A gauge field treatment of a current oscillating at frequency *ν* of interacting neutral atoms leads to a set of matter-wave duals to Maxwell's equations for the electromagnetic field. In contrast to electromagnetics, the velocity of propagation has a lower limit rather than upper limit, and the wave impedance of otherwise free space is negative real-valued rather than 377 Ω. Quantization of the field leads to the matteron, the gauge boson dual to the photon. Unlike the photon, the matteron is bound to an atom and carries negative rather than positive energy, causing the source of the current to undergo cooling. Eigenstates of the combined matter and gauge field annihilation operator define the coherent state of the matter-wave field, which exhibits classical coherence in the limit of large excitation.

## I. INTRODUCTION

This work considers an oscillating current of interacting ultracold atoms through the lens of non-relativistic field theory. Over the past few decades, relativistic quantum field theory has led to the unification of three of the four fundamental forces and now provides our modern picture of the physics of fundamental particles as the Standard Model.^{1} Non-relativistic systems, in particular the connection between Hamilton–Jacobi theory and quantum mechanics, have been studied using a field-theoretic approach since the early days of quantum mechanics.^{2,3} In line with these early works and following the reasoning of Wheeler,^{4} the problem of interacting identical neutral particles can be treated using classical field theory; utilized in conjunction with a Hamilton–Jacobi formalism, the approach leads naturally to a wave description of particles and their interactions. Indeed, it was a significant revelation of the last century that Maxwell's equations themselves follow from the treatment of electrons as excitations of a field taken together with certain symmetry considerations.^{5,6} Our work draws on electromagnetism as a familiar application of classical field theory and offers a new as well as useful perspective on matter-waves.

The wave characteristics of quantum-mechanical particles, either individual or ensembles, are commonly referred to as “matter waves;” they are characterized by a de Broglie wavelength $ \lambda = h / p$, where *h* is Planck's constant and *p* is the particle momentum. An excellent example of matter-waves is the flux of atoms extracted from a Bose–Einstein condensate (BEC)^{7,8} that was first achieved in 1995 in a diluted gas of rubidium and sodium atoms.^{9,10} Early experiments demonstrated the quantum coherence of these systems.^{11–13} The achievement of BEC prompted many-body physics to shift its focus toward ultracold gases, resulting in various models, analytical techniques, and numerical methods, such as the Gross–Pitaevskii equation (GPE), time-dependent Hartree–Fock–Bogoliubov (TDHFB), Bogoliubov–de Gennes equation, quantum kinetic theory, and stochastic methods, among others.^{14–17}

In parallel with advances in atom cooling and manipulation techniques, so have there been advances in atom interferometry^{18–29} that take advantage of the properties of ultracold atoms, particularly their coherence. Atom interferometry and its use of matter-waves has been very much inspired by its optical interferometry counterpart. At the same time, the nature of BEC systems also leads one naturally to think in terms of atom currents flowing from one part of a system to another, thus arose the field of atomtronics, which refers to an atom analog of electronics in which atom flux and chemical potential substitute for electric current and potential.^{8,30–37}

As fundamentally non-thermal-equilibrium open quantum systems, even rather simple atomtronic circuits prove challenging for standard many-body methods. Yet, we can look more deeply at the analogy between atomtronic currents vs electronic currents and matter-waves vs electromagnetic waves. Maxwell's equations inform us that an oscillating electric current gives rise to a coherent electromagnetic wave. We can ask, therefore, whether an oscillating atomtronic current gives rise to a coherent matter-wave. The answer to this question sets the motivation for our work here. We will find that a field-theoretic description of matter-waves provides an additional set of tools in which to treat atomtronic circuits. The approach leverages the analytical, heuristic, and numerical tools that have been well-developed for electromagnetics. In particular is the appearance of “Maxwell matter waves”—the classical limit of coherent matter-waves governed by matter-wave duals to Maxwell equations. Like their electromagnetic counterparts, Maxwell matter-waves exhibit temporal coherence, and they have several other aspects that are familiar from electromagnetics.

