Topological Casimir effect in a quantum LC circuit: real-time dynamics

# Topological Casimir effect in a quantum LC circuit: real-time dynamics

Yuan Yao Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada    Ariel R. Zhitnitsky Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada
###### Abstract

We study novel contributions to the partition function of the Maxwell system defined on a small compact manifold with nontrivial mappings . These contributions cannot be described in terms of conventional physical propagating photons with two transverse polarizations, and instead emerge as a result of tunneling transitions between topologically different but physically identical vacuum winding states. We argue that if the same system is considered in the background of a small external time-dependent E&M field, then real physical photons will be emitted from the vacuum, similar to the dynamical Casimir effect (DCE) where photons are radiated from the vacuum due to time-dependent boundary conditions. The fundamental technical difficulty for such an analysis is that the radiation of physical photons on mass shell is inherently a real-time Minkowskian phenomenon while the vacuum fluctuations interpolating between topological sectors rest upon a Euclidean instanton formulation. We overcome this obstacle by introducing auxiliary topological fields which allows for a simple analytical continuation between Minkowski and Euclidean descriptions, and develop a quantum mechanical technique to compute these effects.

We also propose an experimental realization of such small effects using a microwave cavity with appropriate boundary conditions. Finally, we comment on the possible cosmological implications of this effect.

###### pacs:
11.15.-q, 11.15.Kc, 11.15.Tk

## I Introduction. Motivation.

It has been recently argued [1, 2, 3, 4, 5] that some novel terms in the partition function emerge when pure Maxwell theory is defined on a small compact manifold. These terms are not related to the propagating photons with two transverse physical polarizations, which are responsible for the conventional Casimir effect (CE) [6]. Rather, they occur as a result of tunneling events between topologically different but physically identical topological sectors. While such contributions are irrelevant in Minkowski space-time , they become important when the system is defined on certain small compact manifolds. Without loss of generality, consider a manifold which has at least one non-trivial direct factor of the fundamental group, e.g., . The topological sectors , which play a key role in our discussions, arise precisely from the presence of such nontrivial mappings for the Maxwell gauge theory. The corresponding physically observable phenomenon has been termed the topological Casimir effect (TCE).

In particular, it has been explicitly shown in [1] that these novel terms in the topological portion of the partition function lead to a fundamentally new contribution to the Casimir vacuum pressure that appears as a result of tunneling events between topological sectors . Furthermore, displays many features of topologically ordered systems, which were initially introduced in the context of condensed matter (CM) systems (see recent reviews [7, 8, 9, 10, 11]): demonstrates the degeneracy of the system which can only be described in terms of non-local operators [2]; the infrared physics of the system can be studied in terms of non-propagating auxiliary topological fields [3], analogous to how a topologically ordered system can be analyzed in terms of the Berry’s connection (also an emergent rather than fundamental field), and the corresponding expectation value of the auxiliary topological field determines the phase of the system. In fact, this technical trick of describing the system in terms of auxiliary fields will play a key role in our present discussions.

As we review in section II.1, the relevant vacuum fluctuations which saturate the topological portion of the partition function are formulated in terms of topologically nontrivial boundary conditions. Classical instantons formulated in Euclidean space-time satisfy the periodic boundary conditions up to a large gauge transformation and provide topological magnetic instanton fluxes in the -direction. These integer magnetic fluxes describe the tunneling transitions between physically identical but topologically distinct sectors. Precisely these field configurations generate an extra Casimir vacuum pressure in the system.

What happens to this complicated vacuum structure when the system is placed in the background of a constant external magnetic field ? The answer is known [1]: the corresponding partition function as well as all observables, including the topological contribution to the Casimir pressure, are highly sensitive to small magnetic fields and demonstrate periodicity with respect to the external magnetic flux represented by the parameter where is the area of the system . This sensitivity to external magnetic field is a result of the quantum interference of the external field with the topological quantum fluctuations. Alternatively, one can see this as resulting from a small but non-trivial overlap between the conventional Fock states, constructed by perturbative expansions around each sector, and the true energy eigenstates of the theory, which are only attainable in a non-perturbative computation that takes the tunneling into account. This strong “quantum” sensitivity of the TCE should be contrasted with conventional Casimir forces which are practically unaltered by any external field due to the strong suppression (see [1] for the details).

What happens when the external E&M field depends on time? It has been argued in [4, 5] that the corresponding systems will radiate real physical photons with transverse polarizations. However, the arguments of Ref. [4, 5] were based on purely classical considerations at small frequencies of the external fields. The main goal of the present work is to the study the quantum dynamics of the topological vacuum transitions between states in the presence of a rapidly time-varying external E&M field.

The fundamental technical difficulty for such an analysis is that the radiation of real physical particles on mass shell is inherently formulated in Minkowski space-time with a well-defined Hilbert space of asymptotic states. At the same time, the vacuum fluctuations (“instanton fluxes”) interpolating between the topological sectors and saturating the path integral are fundamentally formulated in Euclidean space-time111This problem is not specific to our system. Rather, it is a quite common problem when the path integrals are performed in Euclidean space-time, but the relevant physical questions are formulated in Minkowski terms. In particular, the problem is well known in QCD lattice simulations with conventional Euclidean formulations. All questions on non-equilibrium dynamics and particle production represent challenges for the QCD lattice community..

