# The Quantum Optics of Asymmetric Mirrors With Coherent Light Absorption

^{1}School of Chemical and Process Engineering, University of Leeds, Leeds, United Kingdom^{2}School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom

The local observables of the quantised electromagnetic field near a mirror-coated interface depend strongly on the properties of the media on *both* sides. In macroscopic quantum electrodynamics, this fact is taken into account with the help of optical Green’s functions which correlate the position of an observer with all other spatial positions and photon frequencies. Here we present an alternative, more intuitive approach and obtain the local field observables with the help of a quantum mirror image detector method. In order to correctly normalise electric field operators, we demand that spontaneous atomic decay rates simplify to their respective free space values far away from the reflecting surface. Our approach is interesting, since mirror-coated interfaces constitute a common basic building block for quantum photonic devices.

## 1 Introduction

The fluorescence properties of an atomic dipole depend primarily on the so-called *local density of states* of the electromagnetic (EM) field, i.e., on the number of EM mode decay channels available at the same location (van Tiggelen and Kogan, 1994; Sprik et al., 1996; Kwadrin and Koenderink, 2013). For example, inside a homogeneous dielectric medium with refractive index *n*, the spontaneous decay rate

to a very good approximation, where

Taking a different approach, Carniglia and Mandel (Carniglia and Mandel, 1971) modeled semi-transparent mirrors by only considering stationary photon modes which contain incoming as well as reflected and transmitted contributions. Their so-called triplet modes depend on reflection and transmission rates and are a subset of the free space photon modes of the EM field. Unfortunately, this approach can result in the prediction of unphysical interference effects when modeling light approaching a mirror from both sides (Zakowicz, 1995). If one wants to avoid such interference problems, adjustments have to be made (Khosravi and Loudon, 1991; Creatore and Andreani, 2008), for example by doubling the usual Hilbert space of the quantised EM field in the presence of a semi-transparent mirror (Furtak-Wells et al., 2018). However, this immediately raises the question where the doubling of the Hilbert space comes from. For a detailed discussion of this question see a recent paper by Southall et al. (2021) which models two-sided semi-transparent mirrors with the help of locally-acting mirror Hamiltonians and a recent paper by Hodgson et al. (2021) which quantises the electromagnetic field in position space.

In the following we use the quantum mirror image detector method by Furtak-Wells et al. (2018) to obtain the basic observables of the quantised EM field in the presence of a mirror-coated dielectric interface. This method maps light scattering in the presence of a two-sided semitransparent mirror onto two analogous free space scenarios. More concretely, in our model, we choose an initial time *a*) to describe the EM field on the right and another one (labeled *b*) to describe the EM field on the left hand side of the mirror interface. For times *t* in the experimental setup in Figure 1, we notice that this amplitude is a superposition of electric field amplitudes seen in two corresponding free space scenarios. To construct the electric field observable for the above experimental setup, we sum up the signals seen by the original detector and a mirror image detector after placing them at the right positions. Doubling the Hilbert space of the EM field and distinguishing two different types of photons, namely *a* and *b* photons, helps to ensure that wave packets which never meet in real space do not interfere in our model.

**FIGURE 1**. Schematic view of a mirror-coated dielectric medium with air on its right hand side. The coating which can be characterised by its electric field reflection and transmission rates

As illustrated in Figure 1, the experimental setup which we consider here consists of a dielectric medium with refractive index *a* and *b* refer to light approaching the mirror from the left and from the right hand side, respectively.

In the absence of losses, energy is conserved and Stokes relation implies that the reflection rates for both sides of the mirror interface are the same