Here, we consider an oscillating current of neutral atoms, such as ^{87}Rb, which repel each other due to mutual van der Waals interactions. In field theory, atom interactions are accounted for by the introduction of a gauge field. In comparison to the Coulomb forces between electrons, van der Waals forces are weak, and they are short-ranged. One might conclude that, therefore, the gauge field can be dismissed if, say, particles are far apart. However, that conclusion proves to be misleading if not wrong. The gauge field that embodies the interaction between neutral atoms is associated with energy and, generally, the transmission of power. At low temperatures, atoms interact through s-wave collisions, and, thus, the gauge field that characterizes these interactions assumes a relatively simple form. We will sometimes refer to this field as the “matteron field” to provide it with an identity. The matteron field, then, accounts for the interactions among the cold neutral atoms.

With a motivation that parallels the electromagnetic case, we are particularly interested in the notion of a single-mode of the matteron field. Generally, a finite temperature gas, or even a pure BEC, will be comprised of a continuum of modes.

Our theoretical development is carried out in three sections. The first is a classical, field-theoretic derivation of the family of Maxwell equations characterized by a few formally introduced constants, where the classical theory does not constrain the dynamical parameters of wave propagation. While assuming the constants to be the speed of light and free-space impedance leads to the well-known Maxwell equations, the same constants with different meaningful values can represent matter-wave duals to Maxwell's equations. These values are provided by a quantum-mechanical treatment of the fields, which is carried out in Sec. III. This section also establishes the formal connection between the quantum-mechanical coherent state and the classical coherence of matter-waves implied by Maxwell equations duals. With the dynamical parameters in hand, Sec. III returns to the classical treatment, giving the fully constrained Maxwell equation duals from which a quantitative correspondence between experiment and theory can be established. We close with Sec. V to highlight the similarities and differences between coherent matter-wave and electromagnetic wave theories.

## II. THE CLASSICAL MATTER-WAVE FIELD

Our development of classical, non-relativistic field theory follows closely the excellent notes of Wheeler,^{4} which include substantially more discussion concerning the development than we provide. Typically, modern physics treatments of field theory are written to address the domain of high-energy physics, so the constraint of Lorentz covariance and its tie to the speed of light are imposed early on.^{38–41} In the realm of ultracold atomic physics, particle velocities of 10 m/s are already very high, placing us in a non-relativistic realm.

^{4}characterizing a flux of particles having mass

*m*,

*S*is Hamilton's principal function, $ U ( x )$ is an applied particle potential,

*R*has units of $ ( length ) \u2212 3$ and so describes the energy carried by the flux as a density, and $ \u2202 i$ is a partial derivative over time

*t*or spatial coordinates $ i \u2208 { x , y , z}$. (Our symbol choice

*R*for density follows the notation of Wheeler;

^{4}the perhaps more natural

*ρ*is reserved for a later analog with electromagnetism.)

*S*is

*w*is dimensionless.

**A**and $\varphi $ and incorporates the replacements,

*q*. In electromagnetics, the charge indicates the strength of the particle interactions; here, the choice to include the symbol is gratuitous, since the interaction strength per particle or per mass could equally well be incorporated into the units for the gauge field. The Lagrangian density is re-written as

**F**and

**G**of the electric field

**E**and magnetic field

**B**,

*ξ*

_{0}and

*η*

_{0}to the electric permittivity and permeability in Sec. IV.

^{41}Define charge and current densities,

**F**and

**G**,

We have now a complete set of Maxwell's equations: Eq. (18) as Gauss' law for the electric field, Eq. (15) as Gauss' law for the magnetic field, Eq. (16) as Faraday's law of induction, and Eq. (20) as Ampère's circuit law with the addition of Maxwell's displacement current, along with the continuity equation Eq. (14). In Wheeler's words, in fact, we have a family of Maxwell's equations.^{4} Substituting $ v 0 \u2192 c , \u2009 \eta 0 \u2192 \mu 0 , \u2009 \xi 0 \u2192 \epsilon 0$ and, of course, $ F \u2192 E$ and $ G \u2192 B$ returns us to electromagnetics in familiar nomenclature. The charge and current densities are sources for the electric and magnetic fields. Equivalently, we can view them as the source for the scalar and vector potentials.