We overcome this obstacle by introducing auxiliary topological fields to effectively describe the tunneling transitions computed in Euclidean space-time. These auxiliary fields can be analytically continued to Minkowski space-time. After making the connection between the auxiliary topological fields and the Minkowski observables, we proceed using conventional Minkowski-based techniques, including the construction of the creation and annihilation operators, coherent states, the appropriate Hamiltonian describing the coupling of the microwave cavity to the system, etc.

Our presentation is organized as follows. In Section II, we review the relevant elements of the system including the formulation of the magnetic (II.1) and electric (II.2) instanton fluxes. In Section III, we construct the dipole moment operators (electric and magnetic types) using our auxiliary fields continued to Minkowski space-time. In section IV, we formulate the problem of radiation in proper quantum mechanical terms by identifying the quantum “states” of the system and studying the quantum matrix elements between them. In Section V, we discuss quantum transitions in the system in a cavity in the presence of a time-dependent external E&M field. In the concluding Section VI, we speculate that the same “non-dispersive” type of vacuum energy (which cannot be expressed in terms of any propagating degrees of freedom, and which is the subject of the present work) might be responsible for the de Sitter phase of our Universe, where the vacuum energy plays a crucial role in its evolution.

## Ii Topological partition function. Euclidean path integral formulation

Our goal here is to review the Maxwell system defined on a Euclidean 4-manifold with sizes in the respective directions. This construction provides the infrared regularization of the system, which plays a key role in the proper treatment of the topological terms related to tunneling events between topologically distinct but physically identical sectors. We start in section II.1 with the construction of the magnetic instanton fluxes considered in [1] and continue in Section II.2 with the electric instanton fluxes considered in [5]. The construction of the respective instantons (1) and (9) have been discussed in the earlier works [1, 5] and even earlier in the original studies of the Schwinger model in 2d [12, 13], so we leave a review of the relevant details to Appendix A. Discussions on how these instanton fluxes can be generated in experiment with suitable boundary conditions can be found in Appendix B.

### ii.1 Magnetic type instantons

In what follows we simplify our analysis by considering a clear case with topological winding sectors in the -direction only. This simplification can be justified with the geometry , similar to the construction of the conventional CE. In this case, our system resembles the 2d Maxwell theory in [1] by dimensional reduction: taking a slice of the 4d system in the -plane will yield precisely the topological features of the 2d torus. With this geometry, the dominant classical instanton configurations that describe tunneling transitions can be written as

 Aμtop=(0, −πkeL1L2x2, πkeL1L2x1, 0), (1)

where is the winding number that labels the topological sector.

This classical instanton configuration satisfies the periodic boundary conditions up to a large gauge transformation, and provides a topological magnetic instanton flux in the -direction:

 →Btop = →∇×→Atop=(0, 0, 2πkeL1L2), (2) Φ = e∫dx1dx2Bztop=2πk.

The Euclidean action of the system is quadratic and has the form

 12∫d4x{→E2+(→B+→Btop+→Bext)2}, (3)

where and are the dynamical quantum fluctuations of the gauge field, and is classical external magnetic field.

As discussed in detail in [1], the quantum fluctuations of the gauge field decouples from the topological and external fields, allowing us to arrive at a simple expression for the topological partition function

 Ztop(τ,θeff)=√πτ∑k∈Zexp[−π2τ(k+θeff2π)2], (4)

where

 τ≡2βL3/e2L1L2. (5)

is a dimensionless system size parameter, and the effective theta parameter is defined in terms of the external magnetic field . Applying the Poisson summation formula leads to the dual expression

 Ztop(τ,θeff)=∑n∈Zexp[−n2τ+in⋅θeff]. (6)

Eq. (6) justifies our notation for the effective theta parameter as it enters the partition function in combination with integer . One should emphasize that the in the dual representation (6) is not the integer magnetic flux defined in Eq. (2). Furthermore, the parameter which enters (4, 6) is not the fundamental parameter normally introduced into the Lagrangian in front of the operator. Rather, should be understood as an effective parameter representing the construction of the state for each slice with non-trivial in the 4d system. In fact, there are three such parameters representing different slices of the 4-torus and their corresponding external magnetic fluxes. There are similarly three parameters representing the external electric fluxes (in Euclidean space-time) as discussed in [2], such that the total number of parameters classifying the system is six, in agreement with the total number of hyperplanes in four dimensions222Since it is not possible to have a 3D spatial torus without embedding it in 4D spatial space, the corresponding construction where all six possible types of fluxes are generated represents a pure academic interest..

To study the magnetic response of the system under the influence of an external magnetic field, we differentiate with respect to the external magnetic field to obtain the induced magnetic field

 ⟨Bind⟩=−1βV∂lnZtop∂Bext=−eβL3∂lnZtop∂θeff (7) =√τπZtop∑k∈Z(Bext+2πkeL1L2)exp[−τπ2(k+θeff2π)2].

