Making information engines do useful work
In the textbook example of a heat engine, fluid repeatedly flows from a hot reservoir to a cold one, some of the thermal energy is converted to work, and then the cycle repeats. Traditional heat engines rely on a temperature difference between two thermal reservoirs. But a different kind of engine, one based not on heat but on information, doesn’t have that dependence. In a famous thought experiment proposed by James Clerk Maxwell in 1867, a demon with information about a system can cleverly manipulate it so that energy can be harvested without the need for a temperature difference. 1
In the past few decades, some information engines have been experimentally realized, although the next step of harvesting appreciable amounts of energy from them still needs to be demonstrated. 2 Instead of storing energy in a chemical, mechanical, or electrostatic way, information engines use the entropy of information encoded in some memory device.
Discharging such a memory then allows heat from a single reservoir to be converted to work. As opposed to lithium-ion batteries, which get warmer when discharged, batteries based on information would cool down, which could be safer. The total energy that can be stored and extracted using an information engine is proportional to the memory’s number of bits, which in previous experiments has been low.
The scalability of information engines has been difficult because of material limitations. But under the right conditions, the discharging and charging of a memory can be intrinsic to a class of materials known as topological insulators. They feature intrinsic locking of the electron’s spin orientation to its momentum direction. As a result, they can directly link electrical work to a built-in memory comprising the material’s nuclear spins. That mechanism could offer a route toward scalable devices powered by information, and the achievable charging and discharging speeds would be limited only by the material’s rate of heat transfer.
Maxwell’s demon
To understand the new type of information engine made of topological materials, first consider Maxwell’s original proposal. At face value, it seems to break the second law of thermodynamics. At the center of the thought experiment, a demon operates a valve between two adjacent chambers in an isolated system filled with gas particles. By opening the valve only at select moments, the demon permits fast-moving particles in the left chamber to enter the right chamber, where they stay, and slow-moving particles in the right chamber are allowed to cross into the left. Over time, the right chamber will contain higher numbers of energetic particles than the left chamber.
For a gas, that means that one chamber has a higher temperature than the other, and mechanical work can be extracted by letting the higher-temperature chamber expand. That seems to violate the second law of thermodynamics because entropy would decrease when a temperature difference is created between the two reservoirs, without a compensating increase in entropy elsewhere. So what’s missing?
To resolve the apparent inconsistency with the second law, you should consider not only the chambers with gas particles but also the demon’s memory. That is where the entropy of information, which was first introduced in 1948 by Claude Shannon, comes in. A full memory corresponds to a high-entropy, disordered state, whereas an empty memory has a zero-entropy, ordered state. For example, more words are necessary for someone to describe the state 0110010 (“a zero, then two ones, two zeros …”) than the blank one 0000000 (“seven zeros”). (For more on information entropy, see the September 2015 PT article “Information: From Maxwell’s demon to Landauer’s eraser ,” by Eric Lutz and Sergio Ciliberto).
The demon can exploit its sorting valve only if it stores in its memory the positions and velocities of the particles. An information engine, similar to a heat engine, has a cyclical process, and before the cycle repeats, the memory needs to be reset. Landauer’s principle, named after Rolf Landauer, states that resetting information costs energy: at least kBT ln 2 per bit, where kB is Boltzmann’s constant and T is the system’s temperature. 3 So, with the memory reset included, some energy cost is required to operate the demon’s cycle. (For more on thermodynamics and information theory, see the November 2021 PT article “The demons haunting thermodynamics ,” by Katie Robertson.)
Leo Szilard refined Maxwell’s thought experiment in 1929 by simplifying the system to a single particle in a box. 4 If a demon knows the particle’s position (left or right), which corresponds to one binary bit of information, it can extract work from the system by appropriately positioning a piston. The extractable work for this single-bit system is kBT ln 2, which exactly equals the energy cost of erasing the single bit of information in the demon’s mind. The work extracted is, therefore, compensated by the energy cost of memory erasure, and the second law remains inviolate. Over the past few decades, the modern field of information thermodynamics has incorporated information concepts into the laws of thermodynamics. 5
Maxwell, in an 1867 letter to the mathematical physicist Peter Guthrie Tait, originally described the demon as a “very observant and neat-fingered being,” which hints at the challenge of developing an experimental system that could test the thought experiment. Recent advances in nanofabrication, however, have enabled the creation of well-defined and controllable demon systems, based on microbeads 6 and single-electron quantum dots, 7 that operate on a single-particle level.
