Foundational Issues in Dynamical Casimir Effect and Analogue Features in Cosmological Particle Creation

Abstract

Moving mirrors as analogue sources of Hawking radiation from black holes have been explored extensively but less so with cosmological particle creation (CPC), even though the analogy between the dynamical Casimir effect (DCE) and CPC based on the mechanism of the parametric amplification of quantum field fluctuations has also been known for a long time. This ‘perspective’ essay intends to convey some of the rigor and thoroughness of quantum field theory in curved spacetime, which serves as the theoretical foundation of CPC, to DCE, which enjoys a variety of active experimental explorations. We have selected seven issues of relevance to address, starting from the naively simple ones, e.g., why one should be bothered with ‘curved’ spacetime when performing a laboratory experiment in ostensibly flat space, to foundational theoretical ones, such as the frequent appearance of nonlocal dissipation in the system dynamics induced by colored noises in its field environment, the existence of quantum Lenz law and fluctuation–dissipation relations in the backreaction effects of DCE emission on the moving atom/mirror or the source, and the construction of a microphysics model to account for the dynamical responses of a mirror or medium. The strengthening of the theoretical ground for DCE is not only useful for improving conceptual clarity but needed for the development of the proof-of-concept type of future experimental designs for DCE. The results from the DCE experiments in turn will enrich our understanding of quantum field effects in the early universe because they are, in the spirit of analogue gravity, our best hopes for the verification of these fundamental processes.

1. Introduction

In this “viewpoint/perspective” paper, we re-examine some basic issues in the dynamical Casimir effect (DCE) from minor (mostly known and correctly dealt with, yet worth pointing out as some popular papers have missed out) to moderate (technically sound, yet conceptually not understood deeply or thoroughly enough) to major (may not have been taken into account fully, yet important for future explorations) using the more developed and advanced theoretical framework and better conceptual understanding gained from the research on cosmological particle creation (CPC) since the late 1960s to today [1,2,3]. In parallel to the 50 years of theoretical developments for CPC, some key issues of which are highlighted here, on the experimental side, a great variety of DCE experiments have been carried out, or are actively in preparation. We are fortunate to have excellent reviews such as [4] on DCE and [5] for analogue gravity which keep up with the latest developments in both fields. The strengthening of the theoretical ground for DCE is not only useful for better conceptual clarity but needed for the development of the proof-of-concept type of future experimental designs for DCE, which is our best hope for the verification of early universe quantum field processes in the spirit of analogue gravity.

1.1. Three Kinds of Quantum Field Processes

Both CPC and DCE encompass three kinds of quantum field processes, see Table 1 of [6]: (1) Casimir effect [7,8,9,10,11], the static limit of DCE, corresponding to a universe with a fixed and constant scale factor, such as the Einstein universe [12], which by itself is of great interest and importance; (2) particle creation [13,14], time-varying scale factor (dynamical, nonadiabatic process), and (3) trace anomaly. Trace anomaly [15,16] (for its gravity context, see discussions in [1,2,3]) involves higher derivatives of the scale factor, which can play an important role in black hole physics such as the Hawking effect [17,18] and in the dynamics of the early universe, such as altering the behavior of the universe near the singularity [19], engendering inflation [20] and addressing the horizon [21] issues. However, for laboratory DCE experiments, since the speed of the moving mirror cannot easily be increased to the range where the trace anomaly factor becomes important, this effect is justifiably ignored. We are not aware of many theoretical papers on the trace anomaly in the context of DCE. In the few which considered it, e.g., the paper by [22] on DCE in $S^1$, it was misconstrued as a backreaction effect. Interested readers can consult a recent paper [23] which contains a trace anomaly calculation and discussions of its backreaction effects in analogue quantum optomechanics (QOMs) [24,25,26,27,28,29,30,31,32,33].

Since the Casimir effect is familiar to the reader, and trace anomaly is rarely mentioned in DCE, in this paper, we shall focus on the most important one of these three quantum processes, namely, particle creation from vacuum fluctuations of the quantum field in both DCE and CPC. We shall show the primary mechanism they both are based upon, namely, the parametric amplification of vacuum fluctuations of a quantum field, and, noticeably, their differences, which are mainly in the setups: namely, the drive in cosmology is due to the expansion or contraction of the universe, whereas in DCE it comes from an external agent, that which sets the mirror in motion and keeps it in some prescribed trajectory. Notice that we use the word “prescribed" for DCE, but not for CPC. This is because the way that a universe evolves follows the solutions of the semiclassical Einstein equation (SCE) [3] with a given quantum matter field source, and the scale factor is determined by solving the SCE and the field equation self-consistently—this is referred to as the cosmological backreaction problem.

1.2. Seven Issues of Interest

In terms of issues, we have selected seven of relevance to discuss. The first two issues pertain to the setup and kinematics: (a) The first is how the moving boundaries change the quantum field enclosed within as compared to a typical cosmological model. If we use for DCE the simplest set up of a rectangular box, where the sides are allowed to expand or contract, then the analogue cosmological model would be the Bianchi type I universe [34], where the scale factors in different directions expand at different rates. We shall explain some subtleties in their differences, such as what happens near the boundary. (b) If we use mirrors as examples for DCE, both perfect as with a conductor, or imperfect as with a dielectric, one can calculate their differences in, say, the amount of particles created, e.g., in [35]. Is there a cosmological analogy to an imperfect mirror? Not so straightforwardly. One can consider adding a boundary term in the action for the quantum field, the importance of which for the considerations of how the global structures of spacetime manifest in the quantum field effects was emphasized by Gibbons and Hawking [36]. This may be an interesting analogue gravity issue to explore in the future but not in our present considerations. However, for common setups in DCE experiments, it is perhaps more useful to account for the more realistic factors, like mirrors with finite reflectivity. For this purpose, we present the mirror–oscillator–field (MOF) model based on microphysics processes designed as an economic theoretical framework for quantum optomechanics. It is also useful for expounding dissipation, noise, fluctuations, and the fluctuation–dissipation relation, which serves as a requisite condition for self-consistency in backreactions.

The next five issues pertain to dynamical processes, which both DCE and CPC share in an intimate manner: (c) nonadiabaticity and parametric amplification; (d) enhancement in finite temperature effects is really due to stimulated emission; (e) dissipation; (f) backreaction; and (g) fluctuation–dissipation relation.

A word on the style of presentation here: because the scope of our paper spans two important, yet on the surface rather disparate, fields which have been active for the past 50 years, we need to be selective on the issues and the coverage. To provide some ordering, we have arranged these seven issues from the commonly better known, mentioned here for completeness, to the lesser explored yet important issues, and some are currently under investigation. The extent of our coverage on each issue will go in proportion as well. Aiming at a ‘perspective’ for the big pictures, we shall place more emphasis on the discussions of key ideas. To economize valuable space, when we invoke formulas in our discussions, we may invoke a few representative papers worthy of some careful study and refer to the formulas therein. Finally, since this paper is in the nature of a ‘viewpoint’, which is almost opposite to a ‘review’ in terms of span and balance, it is almost unavoidable that it draws on the authors’ past experiences and refers to their own work of importance. For this, we beseech the reader’s patience and forbearance.

2. Setup: Requisite Conditions for the Validation of the Analogies

It has been known for many years that the central mechanism of dynamical Casimir or Casimir–Polder [37] effects in the excitation of the quantum field is the same as that in cosmological particle creation, namely, via parametric amplification, the former is by the movement of a mirror or an atom, while the latter is by the expansion or contraction of the universe. The theoretical developments in particle creation processes in the early universe [13,14], moving mirrors [38,39,40], and black holes [17] have advanced in great strides from the late 1960s through the 1970s in the Hawking–Unruh effects [41] to the beginning of the 1980s in inflationary cosmology [42,43,44], pairing with, as their theoretical foundation, the establishment of quantum field theory in curved spacetime (QFCS) [1,2,45,46,47,48] and semiclassical gravity (SCG) [3]. It would be beneficial to experimentalists and their supporting theorists in the fields of atomic–optical–condensed matter–fluid (AOCF) physics who work on DCE or DCP effects to tap into the useful methodology and important results from the QFCS and SCG theoretical enterprises from the 1970s to the 1990s. It would also be helpful to theorists in gravity/cosmology, the present authors certainly included, to better understand the AOCF experimental requirements, from their setups and their implementations to the interpretation of their findings, and place them in the proper contexts of cosmological particle creation to make the best of the many theoretical results obtained before. This task is what is required in the establishment of a viable analogue gravity program as witnessed by the highly successful classical fluid [49] and BEC experiments [50], to give just two established examples, and many more forthcoming. Please refer to the updated review [5] for the latest developments.

It is a happy situation that many prior and ongoing works exist in moving mirror setups linked to the Hawking effect and in accelerating atoms (called ‘detector’ [41,51] in this community) linked to the Unruh effect [41]. Our emphasis here is on dynamical Casimir (moving mirrors) and Casimir–Polder (moving atoms near a mirror or medium) effects, not requiring the moving atom or mirror to be uniformly accelerating or to follow any particular trajectory. The physics of the Hawking effect in black holes, the Davies–Fulling effect [39,40] in moving mirrors [52,53,54,55,56,57], and the Unruh effect for accelerating detectors (atoms) are of a different nature from cosmological particle creation, as all three cases in the former group focus on the effects of the event horizons on the emitted radiation (Hawking effect) or the radiance (an Unruh–DeWitt detector feels hot at the Unruh temperature). Cosmological particle creation originates from the parametric amplification of vacuum fluctuations of the quantum field which does not invoke event horizons. For de Sitter space, when represented by a spatially flat ($k=0$) Robertson–Walker universe (the ‘Poincare patch’) expanding at an exponential rate in cosmic time, it provides the inflationary cosmology [42] scenario. It falls under our “cosmology or dynamical spacetime” category, whereas an observer in a de Sitter space in the static coordinatization (see [1] for a display of different coordinatizations of de Sitter space) feeling thermal radiance—the Gibbons–Hawking effect [36]—belongs to the “black hole” or “event horizon” category. (Thermal particle creation in both the black hole and the cosmology categories can be understood as sharing a similar origin, see [58,59]). The moving mirror analogue of black hole quantum field processes, the Hawking effect, and its backreaction has a much broader literature, which we will not delve into. Our discussions here shall focus only on problems and issues in dynamical spacetime of the cosmology category, which, by analogue gravity correspondence, relate to dynamical Casimir or Casimir–Polder effects.

