Uncertainty Relations for Mesoscopic Coherent Light

The scope of this paper is to show that coherent effects in the propagation of waves in random media can be quantitatively described using an approach akin to thermal non-equilibrium systems, where fluctuations induced by coherent effects play the role of thermal fluctuations and lead to uncertainty relations. To establish this new kind of uncertainty relations (QMUR), we define a cost function $\Sigma$, analogous to the entropy production $\Sigma_{\rm th}$. Then, we establish an expression for the average cost function $\langle\Sigma\rangle$ and apply it to show the genuine interest of QMUR to optimise quantum mesoscopic features in different setups.

The paper is organised as follows. In section 3, we introduce in layman’s terms ideas underlying coherent effects in quantum mesoscopic physics. In section 3.1, we present basic material on the well accepted diffusive limit for the spatial behavior of the incoherent intensity of the wave field. In section 3.2, we discuss how the microscopic time reversal symmetry lost at the level of the diffusive and incoherent wave propagation is restored perturbatively in the weak scattering limit. This essential idea that interference effects are related to time reversal invariance is further discussed in section 3.4 in the equivalent language of a stochastic Langevin equation where the noise is solely driven by spatially local interference terms.

Section 4 is devoted to a phenomenological description of quantum mesoscopic uncertainty relations (QMUR) using Onsager description so as to provide some physical intuition about their meaning. In section 5, we derive QMUR in the more general framework of statistical field theory. This allows to define a cost function at the trajectory level. A generalised form of QMUR is given in section 5.3. Section 6 contains examples to illustrate the meaning and calculation of QMUR in the geometry of a slab and for fluctuating forces. An alternative derivation of QMUR is presented in section 7 which is based on the Cramer-Rao bound hence unveiling a relation with Fisher information. In section 8 we conclude and discuss further potential applications of our results.

The multiple scattering expansion advocated in the previous section allows to describe the evolution of a plane wave in a random medium, i.e. technically to evaluate the disorder averaged (one-particle) Green’s function of (2). But it does not contain information about the spatial and time evolution of a wave packet. For optically thick media, most physical properties are determined not by the average Green’s function, but rather by the two-particle or intensity Green’s function $P(\mathbf{r},\mathbf{r^{\prime}})$, associated to the behaviour of the radiation intensity $I(\mathbf{r})=|E(\mathbf{r})|^{2}$.

A convenient way to illustrate these ideas Akkermans is to start from the expression of the one-particle Green’s function,

where $A(\mathbf{r},\mathbf{r}^{\prime},{\cal C}_{N})$ is the complex valued amplitude associated to a multiple scattering sequence of length $N$, ${\cal C}_{N}=(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N})$. The accumulated phase $k{\cal L}_{N}$ measures the length ${\cal L}_{N}$ of the multiple scattering sequence in units of the wavelength $k^{-1}$.

The two-particle Green’s function $P(\mathbf{r},\mathbf{r^{\prime}})$ is proportional to $GG^{*}$, hence it involves an accumulated phase given by the length difference ${\cal L}_{N}-{\cal L}_{N^{\prime}}$ between any two multiple scattering sequences. Upon averaging over disorder, it can be anticipated that only identical multiple scattering sequences ${\cal L}_{N}={\cal L}_{N^{\prime}}$ up to a single scattering event $l$, will survive the large phase scrambling in the weak scattering limit $kl\gg 1$. Keeping only these two coupled identical one-particle Green’s functions trajectories leads to an approximate two-particle phase independent intensity Green’s function $P_{D}(\mathbf{r},\mathbf{r^{\prime}})$ as represented in Fig. 1.a.

Building on this result known as the incoherent limit, a wealth of phenomenological descriptions has been proposed. Among them, radiative transfer describes the disorder average macroscopic wave intensity $I_{D}(\mathbf{r})$ and the associated current $\mathbf{j}_{D}$, obtained by keeping only incoherent, i.e. phase independent, contributions Akkermans ; Ishimaru . They are related by a Fick’s law, $\mathbf{j}_{D}(r)=-D\boldsymbol{\nabla}I_{D}(\mathbf{r})$ with $D=vl/3$ being the diffusion coefficient, which together with current conservation $\nabla\cdot\mathbf{j}_{D}=0$, leads to a steady state diffusion equation, $-D\Delta I_{D}(\mathbf{r})=0$ with boundary conditions ensuring the vanishing of the incoming diffusive flux (see appendix I).