*v*

_{0}substitutes for the speed of light

*c*, which is an experimentally determined fundamental physical quantity. The matter-wave dual will become evident as we blindly follow the course as we would for electromagnetics. In particular, we are now in a position to consider an oscillating matter current. Consider the positive frequency component of a one-dimensional oscillating current,

^{36}Let us correspondingly write

*n*

*v*

_{0}, so that the refractive index approaches unity, the field amplitudes diverge. The minus sign associated with the impedance has been explicitly brought out because, as we shall see in Sec. III, the matter-wave speed limit is a lower bound rather than an upper bound, and the refractive index is typically less than unity rather than greater than unity as is the usual case for electromagnetics. The fact that the impedance is negative has important physical significance, namely, the power carried by the field,

To provide some insight into the formalism, let us consider a classical picture of particle dynamics as illustrated in Fig. 1. Plotted along the x-axis is the particle potential energy due to an applied potential. In the case of alkali atoms, sculpting a particular potential is conceptually simple to implement, say, utilizing laser light tuned to the red or blue side of an atomic resonance to either raise or lower, respectively, the atomic potential energy. An ensemble of atoms is initially on the left, biased with high potential energy. They are shoved from the left toward the right, say, by the left-hand wall oscillating at the frequency *ν*. The atoms fall down the potential, propagate along a relatively flat region, and then climb back up the potential on the right. Of course, they have gained speed as they roll down the potential, and, therefore, they are far apart and then move closer together again as they climb the potential.

Van der Waals forces are at play only when atoms are close together; therefore, at the bottom of the potential, their interactions are much weaker than at the top, where they are close together. Yet, both the applied and the interaction potential are conservative, so the atoms will “remember” their interaction energy such that they return to the same configuration on the right as they began on the left (more or less, since we have not given detail on how the walls work). If one imagines a continual stream of atoms from the left, the average flux will everywhere be the same, as is the current oscillating everywhere along the system at frequency *ν*. It is the gauge field that serves as the memory that invokes coherence of the current across the system.

## III. QUANTUM TREATMENT OF THE MATTER-WAVE FIELD

The field theory we have applied treats particles in terms of a delocalized field. This field “knows” about the potential everywhere (say) in the $ + x$ direction, and, although the field is everywhere in the half-space, it is also propagating in the $ + x$ direction. In particular, in some sense, its current position and velocity are dependent on a distribution of the possible past positions at historical times. The view of particles as delocalized entities suits well a transition to a quantum-mechanical description. The notion of a short-range interaction is meaningful in a picture of localized particles undergoing a collision, yet when the particles are delocalized and, in fact, occupy an entire half-space, their interaction is manifest differently: if a steady oscillation is occurring in one location, one should expect it to occur in all locations, loosely in keeping with the depiction in Fig. 1.

As a practical matter, absent from the matter-wave duals of Maxwell's equations are values of the two key parameters, from which all dynamics follow. In electromagnetics, the vacuum speed of light and impedance (or the vacuum permeability and permittivity) are taken as fundamental constants of nature and are determined empirically (as is the charge of the electron). To determine the dynamical constants of our field theory, we must turn to a quantum-mechanical description, in which Hamiltonian steps in for the Lagrangian and operators step in for dynamical coordinates and their corresponding functions and functionals.

The classical theory anticipates that the potential energy associated with the gauge fields is negative. Negative energy associated with quantization of the matter-wave gauge field is in stark contrast with the electromagnetic case. While the physics itself is straightforward, conceptual navigation through negative energy territory is best done with some care. We, therefore, begin with the basics.

^{42}They have the commutation property,

*k*is the deBroglie wavenumber associated with the particle momentum and corresponding particle group velocity,

_{m}*N*of the matter field,

_{m}The Hamiltonian Eq. (37) embodies the energy of a matter field, which is of the particles, alone. It does not account for the particle interactions. The classical development likewise began with the Lagrangian for the particles alone, then treated interactions through the introduction of a gauge field, and, finally, considered the energy of the gauge field alone. The quantization of the gauge field, i.e., the matteron field, is similar to that of the matter field. We introduce the ladder operators $ a \u0302 f \u2020$ and $ a \u0302 f$, which obey commutations relations the same as those above for the matter, while they commute with each other, $ [ a \u0302 f , b \u0302 m ] = [ a \u0302 f \u2020 , b \u0302 m ] = 0$, etc.

