Detection of &#x201C;anomalies&#x201D; inside microcavities through parametric fluorescence: a formalism based on modulated commutation relations and consequences on the concept of density of states

Serge Gauvin; Joseph Zyss; Cory Walker

doi:10.1364/JOSAB.36.000C62

1. INTRODUCTION

On one hand, due to its unique properties that allow for the manipulation of electromagnetic field and its coupling to electronic quantum states, the electromagnetic (here, optical) confinement achieved with cavity-based systems (microcavities) continues to be of great interest in the scientific community. This is true for contexts of both linear and nonlinear quantum optics as well as for condensed matter physics and technological applications [1–113]. Usually, the theoretical approach is based on the elegant Purcell and Kleppner conjectures [114,115]. (The Purcell conjecture consists of replacing the free spectral range by the “line width.” The Kleppner conjecture consists of replacing the electronic density of states by the “photonic density of states.”) As pointed out in Section 2.A.2, these conjectures assume the possibility of continuously modulating (here spectrally, “piling-up” of modes near resonances) the density of modes (or states) in between resonant frequencies and, accordingly, in the vicinity of a single mode. Also discussed in Sections 2.A.2 and 2.A.3, this appears paradoxical [but not contradictorily, as stipulated by (14) and in the text that follows (14)].

On the other hand and since the early sixties [116], the scientific community has maintained a strong interest towards parametric fluorescence, or spontaneous parametric down conversion. (Since it underlines the role of the vacuum field fluctuations, as for “usual fluorescence,” we prefer to use here the term “parametric fluorescence” instead of “spontaneous parametric down conversion.”) This optical process is one of the best-known nonlinear interactions [117–125] and is still actively investigated [126–135], especially in view of its ability to generate high-quality entangled photons pairs. This enables a wide range of experiments, from foundations of quantum mechanics, such as in the case of the present study, to applications in optical communication, computing, and data encryption. Being a trigger of parametric fluorescence, the role of vacuum field fluctuations is central. As shown in section 2.A.4 for the case of electromagnetic confinement, this fundamental noise naturally leads to the existence of a modulated commutator relating creation and annihilation operators. Therefore, an alternative point of view emerges. Through the use of this modulated commutator, it is then possible to circumvent the above paradox.

In this regard, even though the phenomenon of parametric fluorescence has been studied intensively over a long period, a number of fundamental issues remain to be clarified. One of these concerns the consequences of electromagnetic cavity confinement on the basic mechanism of photon generation through parametric fluorescence. Indeed, whereas it was recognized at an early stage that parametric fluorescence was triggered by vacuum field fluctuations [116], the impact of confinement on parametric fluorescence received surprisingly less attention, in spite of the potential consequences of confinement on electromagnetic noise.

Formally, in the “standard model” (for the purpose of this paper) described in Ref. [116], the impact of vacuum field fluctuations on parametric fluorescence originates from the existence of canonical commutation relations, for every mode $l$ , in between the photon creation, ${\hat{a}}_{l}^{†}$ , and annihilation, ${\hat{a}}_{l}$ , operators. Consequently and in order to better understand how parametric fluorescence is altered when it originates from within an open microcavity, our main goal in this work is to reassess parametric fluorescence by investigating the role of confined vacuum field fluctuations, as manifested via the modulated commutation relations. It turns out that various quantum nonlinear optical processes, such as second-harmonic generation [136,137] and laser emission [138,139], are impacted. Therefore, advances in the fields of parametric fluorescence and electromagnetic confinement research could be extended to other areas of nonlinear quantum optics.

In contrast to some earlier works, our contribution is a markedly distinct ab initio approach, which makes very few assumptions. Actually, an intimate link between the modulation of vacuum field fluctuations and the density of states is established. As a consequence, a generalized form of the creation and annihilation operators applicable inside as well as outside open cavities is obtained. As a result, this alternative point of view completes and clarifies the usual approach based on the Purcell and Kleppner conjectures (modulation of the density of modes).

These new concepts lead to a generalized point of view suggesting to revisit some basic quantum optical processes. A better understanding of confined nonlinear quantum optics could emerge and have consequences on the foundations of quantum optics in general, which might suggest new phenomena. Of course, this could have significant impact on the applications side, such as new methods to optimize the performance of parametric fluorescence or others, and perhaps new means of efficiently generating entangled photon pairs.

The rest of this work is organized as follows. First, in Section 2.A, we describe the confinement structure under study, the resulting field enhancement, and the impact of confinement on a revisited definition of the density of states, the commutation relations, and finally on the creation and annihilation operators themselves. Second, in Section 2.B, we describe the effect of confinement on parametric fluorescence, opening ways for this basic quantum optical mechanism to probe anomalies inside optical microcavities. Finally, we conclude this paper with a discussion on its main results, their consequences, and some perspectives.

2. THEORY

A. Fields Confinement

1. System Description

In order to determine the correct commutation relations to be applied in a quantum model of parametric fluorescence, it is mandatory to consider the “modes” of an open cavity. However, the proper definition of such modes is not trivial [38,140], especially due to the non-Hermitian nature of open cavities and its consequences on the non-orthogonality of “eigenmodes,” in fact, quasimodes, as well as complex eigenvalues. Furthermore, frequently, the models call on an approximation of orthogonal modes. Here, in contrast to other models that make use of the Langevin formalism to study open cavities [141–144] and as proposed in earlier models [145–147], we make use of the so-called “modes of the universe” approach. Accordingly and as shown below (16) in section 2.A.4, we consider noise (especially quantum noise) simply by way of cavity resonances. Our method follows that in Ref. [148], albeit with more details and with a reformulated version leading new important consequences, such as the generalization of photon creation, ${\hat{a}}_{l}^{†}$ , and annihilation, ${\hat{a}}_{l}$ , operators. Therefore and as shown in Fig. 1, we consider a simple planar open cavity in the $x - y$ plane. In order to avoid considerations about thermal equilibrium, we assume a temperature of 0 K. A priori, we can also assume any photon numbers distribution among all (quasi)modes. Incidentally, it is worth noting that such an open cavity is basically a Gires–Tournois mirror [149].

Fig. 1. Physical model accounting for the universe as a closed cavity, which includes the open cavity of interest. The perfectly conducting walls (intensity reflectance, $r$ , of 1 and amplitude reflectance, $ρ$ , of $- 1$ ), at $z = - d$ and $z = L$ , define the universe. The space between $z = - d$ and $z = 0$ comprises the cavity, while the rest of the universe is between $z = 0$ and $z = L$ . The semi-reflecting mirror at $z = 0$ is assumed lossless and characterized by $ρ = - \sqrt{r}$ and $τ = i \sqrt{1 - r}$ , with $r \approx 1$ for a high-finesse cavity. The subscripts “in” and “ex” refer to the open cavity and the rest of the universe, respectively. The $\hat{a}$ correspond to annihilation operators for each electric field component, $E$ .

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Here, we consider monochromatic electromagnetic waves, of angular frequency $ω$ , propagating inside and outside the open cavity. Accordingly, $E_{in}^{+}$ and $E_{ex}^{+}$ represent the electric fields’ amplitude of the right propagating, or “outgoing,” waves, while $E_{in}^{-}$ and $E_{ex}^{-}$ describe the left propagating, or “incoming,” fields. The subscripts distinguish the interior of the open cavity (inside), “in,” from its exterior (outside), “ex.” Taking into consideration a barrier at $z = L$ , where eventually $L ≫ d$ , we are then able to implement the “modes of the universe” approach. The infinitely thin semi-reflecting cavity wall (beam splitter) located at $z = 0$ has a reflectance, $r$ , eventually close to unity. We also neglect all absorption losses. When necessary, it is then possible to regard the whole system as a high finesse open cavity coupled to the rest of the universe. Since for such a cavity the spectral profile of every mode can be assimilated to a very narrow Lorentzian function [this can be easily obtained from (6)], under integration, it behaves as a Dirac delta distribution profile. This allows the internal and external “modes” to be approximately orthogonal. For the sake of simplicity and following [148], we make use of the amplitude reflectance, $ρ$ , and transmittance, $τ$ , expressed as $ρ = - \sqrt{r}$ and $τ = i \sqrt{1 - r}$ . Thus, the energy conservation law, $ρ ρ^{*} + τ τ^{*} = 1$ , alongside with Stokes reversibility principle, $ρ τ^{*} + τ ρ^{*} = 0$ , are satisfied and can be used to obtain the relations between the inside and outside fields’ amplitudes.

