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Topological Insulators
Topological Insulators
December 2008
This computer simulation shows the predicted boundaries of a topological insulator in an optical lattice. Elevation above the plane represents higher probable density of atoms in a particular quantum state at each location.TIs form in certain materials that, in bulk, have the distinctive physical signature of insulators: That is, the permitted energy levels (or “bands”) in their component atomic structures are characterized by a full valence band and an empty conduction band, with a substantial gap of forbidden energies in between. However, at the boundaries with other materials or the vacuum, they develop gapless regions that permit conduction along one-dimensional paths or the edges of two-dimensional surfaces – but in a single direction only. These states, discovered only a few years ago, are highly promising candidates for use in quantum computation because they are robust in the face of disorder or inhomogeneities.
The chief reason is their topological nature, which in this context refers to the quantum characteristics of the material as a whole, as distinct from its local properties at any point. TIs belong to one topological set, and conventional insulators to another. PFC theorist Tudor Stanescu puts it this way: “Topologically, a sphere, a cube and a pyramid are identical. That is, they can all be smoothly transformed into one another without ‘tearing’ the surface. Similarly, a torus (donut shape) and a coffee cup are identical. There are many types of standard insulators, but they are all topologically identical, like the sphere, cube and pyramid. Topological insulators belong to a different class, like the torus and coffee cup, and you can’t deform them into the other class.”
At the boundary between the two classes of material, peculiar phenomena arise which are of immediate interest to the PFC because one of its major research activities involves the study of correlated and topological matter in cold atoms, typically arrayed in optical lattices, to model condensed-matter systems. That approach is ideally suited to the investigation of TIs.
Figure 1 – The 2-D lattice potential formed by superposition of three standing waves.
Figure 2.The scheme involves the superposition of three standing waves to create a lattice in which the energy minima form a hexagonal unit cell. [Figure 1]
That phenomenon would be extremely difficult to isolate in a conventional condensed-matter system, and direct observation of the edge states would likely be nearly impossible. The PFC theory team, however, is in the process of completing a model experimental protocol using ultracold atoms suspended in a two-dimensional optical lattice of intersecting laser beams to simulate a graphene-like arrangement which would make topological insulators both detectable and adjustable.
The beams are arranged so that the entire structure has a periodic “vector potential” (that is, an overall directional tendency) that acts, in effect, as a magnetic field. As the atoms interact with both the lattice structure and the vector potential, they take on well-defined boundary shapes that are the equivalent of edge states in bulk-matter TIs. These formations are chiral (left- or right-handed, like a mirror image) in the sense that they permit momentum in only one direction.
Figure 2 (right) shows a plot of atomic states in a TI lattice array. The red and yellow dots represent atoms within ordinary energy bands. The blue dots represent atoms in the band gap, constituting the lattice equivalent of one TI edge state. Note that the in-gap atoms only cross the gap in one direction, owing to the chiral nature of the arrangement.
*JQI/PFC investigators on the TI simulation project are Tudor Stanescu, Victor Galitski, Jay Vaishnav, Charles Clark and Sankar Das Sarma. Galitski, Clark and Das Sarma are JQI Fellows. Related publication: "Topological insulators and metals in atomic optical lattices," T. D. Stanescu, V. Galitski, J. Y. Vaishnav, C. W. Clark,and S. Das Sarma, Phys. Rev. A 79, 053639 (2009).
Quantum Dots in Photonic Crystals
Quantum Dots In Photonic Crystals
Transistors and Entanglement
November 2008
Part of the lab set-up shows photonic crystal (left, red dot) in center of cryogenic enclosure (white). An infrared beam coming from the right is directed through a beamsplitter (cube, center) to the crystal.
Every day, a growing amount of the world’s information moves at the speed of light in the form of photons that fly through optical fibers. And increasingly, society is depending on quantum information science to ensure that critical communications traveling over those lines can be made impregnably secure.
There are many possible ways of reaching that goal, and JQI Fellow Edo Waks is exploring one of the most promising: experimental arrangements that combine the blazing speed of photonic crystals with the remarkable, newly discovered “switching” abilities of tiny imbedded structures called quantum dots.
A primary long-term objective is to create a system for generating “entanglement” – the quantum process whereby the properties of two objects become inextricably linked, no matter how far they are separated – between two quantum dots. The dots interact with light beams channeled through the crystal, thus making the whole assembly well-suited to quantum encryption schemes or to fabrication of “quantum repeaters” that can move encrypted messages over long distances.
A common configuration for photonic crystal fabrication with imbedded dots.
Getting to that point, however, will first require understanding how to precision-tune the interactions among crystal features, dots and photons with high fidelity using nothing but light to manipulate the system.
“It’s a complete quantum operation,” Waks says. “That’s really important to quantum networking and quantum computation. In principle, all quantum computations can be performed using this operation alone. And you can take this idea and extend it to entangle two quantum dots.” That feat has not yet been accomplished, but Waks believes it may soon be possible in photonic crystals (PCs).
A Quantum Light Switch
Dots of indium arsenide in a gallium arsenide matrix.
That effect is strikingly analogous to what electrons experience as they move through the geometry of a semiconductor lattice, and it allows researchers to manipulate the passage of light through PCs in much the same way that electrons are controlled in a transistor. For example, creating a defect in the crystal lattice can strongly confine photons much the same way that electrons are confined in lattice defects of a crystal.