The main drawback of these phenomenological approaches is their neglecting of interference effects washed out by the disorder averaging. Yet, phase coherence is not erased by the disorder average and is at the origin of spectacular measurable effects, e.g. Anderson localization (weak and strong), coherent backscattering, universal conductance fluctuations and Sharvin $\&$ Sharvin effect to cite a few (a selection of the extremely vast literature on these topics is accessible in e.g. Akkermans ).

It is worthwhile discussing the role of time reversal symmetry (TRS) in these interference effects. At the level of the wave equation (2), the multiple scattering amplitude and hence the one-particle Green’s function (3) are invariant under time reversal ²²2The notion of time reversal symmetry is sometimes presented as equivalent to reciprocity. It is not so, e.g. in the presence of absorption, an irreversible process, time reversal symmetry is broken but not reciprocity.. Namely, $G(\mathbf{r},\mathbf{r}^{\prime},\mathbf{k})=G^{*}(\mathbf{r}^{\prime},\mathbf{r},-\mathbf{k})$ Akkermans . TRS is broken in the incoherent diffusive approximation. To see it, note from Fig. 1.a, that TRS can be implemented either by reversing the two conjugated multiple scattering sequences in the two-particle Green’s function $P_{D}(\mathbf{r},\mathbf{r^{\prime}})$ or by reversing only one, leaving the second sequence unchanged. In the latter case, the resulting two-particle Green’s function cannot be written as an incoherent function $P_{D}(\mathbf{r},\mathbf{r^{\prime}})$ hence TRS is broken in the incoherent diffusive limit. In the weak scattering limit $kl\gg 1$, it has been shown that reversing a single sequence leads to a two-particle Green’s function given by $P_{D}(\mathbf{r},\mathbf{r^{\prime}})$ times a small and local correction known as a quantum crossing Akkermans . Therefore, quantum crossings are a signature of a broken TRS.

These results can be made more systematic using a semi-classical description which enables to include coherent effects in the incoherent radiative transfer model.

The semi-classical approach starts from the formal sum (3) over multiple scattering trajectories. As already stated, each phase independent, incoherent trajectory obtained for the diffusive intensity $I_{D}(\mathbf{r})\propto G(\mathbf{r_{0},r})G^{*}(\mathbf{r,r_{0}})$, where $\mathbf{r_{0}}$ is the location of the light source³³3This holds for a point source located at $\mathbf{r_{0}}$. For an extended light source, $I_{D}$ is obtained by performing an integral over $\mathbf{r_{0}}$., is built from the pairing of two identical, but complex conjugated, multiple scattering amplitudes $G$ and $G^{*}$. By construction, these two amplitudes have opposite phases so that the resulting diffusive trajectory is phase independent (Fig. 1.a). Unpairing these two sequences gives access to the underlying phase $k{\cal L}_{N}$ carried by each multiple scattering amplitude and thereby to phase dependent corrections. The semi-classical description makes profit of this remark to evaluate systematically phase coherent corrections which correspond to a local crossing Hikami81 , where two diffusive trajectories mutually exchange their phase so as to form two new phase independent diffusive trajectories (Fig. 1.b). This local crossing – a.k.a quantum crossing – irrespective to the exact nature of waves, is a phase-dependent correction propagated over long distances by means of diffusive incoherent trajectories. Yet, the exact local structure of a quantum crossing depends on the exact nature of the wave, its degrees of freedom and applied fields. This picture for coherent mesoscopic effects is presented at an introductory level in Akkermans (section 1.7). The occurrence of a quantum crossing is controlled by a single dimensionless parameter $g_{\mathcal{L}}$ known as the conductance which depends on scattering properties and on the geometry of the medium. For a three dimensional ($d=3$) setup, the conductance $g_{\mathcal{L}}$ is