One major issue with information engines is scalability. The demons in those types of systems are intelligent: By performing suitable manipulations in a feedback loop after reading out the system’s state, they can harvest energy exactly like the demon in Szilard’s engine. The feedback loops, unfortunately, are often complicated and require high-speed and high-precision electronics to operate at Landauer’s limit. Consequently, increasing the number of bits and thereby the amount of work that can be extracted remains an experimental challenge.
A new information engine
Intriguingly, the solution for scalability might be found in unintelligent demons. As opposed to the Szilard engine’s intelligent demon, which relies on a correct readout of the system, unintelligent demons automatically open and close the door between the chambers depending on the properties of the approaching particle. In such systems, an example of which is illustrated in figure 1 , the process of extracting energy does not rely on a separate feedback loop to read the state of the system after it’s been sorted. The sorting property is intrinsic to a topological insulator, which simplifies the infrastructure of the system and allows for scalability.
Figure 1. Extracting work. In the traditional depiction of James Clerk Maxwell’s thought experiment, an intelligent demon could extract work from a box system with two compartments by attaching a pulley and weight to the compartment with the particle; the weight would be lifted by the expansion of the chamber. Correctly attaching the weight requires information—specifically, knowledge of which compartment the particle resides in. If the demon is unintelligent and lacks any information, the weight can still be lifted so long as the demon allows the particle to pass only from left to right. A natural materials-based solution that implements the unintelligent demon can be found in topological insulators.
Some sort of rectification process could function similarly to a valve and take over the demon’s workload. Our envisioned demon essentially acts as a ratchet: Just as a ratchet is a mechanical gear that’s controlled by a pawl and can only rotate in one direction, a demon in this scenario uses information about the system to allow a particle to move only from one chamber to the other but not in the reverse direction.
The Brownian ratchet, a classic thought experiment first introduced by Marian Smoluchowski in 1912 and later refined by Richard Feynman, offers a useful comparison: Thermal fluctuations cause microscopic particles to move randomly. The Brownian ratchet uses the random motions to drive a gear, and a pawl prevents the gear from rotating backward. 8
A Brownian ratchet by itself, however, cannot extract net work from thermal fluctuations because both the ratcheting gear and the pawl will thermally fluctuate. Sometimes, the gear will slip in the reverse direction, which restores thermodynamic balance. A similar problem occurs in electronics. Take, for example, a diode, which is perhaps the most famous electronic rectifier: As shown by Léon Brillouin, the diode’s own thermal fluctuations balance any apparent rectified signal. 9 The missing ingredient from that simple rectification circuit, and what would allow for energy to be harvested without breaking the second law of thermodynamics, is a memory.
The role of memory can be understood by comparing a conventional heat engine with an information engine. Figure 2 highlights the crucial differences between them. In a conventional heat engine, no work can be extracted if the reservoirs are at equal temperatures or if the engine has only a single reservoir. Extracting work in those situations would lead to a net entropy decrease, which violates the second law of thermodynamics. In an information engine, the memory acts effectively as a second reservoir that can store and supply entropy. While heat flows out of the reservoir, the memory’s entropy increases. But the memory’s internal energy does not change. Imagine the memory as a set of zeros and ones that become randomized: The information changes, but zero and one are energetically degenerate.
The heat received from the reservoir can be extracted as work, and the second law will be satisfied. Researchers have developed information-based ratchets in which information about the state of a system can be used to rectify the system’s dynamics. Much progress, such as a ratchet implementation in a molecular motor, 10 has been made in the field.