*ν*(for the time being, we take the oscillator frequency to be general, $ \nu \u2192 \nu f$). A notable difference between the matter field and the matteron field is that the ground state of the latter has nonzero energy. Moreover, the energy associated with the gauge field is negative; we comment further on this in Sec. V. Glauber has pointed out that an inverted harmonic oscillator can be described by ladder operators identical to the normal oscillator but is characterized by negative rather than positive energies,

_{f}^{43}

The quantum of excitation of the matteron field is the dual of the photon of electromagnetics: a massless gauge boson. We have referred to the dual elsewhere as a “matteron,”^{36} hence the naming of this gauge field. In contrast to the photon, the matteron is associated with negative energy.

*N*matterons,

_{f}As we move to explicitly consider particle interactions, there is a subtle but important departure from the case of electrons interacting through the electromagnetic field. In the latter case, we can, philosophically at least, separate the photons from the electrons and then consider how they interact. We saw from the classical treatment that an oscillating particle current serves as a source for the gauge field-one cannot exist without the other. For the corresponding quantum case, an excitation of a matter field is necessarily accompanied by an excitation of the matteron field. We are, thus, interested in the combined matter and matteron field excitations—in particular, what is appropriate to call “a single-mode excitation of the matter-wave field.” Hereafter, we shall assume the excitation is one-for-one, i.e., *N _{m}* =

*N*. In contrast to the familiar deBroglie matter-wave case, there are two distinct frequencies involved: the Planck–Einstein frequency

_{f}*ω*associated with the particle and the frequency

_{m}*ν*associated with the oscillating field, which is set by some external agent (a circuit, say), applying a time-varying force and causing a propagating oscillating current.

_{f}*v*

_{0}is a lower bound on the particle velocity: First, we can understand that the dynamics of the system must be such that the total energy is non-negative, meaning the particle energy must be greater than or equal to the matteron energy, since a negative energy would imply that the system lies in a bound state. This, in turn, means that the minimum velocity is dependent on the frequency

*ν*associated with the gauge field,

_{f}*x*direction. (On the other hand, particles moving in the negative

*x*direction are associated with plane waves of current and charge density propagating in the negative

*x*direction.) This is the rationale for clarifying the distinction between particle flux and particle current in the IV. Having a classical picture of an oscillating particle in mind, its motion, and hence particle flux, is always in the positive direction as long as its center of mass is moving sufficiently fast in the positive direction. Evidently

*n*= 1, the wavelength $ \lambda = 2 \pi / k = 2 \pi / ( n k 0 )$ associated with the field becomes equal to the particle de Broglie wavelength $ \lambda = n 2 \pi / k 0$.

*m*or another subscript to refer specifically to the matter field, and the operator $ a \u0302 f$ specifically with the subscript

*f*to refer to the matteron (gauge) field. The “matter-wave field” will refer to the two taken together. From here on the quantum-mechanical treatment, we will focus onto the single-mode matter-wave case. That means we shall fix the two frequencies, $ \nu f \u2192 \nu $ and $ \omega m \u2192 \omega $, and drop subscripts when there is no ambiguity. Formally, matter and gauge fields occupy distinct Hilbert spaces. In the single-mode case, however, the two are so tightly entangled that for the purpose of this work we can treat them together. We, thus, use $ a \u0302 \nu , a \u0302 \nu \u2020$ as matter-wave field ladder operators,

*N*of identical particles as such a supply (see Fig. 2). We set as the system single-mode ground state,

_{r}*r*) and matter-wave states (

*ν*), as in practice, the former will involve trapped particles and the latter particles (massive and matterons) that propagate in free space. We introduce joint operators,

Total system energy is, or course, not conserved, since for every particle contributed by the reservoir, $ \u210f \nu $ of energy is removed by the gauge field. We keep in mind that the emitted massive particles themselves each carry the full $ \u210f \omega $ worth of energy of the particle contributed by the reservoir—it is the negative energy carried by the matteron that is responsible for the energy shortage. This indicates that the emission of the matteron field itself induces cooling of the reservoir, although the emission of the particles proves to cause heating^{44} such that the next effect is heat added to the reservoir. We have omitted the detailed physics of these processes.