What we want to do here is to add some solid groundwork to help establish the linkages between DCE/DCP and CPC to strengthen this branch of the analogue gravity program. We try to take the stand of the AOCF in perspective and examine relevant elements in the rich QFCS/SCG theory repertoires, which can reinforce the theoretical foundation of AOCF for the analogue gravity purpose. Thus, we shall start with the most rudimentary questions, such as why we should care about ‘curved’ spacetime effects while all of our experiments are carried out in undeniably flat space, and, seemingly even more remote, how the topology of space would enter. We address these issues in this section. In the next section, we focus on the central mechanism of parametric amplification, resonance and discuss the stimulated emission, thermal effects, and nonequilibrium issues.

In the second part of this paper, in Section 4 and Section 5, we move to slightly more complex issues, namely, fluctuations, backreaction, and dissipation. We shall bring forth what was accomplished in SCG, using open quantum system concepts applied to the universe dynamics with quantum fields to address these theoretical issues in DCE. In Section 5, we introduce a working model called the M (mirror) O (oscillator) F (field) model for quantum optomechanics, invented a decade ago [60,61], where a dynamical degree of freedom is assigned to a mirror so that one can include imperfect mirrors and a dielectric medium based on microphysics dynamics. We identify the dynamical variables in DCE and DCP as cogent examples to illustrate how their interplay provides a fuller description of the various experimentally accessible effects. Then, in the spirit of analogue gravity, we describe a problem in cosmology where a tripartite dynamical variable setup works well, namely, the treatment of preheating in inflationary cosmology. The MOF model is a simple example of a more elaborate and systematic theoretical framework developed earlier by Behunin and Hu [62,63] known as the graded influence action formalism based on a nested set of coarse-grained effective actions [64,65,66]. We give a schematic description without going into the details, but interested readers can employ this and other related work using the functional integral formalism [67,68,69,70,71,72] to explore a host of complex issues in the nonequilibrium dynamics of the ordered strata of effectively open quantum systems, such as those manifested in quantum friction [73,74,75,76,77,78,79,80,81,82]. Finally, we mention some notable recent papers which further develop the theoretical basis of the MOF model, where their results can be used to describe various aspects of quantum optomechanics of the atom/mirror/field/medium interactions, from motional decoherence [83,84,85] and entanglement dynamics [86,87,88] in DCE and in composite particles [89] to the detection of quantum features of gravitational waves [90,91,92,93,94,95,96,97].

2.1. Quantum Fields in $S^1$ and $T^3$ Spaces: Boundary Conditions and Topology

DCE has been studied by many authors for more than half a century; see the excellent review by Dodonov. For expounding the foundational issues, let us begin with the simplest setup of two point mirrors, one located at the origin of a line and the other at distance l from it. If the mirrors are perfectly reflecting, a condition we assume until we reach the last section, then a reflecting (Dirichlet) boundary condition on the quantum field at the two ends of this line segment cavity can be imposed. Field theorists may find it convenient to replace the line segment by a ring of circumference l by imposing a periodic boundary condition $\varphi \left(x\right)=\varphi (x+l)$ on the quantum field living on this ring. The range of summation over field modes differs: the Dirichlet condition afforded by a perfectly reflecting mirror sums from 1 to ∞, while the periodic boundary condition sums from $-\infty$ to $+\infty$. When a line segment is turned into a ring, the topology changes. We can think of two aspects in the physical similarities and differences between a reflective boundary condition on the two end points and a periodic boundary condition, which allows for replacing the line segment by a ring:

(i) Similarities in energy consideration: The normalization of modes differs by a factor of $\sqrt{2}$. So in terms of actual results, these two boundary conditions only differ by a small overall factor.

(ii) Differences in phase consideration: A wave under periodic BC preserves the phase condition, namely, after it traverses the full circle and returns to the origin, it will have the same phase as when it started from the origin. However, this is not the case for a reflective BC: a wave propagating in the vacuum upon reflection by a mirror suffers a $\pi$ phase inversion, meaning a positive crest will turn negative right after reflection (this depends on the relative strengths of the dielectric medium of the two regions: here, the vacuum of course has a weaker strength than the mirror, while in the opposite situation, a wave propagating from a denser medium (with a larger dielectric constant) to a lighter medium will, upon reflection, keep the same positive phase) and this will result in an important difference for phase-sensitive issues, such as entanglement [98].

Allowing l to change with time engenders the dynamical Casimir effect. Let us introduce a scale factor $a\left(t\right)$ which at $t=0$ is set equal to 1. the length of the line segment or the circumference of the circle l is evolved to $L\left(t\right)=a\left(t\right)l$ at time t. We can then describe the metric of this $1+1$ D spacetime by the line element $ds$ given by ($c=\hslash =1$):

$$d{s}^2\equiv g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }=\mathrm{d}{t}^2-a{\left(t\right)}^2\mathrm{d}{x}^2,$$

(1)

where $g_{\mu \nu }$ is the metric tensor. If $a\left(t\right)$ is given a periodic time dependence, it becomes a pulsating ring or sphere. This motion which preserves spherical symmetry is described as a ‘breathing’ mode in DCE.

To pursue this issue further, we can consider twisted field boundary conditions, e.g., $\varphi \left(x\right)=-\varphi (x+l)$, namely, the field amplitudes have a negative sign when they make a full circle around. Quantum theories of fields with this type of anti-symmetric BC have been studied by Avis, DeWitt, Dowker, and Isham et al. [99,100,101,102]. These issues bearing on the topology of space and fields are addressed in the context of entanglement dynamics on an Einstein cylinder with $S^1$ (space) $\times R^1$ (time) topology in [103]. We will not pursue the effects of field configurations with nontrivial topology here.

Moving on to three-dimensional space, we meet the popular setup of two parallel conducting plates with time-varying separation [104,105,106]. Here, we will be considering a rectangular box whose three faces are made of conducting plates. Recognizing the subtle differences between the reflective and periodic boundary conditions on the quantum field, we construct the rectangular box by imposing periodic boundary conditions on the field at both ends of each of the three sides. The simplest setup for DCE would be to fix the two transverse directions with equal length and allow the length of the third side (longitudinal direction) to change with time. A rapid change will engender particle creation from vacuum fluctuations, which is the more ubiquitous and prominent quantum field process in dynamical Casimir effect we choose to focus on here.

The rectangular box configuration with time-dependent sizes mimics that of a symmetric Bianchi Type I universe (with periodic conditions imposed) with line element

$$d{s}^2\equiv g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=d{t}^2-[a_1^2\left(t\right)d{x}^2+a_2^2\left(t\right)d{y}^2+a_3^2\left(t\right)d{z}^2],$$

(2)

where $a_i$ are the scale factors in the three directions. We consider the case with $a_2=a_3=1$, $a_1=a\left(t\right)$, i.e., with only one side moving, in the $\pm x$ direction. Imposing periodic boundary conditions at both ends of each of the three sides, i.e., between x, y, $z=0$ and x, y, $z=l$, where l is the coordinate length, would turn this into a ‘box universe’ with topology $R\times T^3$, where R refers to the time direction and T denotes a torus. In our present setup, the scale factors in the y and z directions are fixed, but the scale factor in the x direction is allowed to change in time.

On the conditions which validate setting up the DCE-CPC analogy, we add a note on the issue of spatial homogeneity: the periodic boundary condition we assume for a system exhibiting DCE preserves spatial homogeneity, which facilitates direct correspondence with the homogeneous cosmology. However, in DCE, it is expected that particle creation would be most pronounced in the spatial region closest to the moving mirror, as that is the region where the vacuum fluctuations are parametrically amplified most strongly. To draw a closer analogy with homogeneous cosmology cases, we need to add this homogeneity assumption for DCE (the cases in DCE with oscillating mirror are probably the most commonly described, and second to it are those with changing properties of the medium. The case with an infinite medium in motion without any boundaries and with time-independent material constants, which seems closer in features to homogeneous cosmology, has also been discussed; see [107]). Doing so is not too contrived when the external drive is periodic, or at times long compared to the transient after a rapid expansion of the space ends. The field in this space can be considered homogenized enough to be effectively described by the cosmological analogues used. The conditions for and the effects of particle creation in both situations (rapidly moving boundary and expanding spacetime) are the same. It is expected that after the initial transient has passed, the field will approach a stationary state and become spatially homogeneous on average, albeit temporally remaining in a nonequilibrium steady state. Detailed studies of both DCE and CPC would require nonequilibrium quantum field theory considerations (see [108]). We shall comment on this in the second part on issues deserving further considerations in future developments.

2.2. Coupling Between Quantum Field and Spacetime

We now focus on how a quantum field is coupled to a background spacetime: first, a quick review of the familiar QFT in flat Minkowski spacetime, followed by a discussion of a massless conformal scalar field in the Bianchi Type I universe.