In the following, we construct the observables of the quantised EM field near a mirror-coated interface with coherent light absorption. To correctly normalise these observables, we demand locality and assume that the spontaneous decay rate of a test atom at a relatively large distance *x* from the reflecting surface equals its free space value. The spontaneous atomic decay rates near highly-reflecting mirrors (Morawitz, 1969; Stehle, 1970; Milonni and Knight, 1973; Arnoldus and George, 1988; Drabe et al., 1989; Meschede et al., 1990; Amos and Barnes, 1997; Matloob, 2000; Beige et al., 2002; Dorner and Zoller, 2002) and near dielectric media with and without losses (Carniglia and Mandel, 1971; Wylie and Sipe, 1984; Khosravi and Loudon, 1991; Snoeks et al., 1995; Yeung and Gustafson, 1996; Urbach and Rikken, 1998; Xu et al., 2004; Wang et al., 2005; Creatore and Andreani, 2008; Eberlein and Zietal, 2012; Falinejad and Ardekani, 2019) have already been studied extensively in the literature and theoretical predictions are generally in very good agreement with experimental findings (Drexhage, 1970; Chance et al., 1975a; Eschner et al., 2001; Creatore et al., 2009). Like these papers, we ignore interactions of the atomic dipole with the quantum matter of the mirror surface. Instead we assume here that the test atom and the atoms inside the mirror surface are strongly detuned. For simplicity, we also neglect the angle-dependence of reflection rates.

Despite taking an alternative approach, our results are in good agreement with previous results. In addition, our approach allows us to model scenarios which are not as easily accessible using alternative approaches. For example, the main difference between the setup considered in Ref (Furtak-Wells et al., 2018) and the setup which we consider here is the presence of a dielectric with

This paper comprises five sections. In Section 2 we quantise the EM field in a homogenous medium with a refractive index

## 2 The Quantised Electromagnetic Field Inside a Dielectric Medium

The purpose of this section is to obtain the Hamiltonian and the electric and magnetic field observables of the quantised EM field inside a dielectric medium with refractive index *n*. To do so, we relate its properties to the properties of the quantised EM field in an analogous free space scenario.

### 2.1 Maxwell’s Equations

Our starting point is classical electrodynamics. In a dielectric medium with permittivity ε and permeability μ and in the absence of any charges and currents, Maxwell’s equations state that (Stratton, 1941)

Here *t*. Moreover, we know that the energy of the EM field inside the dielectric medium equals

As an example, we now have a closer look at horizontally polarised light which propagates along the *x*-axis. In this case, consistency with Maxwell’s equations and with the right hand rule of classical electrodynamics requires that *x* direction. Moreover, *x* direction. Substituting these vectors into Eq. 2, they reduce to the differential equations

where the minus and plus signs correspond to different directions of propagation. The solutions of these equations are wave packets which travel at the speed of light *x* axis and for light traveling in other directions.

A special example of a dielectric medium is air with

with the refractive index, as usual, defined as

For air, we simply have

One difference between electric and magnetic field solutions in a dielectric medium and in air is a re-scaling of field vector amplitudes. Here the factors on the right hand side of Eq. 5 have been chosen such that

are the same,

Moreover, on the right hand side of Eq. 5 there is a re-scaling of the position vector

### 2.2 Field Quantisation in Air

Wave-particle duality suggests that the EM field is made up of particles, i.e., photons (Bennett et al., 2016). In the case of light propagation in three dimensions, we characterise each photon by its polarisation λ and its wave vector k. Moreover, we know from experiments that a photon with wave vector

where

where

with

### 2.3 Field Quantisation in a Dielectric Medium

To obtain the electric and magnetic field observables *free space photons*. To do so, we employ the equivalency relations in Eq. 5 which imply that

with

Using this equation, one can show that the EM field Hamiltonian of the dielectric medium and

as suggested by Eq. 8. In our description, a photon of frequency ω has the energy

## 3 The Quantised Electromagnetic Field in the Presence of a Mirror-Coated Interface

To determine the field Hamiltonian *free space photons traveling in air*. As usual, we characterise each photon by its polarisation λ and by its wave vector *a* and *b* and which describe light on the right and light on the left hand side of the mirror surface, respectively, at a given time

where

### 3.1 Highly-Reflecting Mirrors

However, for simplicity, we first have a closer look at a highly-reflecting mirror. In this case, an incoming wave packet changes its direction of propagation upon reaching the interface such that its angle of incidence equals its angle of reflection. Suppose the mirror is placed in the *y* and the *z* component of the electric field vectors of the incoming light accumulate a minus-sign upon reflection to ensure that they remain orthogonal to the direction of propagation. Now suppose a detector measures the electric field amplitude at a position *minus* the electric field seen by a mirror image detector at

if we assume that the *a* and the *b* photons evolve as they would in air. Here

and the tilde indicates that a minus sign has been added to the *x* component of the respective vector. Moreover,