where the macroscopic length $\mathcal{L}(\gg l)$ depends only on the geometry. In the weak scattering limit $kl\gg 1$, the conductance $g_{\mathcal{L}}\gg 1$ and small coherent corrections generated by quantum crossings show up as powers of $1/g_{\mathcal{L}}$. This scheme allows to expand relevant physical quantities, e.g. spatial correlations of the fluctuating intensity $\delta I(\mathbf{r})\equiv I(\mathbf{r})-I_{D}(\mathbf{r})$ as

where the first contribution $C_{1}(\mathbf{r,r^{\prime}})=\frac{2\pi l}{k^{2}}\delta(\mathbf{r-r^{\prime}})$ is short ranged and independent of $g_{\mathcal{L}}$. The two other contributions are long ranged, and respectively proportional to $1/g_{\mathcal{L}}$ and $1/g_{\mathcal{L}}^{2}$. All three terms contribute to specific features of long ranged interference speckle patterns, and have been measured in weakly disordered electronic and photonic media Scheffold98 ; Scheffold97 .

Essentially, all previous considerations stem from the remark that spatially long ranged coherent effects result from short range phase-dependent quantum crossings occurring at scales $\ll l$ (see Fig. 1.b). Stated otherwise, the large scale coarse grained hydrodynamic description of incoherent light can be modified to include coherent effects by inserting a local, properly tailored, noise function so as to reproduce expected long range coherent effects. Building on this remark, an elegant and systematic description has been proposed Spivak87 , based on the Langevin equation,

for the mesoscopic quantities $I(\mathbf{r})$ and $\mathbf{j(r)}$. Disorder averaging is performed only at large scales $\geq l$ hence the stochastic nature of both quantities. This stochastic approach while phenomenological in nature, is equivalent to a perturbation theory for microscopic quantities with respect to the small and dimensionless parameter $1/g_{\mathcal{L}}$. The time-independent noise $\boldsymbol{\xi}\mathbf{(r)}$ includes all the information relative to phase coherence induced by quantum crossings. Its spatial correlations are systematically calculable as powers of $1/g_{\mathcal{L}}$. The $g_{\mathcal{L}}$-independent behaviour accounts for the incoherent diffusive limit. The details of this generally cumbersome but well understood procedure are presented in Soret19 ThesisSoret19 . The noise has zero mean and to lowest order $1/g_{\mathcal{L}}$ it is Gaussian Soret19 ; Spivak87 ,

with the conductivity

similar to thermal diffusive processes Shpielberg2016 . Note that, to lowest order $1/g_{\mathcal{L}}$, (6) is a weak noise Langevin equation. Namely, relative to the mean current, $\boldsymbol{\xi}$ scales like $1/\sqrt{g_{\mathcal{L}}}\ll 1$ (see Appendix III). The Langevin equation (6) based on the two parameters $D$ and $\sigma$, provides a complete hydrodynamic description of the coherent light flow in a random medium and it extends Fick’s law to the fluctuating mesoscopic quantities $I(\mathbf{r})\equiv I_{D}(\mathbf{r})+\delta I(\mathbf{r})$ and $\mathbf{j(r)}\equiv\mathbf{j_{D}(r)}+\delta\mathbf{j(r)}$.

It is important to emphasize that the noise $\boldsymbol{\xi}\mathbf{(r)}$ accounts for phase-dependent corrections (quantum crossings) and not for the random disorder in (2). Note also that $\boldsymbol{\xi}\mathbf{(r)}$ does not restore TRS.

The form (8) of the noise is appealing since a constant $D$ and $\sigma(I)\propto I^{2}$, correspond to the Kipnis-Marchioro-Presutti (KMP) process – a heat transfer model for boundary driven one dimensional chains of mechanically uncoupled oscillators strongly out of equilibrium Bertini05 ; Shpielberg2016 ; Kipnis82 , well described by the macroscopic fluctuation theory Bertini15 ; Jordan . Hence, the Langevin equation (6) driven by local coherent processes (7) suggests to deepen the analogy with thermal diffusive non-equilibrium steady states. To that aim, we recall in the next section some basic concepts in thermal non-equilibrium physics so as to define a cost function and prove a new type of uncertainty relations (QMUR).

