Photon Flux and Distance from the Source: Consequences for Quantum Communication

Abstract

The paper explores the fundamental physical principles of quantum mechanics (in fact, quantum field theory) that limit the bit rate for long distances and examines the assumption used in this exploration that losses can be ignored. Propagation of photons in optical fibers is modelled using methods of quantum electrodynamics. We define the “photon duration” as the standard deviation of the photon arrival time; we find its asymptotics for long distances and then obtain the main result of the paper: the linear dependence of photon duration on the distance when losses can be ignored. This effect puts the limit to joint increasing of the photon flux and the distance from the source and it has consequences for quantum communication. Once quantum communication develops into a real technology (including essential decrease of losses in optical fibres), it would be appealing to engineers to increase both the photon flux and the distance. And here our “photon flux/distance effect” has to be taken into account. This effect also may set an additional constraint to the performance of a loophole free test of Bell’s type—to close jointly the detection and locality loopholes.

Appendices

Appendix 1: The Modal Functions

Details are given in this appendix for the (two-layered) fibre modal functions based on an analysis of the quantization of the $n$-layered fibre with losses [32]. The electrical modal function is $\varvec{\psi }_{mk}(\varvec{r})=\varvec{\psi }_{mk}(\rho )\text {exp}[\text {i}(kz+m\varphi )]$, with

$$\begin{aligned} \varvec{\psi }_{mk}(\rho )=\left\{ \begin{array}{l} a_{1mk}\varvec{v}_{1mk}(\rho )+a_{2mk}\varvec{v}_{2mk}(\rho ),\rho <a\\ b_{1mk}\varvec{w}_{1mk}(\rho )+b_{2mk}\varvec{w}_{2mk}(\rho ),\rho >a \end{array}\right. . \end{aligned}$$

(28)

The radial basis functions $\varvec{v}_{1mk}(\rho ),\varvec{v}_{2mk} (\rho ),\varvec{w}_{1mk}(\rho ),$ and $\varvec{w}_{2mk}(\rho )$ are expressed in terms of the Bessel function J$_{m}$ of order $m$ and the Modified Bessel function K$_{m}$ of the second kind and order $m$ via

$$\begin{aligned} \left\{ \begin{array}{l} \varvec{v}_{1mk}(\rho )=\hat{\varvec{\rho }}\frac{\text {i}m}{\kappa \rho }\text {J} _{m}(\kappa \rho )-\hat{\varvec{\varphi }}\text {J}_{m}^{\prime }(\kappa \rho ),\\ \varvec{v}_{2mk}(\rho )=\hat{\varvec{\rho }}\frac{\text {i}k}{k_{0}}\text {J}_{m}^{\prime }(\kappa \rho )-\hat{\varvec{\varphi }}\frac{mk}{k_{0}\kappa \rho }\text {J}_{m}(\kappa \rho )+\hat{\varvec{z}}\frac{\kappa }{k_{0}}\text {J}_{m}(\kappa \rho ), \end{array}\right. \end{aligned}$$

(29)

and

$$\begin{aligned} \left\{ \begin{array}{l} \varvec{w}_{1mk}(\rho )=\hat{\varvec{\rho }}\frac{\text {i}m}{q\rho }\text {K}_{m} (q\rho )-\hat{\varvec{\varphi }}\text {K}_{m}^{\prime }(q\rho ),\\ \varvec{w}_{2mk}(\rho )=\hat{\varvec{\rho }}\frac{\text {i}k}{k_{0}}\text {K}_{m}^{\prime }(q\rho )-\hat{\varvec{\varphi }}\frac{mk}{k_{0}q\rho }\text {K}_{m}(q\rho )-\hat{\varvec{z}}\frac{q}{k_{0}}\text {K}_{m}(q\rho ). \end{array} \right. \end{aligned}$$

(30)