The inclusion of a weak second-order nonlinearity is achieved when modulating in time, $t$ , the refractive index, $n$ , of the open cavity by a harmonic pump field, $E_{p} (z, t)$ via the dielectric constant $ε$ and its nonlinear susceptibility, $χ^{(2)}$ , both being wavelength-dependent properties. The pump field has an angular frequency $ω_{p}$ and a spatial mode function profile, $u_{p} (z)$ , such that [150]

E_{p} (z, t) = {\tilde{E}}_{p} u_{p} (z) \cos (ω_{p} t) := E_{p} (z) \cos (ω_{p} t) .

Thus, following [116], for the whole universe, we write

ε (z, t) = {\begin{cases} ε_{ex} + δ ε & - d \leq z \leq 0 \\ ε_{ex} & 0 \leq z \leq L \end{cases},

with

δ ε ≪ ε_{ex}

and

δ ε = ε_{o} χ^{(2)} E_{p} (z, t) = ε_{o} χ^{(2)} E_{p} (z) \cos (ω_{p} t) = ε_{o} χ^{(2)} {\tilde{E}}_{p} \cos (ω_{p} t) u_{p} (z) .

Finally, the value of

δ ε

has to be small enough to ensure that a perturbation treatment is valid for the calculation of the amplitude of the generated nonlinear components.

2. Fields Enhancement or Inhibition

It is well known that for both closed and open cavities, a resonant or natural mode is set when a wave, of angular frequency $ω_{ν}$ , is reproduced in phase after a single round trip. Even though this condition is necessary but not sufficient to create stationary states in the case of open (leaky) cavities, such a definition of resonant mode is clear and unambiguous (see Chapter 9 in [38]). We therefore retain this concept as the primary definition of a mode, which is generalized in Section 2.A.3. In the case of a slight mismatch between the angular frequency (or wavelength $λ$ ) of an arbitrary incident plane wave, $ω$ , and $ω_{ν}$ (or $λ_{ν}$ ), the condition of self-repeating wave cannot be fully satisfied. (As the wave progresses, the temporal attenuation of non-self-repeating modes forbids the existence of stationary states. In the case of closed cavities, these states are then virtual.) For finite-size cavities, the optical path is fixed by the geometry of the cavity, the refractive index of the medium, both phase shifts upon reflection and the phase shift upon transmission (in the case of coupled open cavities). As a consequence, the sequence of angular frequencies (or wavelengths) values that enables self-repeating waves can only be a discrete set, which is the “ensemble of natural frequencies.” (Being linked to stationary states, we prefer to restrict the terminology “eigenfrequencies” to the case of closed cavities.) Since the whole universe is closed and of finite size and whatever the finesse of the open cavity, the natural frequencies of both the open cavity and the rest of the universe always form discrete sets. It is noteworthy that since the open cavity behaves as a perfectly reflecting Gires–Tournois mirror, the spectrum of eigenfrequencies for the whole system (the universe) is also discrete. Also, being a Gires–Tournois mirror that involves a frequency dependent phase shift upon reflection ( $2 π$ radians at each eigenfrequency), the optical path changes with frequency. Accordingly, the separation between adjacent eigenmodes also depends on frequency and so the average density of states (which is strictly defined only over a sufficiently large spectral range). Nevertheless and even in the case of a spectrally modulated free-spectral range, the only way to change the density of natural modes and eigenmodes (stationary states) is through the spectral separation between them via a change in the optical path, resulting from changes in the dimensions of the cavity, the refractive index, or phase shifts. In the sense of natural frequency and eigenfrequency as defined in this paper, which form a set of discrete values separated by the finite value of the free-spectral range, it is not possible to continuously modulate a single mode itself. Hence, in the context of microcavities where the spectral separation between resonant modes is large, we are facing a paradox. Indeed, the present meaning of a mode is markedly distinct from current conjectures [114,115,79] that assume it possible to “continuously modulate (spectrally) the density of modes (or states) in between resonant frequencies and, therefore, in the vicinity of a single mode,” which is paradoxical. However and as discussed below, we are not facing a contradiction, since an alternative and a suitable point of view that leads to a natural generalization of the mode concept is possible. This is the purpose of the current section and the following one.

When $L ≫ d$ , the natural modes of the rest of the universe and the eigenmodes of the whole universe are spectrally closely spaced, and most of them do not match in frequency with the natural modes of the open cavity. Nevertheless and surely for $r < 1$ , every wave in the rest of the universe and related to a mode of the whole universe can penetrate into the open cavity. But, in the case of mismatch between the angular frequency of the natural mode of the cavity and the eigenmode, due to the self-interferences inside the open cavity, the resulting amplitude of the internal wave is slightly lower than for the case of a perfectly self-replicating wave. Therefore, through the amplitude of each component associated to a specific angular frequency $ω$ , inside the open cavity we can assign a relative spectral weight to each of them.

Also, since a self-repeating wave interferes constructively with itself, any natural mode of the cavity entails a resonance phenomenon. For each resonant frequency, the energy accumulates in time inside the cavity toward an asymptotic value. The time constant is defined by the quality factor of the open cavity [151]. In between natural frequencies, the energy is expelled from the open cavity. Thus, relative to an atomic system external to the open cavity perturbed by a harmonic incident (pump) field $E_{ex}^{-} (ω)$ , an atomic system internal to the open cavity is submitted to stronger perturbation when $ω$ is close to a natural angular frequency. Conversely, the perturbation is minimal when $ω$ is exactly in between two natural frequencies. Formally, the perturbation enhancement or inhibition is adequately described by the ratio of the (temporal asymptotic) value of the internal field amplitude, $| E_{in} |$ , to the external one, $| E_{ex} |$ . This defines the “field enhancement factor,” $Λ_{E}$ . Here, according to Fig. 1 and in terms of the incoming fields, we define $Λ_{E}$ as

Λ_{E} (ω) \equiv \frac{| E_{in}^{-} (ω) |}{| E_{ex}^{-} |} .

Likewise, the ratio of the internal intensity,

I_{in}

, to the external one,

I_{ex}

, defines the “intensity enhancement factor,”

Λ_{I}

, which is the so-called “modulation function.” Assuming that the refractive index of both the open cavity and the rest of the universe are practically identical, we thus write

Λ_{I}

as

Λ_{I} (ω) \equiv \frac{I_{in}}{I_{ex}} = \frac{{| E_{in}^{-} (ω) |}^{2}}{{| E_{ex}^{-} |}^{2}} = Λ_{E}^{2} (ω) .

(When considering light composed of photons,

Λ_{1}

should be assumed as a rational number or mean value in the presence of noise.) As fully detailed in Ref. [152] for the confinement structure described in the previous section, with

c

being the speed of light in vacuum,

Λ_{I}

is formally given by

Λ_{I} (ω) = \frac{1 - r}{1 + r - 2 \sqrt{r} \cos (2 d ω / c)} .

Figure 2 shows the shape of

Λ_{I} (ω)

. The peak values for

Λ_{I} (ω)

range from about 10 (for

r \approx 0.7

) to about 40,000 (for

r \approx 0.9999

). The similarity to the transmittance curve of an asymmetric Fabry–Perot resonator is noteworthy. In addition to the resemblance in shape, the peaks of the modulation function correspond to the same free-spectral range. However, due to the phase quadrature of the reflected and transmitted waves at the beam splitter, as from

ρ = - \sqrt{r}

and

τ = i \sqrt{1 - r}

, the spectral locations of the natural frequencies are shifted in between peaks. A

2 π

phase shift must be taken into account when crossing a resonance at an integer value of

π c / d

.

Fig. 2. Shape of the “modulation function” $Λ_{I}$ versus angular frequency, $ω$ , for various reflectance values, $r$ . The intensifications of vacuum field fluctuations are located at the vicinity of eigenmodes, which are located at integer values of $π c / d$ .