Charles Bennett showed that a rectification process can extract work from thermal energy, provided that it is coupled to a memory in which entropy can be stored. 11 In his formulation, the information in the memory becomes a thermodynamic fuel and an information resource. 5 An ordered memory acts as a filled-fuel resource: As work is extracted, the memory’s entropy increases, depleting the resource. Once the memory is fully randomized so that its entropy is maximized, the fuel is depleted, and no more work can be extracted. For the memory to continue to operate, it must be reset to an ordered state, which costs at least kBT ln 2 per bit, according to Landauer’s principle. Once it’s recharged, another extraction cycle can begin.
Scalable information engines may be feasible if researchers can implement rectification processes that do not require a feedback loop. Such demons require an intrinsic rectification mechanism coupled to a memory to absorb the entropy. The challenge now lies in finding a material in which both properties naturally occur.
Figure 2. Comparing a conventional heat engine with an information engine. (a) In a traditional engine, some of the heat Q extracted from a hot reservoir at temperature Thot flows into a cold reservoir at temperature Tcold, and some is converted into work W. (b) In an ideal information engine, a memory with a finite number of bits (white arrows) acts as an entropy reservoir whose internal energy remains unchanged. There is no cold reservoir, so in both the charging and discharging phases, all the heat that’s transferred to or from the sole thermal bath (purple) is balanced by the amount of work done on or by the system. The total amount of extractable work is constrained by the number of memory bits.
Topological materials
In 2018, a team led by İnanç Adagideli proposed topological materials as a quantum platform that combines rectification and memory and can function as a topological quantum information engine. 12 What makes topological materials special is the fact that the spin direction of the conduction electrons is locked to the direction in which they move. Electrons with spin up, for example, may move only to the right, and spin-down electrons move only to the left. That property of spin–momentum locking could simplify the work of a demon: By sorting electrons according to their spin, a demon would simultaneously sort them by their movement direction.
Charge transport in the materials forms a method for the extraction of electrical work. Electrons could be coupled to a memory, which in topological materials consists of the atoms’ nuclear spins or the electron polarizations of magnetic atoms.
Adagideli and colleagues’ original topological quantum information engine was based on the 1D conducting edges of 2D topological insulators; 12 the configuration was later generalized to the surfaces of 3D topological insulators. 13 In 3D topological insulators, spin–orbit coupling causes an inverted-bandgap structure: The material is insulating, but the conduction band is brought to a lower energy than the valence band. 14 At the interface between a 3D topological insulator and a normal insulator, such as a vacuum, the band inversion must be restored, which can be achieved only by closing the bandgap. As a result, 2D conducting states emerge at the interface that are topologically protected, and they exhibit spin–momentum locking.
In 2D topological insulators, spin–momentum locking leads to the material edge having two counterpropagating ballistic modes, each with opposite spin directions. Together, those modes form what’s known as the quantum spin Hall state. 15 (For more on these materials, see the January 2010 PT article “The quantum spin Hall effect and topological insulators ,” by Xiao-Liang Qi and Shou-Cheng Zhang.) Backscattering—reversing the direction of electron motion—is not allowed in those states because the electron would have to change its spin.
But figure 3 shows what happens when the electrons are allowed to interact with nuclear spins. Backscattering is allowed via spin-flip interactions with nuclear spins, and such interactions lead to a finite nuclear spin polarization over time. The two of us and Adagideli have recently made the first experimental observations of that effect. 16
The spin-flip interactions can operate as the information engine’s valve. Think of those interactions as a demon rectifying the current. Without an applied current, the spin-flip interactions will cause the nuclear spins to thermally relax from the ordered, polarized state to the disordered state. On average, electrons will be scattered in one direction—the same direction as the applied charging current that polarizes the nuclear spins and leads to a net discharging current.
The nuclear spins in such a system form a memory. While the system is charging, the spins become aligned, and so the information entropy of the nuclear spin subsystem decreases and the memory gets erased. The energy that’s needed for erasing the memory is supplied by the power source of the circuit that’s surrounding the topological insulator, and that energy is dissipated as heat to the environment. The ideal result of the charging phase is a fully polarized nuclear spin system, for which the entropy is zero: That state is equivalent to the fully sorted state in the Maxwell’s demon thought experiment.