Consider a thermal system, coupled to two reservoirs keeping it in a non-equilibrium steady state and sustaining an energy and particle steady state currents $\mathbf{J}_{u},\mathbf{J}_{\rho}$. We assume that the macroscopic system can be divided into small systems still macroscopic in nature, but that are slightly out of equilibrium. Hence, the entropy density for each subsystem is $ds=\frac{1}{T}du-\frac{\mu_{c}}{T}d\rho$ where $T,\mu_{c}$ are the local temperature and chemical potential, and $u,\rho$ are the energy and particle densities. We further assume that energy and density are locally conserved; $\dot{\rho}+\nabla\cdot\mathbf{J}_{\rho}=0$ and $\dot{u}+\nabla\cdot\mathbf{J}_{u}=0$. The entropy flux is thus $\mathbf{J}_{s}=\frac{1}{T}\mathbf{J}_{u}-\frac{\mu_{c}}{T}\mathbf{J}_{\rho}$. The steady state entropy production rate $\langle\dot{\Sigma}_{th}\rangle_{th}$ in each subsystem is defined as the excess from the conservation equation, i.e. $\langle\dot{\Sigma}_{th}\rangle_{th}=(\dot{s}+\nabla\cdot\mathbf{J}_{s})d\mathbf{r}$. This implies that for the entire system $\langle\dot{\Sigma}_{th}\rangle_{th}=\int d\mathbf{r}\,(\nabla\frac{1}{T})\cdot\mathbf{J}_{u}+(\nabla\frac{-\mu_{c}}{T})\cdot\mathbf{J}_{\rho}$. This result can be generalized to account for other thermodynamic forces.

We focus now on thermal non-equilibrium steady state diffusive systems at uniform temperature here set to unity. The steady state current is then given by Fick’s law $\mathbf{J}_{\rho}=-D_{th}\nabla\langle\rho\rangle_{th}$, where $D_{th}$ is the corresponding diffusion coefficient and $\langle\rho\rangle_{th}$ is the steady state density profile. The fluctuating hydrodynamics Bertini15 ; Lecomte_eqlikefluc describes the fluctuations of the diffusive dynamics

where $\langle\eta\rangle_{th}$ vanishes. Note the analogy between time-independent fluctuations in (9) and the mesoscopic Langevin equation (6).

The conductivity $\sigma_{th}$ and diffusion $D_{th}$ are not independent and abide the Einstein relation $f^{\prime\prime}(\rho)=2D_{th}/\sigma_{th}$, where $f(\rho)$ is the free energy density. Moreover, since $\mu=f^{\prime}(\rho)$, previous considerations allow to identify the entropy production in the thermal system as⁴⁴4$D_{th},\sigma_{th}$ are evaluated at $\langle\rho\rangle_{th}$.

The analogy just mentioned between thermal diffusive fluctuations (9) and quantum mesoscopic fluctuations (6), allows to define a cost function $\langle\Sigma\rangle$ for coherent diffusive quantum mesoscopic systems.

Based on (10) and (6), we propose for the disorder averaged cost function $\langle\Sigma\rangle$, the phenomenological expression,

where $\sigma_{D}\equiv\sigma(I_{D})$. The temporal dependence in (10) is disregarded in the time-independent mesosocpic setup. Note that $\langle\Sigma\rangle$ is dimensionless, i.e. the mesoscopic counterpart of the Boltzmann factor $k_{B}$ is unity.

Equipped with the definition of the disorder averaged cost function (11), we are now in a position to prove the QMUR (1). To that purpose, from the stochastic mesoscopic current density in (6), we define the scalar quantity,

where $\mathbf{\widehat{n}}$ is an arbitrary unit vector. Then, we define the inner product,

which allows to write $\langle f\rangle=\langle\mathbf{j}_{D},\mathbf{\widehat{n}}\sigma_{D}\rangle_{\sigma_{D}}$ and $\langle f^{2}\rangle_{c}=\int d\mathbf{r}\,\sigma_{D}=\langle\mathbf{\widehat{n}}\sigma_{D},\mathbf{\widehat{n}}\sigma_{D}\rangle_{\sigma_{D}}$ (see Appendix II).