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Of course, the field and intensity enhancements apply to each component having an angular frequency equal to an integer number of $π c / d$ , which defines the free-spectral range given by $Δ ω \equiv π c / d$ . As expected, (5) and (6) imply that the addition or removal of one photon (averaged temporal asymptotic value) at $ω$ to the rest of the universe entails the addition or removal of $Λ$ photons (averaged temporal asymptotic value) to the open cavity. Of course, considering the dynamical aspects of the open cavities, the overall (spatial and temporal) energy conservation must be fulfilled. However, since virtual photons are temporarily emitted from the vacuum and real photons are temporarily lost into the vacuum, quantum fluctuations (not involved in the local and/or instantaneous energy conservation law) are permanently present in the whole universe. Therefore, (5) and (6) permanently apply to quantum noise. Accordingly, in the presence of any external noise, whatever its origin but especially when its various components are randomly appearing and disappearing in time, due to the internal superposition of a very large number of distinct components, each having uncorrelated amplitude and temporal phase, one with respect to the other and for every natural mode, the noise inside the cavity is amplified relative to the rest of the universe. On the other hand, in between the natural modes, the noise is attenuated. Even though each spatial mode profile is well determined, the lack of correlation in amplitude and temporal phase between various external noise components is such that (even for a well defined wavelength) the resulting field randomly fluctuates in time. Thus, no function can describe the temporal evolution of the resulting electromagnetic field noise. In terms of amplitude and phase, without any fixed relationship in between noise components from the rest of the universe and in the sense of mathematical distribution theory [153], only distributions of expected values for noise components can be determined. In the presence of nil amplitude waves, we then conceive the zero-point energy characterized by an electric field of zero mean value but non-zero square mean value. This quantum noise defines the vacuum field fluctuations [154]. The noise spectrum contains all frequencies, including all the cavity’s natural frequencies. In conclusion, since quantum mechanics states that there is always an unavoidable minimal amount of noise (amplitude and/or phase noises) in the form of vacuum field fluctuations, which are present on every wave (eigenmode) and independent of their amplitude and for all frequencies [16,154–156], the spectral modulation of vacuum field fluctuations inside confinement structures is then clear. As shown in the next two sections, this simple observation strongly suggests revisiting some fundamental aspects in confined quantum electrodynamics, as recently done for optical forces [157]. In this respect, recognition that the noise spectra in the rest of the universe and inside the open cavity are different, the overall radiation pressure on both sides of the semi-reflecting cavity wall should be different, thus leading to the well-known Casimir effect [38,158–164].

3. Density of States as Modulation of Vacuum Field Fluctuations

When considering a quantum optical process that is triggered by vacuum field fluctuations, such as “usual” and “parametric” fluorescence (both spontaneous processes), which is occurring inside a confinement structure such as an open cavity, the involved atomic systems are more or less perturbed by the noise content of an arbitrary incoming field, depending on the frequency matching of this field and the natural frequencies of the confinement structure. Thus, inside the open cavity, the corresponding perturbation Hamiltonian, ${\hat{H}}_{in}^{'}$ , is also modulated in frequency relative to a perturbation Hamiltonian in free space, here, the rest of the universe, ${\hat{H}}_{ex}^{'}$ . According to the perturbation theory and Fermi’s golden rule, which states that a perturbation is required to trigger an electronic transition [165], we can express the transition probability of per unit of time, $P_{fi}$ , for an electron of the atomic system initially from a state $| i ⟩$ onto a final state $| f ⟩$ [165]:

P_{fi} = 2 π \frac{1}{ℏ} | ⟨ f | {\hat{H}}_{in}^{'} | i ⟩ |^{2} ρ (U_{f}) = 2 π \frac{1}{ℏ} | ⟨ f | Λ_{E} {\hat{H}}_{ex}^{'} | i ⟩ |^{2} ρ (U_{f}) = 2 π \frac{1}{ℏ} | ⟨ f | {\hat{H}}_{ex}^{'} | i ⟩ |^{2} Λ_{I} ρ (U_{f}),

where

U_{f}

stands for the transition energy,

ρ (U_{f})

for the electronic density of states, and

Λ_{I} ρ (U_{f})

for “effective density of states.”

We thus conclude that the function $Λ_{I} (ω)$ modulates perturbations inside the cavity in such a way that we can visualize the whole interaction process as due to an effective density of states that matches the cavity quasimodes, especially when the electronic density of states $ρ (U_{f})$ is a slowly varying function. Consequently, the effective density of states can be higher or lower than in the rest of the universe (free space). This formulation leads to the Purcell and Kleppner conjectures [114,115] about enhancement or inhibition of transition rate inside confined structures. Indeed, the noise intensification described by the modulation function $Λ_{I} (ω)$ enhances the transition rate effectively as if there were a higher density of states, or conversely inhibits the transition rate in the case of low effective density of states. We conclude that while not in contradiction with our point of view based on vacuum field fluctuations, Purcell’s and Kleppner’s conjectures are not requirements, especially concerning the replacement of the electronic density of states by a similar concept about the “photonic density of states,” which was then linked in an elegant but ad hoc way to the natural modes (quasimodes) of an open cavity.

Incidentally, in the context of a weak coupling regime, i.e., moderate quality factor [13,90], the open cavity dynamics is then dominated by damping, such that perturbation theory applies through Fermi’s golden rule. In the case of an atomic system having an emission frequency that matches a maximum of $Λ_{I} (ω)$ , then (7) leads to the Purcell factor, $F_{p}$ , in one dimension. Indeed, with the enhancement or inhibition factor, $F$ , being defined as the ratio of the transition probability per unit of time inside the cavity, $P_{fi}$ , to the same transition probability in the rest of the universe, $P_{fi}^{ex}$ (where $Λ_{I} = 1$ ), and making use of (7), we thus write

F (ω) \equiv \frac{P_{fi}}{P_{fi}^{ex}} = Λ_{I} (ω) .

In order to take into account all noise components, (8) has to be integrated over all frequencies encompassing the emission line profile. However, since in the context of a weak coupling regime, the spectral width of the emission line profile is assumed much narrower (Dirac delta distribution profile) than the width of the peaks of

Λ_{I} (ω)

, we conclude that the Purcell factor is then equal to the maximum value of

Λ_{I} (ω)

,

Λ_{I}^{\max}

. On the other hand, from (6), we have

Λ_{I}^{\max} = \frac{1 + \sqrt{r}}{1 - \sqrt{r}} .

As this value is related to the finesse of the cavity (an asymmetric Fabry–Perot cavity having one perfect mirror [166]),

F

, through

F = π \frac{\sqrt[4]{r}}{1 - \sqrt{r}},

which itself relates to the quality factor,

Q

, through

F = \frac{1}{2} \frac{λ}{d} Q,

we finally end with

Λ_{I}^{\max} = \frac{1}{2 π} \frac{1 + \sqrt{r}}{\sqrt[4]{r}} \frac{λ}{d} Q .

Even for a moderate quality factor cavity,

r \sim 1

. The Purcell factor is then given by

F_{p} \approx \frac{1}{π} \frac{λ}{d} Q .

This result applies for a Gires–Tournois cavity and is isomorphic to the three-dimensional case [114].

A supplementary and convincing indication that vacuum field fluctuations are intimately linked to the photonic density of states is that the integration of the modulation function $Λ_{I} (ω)$ provides the exact number of natural modes, $N$ , in any spectral range of the open cavity defined by $N Δ ω$ , where $Δ ω \equiv π c / d$ . Indeed, whatsoever the finesse of the cavity, it can be shown that

\int_{0}^{N Δ ω} Λ_{I} (ω) d ω = N Δ ω .

Since, once integrated, the ratio

Λ_{I} (ω) / Δ ω

gives the number of natural modes,

N

, we conclude that

Λ_{I} (ω) / Δ ω

is formally identical to a density of modes. This is a striking and important result.