This induced magnetic field can also be interpreted as a magnetic dipole moment

 ⟨mind⟩=−⟨Bind⟩L1L2L3 (8)

### ii.2 Electric type instantons

To study the electric instanton fluxes, we consider two parallel conducting plates which form the boundary in the -direction, endowing the system with the geometry of a small quantum capacitor that has plate area and separation at an ambient temperature of . These two plates are connected by an external wire to enforce the periodic boundary conditions (up to large gauge transformations) in the -direction, and so the system can be viewed as a quantum LC circuit where the external wire forms an inductor L. The quantum vacuum between the plates (where the tunneling transitions occur) represents the object of our studies.

The classical instanton configuration in Euclidean space-time which describes tunneling transitions between the topological sectors can be represented as follows:

 Aμtop(t) = (0, 0, 0, 2πkeL3βt) (9) A3top(β) = A3top(0)+2πkeL3,

where is the winding number that labels the topological sector and is the Euclidean time. This classical instanton configuration satisfies the periodic boundary conditions up to a large gauge transformation, and produces a topological electric instanton flux in the -direction:

 →Etop=˙→Atop=(0, 0, 2πkeL3β). (10)

This construction of these electric-type instantons is in fact much closer (in comparison with the magnetic instantons reviewed in the previous Section II.1) to the Schwinger model on a circle where the relevant instanton configurations were originally constructed [12, 13]. The Euclidean action of the system takes the form

 12∫d4x{(→E+→Etop+→Eext)2+→B2}, (11)

where, as in the magnetic case, and are the dynamical quantum fluctuations of the gauge field, is the topological instanton field and is a classical external field.

Unlike magnetic fields, which remain the same under analytic continuation between Euclidean and Minkowski space-times, an electric field acquires an additional factor of as it involves the zeroth component of four-vectors, i.e. . A detailed treatment is given in [5], and here we only state the final expressions for the partition function:

 Ztop(η,θEeff)=∑k∈Zexp⎡⎣−π2η(k+θEeff2π)2 ⎤⎦, (12)

for an Euclidean source , and

 (13)

for a Minkowski source

 θMeff=eL3βEMinkext=−iθEeff. (14)

We have used the dimensionless system size parameter

 η≡2L1L2e2βL3. (15)

Our interpretation in this case remains the same: in the presence of a physical external electric field represented by the complex source , the path integral (12) is saturated by the Euclidean configurations (10) describing physical tunneling events between the topological sectors .

Now, one can compute the induced Minkowski-space electric field and dipole moment in response to the external source by differentiating the partition function (13) with respect to :

 ⟨EMinkind⟩=−1βV∂ln¯Ztop∂EMinkext=−eL1L2∂ln¯Ztop∂θMeff (16) =1¯Ztop∑k∈Z2πkeL3βe−ηπ2k2sin[πkηθMeff].

The expectation value for the electric dipole moment can be competed in complete analogy with magnetic case (8), and it is given by

 ⟨pMinkind⟩=−⟨EMinkind⟩L1L2L3 (17) =−1¯Ztop∑k∈Z2πkL1L2eβe−ηπ2k2sin(πkηθMeff).

Although (8) and (17) have been derived assuming static external magnetic and electric fields, these expressions still hold when the external fields vary slowly compared to all relevant time scales of the system. In this case, the corresponding dipole moments and also take on time dependence in response to semiclassical time-dependent external sources as (8) and (17) suggest. Hence, one can invoke the laws of classical electrodynamics to study the magnetic and electric dipole radiation as a result of this time dependence. The radiation intensity is given by the classical expressions

 dIM(t)=⟨¨mind(t)⟩2sin2θ16π2c3dΩ, dIE(t)=⟨¨pMinkind(t)⟩2sin2θ16π2c3dΩ, (18)

while the total radiated power assumes the classical form

 IM(t)=16πc3⟨¨mind(t)⟩2,  IE(t)=16πc3⟨¨pMinkind(t)⟩2 (19)

for the magnetic and electric systems respectively. If one is to compute the average intensity over a cycle assuming conventional periodic oscillation for the field, one gets

 ⟨IM⟩=ω412πc3⟨mind⟩2,  ⟨IE⟩=ω412πc3⟨pMinkind⟩2. (20)

A few comments are in order. Firstly, (II.3) and (19) makes the important statement that the system emits physical photons from the vacuum in the presence of time-dependent external fields, in close analogy with the dynamical Casimir effect (DCE). Its difference from the conventional DCE [14, 15, 16] is that the radiation from the vacuum in our system is not due to the conversion of virtual to real photons, as illustrated in the top panel of Fig. 1. Rather, it occurs as a result of tunneling events between topologically different but physically identical vacuum winding states in a time-dependent background, and the physical photons here are emitted from these instanton-like configurations describing the tunneling transitions as illustrated in the bottom panel of Fig. 1.

Secondly, the magnetic dipole radiation can be easily understood in terms of topological non-dissipating currents flowing along the ring [4], while the electric dipole radiation can be understood in terms of fluctuating surface charges on the capacitor plates [5]. When the external field fluctuates, the induced non-dissipating currents and surface charges follow suit. This obviously leads to the radiation of real photons as formulae (II.3), (19) imply, which we call the non-stationary TCE. One should emphasize that the interpretation of the TCE (as well as non-stationary TCE, which is the subject of the present work) in terms of topological non-dissipating currents and topological surface charges is the consequential, rather than fundamental, explanation. The fundamental explanation is still the instantons tunneling between the topological sectors, which occur in the system even when topological boundary currents and charges are not generated (for example, in the absence of external fields).

