able to produce large quantities of positrons by concentrating the energy given off by electrons in space and time.

This new ability to create enormous amounts of positrons in the lab is significant-it could one day lead to discoveries explaining why more matter than anti-matter survived the Big Bang at the nascent of the universe. That is, answer the question of why we are made of matter and not anti-matter!

www.physicscentral.com

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