In addition, we can see from (14) that for a given spectral range, the confinement does not add noise to the open cavity. It simply modulates its intensity versus frequency. In brief, the modulation of vacuum field fluctuations and that of the density of states are necessarily concomitant. Indeed, from the point of view of the cavity, the rest of the universe simply behaves as a Gires–Tournois mirror. Then, in terms of modes (self-repeating conditions) and considering that a Gires–Tournois mirror is a perfectly reflecting mirror that causes a frequency-dependent phase shift upon reflection that modulates the optical path versus frequency, the modes inside the cavity are then spectrally and periodically shifted towards each other in the vicinity of natural frequencies, $ω_{ν}$ . As they get closer to each other, their noise contents superpose and add. Accordingly, the resulting noise level decreases between natural frequencies and increases in spectrally narrow regions, thus forming the quasimodes, which are in nature non-orthogonal. For that reason, it appears that (6) has to be linked to the Petermann excess noise factor, which is basically an excess spontaneous emission process [167,168]. In brief, the points of view based on the “modulation of vacuum field fluctuations” or on the “modulation of the density of sates” are conceptually different and opposed in the sense of being complementary, but they formally lead to equivalent results.

4. “Anomalous” Commutation Relations

Referring to Schwarz inequalities [165], quantum noise is usually formally described with the help of commutation relations. Incidentally, as previously discussed in the case of the confinement structure described in Section 2.A.1 and in divergence with the canonical commutation relations for free space, written as

[{\hat{a}}_{ex}^{+} (ω), {\hat{a}}_{ex}^{+ †} (ω^{'})] = [{\hat{a}}_{ex}^{-} (ω), {\hat{a}}_{ex}^{- †} (ω^{'})] = δ (ω - ω^{'}),

the commutation relations between the photon creation,

{\hat{a}}_{l}^{†}

, and annihilation,

{\hat{a}}_{l}

, operators become “anomalous” (non-canonical) inside open cavities [148,169–171]. Indeed, as shown in Ref. [148], the anomalous commutation relations include a modulation function,

Λ (ω)

, and are to be written as

[{\hat{a}}_{in}^{+} (ω), {\hat{a}}_{in}^{+ †} (ω^{'})] = [{\hat{a}}_{in}^{-} (ω), {\hat{a}}_{in}^{- †} (ω^{'})] = Λ (ω) δ (ω - ω^{'}) .

Thus, in general, creation,

{\hat{a}}_{l}^{†}

, and annihilation,

{\hat{a}}_{l}

, operators are position dependent [74,77].

For the confinement structure described in Section 2.A.1, the modulation function is given by (6), i.e., $Λ (ω) = Λ_{I} (ω)$ [152]. Revealing that cavity resonances spectrally modulate its internal noise, this result is important, as it establishes an intimate link between anomalous commutation relations and the intensity enhancement factor. Incidentally, since $Λ (ω)$ can be linked to the density of states via (14), this result gives a supplementary justification to introduce a (local) density of states from the (local) density of energy [172]. Parenthetically, the result $Λ (ω) = Λ_{I} (ω)$ demonstrates that “anomalous” commutation relations are actually spectrally “modulated” commutators, without any anomaly or inconsistency. It is unfortunate that the name “anomalous” has been used to label the commutators that pertain to open cavities. Of course, despite the clear picture it involves, these commutators remain hypothetical and need experimental verification.

According to the definition of eigenmodes given above, it can also be demonstrated (Appendix A of [38]) that every closed cavity of finite size has a discrete modal spectrum. However, conversely to the case of open cavities, in this case, the modes of the whole universe are formally described by orthogonal functions, which form a Hilbertian basis set. As specified in Section 2.A.1, we assume perfectly conducting boundaries at $z = - d$ and $z = L$ . Formally, we describe the modes of the whole system (eigenmodes of the universe) through a piecewise function separating the open cavity from the rest of the universe, on the basis of orthogonal complex exponentials that are linked to propagating waves. In order to take into account the phase shifts for every eigenmode, the eigenwavevector $k_{l} = n ω_{l} / c$ must be described by a dispersion relation that includes the phase shift induced upon reflection on the Gires–Tournois mirror. In addition, since the potential vector, $\vec{A}$ , contains all the information about both electric and magnetic fields, it is convenient to write the piecewise function as follows (single polarization state):

\hat{A} (z, t) = {\begin{matrix} {\hat{A}}_{in} (z, t) & - d \leq z \leq 0 \\ {\hat{A}}_{ex} (z, t) & 0 \leq z \leq L \end{matrix} .

(Here, assuming spatially uniform amplitude wave components outside the cavity, as the universe becomes of infinite size, its energy content then becomes infinite. It is then preferable not to explicitly include the cavity dimensions

d

and

L

into the definitions of

\hat{A}

. Accordingly, the creation and annihilation operators apply to a square root of unit length. In addition, the total Hamiltonian deduced from (18) and (19) applies to a unit area surface that is perpendicular to the

z

axis. Our formalism is equivalent to assuming ab initio that

L \to \infty

, where the field quantization scheme is based on a Fourier integral instead of a Fourier series {see Section 10.2.1 in Ref. [173]}, but still allowing to keep the discrete picture of eigenstates, and thus avoiding the problem of a zero measure of eigenstates {see Section 10.10 in Ref. [173]}. As usual, h.c. stands for “Hermitian complex of the previous term.”) We have

{\hat{A}}_{in} (z, t) = \sum_{l} \sqrt{\frac{1}{4} \frac{ℏ}{ε_{o}} \frac{1}{ω_{l}}} {[{\hat{a}}_{in l}^{+} e^{i (k_{l} z - ω_{l} t)} + h.c.] + [{\hat{a}}_{in l}^{-} e^{- i (k_{l} z + ω_{l} t)} + h.c.]}

and

{\hat{A}}_{ex} (z, t) = \sum_{l} \sqrt{\frac{1}{4} \frac{ℏ}{ε_{o}} \frac{1}{ω_{l}}} {[{\hat{a}}_{ex l}^{+} e^{i (k_{l} z - ω_{l} t)} + h.c.] + [{\hat{a}}_{ex l}^{-} e^{- i (k_{l} z + ω_{l} t)} + h.c.]} .

It can be shown [174] that the modes of the whole system having a wavevector

k_{l}

must fulfill the following condition:

\tan (k_{l} d) \tan (k_{l} L) = \frac{1 - \sqrt{r}}{1 + \sqrt{r}} .

Accordingly, here

l

refers to the modes of the universe, which satisfy the self-repeating wave conditions that are common to both the open cavity, which behaves as a Gires–Tournois mirror, and the rest of the universe. A priori, these conditions are not the same as the ones for the (internal) modes of the open cavity and the (external) modes of the rest of the universe when these subsystems are regarded independently. These internal and external natural modes depend on the optical path fixed by the size of the open cavity and the rest of the universe, the refractive index of the medium, and the phase shifts upon reflection and transmission at the semi-reflecting cavity wall (since the open cavity and the rest of the universe are coupled). Incidentally, in (20), one notices the similarities with the phase-shift equation for reflection from the Gires–Tournois mirror.

Making use of the usual ${\hat{p}}_{l}$ and ${\hat{q}}_{l}$ operators, it is also possible to determine the Hamiltonian of the whole system, ${\hat{H}}_{tot} (t)$ , in terms of only the left propagating (incoming and outgoing) mode field amplitude operators, ${\hat{a}}_{l}$ , as follows (here, $d$ and $L$ arise from calculations, since the creation and annihilation operators in (18) and (19) apply to a square root of unit length):

{\hat{p}}_{in l} (t) = - \sqrt{\frac{ℏ}{2} ω_{l} d} [{\hat{a}}_{in l}^{-} e^{i (k_{l} d - ω_{l} t)} + h.c.],

{\hat{q}}_{in l} (t) = - i \sqrt{\frac{ℏ}{2} \frac{d}{ω_{l}}} [{\hat{a}}_{in l}^{-} e^{i (k_{l} d - ω_{l} t)} - h.c.],

{\hat{p}}_{ex l} (t) = - \sqrt{\frac{ℏ}{2} ω_{l} L} [{\hat{a}}_{ex l}^{-} e^{- i (k_{l} L + ω_{l} t)} + h.c.],

{\hat{q}}_{ex l} (t) = - i \sqrt{\frac{ℏ}{2} \frac{L}{ω_{l}}} [{\hat{a}}_{ex l}^{-} e^{- i (k_{l} L + ω_{l} t)} - h.c.] .