The Cauchy-Schwarz relation associated to the inner product (13) implies the inequality

which together with $2\langle\mathbf{j}_{D},\mathbf{j}_{D}\rangle_{\sigma_{D}}=\langle\Sigma\rangle$ leads to the QMUR (1).

The linear dependence of $f$ upon $\mathbf{j}$ may appear restrictive. Yet, it corresponds to a wealth of physically relevant mesoscopic quantities often considered, e.g. the force induced by coherent light fluctuations recently studied Soret19 . A generalised expression (32), for $f$ and the QMUR will be proposed in section 5. We wish now to obtain an expression of the mesoscopic cost function $\Sigma$ at the stochastic level and not only as a disorder averaged quantity. It will allow to generalize the QMUR (1) and to include a corresponding large deviation bound.

The Langevin equation (6) allows for a stochastic coarse grained approach of quantum mesoscopics, obtained by associating to each realization of the noise $\boldsymbol{\xi}(\mathbf{r})$, a path $\Gamma=\{\mathbf{j}(\mathbf{r}),I(\mathbf{r})\}$ with a divergence free current $\nabla\cdot\mathbf{j}=0$ and appropriate boundary conditions ⁵⁵5$\nabla\cdot\mathbf{j}$ does not necessarily vanish on the boundary. (see appendix I). It would be tempting to identify the stochastic paths $\Gamma$ to the multiple scattering sequences ${\cal C}_{N}=(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N})$ defined in section 3.1. This identification does not hold since ${\cal C}_{N}$ are microscopic scattering sequences obtained from a formal expansion of the Green’s function of the Helmholtz equation (2) for a given disorder configuration, while paths $\Gamma=\{\mathbf{j}(\mathbf{r}),I(\mathbf{r})\}$ are generated by the local stochastic noise (7), associated to quantum crossings, and correspond to coarse grained trajectories.

It is useful to switch from the Langevin description to an equivalent statistical field theory. To that purpose, we employ the Martin-Siggia-Rose technique Martin73 ; tauber2014critical to express the probability $P(\Gamma)$ of a path as

The quadratic form of $\mathfrak{L}(\Gamma)$ results from the (multiplicative) white noise $\boldsymbol{\xi}$ and from $\nabla\cdot\mathbf{j}=0$ implicitly assumed in (15).

The path probability (15) (as well as the Langevin equation (6)) is valid to leading order in $g_{\mathcal{L}}^{-1}\ll 1$ ⁶⁶6The path probability (15) is exact to leading order if $g_{\mathcal{L}}\gg 1$. Otherwise, for $g_{\mathcal{L}}\sim 1$, subleading corrections to the quadratic $\mathfrak{L}(\Gamma)$ become relevant gardiner1985handbook ; Lubensky_fieldTheory . . Moreover, in that limit, the path probability is dominated by a saddle point solution, so that for any observable $O$

Using (15) and (16), it is now possible to define $\Sigma$ and show that $\langle\Sigma\rangle$ is given by (11).

We start by recalling some known results on the thermodynamic definition of the entropy production rate seifert2012stochastic ; van2015ensemble . Then, taking advantage of the analogy between non-equilibrium thermodynamics and quantum mesoscopics, we use the path probability to define the cost function.

Denoting by $P_{th}(\Gamma_{th})$ the path probability of a stochastic process, it is completely reversible if for any path $\Gamma_{th}$, the time-reversed path $\theta\Gamma_{th}$ is equally likely. With this intuition in mind, one can define the (dimensionless) entropy production variable

While $\Sigma_{th}$ can be negative, its average is non-negative,

a result which stems from the non-negativity of the integrand, i.e. $(x-y)\log\frac{x}{y}>0$ for $x,y>0$.