The expansion coefficients $a_{1mk},a_{2mk},b_{1mk},$ and $b_{2mk}$ with the modal frequency $\omega _{mk}$ solving the dispersion relation $G_{m} (\omega _{mk},k)=0$ are determined to meet the following conditions: $\nabla \times \nabla \times \varvec{\psi }_{mk}(\varvec{r})=k_{0}^{2}\epsilon (\rho ) \varvec{\psi }_{mk}(\varvec{r})$ and $\nabla \times \varvec{\psi }_{mk}(\varvec{r})=0$ when $\rho \ne a,$ continuity of the tangential components of $\varvec{E}=\varvec{\psi }_{mk}$ and $\varvec{H}=\left( \text {i}\omega _{mk}\sqrt{\epsilon _{0}\mu _{0}}\right) ^{-1}\nabla \times \varvec{E},$ regularity at $\rho =0,$ and the Silver–Müller radiation condition at infinity. $\omega _{mk}$ should be used for $\omega$ in $k_0,\kappa , q, \epsilon _1$ and $\epsilon _2$. Uniqueness for the coefficients is reached by the normalization condition

$$\begin{aligned} \int _{\mathbb {R}^{3}}\epsilon (\rho )\varvec{\psi }_{mk}(\varvec{r})\cdot \varvec{\psi }_{m^{\prime }k^{\prime }}^{*}(\varvec{r})\text {d}V=2\pi \delta _{mm^{\prime }}\delta (k-k^{\prime }), \end{aligned}$$

(31)

or equivalently

$$\begin{aligned} 2\pi \int _{0}^{2\pi }\rho \text {d} \rho \,\epsilon (\rho )\varvec{\psi }_{mk}(\rho )\cdot \varvec{\psi }_{mk}^{*}(\rho )=1. \end{aligned}$$

(32)

Here $\epsilon (\rho )=\epsilon _{1}$ for $\rho <a$ and $\epsilon _{2}$ for $\rho >a.$ Note that the modal functions do not form a basis since the non-discrete contributions are omitted.

Appendix 2: Derivation of the Asymptotic Results

The asymptotic results for the moments in (18) are based on two asymptotic theorems that are presented next. The first theorem is [40]

Theorem 1

Let $\varphi (t)$ be sectionally continuous in $(0,\infty )$ except at a finite number of critical points. Then

$$\begin{aligned} \int _{0}^{\infty }\text {e}^{\text {i}vx}\varphi (v)\text {d}v=o(1), \quad \text { when }x\rightarrow \infty , \end{aligned}$$

if the integral converges uniformly at $0,\infty$ and the critical points for all $x$ that are large enough.

A variant of Theorem 1 is

Theorem 2

(Riemann–Lesbesgue’s lemma) Let $\varphi \in \mathrm{L}^1$, then

$$\begin{aligned} \int _{0}^{\infty }\text {e}^{\text {i}vx}\varphi (v)\text {d}v=o(1), \quad \text { when }x\rightarrow \infty . \end{aligned}$$

Based on Theorem 1 the following main asymptotic theorem is proved.

Theorem 3

Let $\phi (v)=\frac{\text {d}}{\text {d}v}\left[ \ln v\,\varphi (v)\right]$ and assume that

• (i) $\varphi (v)$ and $\varphi ^{\prime }(v)$ are continuous in a right neigbourhood of $0$

• (ii) $\varphi (v)$ is sectionally continuous in $[0,\infty )$ except at a finite number of critical points

• (iii) $I(x)=\int _{0}^{\infty }$e$^{\text {i}vx}\ln v\,\varphi (v)$d$v$ converges

• (iv) $\int _{0}^{\infty }$e$^{\text {i}vx}\,\phi (v)$d$v$ converges uniformly at $0,\infty$ and the critical points.

Then

$$\begin{aligned} I(x)=\dfrac{-\varphi (0)\text {i}}{x}\left[ \ln x+\gamma -\text {i}\pi /2\right] +o(1/x), \quad x\rightarrow \infty . \end{aligned}$$

(33)

Comment 1

A sufficient condition for (iv) to hold is that the integral converges absolutely. However, this condition is not necessary.