Following straightforward developments, it can be found that

{\hat{H}}_{tot} (t) = \frac{1}{2} \sum_{l} [{\hat{p}}_{in l}^{2} (t) + ω_{l}^{2} {\hat{q}}_{in l}^{2} (t)] + \frac{1}{2} \sum_{l} [{\hat{p}}_{ex l}^{2} (t) + ω_{l}^{2} {\hat{q}}_{ex l}^{2} (t)] \binom{\underset{⏟}{- {\begin{cases} \frac{1}{2} \sum_{l} sinc (2 k_{l} d) [{\hat{p}}_{in l}^{2} (t) - ω_{l}^{2} {\hat{q}}_{in l}^{2} (t)] \\ + \frac{1}{2} \sum_{l} sinc (2 k_{l} L) [{\hat{p}}_{ex l}^{2} (t) - ω_{l}^{2} {\hat{q}}_{ex l}^{2} (t)] \end{cases}}}}{Correcting terms} .

One notices the presence of correcting terms that account for the energy exchanges between the open cavity and the rest of the universe. Indeed and as shown below, they disappear in the case of uncoupled cavities. We underline here that this Hamiltonian refers to the eigenmodes of the universe, which is not the case for the Gardiner–Collett Hamiltonian, as it refers to open cavity quasimodes coupled to a continuum of modes outside the cavity through an interaction Hamiltonian [38,141]. We also underline that in (25),

r

is arbitrary and no hypothesis is made about the quality factor of the open cavity, or about the internal modes of the open cavity and those of the rest of the universe. As seen in (20), the coupling strength expressed in the sine cardinal function of (25) is determined by

r

through

k_{l}

. Finally, we also underline the close resemblance of this coupling strength with the one given in Ref. [38]. Parenthetically, it is noteworthy that the minus sign separating

{\hat{p}}^{2}

and

{\hat{q}}^{2}

in the correcting terms of (25) reveals the structure of a Lagrangian. As is the case for the reflection of light at an interface, this suggests that the principle of least action rules the interaction between the cavity and rest of the universe. This principle minimizes the difference in energy contained in both the electric and magnetic field components and leads to an equipartition of electromagnetic energy into electric and magnetic field components, as expected from classical results [175]. Actually, inserting (21) to (24) into the correcting terms of (25) and making use of canonical and modulated commutation relations leads, for every mode, to a mean value for the Lagrangian of 0. Further considerations on this issue are postponed to a forthcoming work.

In the presence of a high finesse open cavity, for which $r \to 1$ , the open cavity and the rest of the universe are then weakly coupled (except for some specific modes, those that are accidentally, or not, frequency matching). As shown below and since it can still retain the coupling, this specific case is especially useful to simplify the formalism of parametric fluorescence in the cavity. Indeed, from (20) with $N_{in}$ and $N_{ex}$ being integer numbers, we then have $k_{l} = N_{in} π / d$ or $k_{l} = N_{ex} π / L$ , or both. Thus, the terms $sinc (2 k_{l} d)$ and $sinc (2 k_{l} L)$ can both be set to zero (except for $k_{l} = 0$ , a solution that can be omitted). Therefore, the total Hamiltonian takes the following simple form (neglecting $k_{l} = 0$ ), which, however, retains the coupling as

{\hat{H}}_{tot} (t) = \frac{1}{2} \sum_{l} [{\hat{p}}_{in l}^{2} (t) + ω_{l}^{2} {\hat{q}}_{in l}^{2} (t)] + \frac{1}{2} \sum_{l} [{\hat{p}}_{ex l}^{2} (t) + ω_{l}^{2} {\hat{q}}_{ex l}^{2} (t)] .

Formally, the whole system is then almost identical to the case of two independent cavities. The coupling between the open cavity and the rest of the universe is seen when considering (21) to (24) in addition to [148]

{\hat{a}}_{in l}^{-} = i \frac{\sqrt{1 - r}}{1 - \sqrt{r} e^{i 2 k_{l} d}} {\hat{a}}_{ex l}^{-}

and

{\hat{a}}_{in l}^{+} = - i \frac{\sqrt{1 - r}}{e^{- i 2 k_{l} d} - \sqrt{r}} {\hat{a}}_{ex l}^{+} .

These equations are in accordance with (16), when adapted to the discrete case.

Incidentally, defining the “partial modulation functions” [the index “E” stands for electric field amplitude, as in (4)]

Λ_{E in l}^{-} \equiv i \frac{\sqrt{1 - r}}{1 - \sqrt{r} e^{i 2 k_{l} d}}

and

Λ_{E in l}^{+} \equiv - i \frac{\sqrt{1 - r}}{e^{- i 2 k_{l} d} - \sqrt{r}},

we then have the following interesting result [see (4) to (6)]:

Λ_{E in l}^{-} Λ_{E in l}^{- †} = Λ_{E in l}^{+} Λ_{E in l}^{+ †} = Λ_{I} (ω_{l}) .

In summary, retaining the coupling for

r \to 1

through the above equations, the total energy of the system can be approximated by adding

{\hat{H}}_{in}

and

{\hat{H}}_{ex}

:

{\hat{H}}_{tot} = {\hat{H}}_{in} + {\hat{H}}_{ex} .

Finally, assuming

L \to \infty

, (26) becomes [176]

{\hat{H}}_{tot} (t) = \frac{1}{2} \int [{\hat{p}}_{in}^{2} (ω, t) + ω^{2} {\hat{q}}_{in}^{2} (ω, t)] d ω + \frac{1}{2} \int [{\hat{p}}_{ex}^{2} (ω, t) + ω^{2} {\hat{q}}_{ex}^{2} (ω, t)] d ω .

(Here, the operators apply to a square root of unit energy, when energy is expressed in unit of angular frequency.) Unfortunately, it can be shown that the operators

\hat{p}

and

\hat{q}

in (33) do not fulfill the Hamilton canonical equations. Having in view (26) or (33) and in order to correctly quantize the total radiation field of the whole system and so recover the canonical quantum mechanical procedure for closed systems by making use of proper generalized variables that obey Hamilton canonical equations [38,177], it is valuable to introduce the following operators:

{\hat{P}}^{2} (ω, t) \equiv {\hat{p}}_{in}^{2} (ω, t) + {\hat{p}}_{ex}^{2} (ω, t)

and

{\hat{Q}}^{2} (ω, t) \equiv {\hat{q}}_{in}^{2} (ω, t) + {\hat{q}}_{ex}^{2} (ω, t) .

Fortunately, it can also be shown that the operators

\hat{P}

and

\hat{Q}

do fulfill the Hamilton canonical equations. In addition, we can write

\hat{P} (ω, t) \equiv \sqrt{1 + \frac{d}{L} Λ (ω)} {\hat{p}}_{ex} (ω, t),

\hat{Q} (ω, t) \equiv \sqrt{1 + \frac{d}{L} Λ (ω)} {\hat{q}}_{ex} (ω, t) .

According to (33),

{\hat{H}}_{tot}

then becomes

{\hat{H}}_{tot} (t) = \frac{1}{2} \int {\hat{P}}^{2} (ω, t) + ω^{2} {\hat{Q}}^{2} (ω, t) d ω = \frac{1}{2} \int [1 + \frac{d}{L} Λ (ω)] [{\hat{p}}_{ex}^{2} (ω, t) + ω^{2} {\hat{q}}_{ex}^{2} (ω, t)] d ω .

At this point, it is worth recalling that in our model, the cavity and the rest of the universe are two distinct but coupled open cavities, which form a unique closed cavity representing the whole universe. Consequently, the process of canonical quantization can be used. The operators

\hat{P} (ω, t)

and

\hat{Q} (ω, t)

can be regarded as Hermitian, and the canonical commutation relations must prevail for the whole closed system (the entire universe). Then,

[\hat{Q} (ω, t), \hat{P} (ω^{'}, t)] = i ℏ δ (ω - ω^{'}),

and

[\hat{Q} (ω, t), \hat{Q} (ω^{'}, t)] = [\hat{P} (ω, t), \hat{P} (ω^{'}, t)] = 0 .