The energy–momentum tensor of a massive scalar field $\varphi$ in Minkowski space with metric ${\eta }_{\mu \nu }$ is

$$T_{\mu \nu }^{\mathrm{Mink}}={\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -{\displaystyle \frac{1}{2}}{\eta }_{\mu \nu }{\nabla }^{\rho }\varphi {\nabla }_{\rho }\varphi -{\displaystyle \frac{1}{2}}{\eta }_{\mu \nu }{m}^2{\varphi }^2.$$

(3)

Its energy density is given by the expectation value of the 00 component of $T_{\mu \nu }$ with respect to the Minkowski vacuum, i.e.,

$${\rho }_0^{\mathrm{Mink}}\equiv \langle 0|T_{00}|0\rangle =\int {\displaystyle \frac{d^3\mathbf{k}}{2{\left(2\pi \right)}^3}}\left({\left|{\dot{f}}_{\mathbf{k}}\right|}^2+{\omega }_{\mathbf{k}}^2{\left|f_{\mathbf{k}}\right|}^2\right)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left(2s_{\mathbf{k}}+1\right){\displaystyle \frac{\hslash {\omega }_{\mathbf{k}}}{2}}.$$

(4)

where $f_{\mathbf{k}}$ are the amplitude functions of the $\mathbf{k}$th normal mode of the field. They obey an equation of motion like that of the parametric oscillators, meaning whose frequencies are time-dependent. The Hamiltonian for the dynamics of a collection of parametric oscillators is given by

$$H^{\mathrm{Mink}}\left(t\right)={\displaystyle \frac{1}{2}}\sum _{\mathbf{k}}\left({\pi }_{\mathbf{k}}^2+{\omega }_{\mathbf{k}}^2{q}_{\mathbf{k}}^2\right)=\sum _{\mathbf{k}}\left(N_{\mathbf{k}}+{\displaystyle \frac{1}{2}}\right)\hslash {\omega }_{\mathbf{k}},$$

(5)

Comparing this with Equation (4), one can identify ${\left|f_{\mathbf{k}}\right|}^2$ and ${\left|{\dot{f}}_{\mathbf{k}}\right|}^2$ with the canonical coordinates $q_{\mathbf{k}}^2$ and momenta ${\pi }_{\mathbf{k}}^2$, and the eigenvalue of $H_0$ is the energy $E_{\mathbf{k}}=(N_{\mathbf{k}}+1/2)\hslash {\omega }_{\mathbf{k}}$. The analogy of particle creation with parametric amplification is formally made clear

$$N_{\mathbf{k}}\left(t\right)+{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2\hslash {\omega }_{\mathbf{k}}}}\left({\pi }_{\mathbf{k}}^2+{\omega }_{\mathbf{k}}^2{q}_{\mathbf{k}}^2\right)\propto s_{\mathbf{k}},$$

(6)

Please refer to Equation (19), which gives the number density, and Equation (4) says that the energy density of vacuum particle creation comes from the amplification of vacuum fluctuations $\hslash {\omega }_{\mathbf{k}}/2$ by the factor ${\mathcal{A}}_{\mathbf{k}}=2s_{\mathbf{k}}+1$. Now, it is easy to recognize the Minkowski result: since there is no particle creation $s_{\mathbf{k}}=0$, there is no amplification ${\mathcal{A}}_{\mathbf{k}}=1$. In the next section, we shall see how particle creation arises when the background spacetime becomes dynamical.

We now consider a massless conformal quantum field in the rectangular box with one face moving, in analogue to a symmetric Bianchi Type I universe with periodic boundary conditions imposed on three sides as described earlier (due to the imposition of boundary conditions, the global topology of our setup is not the same as the Bianchi Type I Universe; it becomes $T^3$, a three-torus). We wish to explain why the curvature scalar R appears in the field equation. This is a simplified model of the electromagnetic field, which is conformal (photons are massless, and the Maxwell equation in four spacetime dimension is conformally invariant)—simplified here in our treating of just one of the propagating components of a vector field, ignoring the polarizations, which does not alter the parametric amplification mechanism in any significant way (this may affect the kind of DCE that arises from an atomic polarizability-dependent parametric interaction with the field; see [109]). The answer to this query is, indeed the space is flat, so the intrinsic curvature is zero, but because one or more faces are moving, the extrinsic curvature term is nonzero. We shall explain some more after we show the expression for the wave equations governing the normal modes of the field.

Consider a massless conformal scalar field $\varphi$ in this box ‘universe’ with action

$$S_f={\displaystyle \frac{1}{2}}\int d{x}^4\sqrt{\left|g\right|}(g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{\displaystyle \frac{R}{6}}{\varphi }^2),$$

(7)

obeying the Klein–Gordon equation,

$$\begin{array}{ccccccc}\hfill \square \varphi +{\displaystyle \frac{R}{6}}\varphi & =0,\hfill & \hfill & \Rightarrow \hfill & \hfill & \hfill {\partial }_t^2\varphi +{\displaystyle \frac{\dot{a}}{a}}\dot{\varphi }-\sum _i{\displaystyle \frac{{\partial }_i^2\varphi }{a_i^2}}+{\displaystyle \frac{R}{6}}\varphi & =0,\hfill \end{array}$$

(8)

where R is the scalar curvature of spacetime the field lives in. Here comes perhaps the first legitimate question: For the box universe, the space is flat, so where could the curvature enter?

Examining the form of this curvature scalar for a symmetric Bianchi Type I universe, we can see it is proportional to $\ddot{a}\left(t\right)$, the acceleration in time of the scale factor. This is the extrinsic curvature we referred to above, which is nonzero for a time-dependent space, as in the dynamical Casimir and Casimir–Polder effects. There is no such contribution if the space is static and flat (zero intrinsic curvature), as in the Minkowski space shown before. (When a researcher in general relativity mentions quantum fields in a “curved” spacetime, that context includes also dynamical flat spaces. The scalar curvature has two contributions: even though the intrinsic curvature of flat space is zero, the extrinsic curvature measuring the time rate of change of the scale factor (here the moving boundary) is nonzero. This is the same situation as a spatially flat Robertson–Walker (RW) universe being considered as a curved spacetime: spatially flat has zero intrinsic curvature, yet expanding space makes it “curved”, whereas a closed or open RW universe has both intrinsic and extrinsic curvatures.) Proceeding forward, the energy–momentum tensor of this scalar field is given by

$$\begin{array}{cc}\hfill T_{\mu \nu }\equiv {\displaystyle \frac{2}{\sqrt{\left|g\right|}}}{\displaystyle \frac{\delta S}{\delta g^{\mu \nu }}}& =\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{g}^{\lambda \sigma }\left({\partial }_{\lambda }\varphi \right)\left({\partial }_{\sigma }\varphi \right)\hfill \\ \hfill & -{\displaystyle \frac{1}{6}}\left[{\nabla }_{\mu }{\partial }_{\nu }\left({\varphi }^2\right)-g_{\mu \nu }{g}^{\lambda \sigma }{\nabla }_{\lambda }{\partial }_{\sigma }\left({\varphi }^2\right)+{\varphi }^2{G}_{\mu \nu }\right],\hfill \end{array}$$

(9)

where $G_{\mu \nu }\equiv R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R$ is the Einstein tensor. This expression for the stress–energy tensor, introduced by Callen, Coleman, and Jackiw [110] (who called it the ‘new and improved’), is the correct one to use for conformal fields. Notice that it contains the Einstein tensor. Even though in the common representation of electromagnetic theories, the background spacetime structure is rarely mentioned, it actually plays an important dynamical role in the determination of the stress–energy tensor, even in flat spacetimes with nonzero extrinsic curvature.

For a quantum field, the stress–energy tensor is upgraded to be an operator. Taking the vacuum expectation value of the 00 component gives the energy density, which is the quantity of interest for particle creation studies:

$$T_{00}={\displaystyle \frac{1}{2}}{\left(\dot{\varphi }\right)}^2+{\displaystyle \frac{1}{3}}{\displaystyle \frac{\dot{a}}{a}}\varphi \dot{\varphi }+{\displaystyle \frac{1}{6}}\sum _i{\left({\displaystyle \frac{{\partial }_i\varphi }{a_i}}\right)}^2-{\displaystyle \frac{1}{3}}\sum _i\left({\displaystyle \frac{\varphi {\partial }_i^2\varphi }{a_i^2}}\right)-{\displaystyle \frac{1}{6}}{G}_{00}{\varphi }^2,$$

(10)

where an overdot indicates differentiation with respect to t. We shall continue this exposition in the next section when we discuss particle creation as a nonadiabatic process.

2.3. Factors in DCE and Cases in Cosmology with Finer Correspondences

2.3.1. Cosmology of and DCE in 3D Space

Before closing this section, we mention cases where the correspondences between DCE in conventional setups (such as parallel plates or conducting spheres) and CPC are not so well placed. We then suggest alternative analogue setups which we see as having better promise for continual developments.

For example, the closed FLRW universe is an important member of the Standard Model of cosmology, but there is no such correspondence in the set up of DCE in 3-dim space. In DCE, a popular model is that of a pulsating spherical shell, but (in the limit the shell is infinitely thin) that is really a $S^2$ geometry with a time-dependent scale factor. The mathematical definition of a sphere refers to its surface, not the interior, known as a ‘ball’ (or a disk in 2D, different from its perimeter, the ring, in 1D). DCE is interested in the effects of quantum fields in the space contained within the moving shell, namely, the three-dimensional ball $B^3$ (B for ball). The closed FLRW universe refers to the whole spatial extent of a three sphere $S^3$ with a time-dependent radius, not the space within a spherical shell $S^2$, which is a three-dimensional ball $B^3$.