Analogously to (17), we define the cost function variable

where $\Theta\Gamma\equiv\{-\mathbf{j},I\}$ is the reversed path. For the path $\Gamma$ to exist with non-vanishing probability, it needs to correspond to some realization of the noise $\boldsymbol{\xi}$ satisfying (6), to hold the boundary conditions (see appendix I) and maintain $\nabla\cdot\mathbf{j}=0$. If $\Gamma$ exists, so does $\Theta\Gamma$: $\nabla\cdot(-\mathbf{j})$ vanishes, $\boldsymbol{\xi}\rightarrow\boldsymbol{\xi}-2\mathbf{j}$ satisfies (6) and the boundary conditions apply (see appendix I).

From the path probability (15) and the cost function variable (19) we find $\Sigma=-\int d\mathbf{r}\,\frac{2\mathbf{j}\cdot D\nabla I}{\sigma(I)}$. Using the saddle point approximation (16),

Therefore, the disorder averaged cost function variable corresponds to the cost function defined in (11).

While we have discussed so far the analogs between thermal and quantum mesoscopic systems, it is important to note that the underlying physics is quite different. For example, entropy production is notoriously hard to measure in thermal systems martinez2019inferring . Here, we want to argue that the cost function is accessible experimentally. To do so, note that $\langle\Sigma\rangle$ depends on $I_{D}$ alone (through $D,\sigma$). Since $I_{D}$ is a solution of a Laplace equation, it is completely determined by the boundary conditions and therefore so does $\langle\Sigma\rangle$. Let us express the relation of the cost function (20) to the boundary conditions;

with $c_{0}=\frac{2\pi lv^{2}}{3k^{2}}$. Further integration by parts yields

where $d\mathbf{S}=dS\,\mathbf{\widehat{n}}(\mathbf{r})$ with $\mathbf{\widehat{n}}(\mathbf{r})$ the normal vector to the infinitesimal surface $dS$ located at the point $\mathbf{r}$ on the boundary. The second term of the right hand side of (22) vanishes since $-\Delta I_{D}=0$. Let us rescale the surface integral in the right hand term of (22) by the characteristic length of the system $\mathcal{L}$, i.e. $\mathbf{\tilde{r}}=\mathbf{r}/\mathcal{L}$ and $\mathbf{\tilde{S}}=d\mathbf{S}/\mathcal{L}^{2}$. We find

Here $\mathcal{B}$ depends only on the boundary conditions. Note that $\Theta\Gamma$ is not a time reversed path. Indeed, in the case of a uniformly illuminated and symmetric sample, $I_{D}$ is uniform and therefore $\langle\Sigma\rangle=0$. However, quantum crossings still occur and TRS is still broken.

Having obtained expression (20) for the cost function before disorder averaging, by means of the path probability (15), we are now in a position to generalize the QMUR in (1) by relaxing the linear dependence of $f$ defined in (12). To that purpose, we consider the generalized expression

for $f$ with $z$ an arbitrary function. We wish to explore how the fluctuations of $f$ are bounded. To that end, we define the cumulant generating function of $f$,

Next, we derive in the spirit of Sasa18 , the QMUR and its generalization to the cumulant generating function. To do so, we consider another path probability defined with the tilted Lagrangian $\mathfrak{L}_{\mathbf{Y}}=(\mathbf{j}+D\nabla I-\mathbf{Y})^{2}/2\sigma$, where $\mathbf{Y}$ is a divergence free field. The tilted path probability corresponds to the Langevin dynamics $\mathbf{j}=-D\nabla I+\mathbf{Y}+\boldsymbol{\xi}$, with the same noise $\boldsymbol{\xi}$ defined in (6). The tilted process disorder average is denoted by $\langle\cdot\rangle_{\mathbf{Y}}$, such that $\langle\cdot\rangle_{\mathbf{0}}=\langle\cdot\rangle$. The usefulness of the tilted dynamics comes from the fact that under the tilted disorder average, the intensity remains unchanged, i.e. $\langle I\rangle_{\mathbf{Y}}=I_{D}$ for any divergence free field $\mathbf{Y}$, but the average current gets a tilt, i.e. $\langle j\rangle_{\mathbf{Y}}=\mathbf{j}_{D}+\mathbf{Y}$. This tilting dynamics has been used to create a mapping to equilibrium and to generate the time-reversed dynamics in the thermal case Sasa18 ; Shpielberg_Pal . Here we use it to optimize a bound on $\mu(\lambda)$. Using the identity