This localization can enhance the interaction of light inside the cavity with an atomic system, such as a quantum dot, by both holding the light in place, which increases the interaction time, and tightly localizing the light which leads to an increase of electric field intensity.
Three years ago, Waks and Jelena Vuckovic at Stanford University discovered that the enhanced atom-photon interactions in a photonic crystal cavity can lead to transistor action at the quantum level. A quantum transistor could be made using the device shown below. The transistor is composed of a photonic crystal waveguide (formed by removing a row of holes) that is closely coupled to a cavity. If the cavity is empty, light injected into the input port of the waveguide would be completely reflected due to the coupled cavity. By placing a quantum dot inside the cavity, one can disable the coupling between the cavity and waveguide and the light is instead transmitted.
This outcome – called “dipole induced transparency” (DIT) – has obvious utility as a means of controlling the movement of light in a PC. Of course, it is physically impossible to implant or remove a dot every time it is desirable to switch the cavity from transmitting to reflecting or vice versa. Instead, the QD can be effectively removed by exciting it to an optically inactive quantum state, which can be achieved by exciting it with a laser pulse of appropriate wavelegth and duration. In this way, the QD behaves as a switch which turns the transparency of the waveguide on or off.
Entangling Dots?
Orange: Initial light path on PC waveguide. Blue: Reflected path. Red: Transmitted path. Small triangle marks the dot location. The dotted oval marks the cavity area. Yellow arrow indicates interaction with light beam. When the dot couples to the cavity, the beam is transmitted. When the dot is "switched off," and does not couple to the cavity, then the beam is reflected as if no dot were present.
The switching action of the quantum dot in DIT is and does not couple to the cavity, then the reminiscent of transistor action where the logical value beam is reflected as if no dot were present. of a bit register (in this case the QD state) modifies the flow of current (in this case photons). There is one important distinction, however. Unlike classical transistors the QD is a quantum object, and therefore behaves like a quantum bit.
Quantum bits, or “qubits,” are the quantum-mechanical analogue of the minimum information unit in a classical computer. A conventional electronic binary digit, abbreviated “bit,” can have only one of two values: on or off, 0 or 1, as represented by electrical charges, voltages or magnetization. A quantum bit, however, can be 0, 1, or a “superposition” of both at once, owing to the inherently indeterminate nature of unmeasured quantum states.
Certain kinds of quantum operations produce superposition outcomes in which the states of two objects are inseparably entangled, no matter how far apart they are moved. Until some measurement is taken, both objects exist in a superposition of possible states. But once one object is measured, the state of the other is instantly known. That’s the hallmark of entanglement, and it is an essential component of systems that can send and transfer quantum-encrypted messages.
Waks and UMD graduate student Deepak Sridharan have devised a protocol*, based on the earlier Stanford DIT work, to generate entanglement between two dots, and are now testing parts of it in various ways, using nanometer dots of indium arsenide (InAs) embedded in cavities of photonic crystals made of gallium arsenide. Photonic crystal cavities are formed around the quantum dots by patterning the GaAs host substrate using electron beam lithography.
Two Kinds of Transistors: In a field-effect transistor, charges cannot move from the source to the drain if no voltage is applied to the gate. When a voltage is applied, the semiconductor region under the gate is affected by the applied field, and current flows. Bottom: In a photonic crystal, when a light beam (yellow) moving down a waveguide encounters a cavity without an embedded dot -- or when the dot is not coupled to the cavity -- the beam is reflected. When the dot is coupled, the beam is transmitted.
Each dot-and-cavity unit controls the path of a light beam, as in conventional DIT. But instead of switching the dot definitively to the “on” or “off” state, the researchers use a laser pulse adjusted to have an equal probability of putting the dot in either the coupled (“on,” transparent cavity) state or the decoupled (“off,” reflecting cavity) state.
For a simple single-dot system comprising one dot/ cavity and one light beam, the uncertainty of the 50-50 chance adds no ambiguity to the outcome: If the beam is transmitted, the researcher knows for certain that the dot was coupled to the cavity; if the beam is reflected, the dot was decoupled. But a much more interesting situation arises in a system of two dot/cavity pairs. (Call them left and right to match the diagram above.)
In the two-dot entanglement protocol, a coherent light beam is split into two parts, here labeled |a> and |B>. One travels to the left dot, one to the right. Each dot is exposed to an identical laser pulse producing the 50-50 condition, and the traveling light beam is either reflected or transmitted. If it is transmitted, it is discarded for purposes of the experiment. If it is reflected, it is directed to an optical device called a beamsplitter, which divides the incoming beam and sends it to one or the other of two separate detectors labeled d1 and d2 in the diagram.
There are four possible outcomes for this arrangement: Both dots reflect; both dots transmit; the left dot reflects while the right transmits; and the left transmits while the right reflects.
Only two of those outcomes can produce light that arrives at the beamsplitter: Either both dots reflected, or only one did. If both of them reflected, then the combined beam will constructively interfere at detector 1. If only one dot reflected the light, there is a probability that the beam will register on detector 2.
At this point, the peculiar nature of quantum mechanics comes into play. If detector 2 registers a hit, it is certain that only one dot reflected. But it is impossible to tell which of the two dots did so. So the states of the two dots are entangled – each is in a superposition of the reflecting and transmitting states.
* "Generating entanglement between quantum dots with different resonant frequencies based on Dipole Induced Transparency," Deepak Sridharan and Edo Waks. Accepted for Physics Review A. See: arXiv:quant-ph/0703089v2. 9 Oct. 2008.
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