Proof

Let $\varphi _{0}(v)=\varphi (0)$e$^{-v},\xi (v)=\varphi (v)-\varphi _{0}(v),$ and $I_{0}(x)=\int _{0}^{\infty }$e$^{\text {i}vx}\ln v\,\varphi _{0}(v)$d$v.$ The variable transformation $xv=t$ and the Laplace transform with $s=1/x-\text {i}$ then yields

$$\begin{aligned} I_{0}(x)&=\frac{\varphi (0)}{x}\int _{0}^{\infty }e^{-(1/x-\text {i})t}\ln t\,dt-\frac{\varphi (0)\ln x}{x}\int _{0}^{\infty }e^{-(1/x-\text {i} )t}\,dt\end{aligned}$$

(34)

$$\begin{aligned}&=\frac{-\varphi (0)}{1-\text {i} x}\left[ \gamma +\ln (1/x-\text {i})+\ln x\right] . \end{aligned}$$

(35)

Now,

$$\begin{aligned} I(x)-I_{0}(x)&=\int _{0}^{\infty }e^{\text {i}vx}\ln v\,\xi (v)\text {d} v\end{aligned}$$

(36)

$$\begin{aligned}&=\frac{\text {i}}{x}\int _{0}^{\infty }e^{\text {i}vx}\frac{\text {d}}{\text {d} v}\left[ \ln v\,\xi (v)\right] \text {d}v, \end{aligned}$$

(37)

due to (iv) and that $\xi (v)$ is continuous and vanishes at $v=0.$ From Theorem 1 follows that $I(x)-I_{0}(x)=$ o$(1/x),$ when $x\rightarrow \infty$ and an asymptotic expansion of $I_{0}(x)$ gives (33) ending the proof of the theorem. $\square$

Finally in this appendix, we present some details for the derivation of (17, 18). Based on (15),

$$\begin{aligned} \left\{ \begin{array}{l} \tau _{nm}(z)=\tau _{nm}^\mathrm{P}(z)+\tau _{nm}^{\delta }(z)\\ \tau _{nm}^{\delta }(z)=\tau _{nm}^{\delta +}(z)+\tau _{nm}^{\delta -}(z) \end{array} \right. , \end{aligned}$$

(38)

is written as the sum of three terms: the first term $\tau _{nm}^\mathrm{P}(z)$ originates from the Cauchy principle value integration based on the first term in (15), the other two terms $\tau _{nm}^{\delta \pm }(z).$ come from integration of $\delta ^{(n)}(\omega _{mk^{\prime }\varepsilon }-\omega _{mk\varepsilon })$ giving contributions at $k=\pm k^{\prime }$ that is indicated in the super index $^{\delta \pm }$.

Starting with $\tau _{nm}^{\delta }(z)$, it is first observed that $\tau _{1m}^{\delta }(z)$, being one half of the value of (15), vanishes. This can be seen by extending the integration in $t$ in (15) from $(0,\infty )$ to $(-\infty ,\infty ).$ By changing the sign of the integration variables $t, k$ and $k'$ and utilizing the reality condition $f_{m,-k\varepsilon }(\rho )=f_{mk\varepsilon }^{*}(\rho ),$ it is found that $\tau _{1m}^{\delta }(z)=-(\tau _{1m}^{\delta }(z))^*$ that vanishes being real by construction.

Thus, only $n=0$ and $2$ are required for $\tau _{nm}^{\delta }(z)$ and the following general result is used; see [41] for similar results. Let $f(x)$ have simple zeros at $x_{m},$ $m=1,2,$ $f^{\prime }(x_{m})\ne 0$ and $f^{\prime \prime }(x)$ as well as $f^{\prime \prime \prime }(x)$ be continuous in a surrounding of $x_{m}.$ Then, we have