Finally, one should note that the above analysis of dipole radiation is purely classical: the induced dipole moments (8) and (17) are treated as classical dipoles and then varied in the semiclassical limit (such that the expressions (8) and (17) remain valid) to yield electromagnetic radiation.

The new contribution of this paper will be presented in the following sections, where we develop quantum mechanical machinery with which to study the emission of photons from the topological vacuum (). This goal calls for a transition from the classical description (II.3), (19) of emission in terms of dipole expectation values and to a Minkowski description based on quantum mechanical operators, quantum states, and transition matrix elements. We already mentioned the fundamental obstacle in developing such a technique, see Footnote 1 and the corresponding paragraph. Formula (II.3), (19) will serve as the consistency check between the classical and quantum descriptions: it will provide some confidence that the quantum mechanical description (based on auxiliary topological fields developed in the next sections) reproduces the classical formulae (II.3), (19) in the low frequency limit as it should according to the correspondence principle.

## Iii Dipole moment operators

In this section, we use auxiliary fields to construct dipole moment operators in terms of quantum mechanical operators. These operators can be analytically continued to Minkowski space-time. They will play a crucial role in Section V where we study the quantum transitions in the system using quantum mechanical Hilbert states formulated in Minkowski space-time.

The expectation values for the induced electric field and dipole moment in Eq. (16) and (17) were calculated in the Euclidean path integral approach at nonzero temperature . In what follows we wish to formulate the topological features of our system using topological auxiliary fields and topological action. This technique is well known to the particle physics and CM communities. In particular, it was exploited in [17] for the Higgs model in CM context and in [18] for the so-called weakly coupled “deformed QCD”. In the present context of the Maxwell system, this technique was developed in [3], and we follow the notations from that paper.

We first illustrate how to obtain the dipole moment operator for the magnetic system reviewed in Section II.1, as it avoids the potentially confusing analytic continuation between Minkowski and Euclidean space-times. The same procedure can then be easily applied to the electric case reviewed in Section II.2.

### iii.1 Magnetic dipole moment

We follow [3] and insert in the original path integral (4) the following delta functional:

 δ[Bz−ϵzjk∂jak(x)] (21) ∼∫Dbzexp[i∫d4xbz(x)(Bz−ϵzjk∂jak(x))],

where . Here, is treated as the original magnetic field operator entering the action (3), including both the classical -instantons and the quantum fluctuations around them. Therefore, we treat as fast degrees of freedom. In comparison, the auxiliary fields and should be considered slow-varying external sources that effectively describe the large distance physics which results from tunneling transitions. We proceed by summing over all instanton configurations as before and integrating out the original fast degrees of freedom in the presence of the slow fields and . The effective Lagrangian can then be expressed in terms of these auxiliary fields.

Fortunately, the derivations can be performed as before since the Lagrange multiplier field enters (21) in exactly the same manner that the external magnetic field enters the action (3). Therefore, we arrive at

where . One can see from (22) that the topological term is explicitly generated in this effective description. This term has Chern-Simons structure which normally appears in many similar CM commutations (see e.g. [19, 17]), and one should therefore anticipate a number of topological phenomena as a result of this Chern-Simons structure. Furthermore, one can show [3] that the auxiliary field written in momentum space strongly resembles Berry’s connection in CM physics333 In fact, one can argue that the auxiliary fields in our framework play the same role as Berry’s connection in CM physics. In particular, as it is known Berry’s phase in CM systems effectively describes the variation of the parameter as a result of the coherent influence of strongly interacting fermions that polarize the system, i.e., , see e.g. [19]. Our auxiliary fields essentially describe the same physics. Therefore, it is not a surprise that the induced dipole moment , to be discussed below, can be explicitly expressed in terms of these auxiliary fields. .

The integration over is Gaussian, and can be explicitly executed with the result:

 Ztop=∑k∈Z√πτ∫Daexp[−∫d4xL] (23) L=−12(ϵzjk∂jak(x))2+2πk+θeffeL1L2(ϵzjk∂jak(x)).

A few comments are in order. Firstly, the negative sign in Eq. (23) should not be considered as any inconsistency or violation of unitarity. Indeed, the field is an auxiliary non-propagating field introduced into the system to simplify the analysis, and any observable could be computed without it. Instead, this field should be considered as a saddle point saturating the Euclidean partition function in the path integral approach444In many respects this negative sign in Eq. (23) resembles the negative sign for the so-called Veneziano ghost in the course of the resolution of the problem in QCD, see [18] for references and details in the given context. One can explicitly see from the computations in [18] how the negative kinetic term for the Veneziano ghost is generated due to tunneling transitions between different topological sectors in very much the same way as it occurs in our system represented by the effective Lagrangian (23). Precisely this “wrong sign” in the effective Lagrangian might be a key element in understanding the new type of cosmological vacuum energy known as the dark energy, see comments in the concluding section VI..

Secondly, the term in the above Lagrangian couples to both the instanton field expressed in terms of fluxes, and the external field formulated in terms of . The physical meaning of this operator can be easily understood by noticing that it enters the Lagrangian precisely as how a magnetic dipole moment density couples to the external magnetic field. Therefore, we identify with the magnetization of the system.