Having said this, one may be able to, but not necessarily conveniently, set up some analogy between a pulsating shell in DCE with gravity, not with cosmology per se but with black holes. The membrane paradigm [111] may be a good entry point because the intricacies of a black hole related to its event horizon are mapped into the physical properties of a membrane. For massive stars, observing DCE originating from a vibrating boundary in the exterior of a Schwarzschild metric is mentioned in [112].

If one does not insist on moving plates or pulsating shells, the Bose–Einstein condensate (BEC) is a more malleable and richer medium. Indeed, serious attempts have been made to set up correspondences between what goes on in the early universe and the specific dynamics of BEC, leading to what is called ‘laboratory cosmology’, e.g., [113,114,115]. Many fine experiments have mapped out the controllable parameters, e.g., the time-varying quantity is the scattering length via a Feshbach resonance. Because BEC is formed from atoms, theoretical analyses on its dynamical responses can reveal hard-to-access information about its underlying substructure, e.g., the quantity in BEC corresponding to the Planck length in cosmology is the healing length. There are many DCE experimental proposals using changing or ‘modulated’ BEC; see [116,117,118]. A more complete list of references on DCE experiments with BEC can be found in [4], Section 4.2. Please also consult updates of the reviews on analogue gravity [5] for the latest developments.

2.3.2. Imperfect Mirror

The reflectance of a mirror certainly is a very important factor to consider in the design of quantum optics experiments to meet high-precision demands. For a perfect conductor, one can just impose a boundary condition on the field at the position of the mirror. What are the analogues in cosmology? In Euclidean quantum gravity, Gibbons and Hawking asserted the importance of boundary terms in the functional integration over equivalent classes of conformal spacetimes. In particle physics, higher-dimensional, especially braneworld cosmology, what happens in the boundary occupies the main stage. The essence of AdS (anti-de Sitter space)-CFT (conformal field theory) physics hinges on the existence of a duality relation between gravity in the bulk and gauge theories on the boundary. One can, in principle, think of atomic–optical–condensed matter analogue experiments to check on the theoretical predictions by these theories. We will not pursue these fancier lines of thought here but just mention a down-to-earth model known as the mirror–oscillator–field (MOF) model for the description of imperfect mirrors. There, the internal degrees of freedom of an atom or a mirror is made a dynamical variable which can interact with the field. How the dynamics of this variable determines the reflectivity of a mirror has been calculated in these original papers [60,61,119]. We shall return to the structure and functions of the MOF model in the last section.

3. Mechanism, Dynamics, and Processes

The above discussions are about the setups, from the conditions on the quantum fields (boundary conditions) to the characters of spacetimes (curvature and topology) and the nature of their couplings. We now turn to dynamics, first focusing on the key mechanism of particle creation, namely, nonadiabatic processes and parametric amplification, then generalize from vacuum to finite particle initial states in stimulated particle production, and to finite temperature effects, ending with consideration of the fully nonequilibrium dynamics.

3.1. Particle Creation Is a Nonadiabatic Process, Negligible Under Adiabatic Approximations

The equation of motion for the amplitude function $f_{\mathbf{k}}$ of the $\mathbf{k}$th normal mode of a quantum field in a dynamical spacetime written in a generic form is that of a harmonic oscillator with time-dependent frequency ${\omega }_{\mathbf{k}}\left(t\right)$:

$${\ddot{f}}_{\mathbf{k}}+{\omega }_{\mathbf{k}}^2\left(t\right)f_{\mathbf{k}}=0.$$

(11)

(An overdot denotes differentiation with respect to time t). For the treatment of particle creation in a symmetric Bianchi Type I universe with periodic boundary conditions, as the analogue of DCE in a rectangular box with one moving face, it is more transparent to use the conformal time $\eta$ and the conformally related field $\chi \left(\eta \right)$

$$\begin{array}{cccc}\hfill \chi & =a^{1/3}\varphi ,\hfill & \hfill \mathrm{d}\eta & =a^{-1/3}\mathrm{d}t,\hfill \end{array}$$

(12)

resulting in an equation for the mode function ${\chi }_{\overrightarrow{k}}\left(\eta \right)$

$${\chi }_{\mathbf{k}}^{″}+({\Omega }_{\mathbf{k}}^2+Q){\chi }_{\mathbf{k}}=0.$$

(13)

(A prime refers to taking the derivative with respect to the conformal time $\eta$) with a time-dependent frequency

$${\Omega }_{\mathbf{k}}^2={\left(a^{1/3}{\omega }_{\mathbf{k}}\right)}^2=a^{2/3}\sum _i({\displaystyle \frac{k_i^2}{a_i^2}}+m^2).$$

(14)

Here, $Q\left[a_i^{\prime }\left(\eta \right)\right]$ is defined by

$$Q={\displaystyle \frac{1}{18}}\sum _{i<j}{({\displaystyle \frac{a_i^{\prime }}{a_i}}-{\displaystyle \frac{a_j^{\prime }}{a_j}})}^2.$$

(15)

It measures the degree of anisotropy in the expansion of the spacetime and is notably independent of $\mathbf{k}$.

The region in $\mathbf{k}$-space which yields the most particle production contains normal modes which are subjected to strong nonadiabatic parametric amplification. As a qualitative measure for each $\mathbf{k}$ mode, one can use the nonadiabaticity parameter introduced in [120] (Equations (73) and (74)): $\overline{\omega }\equiv {\dot{\omega }}_{\mathbf{k}}/{\omega }_{\mathbf{k}}^2$ in cosmic time (Minkowski time here) or $\overline{\Omega }\equiv {\Omega }_{\mathbf{k}}^{\prime }/{\Omega }_{\mathbf{k}}^2$ in conformal time. Those fulfilling the condition $\overline{\omega }\gg 1$ in cosmic time or $\overline{\Omega }\gg 1$ in conformal time are the modes which contribute most to particle creation. Integrating over these nonadiabatic modes in the energy density expression yields the dominant contribution to particle creation.

This true statement is probably well known amongst the practitioners. But in the literature, there are cases where processes based on the same mechanism as DCE or CPC were treated by adiabatic approximations, which led to qualitatively opposite results and wrong estimates. An example is the proposal of sonoluminescence (SN) by Schwinger [121,122,123,124,125] as quantum vacuum radiation, followed by the work of Eberlein [126,127] and others. (Comments doubting the viability of DCE being the primary cause of SN followed soon after, e.g., [128] claiming that the fundamental physical limitations on the bubble surface velocity limit the predicted number of photons per flash to be much smaller than unity; and [129] pointing out the importance of the details of the trapped gas, that there is no SN if there is no trace of inert gases in the trapped air, and the dependence of the intensity on the ambient temperature.) The prevailing explanation of this interesting effect today seems to lean towards a plasma physics phenomenon. The controversy of Schwinger’s original proposal, see [130,131,132], is not what we want to delve into here. There have been efforts to distinguish the mechanism responsible for SN. For example, Sciama and co-workers [133] pointed out the difference between what they call ‘real’ thermal light and the ‘pseudo’-thermal squeezed-state photons typical of models based on the dynamic Casimir effect. They showed that two-photon correlations provide a powerful experimental test for various classes of sonoluminescence models. Granting, for the sake of argument, that SN is a kind of quantum radiation resulting from the parametric amplification of vacuum fluctuations, with the same mechanism as in DCE, the point we are making here is that, using “standard perturbation theory with an adiabatic approximation" is not the way to go: particle creation is a nonadiabatic process. Also, it is a rather generic feature that for high-frequency $\mathbf{k}$ modes, the number of particles produced falls off exponentially with $\mathbf{k}$ (see Equation (3.125) of [1]), which explains why “the emitted light resembles a black body spectrum”. A recent paper [134] calls for a reexamination using exact solutions, which we support—the methodology has been laid out much earlier in the study of cosmological particle creation.

A common reason for such missteps probably comes from the habitual practice of invoking the WKB approximation as the first line of attack to solving a differential equation with no exact analytic solutions. Let us try this out, i.e., invoke a WKB approximation to the equation of motion for the amplitude functions of the normal modes and see what we come up with. We first derive a formal expression for the number of particles $s_{\mathbf{k}}\equiv {\left|{\beta }_{\mathbf{k}}\right|}^2$ (where ${\beta }_{\mathbf{k}}$ is the Bogoliubov coefficient measuring the portion of the negative frequency component evolved from an initial state containing only the positive frequency component) produced in terms of the amplitudes of the wave function $\left|f_{\mathbf{k}}\right|$ and $\left|{\dot{f}}_{\mathbf{k}}\right|$. We then carry out a WKB expansion to arrive at expressions for $s_{\mathbf{k}}$ in successive adiabatic orders. The following is excerpted from [108] which is based on [120,135].