During the discharge phase, electrical work can be extracted by connecting a load to the topological insulator, equivalent to attaching a weight to the piston in Maxwell’s thought experiment. The power required to drive the current through that load is supplied by the nuclear spin subsystem, which absorbs heat from the environment and gets randomized. Thanks to the information entropy of the nuclear spin subsystem, electric work can be extracted without violating the second law of thermodynamics.
Although electron transport through topological matter is inherently quantum mechanical, the demon’s action in the information engines discussed above relies on coupling between a quantum state (the electron spins) and the environment. Other techniques also can couple a quantum state to the environment to harvest energy. In asymmetric mesoscopic rings, for instance, electrons coherently propagate as waves through the structure until they inelastically scatter into lattice vibrations, and the scattering probability can be linked to the waves’ direction of travel. 17
Figure 3. Spin–momentum locking could enable scalable information engines. In topological insulators, an electron’s spin S is coupled to its momentum p. In the scenario shown in the figure, a right-moving electron (bottom blue shape) is always spin down, and a left-moving electron (top blue shape) is always spin up. If an electron interacts with another magnetic moment, such as the nuclear spin of an atom (white arrows), it can backscatter, and its spin will flip. An electron current, therefore, generates a nuclear spin polarization during the charging phase. During the system’s discharge phase (not shown), thermal relaxation of a finite nuclear spin polarization drives the electrons in one direction. The nuclear spin system’s entropy increases, and environmental heat is converted into usable electrical work without violating the second law of thermodynamics.
Future energy storage
Batteries based on topological materials and their quantum properties have advantages and disadvantages compared with traditional batteries. The long-term challenge for researchers constructing bulky batteries for phones, cars, and other large power applications based on quantum information is to make systems in which every atom serves as an information resource. The amount of energy stored per bit of information is extremely small. But if every atom in an energy storage medium contains information, then the energy density of the battery can become close to that of chemical batteries. 12 Still, the energy density will likely not surpass chemical batteries, such as those based on lithium ions.
But the charging and discharging speed of information-based batteries depends on only how quickly heat can be exchanged between a memory and a reservoir; a sufficiently fast rate would potentially allow such batteries to achieve world-record power densities. High power densities would be especially useful for on-chip devices that require fast charging–discharging speeds and low amounts of waste heat. An information-based battery, which would cool upon discharge, may be safer than the existing batteries that heat up upon discharge. But making an information-based battery with useful power density is still probably many years away. A nearer-term goal of researchers is to develop electronic applications that can be powered by information-based batteries that have a small number of atoms.
Another driving force that’s behind the development of information-powered technology is the potential for inductive energy storage. 13 Transistors and capacitors already can be miniaturized in modern semiconductor technology, but traditional inductors depend on the enclosed area, which makes their miniaturization a challenge. A topological quantum information engine, on the other hand, can have an entropy-driven inductance that is sizable for microscopic devices because of the intrinsic atomic scale of the nuclear spin memory. The experimental realization of an entropy-driven inductor would provide an extra degree of freedom for semiconductor chip design and electronic power engineering.
Information engines based on topological materials have additional benefits. In certain device geometries, an electron can be transferred through the device only if a spin is flipped in a memory. The device’s electrical resistance depends on the amount of current that has passed. 18 Such a resistor with a memory is called a memristor and is a potential component for implementing neural networks in quantum computing and other applications. When used for energy storage, the device efficiency of memristors is predicted to approach Landauer’s limit, where a full kBT ln 2 of energy can be retrieved from every bit in the memory.
The foundations of classical thermodynamics paved the way for the development of chemical, mechanical, and electrostatic engines and batteries, which have served humanity for centuries. In the coming decades, advances in quantum thermodynamics could make possible novel engines and batteries based on information. Topological materials show possibilities for overcoming the challenge of scalability. As experiments reveal more about the fundamental requirements of information engines, the prospects for successful power extraction from a scalable Maxwell demon device will continue to grow.
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More about the authors
Sofie Kölling is a postdoc at ETH Zürich in Switzerland.
Alexander Brinkman is a professor at the University of Twente in The Netherlands.