*ν*. The expectation of the energy carried by the coherent state is

Our theoretical development has involved no reference to the origin of the particle interaction—only that it exists and can be characterized in terms of particle mass *m* and charge *q*. The pivotal requirement is that the interaction energy is sufficient to supply a matteron having energy $ \u210f \nu $. In Fig. 2, this corresponds to a BEC having an interaction mean-field contribution to its chemical potential $ \mu B \u2265 \u210f \nu $ in order to supply matterons to the emission.

*q*is convenient for the sake of analogy and for dimensional bookkeeping. In practical applications, however, one will find it convenient to set the charge either to the particle mass or to unity, corresponding to the impedance given in units of (energy-per-mass)-per-(mass-per-second) or in units of (energy-per-particle)-per-(particle-per-second).

## IV. COHERENT MATTER-WAVE PARAMETERS

The analogy with electromagnetics can provide insight into the matter-wave system through a simple radio frequency circuit. Figure 3 shows an oscillating microwave voltage source delivering power to a load *R _{L}* through an electromagnetic transmission line.

^{45}The voltage and current waves carry energy, and if the source, load resistance, and transmission line impedance are equal, all the energy is delivered to the load. Of note is the fact that the electric current oscillates between negative and positive values and, thus, flows in both directions. The voltage oscillates in phase with the current; as a consequence, the power flows continuously from left to right.

The example highlights an important distinction between current and flux: We will use the term “flux” to refer to the flow of power, equivalently in the case of atomtronics the flow of particles, or quantum-mechanically to the flow of probability. As with the transmission line example, we consider a case in which ultracold atom flux is continuous, while atomtronic current and voltage are oscillating amplitudes. By contrast, a BEC whose center-of-mass is oscillating within a trap is described by an oscillating flux.

*J*

_{0}. In particular, the potential is given by Eq. (33), while the remaining field amplitudes are

*Z*, which, in this context, is referred to as the characteristic impedance

_{f}^{45,46}such that

In circuit applications, it will often be convenient to consider the energy of the particles $ \u210f \omega $ carrying the currents as fixed, while the oscillation frequency *ν* is taken to be variable. In such cases, it will be convenient to make the substitution $ k 0 \u2192 k / n$ in the various formulas for fields and impedance.

## V. REMARKS

Through field theory, we have derived a set of matter-wave duals to Maxwell's equations describing electromagnetic waves. We have intentionally cast the theory to resemble electromagnetic theory. While valuable from an intuition standpoint, doing so also obscures the contrasts between the two.

Conspicuously absent in the treatment are the duals to the constitutive relations corresponding to $ D = \epsilon E$ and $ H = B / \mu $. In particular, for example, the gauge field energy in the Lagrangian Eq. (10) uses the duals to the vacuum permeability and permittivity. That is because here there are no duals to polarization or magnetization of a medium. The velocity associated with the gauge field is tied to that of the atoms, which, in turn, is influenced by an external potential that acts directly on the atoms. By contrast, with electromagnetism, the gauge fields **E** and **B** themselves interact with a medium in which they propagate.

We have shown that an oscillating matter current gives rise to a coherent matter-wave that obeys the Maxwell equations duals. Of dramatic difference from the electromagnetic counterpart is that the oscillating matter current does not produce a radiated field that exists in a source-free region. On the contrary, through the negative sign in the Hamiltonian Eq. (43) for the matteron field, we see that the matter and its gauge field are bound together. The quantization of the field introduces the matteron, a massless gauge boson that is dual to the photon. In a particle picture, the matteron is bound to the massive particle, while the classical particle can be pictured as undergoing longitudinal oscillation. It is worth noting that if the matteron energy were taken to be positive, it would not be bound. As a massless boson, the matteron would then necessarily travel at the speed of light (see, for example, Schwichtenberg^{41}) and be indicative of a long-range force, not of the short-ranged van der Waals forces at play in our particle interactions.