To confirm this conjecture, we should compute the expectation value of to reproduce the magnetic dipole moment derived in the Euclidean path integral approach (8). This task can be easily performed because the integration over is Gaussian and can be carried out by a conventional change of variables

 ϵzjk∂ja′k(x)=ϵzjk∂jak(x)−2πk+θeffeL1L2, (24)

after which the Lagrangian becomes

 L=−12(ϵzjk∂ja′k(x))2+12(2πk+θeffeL1L2)2. (25)

The expectation value of is then given by

 ⟨mind⟩=⟨[−ϵzjk∂jak(x)]⟩V = ∑k∈Z∫Dae−∫d4xL(−ϵzjk∂ja′k(x)−2πk+θeffeL1L2)V∑k∈Z∫Dae−∫d4xL = −L3∑k∈Z(θeff+2πke)exp[−τπ2(k+θeff2π)2]∑k∈Zexp[−τπ2(k+θeff2π)2].

Eq. (III.1) exactly reproduces our previous expectation value of the magnetic dipole moment (8), thereby confirming the identification of the operator with the magnetization of the system.

We would like to mention here that this identification should not surprise the reader. Indeed, it has been previously argued [3] that the auxiliary field can be thought of as Berry’s connection555These similarities, in particular, include the following features: while and are gauge-dependent objects, the observables, such as polarization or magnetization (III.1) are gauge invariant (modulo ) characteristics. Furthermore, the main features of the systems in both cases are formulated in terms of global rather than local characteristics. . The polarization properties of a CM system can be computed in terms of Berry’s connection and Berry’s curvature, see Footnote 3 with relevant references. In our case, the magnetization of the system is also expressed in terms of auxiliary fields. Therefore, Eq. (III.1) is in fact fully anticipated.

### iii.2 Electric dipole moment

The similar procedure can be applied to the electric system to obtain an electric dipole moment operator. The delta functional we insert into (12) is

 δ[Ez−ϵ12jk∂jak(x)] (27) ∼∫Dbzexp[i∫d4xbz(x)(Ez−ϵ12jk∂jak(x))],

where , and is taken to be the Euclidean quantum field including the instanton configurations (10) and quantum fluctuations around them.

We follow the same procedure as before by integrating out the auxiliary field . It leads to the following Euclidean Lagrangian density analogous to Eq. (23) describing the magnetic case

 L=−12(ϵ12jk∂jak(x))2+2πk+θEeffeβL3(ϵ12jk∂jak(x)). (28)

All the comments after Eq. (23) also apply here for the electric case (28). Furthermore, there is an additional complication for the electric case due to the necessity for a transition to physical Minskowski space-time, i.e., we have to replace the Euclidean in (28) by the Minkowski expression according to relation (14):

 L=−12(ϵ12jk∂jak(x))2+2πk+iθMeffeβL3(ϵ12jk∂jak(x)), (29)

where represents the physical electric field. The only difference from the magnetic case is the emergence of the factor in front of the effective theta parameter. Thus, we identify the electric dipole moment operator in Minkowski space-time with . In what follows we confirm this conjecture by explicit computation of the corresponding expectation value.

To proceed with this task we make a shift

 ϵ12jk∂ja′k(x)=ϵ12jk∂jak(x)−2πk+iθMeffeL3β, (30)

such that the Lagrangian (29) in terms of the new variable becomes

 L=−12(ϵ12jk∂ja′k(x))2+12(2πk+iθMeffeL3β)2. (31)

We can now calculate the expectation value of the electric dipole moment:

 ⟨pMinkind⟩=⟨[−iϵ12jk∂jak(x)]⟩V = ∑k∈Z∫Dae−∫d4xL(−iϵ12jk∂ja′k(x)−i2πkeL3β)V∑k∈Z∫Dae−∫d4xL = ∑k∈Z(−i2πL1L2keβ)exp[−η(π2k2+iπkθMeff)]∑k∈Zexp[−η(π2k2+iπkθMeff)] = −1¯Ztop∑k∈Z2πkL1L2eβe−ηπ2k2sin(πkηθMeff),

where is defined in (13). Here, we have removed the constant external term to keep only the truly induced contribution to the dipole moment, consistent with our previous definition in Section II.2. Eq. (III.2) exactly reproduces our previous expression (17) which was originally derived without even mentioning any auxiliary fields. This supports once again our formal manipulations with the auxiliary fields, and it also confirms our interpretation of the operator as the quantum polarization operator of the system. All the comments we have made in Section III.1 regarding the physical meaning of this operator also apply here to the electric case, including the connection with Berry’s phase, which we will not repeat here.

To study the quantum mechanical dipole transitions, we must work in Minkowski space-time where the metric signature allows for propagating on-shell photons. Although the original derivation in this section is performed in Euclidean space-time, we claim that the dipole moment operator represents an operator in Minkowski space-time, as confirmed by the explicit expectation value calculation (III.2).

Our next task is to infer from our previous Euclidean path integral computations the structure of the quantum states, which can then be employed for conventional quantum dipole transitions in Minkowski terms, see Footnote 1 and the related paragraph for explanation of the source of this technical subtlety.

