2.1.1. Rectification of Zero-Point Fluctuations in a Diode
Several suggested approaches to extracting energy from the vacuum involve nonlinear processing of the ZPF. One particular nonlinear process is electrical rectification, in which an alternating (AC) waveform is transformed into a direct (DC) one.
] describes the electrical noise in resistors and diodes that results from zero-point fluctuations. He discusses the use of diodes to extract power from these ambient fluctuations, and compares this to diodes used for thermal energy conversion. For example, in thermophotovoltaics radiation from a heated emitter is converted to electricity. In Valone’s case, however, the source is under ambient conditions. In support of Valone’s approach, a straightforward analysis of diode rectification in the presence of thermal noise does appear to produce a rectified output from that noise unless a compensating current is added to the system [16
]. Valone is particularly interested in the use of zero-bias diodes for zero-point energy harvesting, so as to rectify the ambient fluctuations without having to supply power in providing a voltage bias to the diodes.
This nonlinear extraction represents a sort of Maxwell’s demon [17
]. In 1871 Maxwell developed a thought experiment in which a tiny demon operates a trapdoor to separate gas in equilibrium into two compartments, one holding more energetic molecules and the other holding less energetic ones. Once separated, the resulting temperature difference could be used to do work. This is a sort of nonlinear processing, in which the system, consisting of the demon and the compartments, operates differently on a molecule depending upon its thermal energy. In the nearly fifteen decades since its creation, innovative variations on the original demon have been proposed and then found to be invalid. Despite the best efforts of Maxwell’s demon and his scrutineers [18
] there still is no corroborated experimental evidence for the demon’s viability [20
], and for the current analyses I will assume that he cannot assist us in extracting energy from a system in equilibrium.
In the absence of such a demon, thermal noise fluctuations cannot be extracted without the expenditure of additional energy, because such fluctuations are in a state of thermal equilibrium with their surroundings [21
]. These thermal fluctuations are described by the first term on the right-hand side of Equation (2). In equilibrium, the second law of thermodynamics applies and no system can extract power continuously. All processes in such a system are thermodynamically reversible. A detailed balance description of the kinetics of such a situation was developed by Einstein to explain the relationship between the emission and absorption spectra of atoms [22
], and discussed as a manifestation of equilibrium by Bridgman [23
If the equilibrium is altered, for example by the addition of a non-equilibrium radiation field, then the detailed balance is replaced by a less restrictive steady state condition and it becomes possible to extract power. The difference between detailed balance and steady state is illustrated with the three-state system shown in Figure 1
. Each arrow represents a unit of energy flux. In the steady-state case shown in Figure 1
a, the total flux into any state equals the total flux out of it. Under equilibrium, however, a more restrictive detailed balance must be observed, in which the flux between any pair of states must be balanced. This is depicted in Figure 1
This concept of detailed balance can be applied to the extraction of thermal noise from a resistor at ambient (equilibrium) temperature. To optimally transfer power from a source, in this case the noisy resistor, to a load the load resistance should be adjusted to match that of the source. In that case, the load generates an equal noise power to that of the source, and an equal power is transferred from the load to the source as was transferred from the source to the load. Because of this detailed balance, no net power can be extracted from a noisy resistor.
To analyze the case of extracting energy from thermal noise fluctuations in a diode, consider the energy band diagram for a diode shown in Figure 2
, where transitions among three different states are shown. For simplicity, five other pairs of transitions are not shown and are assumed to have negligible rates. (Not shown are (i) direct transitions between the p-type region valence band and the n-type region conduction band; (ii) transport of valence-band charge carriers (holes) through the junction region; (iii) generation and recombination in the n-type region; (iv) direct transitions between the n-type region valence band and the p-type region conduction band; and (v) generation and recombination in the junction region. A completely parallel set of processes could be added for these transitions, and would not change the physical principles involved, or the conclusions drawn.)
First consider the case of photovoltaic power generation. If the diode operates as a solar cell, light absorbed in the p-type region generates electron-hole pairs, promoting electrons to the conduction band at a rate g that depends upon the light intensity and other factors. The photogenerated electrons diffuse to the junction region, where the built-in electric field causes them to drift across the junction to the n-type region at rate d. The recombination rate, r, and the excitation rate, e, are also shown. Because g >> r and d >> e under solar illumination, i.e., the system is far from equilibrium, there is a net flow of electrons to the n-type region, where they are collected to provide power.