As mentioned already above, the constants

As we shall see below, doing so we find that

### 3.2 Mirror-Coated Dielectric Media

To obtain the electric field observable

As before, the superscripts

In the absence of absorption, energy conservation implies

## 4 Atomic Decay Rates in the Presence of a Mirror Interface

In this section, we finally determine the normalisation constants

### 4.1 Derivation

As usual in quantum optics, we describe the dynamics of a two-level atom with ground state

with

up to terms in second order in

For example, for an atomic dipole inside a dielectric medium with refractive index *n*, the above interaction Hamiltonian

in the usual dipole and rotating wave approximations and with respect to the free energy of the atom and the quantised EM field near the mirror interface. Here *e* is the charge of a single electron,

For

It must be noted that in most dielectric media, *μ* and *n* (Scheel et al., 1999).

To derive the interaction Hamiltonian

Here

for an atomic dipole in front of a mirror-coated dielectric medium (cf. Figure 1).

To calculate its spontaneous decay rate

Before performing any time integrations, we substitute

are independent of *t* and always real. Moreover we know that

up to an imaginary part which does not contribute to later integrals. To perform the remaining

with

Using the above equations and performing time and frequency integrations, while denoting the atom-mirror distance by *x* such that

with *φ* integration, substitute

with *u* integration in Eq. 4.1, we obtain the spontaneous decay rate

for *x* axis. For *a* and *b* interchanged and with

The only other simplification which has been made in the derivation of Eq. 34 is the negligence of surface plasmons and evanescent modes. These modes can provide an additional decay channel for atomic excitation and their presence can lead to an increase of emission rates. However, here we assume that *x* should be large enough for interactions with surface plasmons and evanescent modes not to become important.

### 4.2 The Normalisation Constants ${\eta}_{a}$ and ${\eta}_{b}$

However, before we can make more quantitative predictions, we need to determine the normalisation factors

which implies

Both normalisation factors

For *symmetric mirrors*, we have

which can assume any value between 1 and 2. For example, for highly-reflecting symmetric mirrors with *asymmetric mirrors*,

For example, suppose the reflection rate *b* rather than the *a* photons. This can be understood by taking into account that the *b* photons are present on both sides of the mirror interface in this case while, to a very good approximation, the *a* photons can only be seen on one side.

**FIGURE 2**. The normalisation factors

### 4.3 Discussion

In this subsection, we have a closer look at the spontaneous decay rates *x* is positive. Using Eq. 35,

with the mirror parameter

This equation shows that the difference between the spontaneous decay rates

However, a closer look at Eq. 39 also shows that the spontaneous decay rate *all* the reflection and transmission rates of the mirror interface. This might seem surprising but remember that the dipole interaction between the atom and the surrounding free radiation field plays an integral role in the spontaneous emission of a photon (cf. Eq. 25). In the experimental setup in Figure 1, the atom couples to incoming, reflected and transmitted photon modes which leads to interference effects and the strong dependence of *x*. Moreover, the strength of the atom-field interaction depends on the magnitude of the electric field observable *a* and the *b* photons is shared between the quantised EM field *and* the mirror interface (Furtak-Wells et al., 2018).

The reason for the dependence of

**FIGURE 3**. The mirror parameter

**FIGURE 4**. The spontaneous decay rate *x* for different mirror parameters *x* becomes much larger than the wave length of the emitted light.

##### 4.3.1 Dielectric Media Without Mirror Coatings

In the absence of any coating, energy is conserved and the overall transition matrix for incoming photons needs to be unitary. Taking this into account one can show that (Degiorgio, 1980; Zeilinger, 1981)

in this case. As a result, the mirror constant

and depends only on r and *r* and *x* for two different orientations of the atomic dipole moment *r* of a dielectric medium on its refractive index *n* (Novotny and Hecht, 2006),

the spontaneous decay rate

**FIGURE 5**. The spontaneous decay rate *x* for **(A)** Here we ignore the mirror-coating and the reflection and transmission rates are chosen as suggested in Eq. 41. **(B)** Here the mirror coating is taken into account. Again we assume that **(A)** and **(B)** have many similarities.