## Iv Metastable quantum states in the Maxwell system

The main goal of this section is to identify quantum mechanical states in Hilbert space in Minkowski space using the operators constructed in previous section III. These quantum states have never been explicitly constructed in the previous path integral treatment of this model [1, 2, 3, 4, 5]. We substantiate our identification by reproducing the computed transition matrix elements with corresponding path integral computations in Euclidean space.

Before we proceed we would like to overview a well-known formal mathematical analogy between the construction of the vacuum states in gauge theories and Bloch’s construction of the allowed/forbidden bands in CM physics (see e.g. [20]). The large gauge transformation operator plays the role of the crystal translation operator in CM physics. commutes with the Hamiltonian and changes the topological sector of the system

 T|m⟩=|m+1⟩,  [H,T]=0, (33)

such that the -vacuum state is an eigenstate of the large gauge transformation operator :

 |θ⟩=∑m∈Zeimθ|m⟩,   T|θ⟩=e−iθ|θ⟩.

The parameter in this construction plays the role of the “quasi-momentum” of a quasiparticle propagating in the allowed energy band in a crystal lattice with unit cell length .

An important element, which is typically skipped in presenting this analogy but which plays a key role in our studies is the presence of the Brillouin zones classified by integers . Complete classification can be either presented in the so-called extended zone scheme where , or the reduced zone scheme where each state is classified by two numbers, the quasi-momentum and the Brillouin zone number .

In the classification of the vacuum states, this corresponds to describing the system by two numbers , where is assumed to be varied in the conventional range , while the integer describes the ground state (for ) or the excited metastable vacuum states (). In most studies devoted to the analysis of the vacua, the questions related to the metastable vacuum states have not been addressed. Nevertheless, it has been known for some time that the metastable vacuum states must be present in non-abelian gauge systems in the large limit [21]. A similar conclusion also follows from the holographic description of QCD as originally discussed in [22]. Furthermore, the metastable vacuum states can be explicitly constructed in a weakly coupled “deformed QCD” model [23].

Such metastable states will also emerge in our Maxwell systems defined on a compact manifold. Thus, the complete classification of the states in our system is , where the integer plays a role similar to the -th Brillouin zone in the reduced zone classification as we discussed above.

### iv.1 Identification of quantum states: magnetic system

Through the formal manipulation in Section III.1 we have identified the magnetic dipole moment operator . We have also seen that the quantum mechanical expectation value of reproduces the expectation value computed using Euclidean path integrals (8), i.e.

 ⟨Mz⟩=−(2πL3e)∑k∈Zkexp[−τπ2(k+θeff2π)2]∑k∈Zexp[−τπ2(k+θeff2π)2]. (34)

Formula (34) determines a truly induced magnetic moment when the trivial constant contribution (related to the external magnetic field) is removed from the corresponding expression (8). Formula (34) was derived using conventional path integrals in Euclidean space-time without interpreting it in terms of any physical states.

Now we interpret the result (34) in terms of quantum mechanical states in Hilbert space. Firstly, the factor originates from the partition function (4). This exponential form in Euclidean space suggests that the combination

 ϵ(k)=τπ2k2β=2L3π2k2e2L1L2 (35)

can be interpreted as the energy of state for in Minkowski space. In the case of a non-zero external field , the corresponding energy levels get shifted accordingly as in the well-known problem for a particle on a circle,

 ϵ(k,θeff)=τπ2(k+θeff2π)2β. (36)

We identify the parameter with the label of the metastable vacuum state , similar to the classification of the -th Brillouin zone in CM systems mentioned above. This interpretation is supported by the observation that for the energy is precisely the magnetic energy of the external field, while quantum tunneling generates the excited states with energies (36). In contrast to conventional quantum states in the context of dipole transitions, the “states” in our system are the -instantons that describe tunneling transitions between the infinitely many degenerate vacuum winding states.

Once we accept this interpretation along with the identification of the magnetic dipole moment operator , we then proceed to interpret the corresponding factor in (34) as the non-vanishing transition matrix element rather than a diagonal expectation value .

To simplify notations in what follows, we consider vanishing external field and the lowest excited metastable state , which can be formally achieved by considering the limit . In this case we can interpret (34) as the transition matrix element between the first excited state and the ground state.

 ⟨k=0|Mz|k=1⟩≃−2πL3e⋅e−τπ2. (37)

The main argument behind this interpretation is the observation that the integer parameter which enters (34) originally appeared in the Euclidean path integrals as the instanton action describing the interpolation between two topologically distinct states according to Eq. (4). The same interpretation also follows from the boundary conditions (1) such that (37) can be thought of (in Minkowski terminology) as the configuration describing the transition matrix element between the states which satisfy the non-trivial boundary conditions (1) with and states which satisfy the trivial boundary conditions with .

Yet another argument supporting the Hamiltonian interpretation in terms of the transition matrix elements (37) is the successful matching of our final formula for the intensity of radiation with the classical expression for emission (20) discussed in Section II.3. Indeed, the conventional quantum mechanical formula for the probability for the quantum transition per unit time is known to match well with the classical formula (20) for the intensity of radiation. This spectacular example of classical correspondence implies that the probability for the quantum emission is expressed in terms of the transition matrix element (37) to match the classical formula (20)

 R1→0=ω3μ03πℏc3|⟨k=0|Mz|k=1⟩|2, ℏωR1→0→⟨IM⟩. (38)

In this well known correspondence the magnetic moment as usual is identified with the time dependent transition matrix element . In this case the magnetic moment entering formula (20) for the classical emission should be identified with while the magnetic moment entering the quantum mechanical expression (38) should be identified with transition matrix element (37). This well-known correspondence between classical and quantum descriptions once again supports our interpretation of (37) as the transition matrix element between the excited and ground states, though the original computations (34) from which formula (37) was inferred were performed in the Euclidean path integral approach without any notions of the Hamiltonian formulation.