We seek a solution in the form

$$\begin{array}{ccc}\hfill f_{\mathbf{k}}\left(t\right)& =\sqrt{{\displaystyle \frac{\hslash }{2{\omega }_{\mathbf{k}}}}}\left({\alpha }_{\mathbf{k}}{e}_{\mathbf{k}}^{-}+{\beta }_{\mathbf{k}}{e}_{\mathbf{k}}^{+}\right),\hfill & \hfill e_{\mathbf{k}}^{\pm }\equiv exp\left(\pm i\int dt{\omega }_{\mathbf{k}}\right).\end{array}$$

(16)

The two functions ${\alpha }_{\mathbf{k}}$, ${\beta }_{\mathbf{k}}$ are the positive and negative frequency components of a formal solution $f_{\mathbf{k}}$, but without a well-defined vacuum, they do not convey the meaning of particles and antiparticles as we forewarned with regard to the Bogoliubov coefficients. Since the single Equation (16) does not determine the coefficients ${\alpha }_{\mathbf{k}}$ and ${\beta }_{\mathbf{k}}$ uniquely, we need another condition. We shall choose it such that the Wronskian condition ${\left|{\alpha }_{\mathbf{k}}\right|}^2-{\left|{\beta }_{\mathbf{k}}\right|}^2=1$, which guarantees the unitarity of the system dynamics under Bogoliubov transformations, is satisfied. The auxiliary condition imposed on ${\dot{f}}_{\mathbf{k}}$ is

$${\dot{f}}_{\mathbf{k}}\left(t\right)=-i\sqrt{{\displaystyle \frac{\hslash {\omega }_{\mathbf{k}}}{2}}}\left({\alpha }_{\mathbf{k}}{e}_{\mathbf{k}}^{-}-{\beta }_{\mathbf{k}}{e}_{\mathbf{k}}^{+}\right).$$

(17)

Inverting these two equations, we can express the complex function ${\beta }_{\mathbf{k}}$ in terms of ${\left|f_{\mathbf{k}}\right|}^2$, ${\left|{\dot{f}}_{\mathbf{k}}\right|}^2$ as follows:

$$\begin{array}{cccc}\hfill {\alpha }_{\mathbf{k}}& =\sqrt{{\displaystyle \frac{{\omega }_{\mathbf{k}}}{2\hslash }}}\left(f_{\mathbf{k}}+{\displaystyle \frac{i}{{\omega }_{\mathbf{k}}}}{\dot{f}}_{\mathbf{k}}\right)e_{\mathbf{k}}^{+},\hfill & \hfill {\beta }_{\mathbf{k}}& =\sqrt{{\displaystyle \frac{{\omega }_{\mathbf{k}}}{2\hslash }}}\left(f_{\mathbf{k}}-{\displaystyle \frac{i}{{\omega }_{\mathbf{k}}}}{\dot{f}}_{\mathbf{k}}\right)e_{\mathbf{k}}^{-}.\hfill \end{array}$$

(18)

Making use of the Wronskian condition, we obtain

$$s_{\mathbf{k}}\equiv {\left|{\beta }_{\mathbf{k}}\right|}^2={\displaystyle \frac{1}{2\hslash {\omega }_{\mathbf{k}}}}\left({|{\dot{f}}_{\mathbf{k}}|}^2+{\omega }_{\mathbf{k}}^2{\left|f_{\mathbf{k}}\right|}^2\right)-{\displaystyle \frac{1}{2}}.$$

(19)

It is tempting to regard $s_{\mathbf{k}}={\left|{\beta }_{\mathbf{k}}\right|}^2$ as the amount of particle production. However, we need to be careful that the vacuum state is well defined to make sense of particles. We shall leave out the construction of the nth-order adiabatic vacuum, as it is well treated in textbooks where one can trace back to the original papers.

Now, we perform a WKB expansion on $s_{\mathbf{k}}$ and obtain the second- and fourth-order adiabatic expressions for $s_{\mathbf{k}}$:

$$\begin{array}{cc}\hfill s_{\mathbf{k}\left(2\right)}& ={\displaystyle \frac{1}{16}}{\overline{\omega }}^2,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill s_{\mathbf{k}\left(4\right)}& ={\displaystyle \frac{1}{16}}\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\overline{\omega }{\overline{\omega }}^{″}}{{\omega }^2}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\overline{\omega }}^{\prime 2}}{{\omega }^2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\overline{\omega }}^{\prime }{\overline{\omega }}^2}{\omega }}+{\displaystyle \frac{3}{16}}{\overline{\omega }}^4\right).\hfill \end{array}$$

(21)

In a suitably defined adiabatic vacuum, the above expressions provide the number of particles created from vacuum fluctuations under the adiabatic approximation, which corresponds to letting $\overline{\omega }\ll 1$. We can see that particle creation obtained under the adiabatic approximation is very small. (The adiabatic expansion for particle production in the high-frequency range at the zeroth, second and fourth adiabatic orders above matches the quartic, quadratic, and logarithmic divergences in the vacuum energy density, respectively. Substituting these expressions for $s_{\mathbf{k}\left(\mathrm{div}\right)}=1+s_{\mathbf{k}\left(2\right)}+s_{\mathbf{k}\left(4\right)}$ for each $\mathbf{k}$ mode into the expression for the vacuum energy density, one can identify the divergent contributions. This is the basis for adiabatic regularization [136,137,138,139,140,141]. Note that it is letting $\mathbf{k}$ go to infinity that causes the UV divergence, not because the particle number becomes very high. The identification of successive divergent terms is facilitated by an adiabatic expansion of the particle production expression). This is the reason why one should not invoke any adiabatic approximation for the calculation of particle production: because it is intrinsically a nonadiabatic quantum process.

3.2. Enhancement at Finite Temperature Comes from Stimulated Emission, True Only for Bosons

What we have discussed so far concerns particle creation from the vacuum. We learned that it is due to the parametric amplification of vacuum fluctuations of the quantum field at zero temperature. What happens at a finite temperature? Plunien et al. [142] studied the thermal effects on the creation of particles under the influence of time-dependent boundary conditions. For a resonantly vibrating mirror cavity, the thermal effect on the number of created photons is obtained nonperturbatively. These authors concluded that finite temperatures can enhance the pure vacuum effect by several orders of magnitude.

Finite temperature effects certainly are important because most experiments are not carried out at zero temperature. We wish to add some comments to (a) identify the root cause of this effect and (b) reveal the crucial condition their claim depends on.

(a) A first discriminant: enhancement is possible only for bosons, not for fermions. Fermions may see an attenuation [143,144,145,146,147].

(b) The root cause and more general underlying mechanism is stimulated emission, namely, the parametric amplification of a state already with N particles present. How these particles, as long as they are bosons, are initially distributed is secondary. The greater amount of particles produced found by Plunien et al. came about not because they obey a thermal distribution and thus are regarded as a finite temperature effect but because they are parametrically amplified in a stimulated production process.

Let us first discuss the stimulated production, then the thermal effects. If, at the initial time, there are already $N_{\mathbf{k}}\left(t_0\right)$ numbers of particles present, then the same parametric amplification mechanism, which, when applied to vacuum fluctuations, gives rise to spontaneous particle production as we saw before, now gives rise to stimulated production. It yields an energy density ${\rho }_n$ with respect to the n-particle state at $t_0$ given by

$$\begin{array}{cc}\hfill {\rho }_n^{\mathrm{Mink}}=\langle n|T_{00}|n\rangle & =\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left({|{\dot{f}}_{\mathbf{k}}|}^2+{\omega }_{\mathbf{k}}^2{\left|f_{\mathbf{k}}\right|}^2\right)\langle {\widehat{a}}_{\mathbf{k}}^{†}{\widehat{a}}_{\mathbf{k}}\rangle ,\hfill \\ \hfill & =\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left(2s_{\mathbf{k}}+1\right)\hslash {\omega }_{\mathbf{k}}\langle {\widehat{N}}_{\mathbf{k}}\left(t_0\right)\rangle .\hfill \end{array}$$

(22)

Comparing this expression with Equation (4), we see that the ${\displaystyle \frac{1}{2}}\hslash {\omega }_{\mathbf{k}}$ term is replaced by $N_{\mathbf{k}}\left(t_0\right)\hslash {\omega }_{\mathbf{k}}$, the number of particles originally present in the $\mathbf{k}$th mode. It is multiplied by an amplification factor. As mentioned earlier, in a Minkowski spacetime, there is no particle creation, $s_{\mathbf{k}}=0$, and the amplification factor is equal to one. An expanding spacetime would produce particles from the vacuum fluctuations of the field and from the particles originally present. Combining Equations (4) and (22), for a density matrix diagonal in the number state, the total energy density of particles created from the vacuum and from those already present in the n-particle state is given by

$${\rho }^{\mathrm{Mink}}={\rho }_0^{\mathrm{Mink}}+{\rho }_n^{\mathrm{Mink}}=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left(2s_{\mathbf{k}}+1\right)\hslash {\omega }_{\mathbf{k}}\left[{\displaystyle \frac{1}{2}}+\langle N_{\mathbf{k}}\left(t_0\right)\rangle \right].$$

(23)

For bosons obeying a thermal distribution at temperature $T={\beta }^{-1}$, magnification of the n-particle thermal state gives the finite-temperature contribution to particle creation, with energy density

$${\rho }_T^{\mathrm{Mink}}=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left(2s_{\mathbf{k}}+1\right){\displaystyle \frac{\hslash {\omega }_{\mathbf{k}}}{e^{\beta \hslash {\omega }_{\mathbf{k}}}-1}}.$$

(24)

We can see that the large enhancement in the amount of particles produced as reported by Plunien et al. is more due to the stimulated amplification of existing particles than as a finite temperature effect per se. The former refers to the parametric amplification of a state already with N particles present, while the latter refers to the particular distribution they find themselves in, such as the B-E distribution for Bose particles. Finding a greater number of particles produced as compared to the vacuum initial state applies to any state with a finite number of bosons present initially; it does not matter whether the n particles are in equilibrium, obeying a thermal distribution such as the Bose–Einstein distribution. Hence, it is not a thermal effect per se.

As a matter of fact, the thermal equilibrium condition these particles find themselves in originally may not hold in a dynamical setting. This is because at every moment, there are newly produced particles which will be parametrically amplified. The system at any moment is in a nonequilibrium condition. One needs to examine scattering processes amongst the produced particles to assess how fast the system can settle into equilibrium, and that depends on how fast the system is being driven compared to the rate of these processes, and many other factors. Equilibrium condition cannot be assumed to exist or be maintained without such scrutiny.

