Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Decoherence modeling of polarization mode dispersion for one- and two-photon states in optical fibers

Yiwen Liu and Yian Lei

Open Access Open Access

Abstract

We investigate the effects of polarization mode dispersion (PMD) on the propagation of single- and two-photon states in single-mode fibers. We look at PMD as decoherence due to the system's coupling with the bosonic environment involved in optical birefringence and a thermal bath, respectively. The evolution of the polarization density matrix is described by a master equation, and its solution is used to calculate the differential group delay (DGD) due to the fiber. The PMD effects on the dynamics of EPR states in optical fibers are investigated by using the same method. In both the short-time and long-time regimes, we find that PMD-induced and thermal-induced decoherence behave similarly.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Decoherence of open quantum systems is the main cause of errors in quantum communication and quantum computing experiments [1,2,3]. The phase and amplitude of superposition states are frequently degraded due to the unavoidable interaction between system and environment. In turn, most of the loss of information in optical communication systems may be assessed by proper modeling of decoherence.

In long-distance optical fiber transmission systems, polarization mode dispersion (PMD) is perhaps one of the most detrimental phenomena reducing the system performance [4,5,6]. PMD is mainly due to the random birefringence caused by imperfections and asymmetries of the structure and shape of the fiber. Birefringence leads to different group velocities for orthogonal polarization modes, resulting in different phases and arrival times. The time-delay difference between the slow and fast modes is usually referred to the differential group delay (DGD).

Mueller matrix and PMD vector dynamic equation are used to describe the statistical properties of the PMD vector [7]. On the other hands, a lot of studies have been conducted to characterize the effect of PMD on entangled states [8,9,10]. In general, these works have adopted a common theoretical model, which was first proposed by Phoenix S. Y. Poon and C. K. Law in 2011 [10,11,12]. This model uses the density matrix as a function of the arbitrary values of DGD to calculate the dynamics of decoherence between polarization-entangled photons. In the latest investigation, the approach has been able to simulate how PMD reduces entanglement with mode filtering element [13,14].

But there is still a large gap between the study of PMD-induced decoherence and other types of decoherence. First, the traditional master equation approach uses the evolution time as the parameter to describe the density matrix, rather than the values of DGD; Next, little attention has been paid to compare the dynamical process of PMD-induced decoherence with the influence of noise from the other sources, such as thermal noise.

In this paper, we propose a master equation approach to characterize PMD-induced decoherence, and quantitatively analyze its relevance in the dynamics of one- and two-photon polarization states in optical fibers. We compare the dephasing of polarization states due to distinct environments by splitting the degrees of freedom interacting with polarization into the “internal” and “external” environments.

The paper is structured as follows. In Section 2 we review the description of the coupling between polarization with the “internal” and the “external” environment, as well as the specific definitions of each, whereas in Section 3 and 4 we use a master equation for the polarization density matrix to describe the dynamics of single-photon states. In Section 5 we investigate the effects of polarization mode dispersion on two-photon entangled states. Section 6 closes the paper with some concluding remarks.

2. Coupling between polarization with the “internal” and the “external” environment

The closed quantum system represents an idealization. A realistic quantum system is always open, that is, it is coupled to another system, possibly multipartite, which is usually referred to as the environment (or the bath). The interaction with the environment leads to the loss of coherence and information [15]. A system initially prepared in a quantum superposition and then interacting with its environment can hardly maintain its coherence. If the system and the environment exchange energy, or just because of dephasing, the initial coherence superposition is converted into a mixed-state [16], with a rate that depends on the structure of the environment [17,18,19].

In many decoherence processes, the environment can be regarded as a bosonic bath of oscillators at thermal equilibrium (a zero-temperature environment may be often assumed, especially when working at optical frequencies) [20]. The Hamiltonian of a collection of non-interacting harmonic oscillators is as follows:

$${H_\textrm{b}}\textrm{ = }\sum\limits_i^\infty {\hbar {\omega _i}} {b_i}^ + {b_i}$$
$\textrm{b}_\textrm{k}^\textrm{ + }$, ${\textrm{b}_\textrm{k}}$ are the creation and annihilation operators of harmonic oscillators. The system-bath interaction describes the microscopic origin of the dephasing and dissipative phenomenon at hand, such as spontaneous decay of two-level atomic system, information loss in a quantum register, and collective decoherence of superconductivity quantum circuit. A general feature of this type of decoherence process is that the survival probability of the initial state decreases exponentially with time, and the spontaneous decay rate, $\gamma$, is a constant.