In the special case of a highly-reflecting mirror, which adds a minus sign to the electric field amplitude upon reflection (Morawitz, 1969; Stehle, 1970; Milonni and Knight, 1973; Arnoldus and George, 1988; Drabe et al., 1989; Meschede et al., 1990; Amos and Barnes, 1997; Matloob, 2000; Beige et al., 2002; Dorner and Zoller, 2002), we have *y* and the *z* components of the electric field vanish along the mirror surface. If there is no electric field to couple to, then there is no atom-field interaction and the atom cannot decay. In contrast to this, an atomic dipole which aligns parallel to the mirror surface couples only to the *x* component of the electric field. This component is now

##### 4.3.2 Dielectric Media With Mirror Coatings

In the presence of mirror coatings, the possible absorption of light in the interface needs to be taken into account. As we have seen in Section 2, in the quantum mirror image detector method (Furtak-Wells et al., 2018), this is done by evolving photon states in exactly the same way as they would evolve in free space, i.e. without reducing their energy in time. However, as one can see from Eq. 19, photons which have either been transmitted or reflected by the mirror interface contribute less to the electric field observable *x*. However, as Figure 5B shows, this is not the case. The calculations in Section 4.2 show that the presence of non-zero loss rates,

changes the normalisation constant *x*. The only difference is that, while

Figure 6 shows cases, where the reflection rates *l* changes between 0 and its maximum possible value of *r*. As in Figure 5B, we observe a relatively weak dependence on the spontaneous decay rate *x* which matches the results presented for example in Refs. (Yeung and Gustafson, 1996; Eberlein and Zietal, 2012).

**FIGURE 6**. The spontaneous decay rate *x*. Here

Finally, Figure 7 shows that the spontaneous decay rate *x*. This occurs due to a reduction of the normalisation constant *b* photons.

**FIGURE 7**. The spontaneous decay rate *x*. As in Figure 6, **(A)** For example, varying **(B)** However, changing

## 5 Conclusion

The fluorescence properties of an atomic dipole depend on the so-called local density of states of the quantised EM field (van Tiggelen and Kogan, 1994; Sprik et al., 1996) which itself depends in a complex way on the properties of all of its surroundings. For example, as this paper illustrates, the spontaneous decay rate of an atom near a mirror-coated interface depends on the reflection and transmission rates,

To obtain an expression for the electric field observable

The main difference between the current paper and earlier work (Furtak-Wells et al., 2018) is that this paper considers a more general scenario. It is emphasised that the quantum optical properties of the atom depend on the characteristics of the media on both sides of the mirror interface. It is also shown that non-zero loss rates do not necessarily reduce the effect of the mirror by as much as one might naively expect. For example, the spontaneous decay rate of an atom can exhibit a relatively strong dependence on the atom-mirror distance *x* even for loss rates

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

## Author Contributions

All authors contributed to the conception and design of this study. BD and AB wrote the first draft of the manuscript. BD, NF-W and AB performed and checked the analytical calculations. BD made all the figures in the manuscript with the help of NF. All authors contributed to manuscript revision and read and approved the submitted version.

## Funding

We acknowledge financial support from the Oxford Quantum Technology Hub NQIT (Grant number EP/M013243/1).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: quantum photonics, quantum optics, macroscopic quantum electrodynamics, open quantum systems, spontaneous photon emission

Citation: Dawson B, Furtak-Wells N, Mann T, Jose G and Beige A (2021) The Quantum Optics of Asymmetric Mirrors With Coherent Light Absorption. *Front. Photonics* 2:700737. doi: 10.3389/fphot.2021.700737

Received: 26 April 2021; Accepted: 25 June 2021;

Published: 12 July 2021.

Edited by:

Eilon Poem, Weizmann Institute of Science, IsraelReviewed by:

Gabriel Hetet, École Normale Supérieure, FranceZhichuan Niu, Institute of Semiconductors (CAS), China

Copyright © 2021 Dawson, Furtak-Wells, Mann, Jose and Beige. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Benjamin Dawson, py13bhd@leeds.ac.uk