Working back and expressing

\hat{P}

and

\hat{Q}

in terms of the corresponding field amplitude operators, the following commutation relations that show the same modulation function as given in (6),

Λ (ω) = Λ_{I} (ω)

, are coming up:

[{\hat{a}}_{ex}^{+} (ω), {\hat{a}}_{ex}^{+ †} (ω^{'})] = [{\hat{a}}_{ex}^{-} (ω), {\hat{a}}_{ex}^{- †} (ω^{'})] = \frac{1}{L + Λ (ω) d} δ (ω - ω^{'}),

and

[{\hat{a}}_{in}^{+} (ω), {\hat{a}}_{in}^{+ †} (ω^{'})] = [{\hat{a}}_{in}^{-} (ω), {\hat{a}}_{in}^{- †} (ω^{'})] = Λ (ω) [{\hat{a}}_{ex}^{-} (ω), {\hat{a}}_{ex}^{- †} (ω^{'})] .

Thus, in (41) and (42), we obtain similar modulated commutation relations to (15) and (16), as found in Ref. [148]. However, it is worth noting the differences. Indeed, the inclusion of the barrier at

z = L

, which was omitted in [148], leads to the distinctive

1 / [L + Λ (ω)]

factor that appears in (41). From a strict point of view, the commutation relations in (41) are thus also non-canonical and apply only to the rest of the universe, which in principle can be comparable in size with the open cavity itself. In fact, here we are dealing with two distinct coupled cavities, together forming a single closed cavity. From (41), we also understand that the effect of the modulation function,

Λ (ω)

, in the rest of the universe becomes negligible for sufficiently large value of

L

. Let us underline the fact that the current model represents an approximation for an open cavity. Indeed, the whole system composed of the cavity and the rest of the universe must be seen as a closed system. It is to be noted that when

L \to \infty

, the rest of the universe behaves effectively as an unvarying reservoir to which the cavity represents only a negligible perturbation. Since the rest of the universe becomes equivalent to the entire universe itself, the commutation relation becomes then canonical. It is noteworthy that redefining the above external field operators (applying per square root of unit length) via the explicit inclusion of the constant

L

in (41) into (18) and (19), as it is frequently done when dealing with a closed cavity that contains a finite amount of energy [155], and neglecting

d

, we then recover (15). In other words, since the microcavity has a negligible effect in the case of

L \to \infty

, very much like we assume that the rest of the universe (the reservoir) acts effectively as a “canonical” cavity, we can consistently assume that the canonical commutation relation is applicable outside the microcavity. Reached through a systematic formal approach, this fundamental postulate about the reservoir should be recognized. It allows the elimination of ambiguities about the justifications for taking the canonical commutation relations (15) for the outside fields of an open cavity. However, it is important to note that the commutation relations

[{\hat{a}}_{in}^{-} (ω), {\hat{a}}_{in}^{- †} (ω^{'})] = Λ (ω) [{\hat{a}}_{ex}^{-} (ω), {\hat{a}}_{ex}^{- †} (ω^{'})]

and

[{\hat{a}}_{in}^{+} (ω), {\hat{a}}_{in}^{+ †} (ω^{'})] = Λ (ω) [{\hat{a}}_{ex}^{+} (ω), {\hat{a}}_{ex}^{+ †} (ω^{'})]

always hold without having to resort to the concept of reservoir.

5. Creation and Annihilation Operators inside Open Cavities

The necessity to fulfill (16) inside an open cavity leads to infer that the creation and annihilation operators that apply in such an environment, for every eigenmode $l$ and its corresponding Fock state $| n ⟩$ , are

{\hat{a}}_{in}^{\pm †} | n ⟩ = \sqrt{n + Λ} | n + Λ ⟩

and

{\hat{a}}_{in}^{\pm} | n ⟩ = \sqrt{n} | n - Λ ⟩ .

As expected, these generalized operators recover their canonical form whenever

Λ = 1

, as in free space. In agreement with (6), for every eigenmode an amplitude change of 1 photon in the rest of universe (free space) corresponds to a change of

Λ

photons inside the microcavity. However and in agreement with the intuitive classical picture of electromagnetic resonator dynamics, the addition (removal) of one input photon at

ω

, which corresponds to

Λ

photons to accumulate into (to be expelled from) the resonator, is linked to a new stationary state that is reached only asymptotically in time, and requires a total addition (removal) of

Λ + 1

photons to the whole universe. Actually, in (43) and (44), time implicitly tends to infinity,

t \to \infty

.

The above generalized operators underline the fact that in the same way as in free space, at least one photon must be added to, or removed from, a mode in order to detect a noticeable change in the electric field amplitude; in the presence of a spectrally modulated quantum noise this number becomes $Λ$ . In other words, for the fields inside the cavity that are described by (18) and buried in quantum noise, the smallest detectable intensity change is $Λ$ . Accordingly, inside the cavity, the states between $| n ⟩$ and $| n + Λ ⟩$ are not directly physically detectable, as is the case for the states between $| n ⟩$ and $| n + 1 ⟩$ in free space. These intermediate states cannot be stationary but rather transitory, hence virtual.

Also, it is noteworthy that these operators give the correct number of photons in the eigenmodes. Thus, they still define the photon number operator, ${\hat{N}}_{in}$ , by way of ${\hat{a}}_{in}^{\pm †} {\hat{a}}_{in}^{\pm}$ . Indeed, we can write

{\hat{N}}_{in} | n ⟩ = {\hat{a}}_{in}^{\pm †} {\hat{a}}_{in}^{\pm} | n ⟩ = \sqrt{n} {\hat{a}}_{in}^{\pm †} | n - Λ ⟩ = \sqrt{n} \sqrt{n - Λ + Λ} | n - Λ + Λ ⟩ = n | n ⟩ .

As required, this means that

Λ

does not describe the number of photons in a mode, but instead a measure of its fluctuation, which is its variance in the zero-point energy state. Actually, making use of (18) and

E = - \partial_{t} A

, for every eigenmode

l

and its corresponding Fock state

| n ⟩

, we can easily determine that inside the cavity and for the forward propagating component we have (compare with equation 5.2.9 in Ref. [155])

⟨ n | {\hat{E}}_{in l}^{+ 2} | n ⟩ = \frac{1}{2} \frac{ℏ}{ε_{o}} ω_{l} (n + \frac{Λ}{2}) .

(Since we do not include the cavity dimensions in (18) and (19), here, the units of the squared electric field are

V^{2} m

.) The case of a coherent state is postponed to a forthcoming work.

Also, for the same eigenmode $l$ and as indicated by (5), but here from a quantum point of view, the ratio of the intensity inside the cavity and the rest of the universe determines the relative probability per unit of length, $P_{Λ}$ , to detect a photon inside the cavity comparatively to the rest of the universe. Assuming that the refractive index of both the open cavity and the rest of the universe are practically identical, for the forward propagating component, we then write

P_{Λ}^{+} = \frac{⟨ n | {\hat{E}}_{in l}^{+ †} {\hat{E}}_{in l}^{+} | n ⟩}{⟨ n | {\hat{E}}_{ex l}^{+ †} {\hat{E}}_{ex l}^{+} | n ⟩} .

A straightforward calculation gives

P_{Λ}^{+} = \frac{n + Λ / 2}{n + 1 / 2} .

This result provides an additional physical interpretation of

Λ

. Indeed, when

n

is very large (

n ≫ Λ

), this probability is always close to unity. Then, the effect of confinement vanishes, which might make physical consequences difficult to observe. On the other hand, when

n

is sufficiently small, this probability depends on the value of

Λ

. Still, in this case and when the cavity is strongly confining (in resonance),

Λ ≫ 1

and

P_{Λ}^{+}

\approx Λ / n \sim Λ

. A contrario, when the cavity is expelling (out of resonance), this probability is lower than 1 when

Λ < 1

, as required. Finally and as expected, (48) shows that this probability is exactly

Λ

for the zero-point energy state (

n = 0

), the case of virtual photons.