To conclude our discussion on this topic of thermal particle production in DCE and cosmology, what was found by Plunien et al., namely, the enhancement of particles produced at finite temperature compared to that from the vacuum, is not a thermal effect per se; the root cause of enhancement is the stimulated emission of bosons, and it is not true for fermions.

Now, let us see how this formulation of finite temperature field theory appears in cosmology. For a conformally invariant field like photon in a conformally flat spacetime like the Friedmann–Lamaitre–Robertson–Walker (FLRW) universe, we would obtain the same expressions for the conformally related field $\chi$ in conformal time $\eta$ (except for a scaling factor of $1/a^4$)

$${\rho }_T^{\mathrm{conf}}={\displaystyle \frac{1}{a^4}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left(2s_{\mathbf{k}}+1\right){\displaystyle \frac{\hslash {\omega }_{\mathbf{k}}}{e^{\beta \hslash {\omega }_{\mathbf{k}}}-1}}.$$

(25)

If $s_{\mathbf{k}}=0$, meaning, if there were no new particles created during the evolution—and this is related to the condition for maintaining thermal equilibrium, the Stefan–Boltzmann relation holds for a massless conformal field in the FLRW universe

$${\rho }_T^{\mathrm{conf}}={\displaystyle \frac{{\pi }^2}{30{\hslash }^3}}{T}^4.$$

(26)

Thus, $Ta$ is a constant throughout the evolution of the radiation-dominated FLRW universe. $N_{\gamma }\sim {\left(Ta\right)}^3$ is proportional to the number of relativistic particles present, which provides the entropy content of the universe [148].

Further discussions of finite-temperature particle creation, nonequilibrium conditions [149], and the related entropy generation problem can be found in [150,151].

4. Backreaction: Vacuum Viscosity, Quantum Lenz Law and Fluctuation–Dissipation Relation

4.1. Dissipation Due to the Backreaction of Particles Created by the Parametric Amplification of Quantum Fluctuations Has Memory

This long title summarizes the logical sequence of events and effects (particles are created) and processes (via parametric amplification), beginning with the quantum fluctuations of the field. In this section, we consider the eventful duo-elements: backreaction (process) and dissipation (effect).

We learned from the last section that a rapid expansion of the universe can parametrically amplify the vacuum fluctuations of a quantum field into particle pairs. Here, we study their effects on the dynamics of the early universe. This is known as the backreaction problems. The theoretical basis is semiclassical gravity [3], where both the semiclassical Einstein (SCE) equation and the quantum field equation of motion need be solved in a self-consistent manner. The SCE equation has as a source, in addition to the classical stress–energy tensor which we, focusing on purely quantum effects, will ignore, also the expectation values of the stress–energy tensor operator of a quantum field $\varphi$, in the form of the Einstein tensor $G_{\mu \nu }\left[g_{\alpha \beta }\right]=8\pi \mathrm{G}\langle {\widehat{T}}_{\mu \nu }\left(\varphi \right)\rangle$, where $\mathrm{G}$ is Newton’s constant.

In most situations, the backreaction of quantum processes like particle creation engenders dissipation in the system’s dynamics (an interesting exception is the quantum effects of trace anomaly; see [23]). To address this issue, it is useful to adopt some concepts and techniques in open quantum systems. For cosmology, the open system is the evolving universe, while for DCE, it is the moving mirror; in both cases, their environment is the quantum field. Since backreaction effects in DCE [152,153,154,155] and related phenomena in analogue black hole experiments [50] are gradually surfacing to the fore, it may prove useful to describe what was performed in early cosmology so we can benefit from the wisdom gained there to better formulate accessible problems of interest for the exploration of similar issues in analogue gravity experiments.

We shall present the results from one of the best studied cases of the cosmological backreaction problem, that of the particle creation of massless conformal scalar fields in a Bianchi Type I universe, for two reasons: (1) The effect is strong and can lead to a rapid isotropization of the early universe near the Planck time. (2) Many methods have been developed and applied to this problem, e.g, it has been treated via (a) both the canonical quantization [156] and the path-integral methods [157,158], (b) both the in–out and in–in or closed-time path, Schwinger–Keldysh [159,160] methods [161] and (c) the Feynman–Vernon formalism [162] open quantum system methods [163], which is particularly useful for understanding the properties of quantum noise. With part (c), we begin to enter into the realm of semiclassical stochastic gravity [3], where the two point correlations of the stress–energy tensor become the centerpiece. If one’s interest is in nonlocal dissipative and non-Markovian (memory) effects, one can stay within the realm of semiclassical gravity and focus on Equations (3.18) or (3.24) in [161] for closer scrutiny. To see how colored (some communities call this non-Markovian) noises associated with quantum fluctuations of the field enter, focus on Equation (5.13) of [163], where a fluctuation–dissipation relation for particle creation from the parametric amplification of the quantum fluctuations of the matter field and its dissipative backreaction on spacetime dynamics is derived for semiclassical cosmology.

With this background, we can resume our studies of DCE in a symmetric rectangular box with one face moving, starting with the formula for the stress–energy tensor of a massless conformal scalar quantum field Equation (9). Of special interest is the energy density Equation (10). These expressions have UV divergences and require viable schemes of regularization or renormalization. This quest occupied the attention of most researchers in QFCS in the 5 years from 1974 to 1978, covering half a dozen methods, four of which remain in continual usage. They are the dimensional, zeta-function, point-separation, and adiabatic regularizations, the last one being the most adept for cosmological particle creation considerations. Serious studies of backreaction began in 1978, continued on in the next decade, and finished off with the establishment of semiclassical gravity.

We shall not repeat describing the procedures of regularization nor the assortment of different parts of the energy density responsible for the different physical processes involved. Please read Section 3 of [23] for the details. Noteworthy is the general principle evoked from these results, first discovered for the antecedent cosmological backreaction problems [149,164]. Called the ‘Quantum Lenz Law’ by Hu, it says, similar to the classical law of magnetic induction, that the backreaction will act in such a way as to resist further changes. The findings in [23] validate its manifestation for DCE, at least for the case of a moving rectangular box studied there in detail, and we believe it applies broadly as a law.

4.2. Quantum Lenz Law, Vacuum Viscosity, Energy Balance, Self-Consistency Condition and Fluctuation–Dissipation Relation

The quantum Lenz law encapsulates several interesting features of the backreaction effects of quantum processes, here exemplified by the effects of particle creation from vacuum fluctuations in DCE or cosmology. We will mention three. First, the backreaction force exerted by the created particles, which results from the quantum field being parametrically amplified, is not just a dissipative force; it is also reactive, like the combined effects of resistance and inductance. Second is non-Markovian dynamics, memory, where the response has a time lag or phase difference. Here, the nonlocal dissipation kernel on the LHS of the Langevin equation is a clear sign of the existence of memory: what happens at one particular moment in time depends on the history of its past—how far back is a measure of the degree of non-Markovianity.

Memory effects in an open system are unavoidable when the backreaction is included in the consideration. The physical reason is easy to see: the system, be it the moving mirror or the evolutionary universe, and its environment, here the quantum field, each follows its own dynamics described by an ordinary differential equation with disparate characteristic time scales. When one coarse-grains the information of the environment and absorbs its overall effect into the system, the open system will be described by an integral–differential equation with a nonlocal kernel—nonlocal because it contains the time scales of both subsystems to begin with.

Third is the energy balance. The energy extracted from the field derived from taking the energy density of particles produced and integrating over all modes covering all space is supplied in the case of cosmology by the expansion of the universe, according to Einstein’s equations. The semiclassical Einstein equation which governs the dynamics of both the spacetime and the quantum field is self-regulating because its driving source, the expectation value of the stress–energy tensor of the scalar field, depends on how the field changes, which in turn obeys a field equation on a background spacetime whose dynamics is governed by the SCE. The partitioning of the energy fraction in spacetime and field dynamics can change in time, but the self-consistency condition between the system (spacetime) and its environment (quantum field) ensures energy conservation. The Lenz law captures the essence of this type of backreaction on the system by quantum processes of this kind in the environment. Read these two papers for an explicit demonstration of this relation for quantum fields in the homogeneous yet weakly anisotropic Bianchi Type I universe [161] and for the weakly homogeneous but isotropic FLRW universes [165].

Since particle creation originates from the vacuum fluctuations of the quantum field, the balance between the fluctuation energy in the environment and the dissipative energy in the system dynamics can be regarded as a form of fluctuation–dissipation relation. We do not have the space to delve into how the noises of quantum fields are defined as it begins to enter the realm of stochastic gravity [3], but we refer the reader to these two papers [163,166] which treated this topic in great detail.

The case for DCE is slightly different, in that there is an external agent which controls how the mirror moves, and thus it involves three parties: drive, mirror, and field. If the agent’s function is to set the mirror in motion and after the mirror moves at a constant speed stop the drive, then the energy budget will remain unchanged, and the mirror–field interplay can be studied under a controlled situation. One can then ask the interesting question, how would particle creation from the quantum field affect the motion of the mirror? In classical physics, a moving object encountering a damping force such as air resistance will gradually slow down, unless, as classical mechanics says, it moves in the vacuum. Here now, allowing for the presence of a quantum field, we ask whether even the vacuum can exert some reactive force on the object. And indeed it does: the backreaction of particle creation acts like the vacuum has some viscous property. This has been calculated for several important quantum field processes in [167].