On the other hand, PMD is a unique kind of decoherence process. The whole decoherence process involves at least three interacting subsystems: the polarization state of the photon, which is the observable carrying quantum information, is taken as the reduced system S; the frequency degree of freedom of the photon is defined as the “internal” environment, and the thermal state of the bath, which contains an infinite number of degrees of freedom represents an external source of noise. In other words, this is an example where the interaction between the system and the bath does not only involve two or more particles but also involves two (or more) commuting observables of a single particle [21].

The coupling between the system with the “internal” and the “external” environment leads to non-dissipative decoherence, which introduces phase fluctuations among the components of any superposition of eigenstates [22]. We will use the master equation approach to characterize the phenomenon and demonstrate the peculiarities of this kind of decoherence process.

3. Master equation

The Hamiltonian of photons propagating in a single-mode fiber may be written as the sum of five terms [23]:

$$H = {H_0} + {H_I} = {H_\textrm{s}} + {H_{in}} + {H_{ex}} + {H_I}^1\textrm{ + }{H_I}^2$$
where
$$\begin{array}{c} {H_s} = \Delta {\sigma _z}, {H_{in}} = \hbar \omega \left( {{a^ + }a + \frac{1}{2}} \right), {H_{ex}}\textrm{ = }\sum\limits_k^N {\hbar {\omega _k}} \left( {{b_k}^ + {b_k} + \frac{1}{2}} \right),\\ {H_I}^1\textrm{ = }\hbar \chi ({{a^ + }\textrm{ + }a} ){\sigma _z},{H_I}^2\textrm{ = }\hbar {\sigma _z}\sum\limits_k^N {({{g_k}{b_k}^ +{+} {g_k}^\ast {b_k}} )} \end{array}$$
${H_s}$ is the Hamiltonian of a two-level system, which describes the polarization of the photon. ${H_{in}}$ is the Hamiltonian of the “internal” environment, while ${H_{ex}}$ is the Hamiltonian of the “external” environment. ${H_I}^1$, ${H_I}^2$ are the interaction Hamiltonian of the system with the “internal” and the “external” environment. ${\textrm{a}^ + }$, $\textrm{a}$, $\textrm{b}_\textrm{k}^\textrm{ + }$, ${\textrm{b}_\textrm{k}}$ are the creation and annihilation bosonic operators describing frequency mode. $\Delta$ is the the frequency difference between polarized states which is zero in this case.$\chi$ denotes the coupling constant between the polarization mode and the frequency mode in the birefringence effect, and ${g_k}$ denotes the coupling constant between the polarization mode and the thermal noise in the external field.

The frequency distribution of the internal and external environments differs significantly: in single-mode fiber, the frequency of propagating photons is limited to a fixed frequency ${\omega _0}$, while the thermal noise may be distributed in an infinite number of frequencies ${\omega _k}$. Thus, the Hamiltonian of the “internal” environment is a single oscillator. ${\sigma _z}$ denotes the third spin-1/2 Pauli operator:

$${\sigma _z} = |0 \rangle \left\langle 0 \right|- |1 \rangle \left\langle 1 \right|$$

$|0 \rangle$, $|1 \rangle$ denote the linearly polarized states parallel to or perpendicular to the birefringence axis. In the actual case, the birefringence axes are varied at different frequencies. But the main purpose of this study is to explore the relationship between decoherence and DGD. Thus, we can assume that the pulses in fiber have a small bandwidth and the birefringence axis will not vary.

In the interaction picture, the master equation describing the dynamics of the polarization density matrix is given by

$$\begin{aligned} \frac{{\textrm{d}{\rho _s}(t )}}{{\textrm{d}t}} &={-} \frac{i}{\hbar }T{r_{in}}[{{H_I}^1(t ),{\rho_{in}}(0 )\otimes {\rho_s}(t )} ]- \frac{i}{\hbar }T{r_{ex}}[{{H_I}^2(t ),{\rho_{ex}}(0 )\otimes {\rho_s}(t )} ]\\ &- \frac{1}{{{\hbar ^2}}}\int\limits_0^t \textrm{d} t^{\prime}T{r_{in}}[{{H_I}^1(t ),[{{H_I}^1({t^{\prime}} ),{\rho_{in}}(0 )\otimes {\rho_s}({t^{\prime}} )} ]} ]\\ &- \frac{1}{{{\hbar ^2}}}\int\limits_0^t \textrm{d} t^{\prime}T{r_{ex}}[{{H_I}^2(t ),[{{H_I}^2({t^{\prime}} ),{\rho_{ex}}(0 )\otimes {\rho_s}({t^{\prime}} )} ]} ]\end{aligned}$$
which may be rewritten as
$$\frac{{d{\rho _s}(t )}}{{dt}} ={-} i\alpha \left[ {\frac{{{\sigma_z}}}{2},{\rho_s}(t )} \right] - \int\limits_0^t {dt^{\prime}{f_1}({t^{\prime}} )\left[ {\frac{{{\sigma_z}}}{2},\left[ {\frac{{{\sigma_z}}}{2},{\rho_s}(t )} \right]} \right]} - \int\limits_0^t {dt^{\prime}{f_2}({t^{\prime}} )\left[ {\frac{{{\sigma_z}}}{2},\left[ {\frac{{{\sigma_z}}}{2},{\rho_s}(t )} \right]} \right]}$$