allows to rewrite the cumulant generating function $\mu(\lambda)$ as

The Jensen inequality, $\langle e^{O}\rangle_{\mathbf{Y}}\geq e^{\langle O\rangle_{\mathbf{Y}}}$, then implies

noting that the term $\frac{\mathbf{Y}}{\sigma}\cdot(\mathbf{j}+D\nabla I-\mathbf{Y})$ vanishes under the tilted disorder averaging. Choosing $\mathbf{Y}=\alpha\mathbf{j}_{D}$ and noting that the tilting field leaves the disorder averaged intensity $I_{D}$ unchanged, we find

From (28) and (29), we recover the cumulant generating function bound

This inequality is valid for any $\alpha$ and any choice of $f$. To recover the generalized QMUR, we assume $\alpha\ll 1$ and develop the right hand side of (30) to second order in $\alpha$. The quadratic expression can be optimized by $\alpha=2\lambda\langle\partial_{\mathbf{j}}f\rangle/\langle\Sigma\rangle$, where

Then, the inequality to second order in $\lambda$ implies the generalized QMUR

Using the optimal $\alpha$, we recover a large deviation bound for the fluctuations of $f$. Namely, using $\alpha=2\lambda\langle\partial_{\mathbf{j}}f\rangle/\langle\Sigma\rangle$ in (30). For example, the linear choice $z=\mathbf{j}\cdot\widehat{n}$ leads to

In this section, we have introduced the definition of the stochastic cost function $\Sigma$. We have also proved the QMUR for general fluctuating quantities $f$ in (32) and derived a large deviation bound (30) and (33). Note that $\langle\Sigma\rangle$ arises naturally in the optimization of the QMUR. This comes from the coarse grained level of the Langevin equation, leading to a quadratic Lagrangian $\mathfrak{L}$. We note that for a microscopic theory with non-quadratic Lagrangian, e.g. for a master equation, the optimal bound is much more cumbersome and currently lacks physical interpretation shiraishi2021optimal . We now apply the QMUR to some physically relevant examples.

First, we present a direct check of the generalised QMUR (32) for the transmission coefficient $T(\theta)$ – the ratio between the light intensity transmitted in the direction $\mathbf{\widehat{s}}$ (see Fig. 1) and the incoming intensity. The transmission coefficient and its fluctuations – which give rise to speckle patterns – have been extensively studied and measured Goodman ; Akkermans ; Boer92 ; Stephen87 . Deciphering the information encoded in fluctuations of the transmission coefficient is still an active field of research, with a broad range of applications such as imaging of biological tissues and turbid media, sensing and information transmission in random media Fayard18 ; Sarma19 . Remarkable progress has also been made recently in the ability to control the transmission of light in random media, with the emergence of wave front shaping techniques Mosk07 ; Rotter17 . In this context, providing a general bound for the fluctuations of $T(\theta)$ using the QMUR is of particular interest.

Let $I(\mathbf{\widehat{s},r})$ be the fraction of light intensity propagating in the direction $\mathbf{\widehat{s}}$. The transmission coefficient is then defined as $T(\theta)=\widehat{s}_{x}I(\mathbf{\widehat{s}},L)/I_{0}$, where $\theta\in[-\pi/2,\pi/2]$ is the angle between $\mathbf{\widehat{s}}$ and the $x$ axis. In the literature, $I(\mathbf{\widehat{s},r})$ is called the specific intensity Ishimaru ; Akkermans . Its angular average gives the total intensity, $I(\mathbf{r})=\overline{I(\mathbf{\widehat{s},r})}$. The specific intensity satisfies the radiative transfer equation. For details on how to obtain this equation, we refer the reader to the section A5.2 in Akkermans . In the absence of light source inside the medium, it is given by