$$\begin{aligned} \left\{ \begin{array}{l} \delta \left\{ f(x)\right\} =\sum \limits _{m=1,2}\frac{\delta (x-x_{m} )}{\left| f^{\prime }(x_{m})\right| }\\ \delta ^{\prime \prime }\left\{ f(x)\right\} =\sum \limits _{m=1,2}\frac{1}{\left| f^{\prime }(x_{m})\right| ^{3}}\left\{ \delta ^{\prime \prime }(x-x_{m})\right. \\ -\left. \frac{3f^{\prime \prime }(x_{m})\delta ^{\prime }(x-x_{m} )}{f^{\prime }(x_{m})}+\frac{\delta (x-x_{m})}{f^{\prime }(x_{m})}\left[ \frac{3\left[ f^{\prime \prime }(x_{m})\right] ^{2}}{f^{\prime }(x_{m} )}-f^{\prime \prime \prime }(x_{m})\right] \right\} \end{array} \right. \end{aligned}$$

(39)

The second observation is that the oscillating factor $\mathrm{exp}[\text {i}(k-k')z]$ in (13) is, due to (36), present in $\tau _{nm}^{\delta +}(z)$ but not in $\tau _{nm}^{\delta -}(z)$. It is therefore expected from Riemann–Lebesgue’s theorem that $\tau _{nm}^{\delta -}(z)$ does not contribute to the dominating behaviour of $\tau _{nm}(z)$. Confirming details for this supposition are, however, not presented.

Using the first equation in (39), we can perform the $k^{\prime }$ integration in (13) to get $\tau _{0m}^{\delta +}(z)=\tau _{0m}^{\sim }$ and using the supposition above we find that $\tau _{0m}^{\delta }(z)=\tau _{0m}^{\sim }+\mathrm{o}(1),z\rightarrow \infty$.

For $\tau _{2m}^{\delta }(z)$ we have that the dominating contribution comes from the first term in the second equation in (36) with $k'=k$. The result is that $\tau _{2m}^{\delta }(z)=\tau _{2m}^{~}z^2+\mathrm{o}(z^2)$. This ends the asymptotic calculations for $\tau _{nm}^{\delta }(z)$.

Next $\tau _{nm}^\mathrm{P}(z)$ is on the cards. After transforming variables in (13) using

$$\begin{aligned} \left\{ \begin{array}{l} \xi =k+k^{\prime }\\ \eta =k-k^{\prime } \end{array} \right. , \end{aligned}$$

(40)

the result is

$$\begin{aligned} \tau _{nm}^\mathrm{P}(z)=\lim _{\varepsilon \rightarrow 0}\frac{n!}{2\text {i}^{n+1}}\int \!\!\!\int _{\mathbb {R}^{2}}\hbox {d} \xi \,\hbox {d}\eta \frac{G_{nm\varepsilon }(\xi ,\eta )}{(\xi \eta )^{n+1}}\text {e}^{\text {i}\eta z}, \end{aligned}$$

(41)

where

$$\begin{aligned} G_{nm\varepsilon }(\xi ,\eta )=\frac{2\pi }{\left[ F_{m\varepsilon }(\frac{\xi +\eta }{2},\frac{\xi -\eta }{2})\right] ^{n+1}}\int _{0}^{a}\rho \,\text {d}\rho f_{m\frac{\xi +\eta }{2}\varepsilon } (\rho )f_{m\frac{\xi -\eta }{2}\varepsilon }^{*}(\rho ). \end{aligned}$$

(42)

The Cauchy principle integrals for $\xi$ and $\eta$ in (41) are calculated using the regularization scheme [41]

$$\begin{aligned} \int _{-\infty }^{\infty }x^{-n-1}\gamma (x)\text {d}x=\frac{-1}{n!}\int _{-\infty }^{\infty }\gamma ^{(n+1)}\ln |x|\text {d}x, \quad n=0,1,2, \end{aligned}$$

