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Collective Electrodynamics I (Superconductor͞quantum Coherence͞vector Potential͞magnetic Interaction)

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Collective Electrodynamics I (Superconductor͞quantum Coherence͞vector Potential͞magnetic Interaction) Proc. Natl. Acad. Sci. USA Vol. 94, pp. 6013–6018, June 1997 Physics Collective electrodynamics I (superconductoryquantum coherenceyvector potentialymagnetic interaction) CARVER A. MEAD California Institute of Technology, MS 136-93, Pasadena, CA 91125 Contributed by Carver A. Mead, April 2, 1997 ABSTRACT Standard results of electromagnetic theory B, that has a natural connection with the quantum nature of are derived from the direct interaction of macroscopic quan- matter—as highlighted by Aharonov and Bohm (3). tum systems; the only assumptions used are the Einstein– Hamilton’s formulation of classical mechanics was—and deBroglie relations, the discrete nature of charge, the Green’s remains—the starting point for the concepts underlying the function for the vector potential, and the continuity of the wave quantum theory. The correspondence principle would have function. No reference is needed to Maxwell’s equations or to every quantum system approach the behavior of its classical- traditional quantum formalism. Correspondence limits based mechanics counterpart in the limit where the mechanical on classical mechanics are shown to be inappropriate. action involved is large compared with Planck’s constant. Although superconductivity was discovered in 1911, the recognition that superconductors manifest quantum phe- But the real glory of science is that we can find a way of nomena on a macroscopic scale (4) came too late to play a thinking such that the law is evident. – R. P. Feynman role in the formulation of quantum mechanics. Through modern experimental methods, however, superconducting Foundations of Physics structures give us direct access to the quantum nature of matter. The superconducting state is a coherent state formed Much has transpired since the first two decades of this century, by the collective interaction of a large fraction of the free when the conceptual foundations for modern physics were put electrons in a material. Its properties are dominated by in place. At that time, macroscopic mechanical systems were known and controllable interactions within the collective easily accessible and well understood. The nature of electrical ensemble. The dominant interaction is collective because the phenomena was mysterious; experiments were difficult and properties of each electron depend on the state of the entire their interpretation was murky. Today, quite the reverse is ensemble, and it is electromagnetic because it couples to the true. Electrical experiments of breathtaking clarity can be charges of the electrons. Nowhere in natural phenomena do carried out, even in modestly equipped laboratories. Electronic the basic laws of physics manifest themselves with more apparatus pervade virtually every abode and workplace. Mod- crystalline clarity. ern mechanical experiments rely heavily on electronic instru- This paper is the first in a series in which we start at the mentation. Yet, in spite of this reversal in the range of simplest possible conceptual level, and derive as many experience accessible to the average person, introductory conclusions as possible before moving to the next level of treatments of physics still use classical mechanics as a starting detail. In most cases, understanding the higher level will point. allow us to see why the assumptions of the level below were Ernst Mach wrote (p. 596 in ref. 1), ‘‘The view that makes valid. In this stepwise fashion, we build up an increasingly mechanics the basis of the remaining branches of physics, and comprehensive understanding of the subject, always keeping in view the assumptions required for any given result. We explains all physical phenomena by mechanical ideas, is in our avoid introducing concepts that we must ‘‘unlearn’’ as we judgment a prejudice. The mechanical theory of nature, is, progress. We use as our starting point the magnetic inter- undoubtedly, in a historical view, both intelligible and pardon- action of macroscopic quantum systems through the vector able; and it may also, for a time, have been of much value. But, and scalar potentials AW and V, which are the true observable upon the whole, it is an artificial conception.’’ quantities. For clarity, the brief discussion given here is Classical mechanics is indeed inappropriate as a starting limited to situations where the currents and voltages vary point for physics because it is not fundamental; rather, it is the slowly; the four-vector generalization of these relations not limit of an incoherent aggregation of an enormous number of only removes this quasi-static limitation, but gives us elec- quantum elements. To make contact with the fundamental trostatics as well (5, 6). nature of matter, we must work in a coherent context where the quantum reality is preserved. Model System R. P. Feynman wrote (p. 15–8 in ref. 2), ‘‘There are many changes in concepts that are important when we go from Our model system is a loop of superconducting wire—the two classical to quantum mechanics. Instead of forces, we deal ends of the loop being colocated in space and either insulated with the way interactions change the wavelengths of waves.’’ or shorted, depending on the experimental situation. Experi- Even Maxwell’s equations have their roots in classical me- mentally, the voltage V between the two ends of the loop is chanics. They were conceived as a theory of the ether: They related to the current I flowing through the loop by express relations between the magnetic field B and the electric field E, which are defined in terms of the classical force F 5 LI 5 Vdt5F. [1] q(E 1 v 3 B) on a particle of charge q moving with velocity v. E But it is the vector potential A, rather than the magnetic field Two quantities are defined by this relationship: F, called the The publication costs of this article were defrayed in part by page charge magnetic flux*, and L, called the inductance, which depends payment. This article must therefore be hereby marked ‘‘advertisement’’ in on the dimensions of the loop. accordance with 18 U.S.C. §1734 solely to indicate this fact. © 1997 by The National Academy of Sciences 0027-8424y97y946013-6$2.00y0 *This definition is independent of the shape of the loop, and applies 6013 Downloaded by guest on October 2, 2021 6014 Physics: Mead Proc. Natl. Acad. Sci. USA 94 (1997) Current is the flow of charge: I 5 dQydt. Each increment of intuition and common sense, gives us ‘‘a way of thinking such charge dQ carries an energy increment dW 5 VdQinto the that the law is evident.’’ Electrons in a superconductor are loop as it enters.† The total energy W stored in the loop is thus described by a wave function that has an amplitude and a phase. The earliest treatment of the wave nature of matter was the 1923 wave mechanics of deBroglie. He applied the 1905 W 5 EVdQ5EVI dt Einstein postulate (W 5\v) to the energy W of an electron wave, and identified the momentum pW of an electron with the propagation vector of the wave: p 5\kW. Planck’s constant h dI 1 W 5 L Idt5L IdI 5 LI2. [2] and its radian equivalent \5hy2pare necessary for merely Edt E 2 historical reasons—when our standard units were defined, it was not known that energy and frequency were the same If we reduce the voltage to zero by, for example, connecting the quantity. two ends of the loop to form a closed superconducting path, The Einstein–deBroglie relations apply to the collective the current I will continue to flow indefinitely: a persistent electrons in a superconductor. The dynamics of the system can current. If we open the loop and allow it to do work on an be derived from the dispersion relation (10) between v and kW. external circuit, we can recover all the energy W. Both v and kW are properties of the phase of the wave function If we examine closely the values of currents under a variety and do not involve the amplitude, which, in collective systems, of conditions, we find the full continuum of values for the is usually determined by some normalization condition. In a quantities I, V, and F, except for persistent currents, where superconductor, the constraint of charge neutrality is such a only certain discrete values occur for any given loop (7, 8). By condition. experimenting with loops of different dimensions, we find the The wave function must be continuous in space; at any given condition that describes the values that occur experimentally: time, we can follow the phase along a path from one end of the loop to the other: the number of radians by which the phase F 5 Vdt5nF0. [3] advances as we traverse the path is the phase accumulation w E around the loop. If the phase at one end of the loop changes relative to that at the other end, that change must be reflected 215 Here, n is any integer, and F0 5 2.06783461 3 10 volt- in the total phase accumulation around the loop. The fre- second is called the flux quantum or fluxoid; its value is quency v of the wave function at any point in space is the rate 9 accurate to a few parts in 10 , independent of the detailed size, at which the phase advances per unit time. If the frequency at shape, or composition of the superconductor forming the loop. one end of the loop (v1) is the same as that at the other end We also find experimentally that a rather large energy— (v2), the phase difference between the two ends will remain sufficient to disrupt the superconducting state entirely—is constant, and the phase accumulation will not change with required to change the value of n. time. If the frequency at one end of the loop is higher than that 3 The more we reflect on Eq.
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