We can therefore write $T(\theta)$ in the form

where $z(I,\mathbf{j})=\delta(x-L)\widehat{s}_{x}(I(x)+\frac{l}{D}\mathbf{\widehat{s}}\cdot\mathbf{j}(x))/I_{0}$, and use (32) to obtain

where the lower bound of the QMUR is given by

Using Fick’s law $\mathbf{j}_{D}=-D\nabla I_{D}$ and the solution to (53) for $I_{D}$ in a slab geometry, we obtain the expression of the lower bound, $\langle\partial_{\mathbf{j}}T(\theta)\rangle=\widehat{s}_{x}^{2}\frac{15(1-e^{-L/l})}{8\pi(u+2)}\simeq\widehat{s}_{x}^{2}\frac{15}{8\pi(u+2)}$ where $u=3L/2l$ and $\widehat{s}_{x}=\cos(\theta)$. The lower bound reaches its maximum for $\theta=0$. In the slab geometry, the correlation function of the transmission coefficient is given by (see section 12.4 in Akkermans )

Reinjecting this expression, together with the lower bound (38) and $\langle\Sigma\rangle=g_{\mathcal{L}}\frac{u^{2}}{(1+u)}$, in the QMUR (37), and rearranging the terms to separate those depending on $u$ and $s_{x}$, we find

For $u>0$ and $\theta\in[-\pi/2,\pi/2]$, we have

Hence (40) is always satisfied, and the QMUR (37) is indeed justified.

We now briefly discuss the recently studied radiative force induced by mesoscopic coherent fluctuations of light Soret19 . The radiative force exerted on a suspended membrane, of surface $\delta S$, immersed in the medium, see Fig. 1.b, is given by $\delta\mathbf{f}=\mathbf{\widehat{n}}\,v^{-2}\int_{\delta S}\mathbf{j}\cdot\mathbf{\widehat{n}}$ where $\mathbf{\widehat{n}}$ is a unit vector normal to $\delta S$ and $v$ is the group velocity. As a result of coherent effects described by quantum crossings, this force displays fluctuations, which typically scale like $\langle\delta f^{2}\rangle_{c}/\langle\delta f\rangle^{2}\sim 1/g_{\mathcal{L}}$ Soret19 . Since $g_{\mathcal{L}}$ is an easily tunable parameter, one can choose a setup where the fluctuations are measurable, and significantly enhanced compared to other forces exerted on the membrane, such as Van der Waals forces Soret19 . The spatially averaged force, $f_{av}=v^{-2}\int_{V}\mathbf{j}\cdot\mathbf{\widehat{n}}$, satisfies the QMUR (1). Indeed using $\langle f_{av}^{2}\rangle_{c}=v^{-4}\int_{V}\sigma_{D}(\mathbf{r})$ and the expressions for $I_{D}$ and $\mathbf{j}_{D}$ in a slab geometry, given earlier, we obtain

where again $u=3L/2l$. Equality in (42) is attained only in the nonphysical $u=0$ value; experimentally, it is reasonable to achieve $u\sim 10$, for which the ratio (42) is $\sim 4$. Indeed we find that the QMUR (42) is a good bound on the fluctuation induced force inside the slab.

Finally, we check numerically the inequality (32) for the radiative forces, $f_{av}=v^{-2}\int d\mathbf{r}\,\mathbf{j}(\mathbf{r})\cdot\mathbf{\widehat{n}}$. To compute (25), we use the fact that, since the noise is weak, the integrand on the r.h.s. of (25) dominated by the saddle point solution (58). We obtain the saddle point solution (58) numerically, and check that the cumulant generating function satisfies (32), see Fig. 2.

Let us define for the function $\zeta(\Gamma)$, $\psi(\theta)\equiv\langle\zeta(\Gamma)\rangle_{\theta}$. Furthermore, we define the inner product $\langle a,b\rangle_{\theta}=\int d\Gamma a(\Gamma)b(\Gamma)P_{\theta}(\Gamma)$. We notice that

Then, applying the Cauchy-Schwarz inequality, we find

Identifying

we recover (43).