Equation (6) is derived from the Redfield equation, which is obtained by applying the Markov approximation to the original master equation and consists in assuming an instantaneous relaxation of the bath. The first term represents the unitary evolution generated by the interaction Hamiltonian. The second and the third ones are irreversible terms, which describe dephasing induced by birefringence and by the thermal bath.

The above master equation is not solvable for every possible forms of photon spectrum and thermal noise spectrum. Therefore, we need to restrict our calculations to specific spectral distributions. We can set the spectral density of the “internal” environment as a Lorentzian spectrum [24]:

$${J_1}(\omega )= \delta ({\omega - {\omega_0}} )\sim \frac{{{\gamma _1}}}{\pi }\frac{{\omega _\textrm{c}^2}}{{{{({\omega - {\omega_0}} )}^2} + \omega _\textrm{c}^2}}$$

We get the correlation function [25]:

$${f_1}({t^{\prime}} )= \int {d\omega } {J_1}(\omega )\textrm{exp} [{i({\omega - {\omega_0}} )t^{\prime}} ]= \frac{{{\gamma _1}}}{{{\tau _1}}}{e^{ - \frac{{t^{\prime}}}{{{\tau _1}}}}}$$

Also, we can set the spectral density of the “external” environment as an ohmic bath [18]

$${J_2}(\omega )= 4\sum\limits_k {{{|{{g_k}} |}^2}\delta ({\omega - {\omega_k}} )} \sim A\omega {e^{ - \frac{\omega }{\Omega }}}$$

We get the correlation function of the “external” environment:

$${f_2}({t^{\prime}} )= \int {d\omega } {J_2}(\omega )\cos ({\omega t^{\prime}} )\coth \left( {\frac{{\hbar \omega }}{{2{k_B}T}}} \right)$$
${\omega _c}$, $\Omega $ are the cut-off frequencies, as required in order to account for fall-off of the coupling at sufficiently high frequencies.

Let us now solve the master equation for a generic pure initial state, that is:

$${\rho _s}(0 )= {|{a|0 \rangle + b|1 \rangle } |^2} = \left( {\begin{array}{{cc}} {{{|a |}^2}}&{a{b^ \ast }}\\ {{a^ \ast }b}&{{{|b |}^2}} \end{array}} \right)$$

The solution of the master equation can be expressed as

$${\rho _s}(t )= \left( {\begin{array}{{cc}} {{{|a |}^2}}&{a{b^ \ast }{e^{i\alpha t - {\Gamma _1}(t )- {\Gamma _2}(t )}}}\\ {{a^ \ast }b{e^{ - i\alpha t - {\Gamma _1}(t )- {\Gamma _2}(t )}}}&{{{|b |}^2}} \end{array}} \right)$$
${\Gamma _1}(t )$ describes the contribution of birefringence environment to polarization decoherence in optical fibers:
$${\Gamma _1}(t )= {\gamma _1}\left( {t + {\tau_1}{e^{ - \frac{t}{{{\tau_1}}}}} - {\tau_1}} \right)$$
${\Gamma _2}(t )$ denotes the contribution of the vacuum fluctuations or the thermal states to decoherence. There is an exact analytic solution for ${\Gamma _2}(t )$[18]:
$$\begin{aligned} {\Gamma _2}(t ) &= \int\limits_0^\infty {d\omega {J_2}(\omega )} \coth \left( {\frac{\omega }{{2{k_B}T}}} \right)\frac{{1 - \cos \omega t}}{{{\omega ^2}}}\\ &= \frac{1}{2}\ln \left( {1 + \frac{{{t^2}}}{{{\tau_2}^2}}} \right) + \ln \left[ {\frac{{\sinh ({{\gamma_2}t} )}}{{{\gamma_2}t}}} \right] \end{aligned}$$
where ${\gamma _1}$ and ${\gamma _2}$ denote effective coupling constants and effective decay rates. The evolution of the density matrix confirms the dephasing nature (no dissipation) of the process since it reveals that the off-diagonal elements decay exponentially, while the diagonal elements do not change with time. When the decoherence time (propagation time) t is much longer than the correlation time ${\tau _1}$, ${\tau _2}$, the density matrix can be approximated as:
$${\rho _\textrm{s}}^\prime \textrm{ = }\left( {\begin{array}{{cc}} {{{|\textrm{a} |}^2}}&0\\ 0&{{{|b |}^2}} \end{array}} \right)$$