The diode would operate in much the same way for rectifying thermal fluctuations under equilibrium, except that now g would represent the thermal generation rate. In this case, however, the generation rate and drift rate across the junction would be much smaller than under solar illumination. Under thermal equilibrium and in the absence of a Maxwell’s demon to influence one of the processes, a detailed balance is strictly observed, such that g = r and d = e. The second law of thermodynamics does not allow for power generation.
Valone’s proposal [9
] makes use of power generation in a diode from ZPE fluctuations described by the second term on the right-hand side of Equation (2). Whether this is feasible becomes a question of whether the zero-point energy in a diode is in a state of true equilibrium with its surroundings. It has been generally accepted that the “’vacuum’ should be considered to be a state of thermal equilibrium at the temperature of T
= 0” [12
]. Recently, using the principle of maximal entropy, Dannon has shown explicitly that zero-point energy does, in fact, represent a state of thermodynamic equilibrium [24
]. Therefore, it is clear that the detailed balance argument presented above for the case of thermal fluctuations also applies to ambient zero-point energy fluctuations, and a diode cannot rectify these fluctuations to obtain power.
2.1.2. Harvesting of Vacuum Fluctuations Using a Down-Converter and Antenna-Coupled Rectifier
A somewhat different approach to the nonlinear processing of the ZPF for extracting usable power would be to use an antenna, diode and battery. The radiation is received by the antenna, rectified by the diode, and the resulting DC power charges the battery. In the microwave engineering domain, this rectifying antenna is known as a rectenna [25
]. Because of its ν3
dependence, shown in Equation (1), the ZPF power density at microwave frequencies is too low to provide practical power. Therefore, to obtain practical levels of power, a rectenna must operate at higher frequencies, such as those of visible light or even higher. There are diodes that operate at petahertz frequencies. One example is a graphene geometric diode [26
] but the rectification power efficiency of optical rectennas at such high frequencies is generally low [27
]. The first question about ZPF rectification is how it can be made practical. The second, and more important question here, is whether this is feasible from fundamental considerations. I address these in turn.
A method to extract ZPE is proposed in a 1996 patent by Mead and Nachamkin [4
], which describes an invention to produce lower beat frequency from high-frequency ZPF to make it more practical to rectify. The invention includes resonant microscopic spheres that intercept ambient ZPF and build its intensity at their resonant frequency. The high-intensity oscillation induces interactions between two spheres of different size such that a lower beat-frequency radiation is emitted from them. This lower beat-frequency radiation is said to be then absorbed by an antenna and rectified to provide DC electrical power. (Note that the beat frequency is not frequency down-conversion, which requires a nonlinear mixer. The down-conversion occurs only after the signals encounter the diode, and so the antenna cannot actually pick up the short-wavelength ZPF, and the diode cannot rectify the high-frequency ZPF.) The invention is depicted in Figure 3
Mead and Nachamkin’s approach can be broken down into three steps:
Producing lower beat frequency radiation from the ambient ZPF;
Collecting the beat-frequency radiation at the diode by the antenna;
Rectification of the concentrated radiation by the diode.
Step (a) is a process akin to that performed by the diode described in the previous section, except that in the current case the ZPF produces an intermediate beat frequency whereas in the previous case the ZPF fluctuations in a diode were down-converted to DC. Therefore, this step operating with incoming radiation under equilibrium must observe a detailed balance of rates. Regarding steps (b) and (c), under equilibrium a source, antenna and load are in detailed balance, such that the power received by the antenna from the source and transferred to a load is equal to the power transmitted back to the source [28
If steps (a) and (b) could provide a greater-than-equilibrium concentration of power to the diode, then the diode in step (c) would no longer be operating under equilibrium. When driven far from equilibrium the harvesting efficiency would be limited to the Carnot efficiency in a quantum heat engine, unless the extra energy somehow created some coherence [29
]. However, as argued above, the concentration of power at the diode cannot occur under equilibrium. In summary, when applied to the harvesting of vacuum fluctuations each of the three steps in the ZPF down-converter system is subject to a detailed balance of rates, and therefore, the system cannot provide power.