Incidentally, when used in conjunction with Fermi’s golden rule, these operators are found in agreement with a former prediction based on the use of Heisenberg’s equation of motion, which indicates that inside a microcavity, a threshold exists regarding the generation of second-harmonic waves (SHG) [136,137]. An experimental verification of such a threshold is currently in progress.

B. Confinement of Parametric Fluorescence: Detection of “Anomalies” inside Microcavities

When the open cavity is of high finesse, its natural modes can be approximated by a set of discrete eigenmodes. With a proper selection of pump wavelength, $ω_{p}$ , open cavity size, $d$ , and nonlinear susceptibility, $χ^{(2)}$ (preferably a weakly dispersive medium), we can assume that only three modes (pump, signal, and idler) are efficiently coupled inside the open cavity to allow the emergence of parametric fluorescence.

The formalism for the confined environment in the presence of an open cavity is easily adapted from the case of a closed cavity. Nonetheless, before considering parametric fluorescence inside an open cavity having a high finesse, it appears useful to recall the major results for the case of a closed cavity, which is based on canonical commutation relations. This last context leads to the formalism of the “standard model” [116]. It is based on the quantum mechanical model accounting for parametric fluorescence in free space, which is assumed as a closed cavity. For the sake of simplicity and as previously assumed in Ref. [116], there are only two modes (“signal”, $ω = ω_{s}$ , and “idler”, $ω = ω_{i}$ ) that are quantum mechanically coupled in the process of parametric fluorescence through the nonlinearity of the medium inside the open cavity. Because of its high intensity, the pump mode is treated as a classical field. Formally, the nonlinearity is contained in the parameter $κ$ , defined in (50) below. Depending on time, when neglecting the pump depletion, the interaction Hamiltonian, ${\hat{H}}_{int}$ , is then written as

{\hat{H}}_{int} (t) = - \frac{1}{2} ℏ κ [{\hat{a}}_{s}^{†} (t) {\hat{a}}_{i}^{†} (t) e^{- i ω_{p} t} + {\hat{a}}_{s} (t) {\hat{a}}_{i} (t) e^{i ω_{p} t}] .

In addition to other parameters found in (2) and (3), the coupling parameter,

κ

, basically depends on the strength of the nonlinearity of the medium through its nonlinear susceptibility,

χ^{(2)}

, and the normalized spatial mode transverse profiles of the coupled modes,

u_{s} (z)

and

u_{i} (z)

. It also depends on the angular frequency of these modes,

ω_{s}

and

ω_{i}

, where

ω_{s} + ω_{i} = ω_{p}

. The coupling parameter can then be written as

κ = \frac{1}{2} \frac{ε_{o}}{ε_{ex}} χ^{(2)} \sqrt{ω_{s} ω_{i}} \int_{- d}^{0} E_{p} (z) u_{s} (z) u_{i} (z) d z .

Considering the nonlinear medium described by (2) and (3), the average photons number in signal mode,

{\bar{N}}_{s} (t)

, and idler mode,

{\bar{N}}_{i} (t)

, can be expressed as a function of the initial photons number in these two modes,

N_{s} (0)

and

N_{i} (0)

, i.e.,

{\bar{N}}_{s} (t) = N_{s} (0) \cosh^{2} (κ t) + (N_{i} (0) + 1) \sinh^{2} (κ t) .

An equivalent equation is obtained for the idler wave by permutating the “s” and “I” indices. The occurrence of parametric fluorescence is seen in this expression, since the average photons number in the signal mode increases with time, and thus in space, even if there were initially no photons in either mode. As for usual fluorescence, parametric fluorescence is then also triggered by vacuum field fluctuations. Formally, one can trace the origin of this statement to the non-vanishing value of the commutation relation for the field amplitude operators. Indeed, following the same approach as in Ref. [116], one can reformulate expression (51) so as to evidence the influence of the commutation relations between the creation and annihilation operators, with main steps as follows. We have to evaluate the average photon numbers

{\bar{N}}_{s} (t) = ⟨ {\hat{a}}_{s}^{†} (t) {\hat{a}}_{s} (t) ⟩

and

{\bar{N}}_{s} (t) = ⟨ {\hat{a}}_{i}^{†} (t) {\hat{a}}_{i} (t) ⟩

. Making use of the Heisenberg equation of motion, we then find the temporal evolution of the creation and annihilation operators for the signal,

{\hat{a}}_{s}^{†} (t)

, and idler,

{\hat{a}}_{i}^{†} (t)

, modes. The Hamiltonian, including the interaction, is then found to be

{\hat{H}}_{tot} (t) = \frac{1}{2} \int_{- d}^{0} ε_{ex} {| {\hat{E}}_{in} (z, t) |}^{2} + \frac{1}{μ_{in}} {| {\hat{B}}_{in} (z, t) |}^{2} d z + \frac{1}{2} \int_{0}^{L} ε_{ex} {| {\hat{E}}_{ex} (z, t) |}^{2} + \frac{1}{μ_{ex}} {| {\hat{B}}_{ex} (z, t) |}^{2} d z + \frac{1}{2} ε_{o} χ^{(2)} {\tilde{E}}_{p} \cos (ω_{p} t) \int_{- d}^{0} u_{p} (z) {| {\hat{E}}_{in} (z, t) |}^{2} d z,

where

B

is the magnetic induction,

μ

the permeability, and

{\hat{E}}_{•}

takes into account both

{\hat{E}}_{•}^{+}

and

{\hat{E}}_{•}^{-}

. The last part of (52) corresponds to the interaction part of the Hamiltonian,

{\hat{H}}_{int} (t)

. In terms of

{\hat{a}}_{s}^{†} (t)

and

{\hat{a}}_{i}^{†} (t)

, the interaction Hamiltonian is found to be

{\hat{H}}_{int} (t) = ℏ \cos (ω_{p} t) \iint κ [{\hat{a}}_{in}^{- †} (ω, t) e^{- i ω \frac{d}{c}} + {\hat{a}}_{in}^{-} (ω, t) e^{i ω \frac{d}{c}}] [{\hat{a}}_{in}^{- †} (ω^{'}, t) e^{- i ω^{'} \frac{d}{c}} + {\hat{a}}_{in}^{-} (ω^{'}, t) e^{i ω^{'} \frac{d}{c}}] d ω d ω^{'},

where

κ = κ (ω, ω^{'}) = \frac{1}{2 π} \frac{ε_{o}}{c} \frac{χ^{(2)}}{ε_{ex}} {\tilde{E}}_{p} L \sqrt{ω ω^{'}} \int_{- d}^{0} u_{p} (z) \sin (ω \frac{z + d}{c}) \sin (ω^{'} \frac{z + d}{c}) d z .

As detailed in [174] [in [174], (43) and (44) were implicitly used] where we made the assumptions that

Λ (ω_{s}) = Λ (ω_{i}) = Λ

and of weak nonlinearity with

κ = κ (ω_{s}, ω_{i}) = κ (ω_{i}, ω_{s})

, we finally express the signal mode inside the open cavity as

{\bar{N}}_{s} (t) = N_{s} (0) \cosh^{2} (Λ κ t) + [N_{i} (0) + Λ] \sinh^{2} (Λ κ t) .

An equivalent equation is obtained for the idler wave through the permutation of indices “s” and “i.”

From (55), one understands that electromagnetic confinement leads to two concomitant effects onto the behavior of parametric fluorescence inside a cavity. Indeed, the modulation function $Λ$ appears in two different and specific locations in the general results (55). Accordingly, along with the effect on noise, where the modulation factor $Λ$ figures in front of the hyperbolic trigonometric function, $Λ$ also appears in its argument, where it is related to the pump field amplitude intensification through resonance [136]. Of course, the intensification of quantum noise is of fundamental importance at this point, in that parametric fluorescence is markedly distinct from many other nonlinear optical processes such as second-harmonic generation, which is not noise driven.