One can also ask, in analogy with the backreaction from cosmological particle creation, whether some form of a fluctuation–dissipation relation exists in DCE. A first attempt was made in [168]. (We feel the treatment of noise therein is incomplete and are currently working on this problem with S. Butera [169]). The situation is more complex since there are three parties involved. In the field, if it is in a steady state, there is a relation between the causal Green function (the antisymmetric two-point function) responsible for dissipation and the Hadamard function, the symmetric part of the two-point function) for the noise. This was the key observation in [170] for black hole thermal radiance associated with the Hawking effect—notice the black hole has to be placed in a box to keep it at quasi-equilibrium, as otherwise there is no such relation. There is also a relation between the energy supplied by an external agent to sustain the movement of the mirror or atom and the energy associated with the particles produced by the excitation of the quantum field fluctuations. The intermediary, and the most interesting party, is the atom whose internal degree of freedom (idf) interacts with the field, and its external degree of freedom (edf) responds to the external agent. How these two interactions play out is the main attraction. This latter kind of relation is closer to the FDR in the expanding universe cases described above. We shall come back to this topic in the next section after we introduce a new model based on microphysics which can accommodate these three-party interactions, the mirror–oscillator–field (MOF) model.

5. MOF Model for Moving Atoms or Mirrors and Analogous Models in Cosmology

In this section, we continue to explore suitable theories or models for a closer and better description of moving mirrors or atoms interacting with a quantum field which can capture the essential quantum field processes, such as particle creation and its backreaction effects. These problems all fall under the new and fast developing field known as quantum optomechanics (QOM). We shall explain the structure of a new theory called the MOF model and how it can be applied to solving practical problems in QOM. We then bring up, in the spirit of analogue gravity, a problem in early universe cosmology which has a similar theoretical structure, namely, preheating the universe after inflation. The setup of the problem and its solution may offer some inspiration in how to tackle challenging problems in quantum optomechanics. We end with a description of the most recent theoretical developments of the MOF model and suggest some problems of interest which it can be most usefully applied to.

The distinct feature of the MOF (mirror–oscillator–field) model is the oscillator: representation of the internal degrees of freedom in an atom by an oscillator, rather than by two levels, and calling it a ‘harmonic atom’ is not the main feature. The main feature is to endow the mirror with an internal degree of freedom represented by a harmonic oscillator (HO), whose dynamics obeys an equation of motion coupled to those of the quantum field. Making the idf of a mirror a dynamical variable gives life to what goes on inside the mirror, similar to the microphysics models of a dielectric plane (e.g., in [171,172,173]) but simpler and more effective (read [119] about how a single HO, the ‘atom mirror’, can be made to represent a broad-band spectrum). This route was taken after Galley and one of the present authors found it rather (a) restrictive to represent a mirror by a boundary condition imposed on the field, usually the Dirichlet condition for a perfect conductor, because it cannot cope with mirrors which are less than perfect; and (b) difficult when using an auxiliary field as a constraint, as one needs to deal with the nonlocal aspects in the equations of motion. Structurally, the MOF model can be linked to many other operating models of mirror–field systems as described in detail in [60]. The immediate benefit is a good representation of the reflectance or transmissivity of an imperfect mirror [61]. But its attractiveness is more in its functionality, in capturing the primary physics of optomechanical systems, which extend from experiments involving atom–mirror–optical systems to LIGO gravitational wave detectors. Let us first look at the structure of the MOF model, how it captures the essence of DCE and DCP (dynamical Casimir–Polder) effects and some generic properties of quantum optomechanical systems. Here, we follow the description of [61].

5.1. Structure of the MOF Model for Quantum Optomechanics (QOM)

Consider a point mirror interacting with a massless scalar field in (1 + 1)-dimensional spacetime. The mirror is described by two separate degrees of freedom (dofs) represented by harmonic oscillators: (i) the mechanical dof or mdf by one oscillator with mass M, representing the center of mass (CoM) of a mirror or atom, is suspended in a harmonic potential of frequency ℧, and (ii) the internal dof or idf is described by another harmonic oscillator of mass m and frequency $\Omega$. While the mdf does not interact with the field directly, the idf is bilinearly coupled to the quantum field and constrained to be at the CoM position, meaning the field takes on values at the location of the atom at a specific time. Thus, it leads to an indirect interaction between the field and the mdf. (The traditional treatment goes by a radiation pressure type of coupling of the form $Nx$, where N is the number of photons and x denotes the displacement of the mirror). The idf–field interaction determines all the optical properties of the mirror as studied in [60,61].

The idf–field dynamics, representing the electronic excitations for the case of an atom, has a much faster time scale compared to that of the mechanical motion of the atomic center of mass such that $\Omega \gg \mho$. This is the case for a mirror moving at nonrelativistic speed. The action in the MOF model is given by

$$\begin{array}{cc}\hfill S& =\int dt\left\{({\displaystyle \frac{1}{2}}M{\dot{Z}}^2-{\displaystyle \frac{1}{2}}M{\mho }^2{Z}^2)+({\displaystyle \frac{1}{2}}m{\dot{q}}^2-{\displaystyle \frac{1}{2}}m{\Omega }^2{q}^2)\right.\hfill \\ \hfill & +\left.\int dx{\displaystyle \frac{{ϵ}_0}{2}}\left[\left({\partial }_t{\Phi }^2\right)-c^2{({\partial }_x\Phi )}^2+\lambda \dot{q}\Phi \delta (x-Z)\right]\right\},\hfill \end{array}$$

(27)

where we denote the center-of-mass position of the mdf by $Z\left(t\right)$, the amplitude of the idf by $q\left(t\right)$ and the scalar field by $\Phi (x,t)$. Noticing that the free space permittivity in (1+1)-dimensions scales as ${ϵ}_0\sim {\left(\mathrm{Charge}\right)}^2{\left(\mathrm{Time}\right)}^2{\left(\mathrm{Mass}\right)}^{-1}{\left(\mathrm{Length}\right)}^{-1}$; the free field Lagrangian in [60] is rescaled here by a factor of ${ϵ}_0$ for dimensional consistency.

For the relativistic motion that is required for the treatment of acceleration radiation such as the Unruh effect, one needs to use the proper time, modify the kinetic terms, and pay attention to the time-slicing scheme. This model will describe a generalization of the Unruh–DeWitt detector theory [41,51,174], an interesting topic which occupies a prominent place in the study of relativistic quantum information theory. See [175,176,177] for further details.

In drawing a correspondence between the scalar field used here and an electromagnetic (EM) field, we observe that the free field Lagrangian would correspond to that of an EM field if $\Phi (x,t)$ is to represent the vector potential A. We choose a form of the bilinear coupling motivated by the electrodynamic form of interaction $(\sim {\displaystyle \frac{e}{mc}}p·A)$, bearing in mind that the mirror’s idf can also be used to represent the electronic level structure inside a ‘harmonic’ atom. This type of coupling is often referred to as ‘derivative’ coupling. A similar model for describing mirror–field interactions has also been proposed in [178]. The original MOF model in [60] has a different form of coupling $(\sim$$\lambda q\Phi )$, called ‘minimal’.

The $\delta (x-Z)$ factor in the coupling restricts the idf–field interaction to the center-of-mass position and the position dependence of the scalar field in turn leads to an effective force on the mdf. We choose the coupling $\lambda$ to have the dimensions of the electronic charge e and $\Phi (x,t)$ to have the dimensions of $A/c$. This is in agreement with the correspondence of the MOF model (see [60]) with the Barton–Calogeracos (BC) model [179,180], where in the limit of adiabatic idf evolution, the coupling $\lambda$ can be physically identified as the surface charge density.

To bring the MOF closer to home, we show how it can describe the classical optical and mechanical properties exhibited by a mirror, leading to the known intensity–position radiation pressure coupling. The coupled equations of motion for the classical amplitudes of the mdf, idf and field ($\{\overline{Z},\dot{\overline{Z}},\overline{q},\dot{\overline{q}},\overline{\Phi },\dot{\overline{\Phi }}\}$ respectively) are obtained from the action in Equation (27), that is, $\delta S=0$,

$$\begin{array}{cc}\hfill \ddot{\overline{Z}}\left(t\right)+{\mho }^2\overline{Z}\left(t\right)& ={\displaystyle \frac{\lambda \dot{\overline{q}}\left(t\right)}{M}}{\partial }_x\overline{\Phi }(\overline{Z},t),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \ddot{\overline{q}}\left(t\right)+{\Omega }^2\overline{q}\left(t\right)& =-{\displaystyle \frac{\lambda }{m}}\dot{\overline{\Phi }}(\overline{Z},t),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {ϵ}_0\left[\ddot{\overline{\Phi }}(x,t)-c^2{\partial }_x^2\overline{\Phi }(x,t)\right]& =\lambda \dot{\overline{q}}\left(t\right)\delta (x-\overline{Z}).\hfill \end{array}$$

(30)

It can be seen that the moving idf acts as a point source for the field, and the idf is in turn driven by $\dot{\Phi }$ at the center-of-mass position $\overline{Z}$. Also, with $\lambda$ representing the charge density, it can be seen from Equation (29) that the force on the surface charge degree of freedom is ∼$\lambda \dot{\overline{\Phi }}$. We assume here that the mirror’s center-of-mass velocity is in the nonrelativistic limit such that $|d\overline{Z}/dt|\ll c$. For a relativistically moving mirror, the idf would more generally observe a Doppler shift in the field with respect to the moving center of mass as Equation (29) becomes

$$\ddot{\overline{q}}+{\Omega }^2\overline{q}=-{\displaystyle \frac{\lambda }{mc}}\left({\dot{\overline{Z}}}^0{\partial }_t+{\dot{\overline{Z}}}^1{\partial }_x\right)\overline{\Phi }\left({\overline{Z}}^{\mu }\right),$$

(31)

where ${\overline{Z}}^{\mu }=({\overline{Z}}^0\left(\tau \right),{\overline{Z}}^1\left(\tau \right))$ is the worldline of the mirror parametrized by its proper time $\tau$, and $\dot{O}\equiv dO/d\tau$. As the motion of the mirror’s center of mass leads to the motion of the charges at the surface interacting with the field, the surface charges experience a Doppler-shifted field, which in turn changes their optical response, leading to dynamically changing boundary conditions observed by the field.