We conclude with the following remarks. As we mentioned previously, the expectation value (34) vanishes when the external field is zero, though we claim that the transition matrix element (which eventually leads to the emission of real photons) does not vanish according to (37). There is no contradiction here as the expectation value (34) vanishes at as a result of cancellation between states, while in our discussions above we selected a single state which obviously must be somehow produced by non-equilibrium dynamics and separated from the state.

Finally, the transitions between the quantum states described here should not be confused with multiple tunneling transitions between the infinitely degenerate vacuum winding states that make up the -vacuum, classified by two parameters as discussed at the very beginning of this section. Unlike the vacuum winding states, these quantum states are separated in energy (35) and the transitions between them form the central subject of this section.

### iv.2 Identification of quantum states: electric system

Through the formal manipulation in Section III.2 we have identified the dipole moment operator , whose quantum mechanical expectation value reproduces the expectation value computed in the Euclidean path integral approach, i.e.,

 ⟨Pz⟩=−1¯Ztop∑k∈Z2πkL1L2eβe−ηπ2k2sin(πkηθMeff). (39)

Following the magnetic system in the previous section, we wish to interpret this expression in terms of quantum states in Hilbert space. In the limit, the energy of each state can be read off the Boltzmann factors:

 ϵ(k)=ηπ2k2β=2π2k2L1L2e2β2L3, (40)

analogous to (35). As in Section IV.1, we work in the reduced zone scheme with and identify the configurations labeled by integers as the quantum states . In particular, is the ground state and represents the excited metastable states. The supporting arguments made in Section IV.1 apply to the electric system as well.

This connection allows us to further identify the transition elements of the matrix from (39) where we keep only the state to simplify the notations:

 ⟨k=0|Pz|k=1⟩=−i2πL1L2eβe−η(π2+iπθMeff) (41)

which is analogous to formula (37) for the magnetic system.

One can repeat the arguments presented in the previous subsection IV.1 to infer that the correspondence formula for the electric dipole transition assumes the form

 R1→0=ω33πℏc3ϵ0|⟨k=0|Pz|k=1⟩|2, ℏωR1→0→⟨IE⟩. (42)

This example of classical correspondence implies that the probability for the electric dipole transition is expressed in terms of the transition matrix element (41) to match the classical formula (20).

Our comments after Eq. (38) for the magnetic case still hold for the electric case, and we shall not repeat them here. The only additional remark we would like to make to conclude this section is as follows. All our results on the identification of the dipole moment operators and their expectation values (34) and (39) are based on the Euclidean path integral approach. We did not and could not construct the corresponding Hilbert space and the corresponding wave functionals in Minkowski space-time which would depend on the E&M field configurations. However, using the correspondence principle (and some other hints and indications) we were able to reconstruct the relevant matrix elements (37) and (41) without complete knowledge of the wave functionals . Fortunately, this is the only information we need in our following studies of quantum dipole transitions in a cavity.

## V Quantum dipole transitions in a cavity

The goal of this section is to construct the effective Lagrangian describing the interaction between the physical E&M fields and the auxiliary fields introduced in Sections III and IV. This coupling will allow us to carry out proper quantum computations for the rate of emission of real physical photons, because the relevant transition matrix elements (37) and (41) have been computed in Minkowski space-time. This puts us in a position to use the well developed procedure to study quantum dipole transitions, such as in the phenomenon of stimulated emission.

Numerically the decay rate (42) is extremely low (see Section V.3 for numerical estimates). It has been known for quite some time that different types of microwave (optical) cavities can drastically increase the sensitivity for photon detection. Due to the smallness of the magnitude of all the topological effects of our Maxwell system, including the intensity of photon radiation, there might be hope that the stimulated emission of photons from the capacitor configuration can be detected using microwave (optical) resonators.

Essentially, we adopt the conventional technique normally used to study a system consisting of an atom in an optical cavity. The role of the atom in our case is played by the topological Maxwell system as described in the previous sections, while the optical cavity is replaced by a microwave cavity as the typical frequencies for our system are much smaller than atomic frequencies.

However, it should be noted that a specific design for microwave cavities in a possible experiment is certainly beyond the scope of this paper, and we shall proceed with only a general sketch of the possible experimental setup for illustrative purposes exclusively. Our numerical estimates given in Section V.3 suggest that the typical sizes where persistent currents have been observed and where coherent Aharonov-Bohm phases can be maintained could be a good starting point for a possible design. However, we are reluctant to put forward a specific experimental setup since our main goal is to describe a new phenomenon, rather than to design a device for its observation or measurement. We leave the questions on possible design for others in the community who can then use their own expertise to devise suitable experimental apparatuses.