3. DISCUSSION

From (55), we understand that parametric fluorescence should be intensified by a significant amount, $Λ$ , inside high finesse microcavities. In fact, for a semi-reflecting cavity wall, here visualized as an output mirror, having a modest reflectance of 0.99, one can estimate from (6) that the maximum value of the modulation factor is about 400. Consequently, the intensification of parametric fluorescence should be relatively easy to detect and might serve to probe the cavity interior, especially in order to check the validity of the modulated commutation relations. Actually and as initially suggested in [148], we can infer from Schwarz inequality that modulated commutation relations should lead to modulated uncertainty relations. Attempts to probe this modulated uncertainty relation came up to the proposition of inserting a beam splitter inside the cavity to extract the internal field [148]. However, further direct calculations disproved the adequacy of this scheme [148]. Therefore, considering the expected strong intensification of parametric fluorescence generated inside a microcavity as compared to free space, an appropriate experiment should be conceived based on parametric fluorescence in order to detect if it can effectively be intensified. The results of such experiments would thus be a step towards the validity, or not, and implications of the modulated commutation relations discussed in this work. Indeed, the meaning of the results presented here is apparent form the fact that parametric fluorescence is a noise-driven nonlinear optical process with the potential to probe inside an open cavity in order to observe the effects of modulated commutation relations.

In summary, from our point of view, an open cavity behaves basically like any resonator by intensifying the amplitude of external signals, which are frequency matched to a mode of the cavity, without regard to the amplitude and phase, so that the noise is also intensified. Since the rest of the universe behaves as reservoir, it can be assumed to be a white noise source with its lowest level being of quantum origin. Accordingly, we conclude that the vacuum field fluctuations in a specific cavity mode can be largely intensified. This appears to be the fundamental meaning of the modulated commutation relations.

Incidentally, from (21), (22), and (26) and also considering only part of an eigenmode, $l$ , inside the open cavity, it can be easily shown that the Hamiltonian per unit of length, ${}^{'}{\hat{H}}_{in l} (t)$ , defined as, ${\hat{H}}_{in l} (t) / d$ , is given as follows, where (45) is used for the right-hand side:

{}^{'}{\hat{H}}_{in l} (t) = ℏ ω_{l} ({\hat{a}}_{in l}^{- †} {\hat{a}}_{in l}^{-} + \frac{Λ_{l}}{2}) = ℏ ω_{l} (n_{in l} + \frac{Λ_{l}}{2}),

with

Λ_{l} = \frac{1 - r}{1 + r - 2 \sqrt{r} \cos (2 k_{l} / c)} .

One then notices that in absence of confinement, where

r = 0

, and as expected, this Hamiltonian recovers its canonical form.

Finally, it is worth noting that the intensification and inhibition of the spontaneous photon emission process by means of parametric fluorescence emerges from the modulation of vacuum field fluctuations, without need to resort to the questionable concept of density of states.

4. CONCLUSION AND PERSPECTIVES

In conclusion, (14) establishes an intimate link between vacuum field fluctuations and the density of states that emphasizes the complementary points of view of modulating the density of sates and the vacuum field fluctuations. This is what we identify as the “concomitance law.” In some circumstances, such as in the present case of parametric fluorescence, a formalism based on the modulation vacuum field fluctuations might be more convenient than one based on the modulation of density of sates. This situation is similar to the Heisenberg point of view of quantum mechanics (matrix mechanics) versus the Schrödinger one (wave mechanics), where two formally equivalent but conceptually different theories give the same predictions. In addition, this point of view also encompasses (and unifies) the point of view of “weak” and “strong” coupling regimes (strong atomic damping implies significant shift in resonant frequency), as it avoids a distinction between both these concepts.

The generalized operators (43) and (44) and the modulated commutation relations (41) and (42) deeply modify the current picture of some quantum optical processes. Indeed, in addition to the appearance of $Λ$ in (43) and (44), the modulated commutation relations indicate that inside cavities, the Planck’s constant is no longer a constant but instead a spectrally modulated value. This is an important consequence. Of course, the physical importance of theses results should not be underestimated but their physical realm must be ascertained. The intensification, or not, of the parametric fluorescence appears as a good test for this purpose. The extension of the present study to various quantum nonlinear optical processes should also be undertaken. Also, since (43) and (44) involve transitions with more than one photon, we anticipate that confinement structures might have impacts on the bunching properties of photons. As a further perspective, we should incorporate the temporal dependence into ${\hat{a}}_{in}^{\pm †}$ . Indeed, in this work, we consider only the temporal asymptotic value of the internal field amplitude, $| E_{in} |$ . Thus, in (43) and (44), time implicitly tends to infinity, $t \to \infty$ . Fortunately, in practice, the asymptotic behavior is quickly reached, in about 100 ns for an empty cavity having a very high quality factor: $Q \sim 10^{8}$ at $λ = 1 μm$ .

Exploring further the consequences in the case of an open cavity coupled to the rest of the universe with $r \to 1$ , one recognizes that as the open cavity gradually closes on itself, it eventually becomes fully isolated. Then, the open cavity tends to befall as a whole universe on its own and could then be detached from the rest of the universe. Concomitantly, the vacuum field fluctuations initially present in a natural mode of the open cavity and coupled to the rest of the universe ultimately concentrate into a single discrete eigenmode. Since the presence of photons must be sustained by eigenmodes for all possible field configurations in space and time (in the sense of distribution theory), it then appears possible that vacuum field fluctuations, being virtual photons, behave as some sort of medium, perhaps upstream to the manifestation of real photons if enough energy is provided. In fact, it seems possible that the vacuum field fluctuations (then acting as the “pixels” of the universe, or a new conception of the ether) represent and stand for the actual definition of any quantum stationary states. According to this point of view and in agreement with the above formalism, we can find photons only where there are vacuum field fluctuations, i.e., where there are eigenmodes. Extrapolating this idea leads us to conceive real photons (and all other particles in general) as being some sort of “shapes” made of vacuum field fluctuations that receive enough energy (linked to $ω$ ) and momentum (linked to $k$ , the effective mass being then determined from the dispersion relation) to emerge from the vacuum and persist, as real photons, in the stationary state defined by the eigenmode forming inside the open cavity. Incidentally, one may notice that in the absence of any dissipation process, such as in the case of closed cavities described here, any oscillator requires only an infinitesimal amount of energy per unit of time to end with an oscillation of finite amplitude. Consequently, since the eigenmodes completely define a quantum system by a complete set of stationary sates (basis), we are directed to question the essential physical meaning of stationary quantum states in general. Then, it seems that we could conceive eigenstates not only as mathematical conditions that correspond to self-repeating waves (electromagnetic waves and wavefunctions) that perpetually interfere constructively with themselves, but also incorporate, or are constituted of, vacuum field fluctuations. It seems that everything is basically made of vacuum field (all fields) fluctuations, when enough energy and momentum are available.

It is possible that some physical concepts are currently missing in the scientific language. In order to correctly describe the full realm of the link between vacuum field fluctuations and quantum eigenstates, we suggest reinvestigating some important quantum optical processes and also to extend and adapt the present work to the case of fermions. For this task, it seems fruitful to circumvent the distinction between occupation number for bosons, $n$ , and presence probability for fermions. The case of fermions with a presence probability, $n$ , smaller than 1 then corresponds to a fractional (mean) occupation number (such as for the 1/2 figuring in the single mode energy $(n + 1 / 2) ℏ ω$ ), and the case of bosons with an occupation number grater than 1 corresponds to a presence probability of $n$ . In addition, this gives a better understanding of (56).

Funding

New Brunswick Innovation Foundation (NBIF); Natural Sciences and Engineering Research Council of Canada (NSERC); Université de Moncton (U de M).

Acknowledgment

The authors declare that there are no conflicts of interest related to this article.

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Detection of “anomalies” inside microcavities through parametric fluorescence: a formalism based on modulated commutation relations and consequences on the concept of density of states

Abstract

1. INTRODUCTION

2. THEORY

A. Fields Confinement

1. System Description

2. Fields Enhancement or Inhibition

3. Density of States as Modulation of Vacuum Field Fluctuations

4. “Anomalous” Commutation Relations

5. Creation and Annihilation Operators inside Open Cavities

B. Confinement of Parametric Fluorescence: Detection of “Anomalies” inside Microcavities

3. DISCUSSION

4. CONCLUSION AND PERSPECTIVES

Funding

Acknowledgment

REFERENCES

Cited By

Figures (2)

Equations (57)

Journal of the Optical Society of America B