It is reasonable to consider whether our matter-waves are, in fact, sound waves, and whether the matteron should, in fact, be called a phonon. Sound waves, such as those that can propagate in a BEC,^{14} arise from the interaction among the condensate atoms, and the propagation velocity depends upon, among other things, the atomic density. We have seen in the discussion at the end of Sec. IV that the interaction energy associated with a matter-wave as it propagates in otherwise empty space can be arbitrarily low, and that the wave propagation speed is given by the particle group velocity. So there is nothing acoustic about the matter-waves under consideration.

The picture of longitudinal oscillation of the massive particle is perhaps at first objectionable because oscillation implies a restoring force. While matterons are massless, they carry momentum $ \u210f k 0$, and they are bound to the massive particle with energy $ \u210f \nu $. Thus, we can think of a massive particle attached to a lighter one with a spring that supplies a restoring force as the two oscillate with respect to each other.

The contrast between matter-waves and electromagnetics is further highlighted, for example, by the dual to Gauss' law, Eq. (18), which might suggest field lines emanating from an isolated charge as it does for the electric case. Yet, the dispersion implicit in the dual to permittivity *ξ* Eq. (60) belies the complexity under the hood, so to speak, particularly at DC. In short, at a distance away from any current, no matteron field is detectable. Yet, within the flux of particles, their effects are measurable: the fact that the energy of the bound particle-plus-matteron state is less than the energy of the massive particle alone reveals the physics behind the flow of negative power carried by the gauge field.

The dynamical parameters for matter-waves, namely, the propagation speed and impedance, are imposed by quantum mechanics rather than by relativity as is the case for electromagnetism. The quantum treatment of the fields provides a clear understanding of the meaning of “classically coherent matter wave” as an eigenstate of the combined matter-field-plus-gauge-field annihilation operator, which has a well-defined phase relative to the oscillation frequency *ν* of the current. Keeping to the classical domain, it seems appropriate to refer to these waves as “Maxwell matter waves.” In this context, our duals to Maxwell's equations were contrived to appear the same as the familiar set, yet the fundamental conclusions are that the propagation velocity has a lower rather than upper limit, and that the impedance of free space has a negative real value for Maxwell matter-waves, compared with the positive 377 ohms for electromagnetic waves.

At the same time, Maxwell matter-waves do obey their dual Maxwell's equations, and thus the heuristic and analytical tools that already exist for electromagnetics can be put to effective use to address matter-wave calculations. Indeed, one can work with the fields **F** and **G**, but it is also natural to work in a circuit theory context involving scalar potential and current, connected to each other through impedance, as we have seen. Step changes in the refractive index, for example, lead to familiar coefficients of wave amplitude and power reflection and transmission that arise in the case of particle wavefunctions, light waves, microwaves, and so on.^{45–47}

It is relevant to note that the interference of Maxwell matter-waves reveals substantially different character than the interference of de Broglie matter-waves. For example, the fringe spacing of interfering de Broglie waves will decrease with increasing particle velocity, while that of Maxwell matter-waves will increase.

Traditional is the de Broglie view in which thinks of matter classically as a particle phenomenon whose wave character is revealed by quantum mechanics. Here, we see that Maxwell matter-waves provide the complementary view in which matter can be viewed classically as a wave phenomenon whose particle aspects are revealed by quantum mechanics.

## ACKNOWLEDGMENTS

The work of K. Krzyzanowska was supported by the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the U.S. Department of Energy (DOE). The work of D. Z. Anderson was supported by Infleqtion. We are grateful to S. Du and V. Colussi for valuable discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

D.Z.A. owns stock in ColdQuanta Inc. and serves on the Board of Directors of ColdQuanta Inc. (doing business as Infleqtion).

### Author Contributions

**Dana Z. Anderson:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Katarzyna Anna Krzyzanowska:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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