### v.1 Coupling with quantum E&M field

First, we want to demonstrate that the quantum propagating E&M field couples to the magnetic and electric dipole moment operators and in exactly the same way as it does to the dipole moment operators in conventional quantum mechanics. Indeed, from (23) one can deduce that the interaction of the quantum field with the auxiliary fields is given by the following extra term in the Lagrangian

 ΔLMint=Bquantz⋅[ϵzjk∂jak(x)], (43)

where is expressed in terms of the conventional quantum propagating field . The relation (43) follows from the fact that the parameter entering (23) represents the total E&M field, including the classical and the quantum parts, i.e. . In our previous discussions we kept only classical, constant, portion of the field. In our present discussions in this section we obviously need the quantum, fluctuating, portion of the field as well.

The expression (43) obviously has the structure of a quantum field interacting with the magnetic moment operator expressed in terms of the auxiliary fields and derived in (III.1) using the Euclidean path integral approach. Precisely the matrix element of this operator has been computed in (37). The operator and its transition matrix element play the same role in our computations as the electron magnetic moment operator and the corresponding matrix elements do in atomic physics with the conventional coupling .

The same arguments also apply to the quantum coupling of the E&M quantum field with the electric dipole moment operator . Indeed, from (29) one can deduce that the interaction of the quantum field with the auxiliary field is given by the following extra term in the Lagrangian

 ΔLEint=Equantz⋅[iϵ12jk∂jak(x)]. (44)

This is because which enters (29) represents the physical electric field, including the constant external part and the fluctuating quantum part. The expression (44) obviously has the structure of the interaction between the quantum field and an electric dipole operator expressed in terms of the auxiliary fields and derived in (III.2) using the Euclidean path integral approach. Precisely the matrix element of this operator has been computed in the previous section (41). The operator and its transition matrix element play the same role as the electron dipole moment operator and the corresponding matrix elements do in atomic physics with conventional coupling .

The essence of the auxiliary fields employed above is that they effectively account for the interaction between nontrivial topological configurations (which themselves describe the tunneling events) and the propagating physical photons. All the relevant information about these auxiliary fields, originally introduced in the Euclidean path integral approach, is encoded now in terms of the matrix elements (37) and (41) in Minkowski space-time such that one can proceed with the computations of the quantum transitions using conventional Hamiltonian techniques, which we shall do in the next section.

### v.2 Jaynes-Cummings Hamiltonian for the topological Maxwell system

We consider the electric system and limit ourselves to two states: an excited state and the ground state . The two levels are separated by an energy difference according to (40). Here we use the notation , where is the number of photons (not to be confused with being the winding states) and indicates the state of the . Suppose we prepare the system in the state and tune the oscillating external field to the resonance frequency . The transition rate from the to the state is determined by the corresponding transition matrix element (41) inferred previously from the Euclidean path integral computations (39).

First, as the energy of the k-states grow quadratically with , , we can neglect highly excited metastable states by considering only the leading contributions to the dipole moment (41) due to the transition from to . To simplify the analysis and to emphasize the basic features of the system, we also neglect the state which is degenerate to the state for vanishing external fields. In principle, it can be easily accounted for. However, we want to make our formulae as simple as possible, and we ignore this extra state for now.

If we assume that only a single cavity mode exists, which is a good approximation in the case of a high Q resonator, the system can be described by the Jaynes-Cummings Hamiltonian:

 H=H0+HI (45) H0=ℏω0a†a+ℏωa2σz, HI=ℏ(gaσ++g∗a†σ−),

coupling a single harmonic oscillator degree of freedom to our two-level system and . Here, and describes the coupling of our two-level system with the quantized E&M field with two transverse polarizations. Assuming the E&M field is polarized in the -direction, reads:

 g=−i√ω2ℏV⟨k=0|Pz|k=1⟩. (46)

One can easily check that on resonance, , the interaction Hamiltonian commutes with the free Hamiltonian , i.e. . Therefore, the eigenstates of the full Hamiltonian can be written as a linear combination of the degenerate eigenstates of . The degenerate eigenstates of are and . Within this degenerate subspace, the state of the system at time can be written and the dressed eigenstates of the full Hamiltonian are . Solving the Schrödinger equation yields the time evolution

 cn,1(t)=cn,1(0)cos(Ωnt)−icn+1,0(0)sin(Ωnt) (47) cn+1,0(t)=cn+1,0(0)cos(Ωnt)−icn,1(0)sin(Ωnt),

where .

In particular, if we prepare the in its excited state and the initial cavity field with photons, i.e. and , then at a later time the probability for finding the vacuum in the state is

 Pk=1=|⟨n,1|Ψ(t)⟩|2=12(1+cos2Ωnt). (48)

The sinusoidal oscillation indicates that energy is constantly exchanged between the and the cavity field. This is of course, the conventional Rabi oscillations with the only difference being that instead of a two-level atomic system, the transitions in our case occur between the metastable and ground states in the , similar to the Brillouin zone classification as discussed at the very beginning of Section IV.

It is particularly interesting to investigate the dynamics of our system ( plus quantum E&M field) when we start with an initial cavity field that is a coherent state of photons:

 |α⟩=e−|α|2/2∑nαn√n!|n⟩. (49)

The time evolution of the state probability is

 Pk=1≈12