(43)

with the result

$$\begin{aligned} \left\{ \begin{array}{l} \tau _{nm}^\mathrm{P}(z)=\lim _{\varepsilon \rightarrow 0}\dfrac{1}{2\text {i}^{n+1}n!}\int _{\mathbb {R}}\,\hbox {d}\xi \ln |\xi |\frac{\partial ^{n+1} J_{nm\varepsilon }(\xi ,z)}{\partial \xi ^{n+1}}\\ J_{nm\varepsilon }(\xi ,z)=\int _{\mathbb {R}}\,\hbox {d}\eta \ln |\eta |\frac{\partial ^{n+1} }{\partial \eta ^{n+1}}\left[ G_{nm\varepsilon }(\xi ,\eta )\text {e}^{\text {i}\eta z}\right] \end{array} \right. . \end{aligned}$$

(44)

The dominating contribution $J_{nm\varepsilon }^{\,\mathrm {dom}}(\xi ,z)$ to $J_{nm\varepsilon }(\xi ,z)$ comes at $\eta =0$ in (44) making $J_{nm\varepsilon }^{\,\mathrm {dom}}(\cdot ,z)$ an even function for $n$ even. The contribution from $J_{nm\varepsilon }^{\,\mathrm {dom}}(\xi ,z)$ to $\tau _{nm}^\mathrm{P}(z)$ is therefore vanishing when $n$ is even. As a result, $\tau _{nm}^\mathrm{P}(z)=\mathrm{o}(\tau _{nm}^{\delta }(z)),z\rightarrow \infty$ for even $n$. The proof of this statement is omitted.

In contrast, $J_{1m\varepsilon }(\xi ,z)$ provides the dominating to the asymptotic behaviour of $\tau _{1m}^{\,\mathrm P}(z)$ and then also $\tau _{nm}(z)$.

With

$$\begin{aligned} \frac{\partial ^2}{\partial \eta ^2}\left[ G_{1m\varepsilon }(\xi ,\eta )\text {e}^{\text {i}\eta z}\right] \!=\! \text {e}^{\text {i} \eta z}\left[ \frac{\partial ^2 }{\partial \eta ^2}G_{1m\varepsilon }(\xi ,\eta )\!+\!2\text {i} z\frac{\partial }{\partial \eta }G_{1m\varepsilon }(\xi ,\eta )\!-\!z^2G_{1m\varepsilon }(\xi ,\eta )\right] , \end{aligned}$$

(45)

it is found that the last term provides the dominating contribution to $\tau _{nm}(z)$. To this end, we define

$$\begin{aligned} J_{nm\varepsilon }^{\,\mathrm{dom}}(\xi ,z)=-z^2 \int _{\mathbb {R}}\,\hbox {d}\eta \ln |\eta | G_{1m\varepsilon }(\xi ,\eta )\text {e}^{\text {i}\eta z}, \end{aligned}$$

(46)

which due to that $G_{1m\varepsilon }(\xi ,-\eta )=G_{1m\varepsilon }^*(\xi ,\eta )$, is

$$\begin{aligned} J_{1m\varepsilon }^{\,\mathrm{dom}}(\xi ,z)=-2z^2\mathrm{Re }\int _{0}^{\infty }\,\hbox {d}\eta \ln {\eta } G_{1m\varepsilon }(\xi ,\eta )\text {e}^{\text {i}\eta z}+o(1), \quad z\rightarrow \infty . \end{aligned}$$

(47)

An application of Theorem 3 then yields that

$$\begin{aligned} J_{1m\varepsilon }(\xi ,z)=J_{1m\varepsilon }^{\,\mathrm{dom}}(\xi ,z)+\mathrm{o}(z)=\pi zG_{1m\varepsilon }(\xi ,0)+\mathrm{o}(z). \end{aligned}$$

(48)

Assume for the asymptotic analysis for large $z$ that Riemann–Lebesgue’s lemma can be used on $\varphi (\cdot )=\ln |\cdot |\frac{\partial ^{2}}{\partial \cdot ^{2}} G_{1m\varepsilon }(\xi ,\cdot )$ and Theorem 3 on $\varphi (\cdot )=G_{1m\varepsilon }(\xi ,\cdot ).$ Note that this puts requirements on the initial state $g_{m\cdot }$ in (11) rather than on $\omega _{m\cdot \varepsilon }$ and $\psi _{m\cdot }$ that are fixed functions.