Using the MOF model to represent an imperfect mirror’s reflectance or transmissivity is considered in detail in [60,61]. With what is described above, it is not difficult to identify the key players—dynamical variables—in DCE, DCPE, and quantum friction. Once the idf–field interactions and the motion of the mdf (CoM) are stipulated, one can set up the coupled equations of motion and solve them for the description of their dynamics. The expanse of potential applications of the MOF model is vast, and investigations have just begun.

5.2. An Interlude: Cosmological Analogue in Preheating After Inflation

In keeping with the aims of this paper, i.e., finding the analogies and defining the correspondences between atom/mirror field interactions and quantum field processes in the early universe, we describe one scenario in inflationary cosmology, the preheating phase after the universe ended inflation and began to enter the reheating phase, eventually warming up to become the radiation-dominated FLRW universe we are at home with. This case study intends to serve two purposes relevant to the themes we have discussed so far:

(A) Backreaction is important. We have discussed the importance of the backreaction of particle creation at the Planck scale on the dynamics of the background spacetime. The backreaction effects of particles produced from the parametric excitation of the fluctuations of the inflaton field at the GUT scale is also of primary importance. Their backreaction on the background inflaton field is responsible for preheating the universe and eventually damping it out, driving the universe to its true vacuum.

(B) Tripartite interaction. The three key players we talked about in QOM have their analogues here, except for the relative importance of the roles they play. The emphasis in QOM is more on the atom or the mirror: how its external or mechanical dof changes the field in the case of moving atom or mirror and how the field affects its idf. In cosmology, the attention is more on the field and its fluctuations.

(1) Classical background spacetime with metric $g_{mn}$, like the mechanical dof Z for the CoM motion of the mirror or atom;

(2) Inflaton quantum field $\Phi \left(t\right)=\varphi \left(t\right)+\tilde{\varphi }\left(t\right)$, where $\varphi$ is the background inflaton field, and $\tilde{\varphi }\left(t\right)$ its fluctuations.

(3) The fluctuations $\tilde{\varphi }$ of inflaton field acts like the idf of a mirror or atom.

The dynamics proceeds in several stages: Notice first that (1) always enters in (2) and (3), but the backreaction effects of (2) and (3) on the background spacetime geometry are insignificant since the GUT energy scale is five orders of magnitude lower than the Planck scale. The main stage is on how particle creation from the parametric amplification of the quantum fluctuations of the inflaton field changes the dynamics of the inflaton field, which in turn changes the cause of the universe.

At the end of the slow-roll (quasi-de Sitter) phase, the inflaton field starts falling into a steep potential well, which is the beginning of the (pre)heating phase. The rapid oscillation of (2) in this phase parametrically amplifies (3) $\tilde{\varphi }$, and ample particle creation ensues. The backreaction of (3) on (2) causes $\varphi \left(t\right)$ to damp away quickly, eventually settling the inflaton field $\Phi \left(t\right)$ to the minimum of the potential well. This particle creation process based on the same parametric amplification mechanism—like that at the Planck scale being responsible for the isotropization and homogenization of the universe—is responsible for the preheating of the universe and the eventual ushering-in of the radiation-dominated FLRW phase.

Thus, preheating after the inflationary stage serves as an illustration of this tripartite dynamical variable scheme analogous to the MOF model. For a thorough treatment using QFTCST and SCG methodologies, see [181].

5.3. Developments and Applications of the MOF Theoretical Framework

Returning to the optomechanical side of the analogue, we wish to end by pointing out some current work furthering the theoretical developments of the MOF model, and we identify interesting problems to which this model can be applied for finding meaningful solutions.

(i) Not long after the first two papers on MOF [60,61] appeared, Lin [119] continued to explore the features of the MOF model in relation to the functionality of the Unruh–DeWitt detector, which in (1 + 1) dimension he referred to as an atom mirror. Lin showed that when coupling between the oscillator representing the mirror’s idf and the quantum field is strong, a broad frequency range of the quantum field can be mostly reflected by the atom mirror at late times. Constructing a cavity model with two such atom mirrors, he demonstrated how the quantum field inside the cavity evolves from a continuous to a quasi-discrete spectrum, which gives a negative Casimir energy density at late times.

The differences between the two types of coupling with regard to the reflectivity properties of an imperfect mirror as a function of the normal mode frequency $\omega$ of the field in ratio to the idf oscillator frequency ${\Omega }_0$ are clearly shown in Figure 1 of his paper [119], based on their late time behavior. In the static limit $\omega \to 0$, the minimal-coupled and the derivative-coupled detectors act like a dielectric and a metal mirror, respectively, as noted earlier in [60] for minimal coupling and in [61] for derivative coupling.

Another reassuring find is that the late-time renormalized energy density of the field inside a cavity of the atom mirrors is always negative, even in the weak oscillator–field coupling regime, where the cavity modes are few.

(ii) Sinha, Lopez and Subaşı (2021) [182] used the MOF theoretical framework and studied the nonequilibrium dissipative dynamics of the center of mass (CoM) of a particle coupled to a field via its internal degrees of freedom. Notice even the field is directly coupled only to the idf of the atom or mirror; the field value is assumed at the spacetime point of CoM. Therefore, there is an indirect nonlinear coupling between the field and the CoM. They used the influence functional approach to account for the self-consistent backaction of the different degrees of freedom on each other, including the coupled nonequilibrium dynamics of the internal degree of freedom and the field, and their influence on the dissipation and noise of the center of mass. They assumed a weakly nonlinear coupling and adopted a functional perturbative method [183] to derive the influence action and from there obtain the effective equation of motion for the CoM of the atom or mirror.

(iii) Motional decoherence: An interesting physics which exemplifies the indirect coupling between the CoM of an atom and its idf via the direct coupling between the atom’s idf and the field is motional decoherence. Movement of the atom’s CoM changes the field, and the disturbance is registered in the idf of atom, causing dephasing or decoherence. Dalvit and Neto treated this problem by way of radiation pressure [83,84]. It would be pleasing to see this re-performed via the MOF model, a prototype of which was studied by Shresta and Hu [85]. The nonequilibrium dynamics formalism of Sinha et al. [182] is expected to provide a more thorough analysis of this problem.

(iv) Butera (2022, 2023). In [184], Butera used the MOF theoretical framework to provide a microphysics upgrade of the traditional radiation–pressure coupling between a mirror’s mechanical dof and the field [179,180,185]. Owing to the nonlinear nature of their coupling, Butera adopted the perturbative functional approach, the same as that of Sinha et al. [182], to derive a second-order effective action for the system of two mirrors. He found that the quantum and thermal fluctuations of the field appear in the form of colored noises acting on the mirrors, whose nonequilibrium dynamics is non-Markovian. He also derived the quantum Langevin equations and showed the existence of a generalized fluctuation–dissipation relation. In [186], using second-order perturbation theory, he derived the master equation governing the mechanical motion of the mirror. His analysis revealed that the mirror experiences colored noise and nonlocal dissipation, which originate from the emission of particle pairs via the dynamical Casimir effect.

(v) Graded influence action. The MOF model is a simplified version of a grander theoretical framework known as graded influence action, which refers to a nested set of coarse-grained effective actions [64,65,66], where one can deal with the interactions of many dynamical variables in an orderly manner. Graded refers to an ordering of which variables are to be coarse-grained first, forming the first tier of an effectively open system, followed by another variable, forming the second tier of effectively open system, and so on. At the end, one would come up with the sum total effect of all the variables with direct or indirect interaction with the one variable of special interest to us, by way of a nested set of influence actions. A vivid demonstration of the application of this formalism is shown in the derivation of the atom–dielectric forces by Behunin and Hu in [62,63] (see Figure 1 of [63] for the orderly coarse-graining of five variables), which the reader is referred to for greater details and insights. A work in the same vein investigating the effects of a medium-altered quantum field on an atom is currently in preparation [187].

(vi) Quantum friction A neutral atom moving at uniform speed slight above a dielectric plate will experience a drag force, which is commonly known as quantum friction (as remarked in our earlier discussion of analogy with the backreaction due to particle creation, a better term would be quantum viscosity, as friction connotes a simple resistive force, but in reality, this is a reactive force with phase dependence and memories, like that in an inductor. For theory work which resonates with our emphasis and concerns, see [188,189,190,191]. This phenomenon is a good arena to see how a graded influence action formalism is played out, similar to the way Behunin and Hu derived the atom–dielectric force: first, calculate the field modified by a medium, then calculate how the idf of the moving atom interacts with the medium-altered field. Noticing that the field value is taken at the spatial point where the CoM of the atom is located at that instant, that is where the movement of the atom enters into the picture. The backreaction of the medium-altered quantum field on the CoM of the atom is what gives the drag force known as quantum friction. Similar to the backreaction of particle creation on the dynamics of the universe, there should also be a fluctuation–dissipation relation depicting the balance between the moving atom and the fluctuations in the quantum field. And, if the atom’s motion is kept at a constant speed by an external agent, the agent should experience a reactive force from the atom, which, when traced back, originates from the medium-altered quantum field fluctuations. This nested set of influences captured by the graded influence action is what makes this effect so interesting and this method so empowering.