This means that there is no longer any coherence between the eigenstates of polarization modes, and the quantum superposition has been converted to the mixed state.

4. Evolution in different time regimes

There are three important time-scales in system-bath interaction: the correlation time of the bath ${\tau _1}$, ${\tau _2}$, which is connected to the spectral width

$${\tau _1} \approx {\omega _c}^{ - 1},{\tau _2} \approx {\Omega ^{ - 1}}$$
the relaxation time ${\tau _{R1}}$, ${\tau _{R2}}$, which depends on the coupling constant
$${\tau _{R1}} \approx \gamma _1^{ - 1},{\tau _{R2}} \approx \gamma _2^{ - 1}$$
and the evolution time of system t, which is also the propagation time of photons in fiber.

In different regimes of time, the dynamics of polarization states are varied.:

For $t \ll {\tau _1},{\tau _2}$, the magnitude of ${\Gamma _1}(t )$,${\Gamma _2}(t )$ increase with the square of t:

$${\Gamma _1}(t )\approx \frac{{{\gamma _1}}}{{2{\tau _1}}}{t^2},{\Gamma _2}(t )\approx \frac{1}{{2{\tau _2}^2}}{t^2}$$

For $t \gg {\tau _{R1}},{\tau _{R2}}$, the magnitude of ${\Gamma _1}(t )$,${\Gamma _2}(t )$ increases linearly with propagation time:

$${\Gamma _1}(t )\approx {\gamma _1}t,{\Gamma _2}(t )\approx {\gamma _2}t$$

Even though the sources of noise for the “internal” and “external” environments are different, we discover that the dephasing mechanism in optical fiber is nearly identical in the short- and long-time regimes. For small $t$, ${\Gamma _1}(t ), {\Gamma _2}(t )$ both decay as $\textrm{exp} ({ - {t^2}} )$, while in long-time regime the results always decay as $\textrm{exp} ({ - t} )$. The Markovian approximation cannot describe the dynamics for times shorter than the correlation time of the bath.

According to previous studies, we can set the coupling constant [24]:

$${\gamma _2} = 8A{k_B}T$$
$A$ is a dimensionless parameter characterizing the strength of the coupling between the polarization and the “external” environment at temperature T.

Next, when thermal noise is ignored and only the birefringence effect is considered, we can obtain the mean square DGD of the fiber $\left\langle {{\tau^2}} \right\rangle$, which can be defined as the group-delay difference between the slow and fast polarization modes $|0 \rangle$, $|1 \rangle$ as follows [26]

$$\left\langle {{\tau^2}} \right\rangle \propto {\Gamma _1}(t )$$

By using a stochastic equation approach [27,28], one may obtain another expression linking the DGD to the length of fiber L and the correlation length ${L_c}$, which has been verified by relevant experiments. This is given by

$$\left\langle {{\tau^2}} \right\rangle \textrm{ = }2{\left( {\frac{{\Delta n}}{c}{L_c}} \right)^2}\left( {\frac{L}{{{L_c}}} + {e^{ - \frac{L}{{{L_c}}}}} - 1} \right),\Delta n = {n_1} - {n_2}$$
where ${n_1}$ and ${n_2}$ are the effective refractive indices of slow and fast polarization modes in the fiber. Let us make links between $L$ and t, ${L_c}$ and ${\tau _1}$:
$$L = \frac{c}{{{n_0}}}t,{L_c} = \frac{c}{{{n_0}}}{\tau _1}$$
${n_0}$ is the effective refractive index in the propagation direction of the optical axis. Then by comparing Eq. (22) and ${\Gamma _1}(t )$, we just set the decay rate ${\gamma _1}$ as:
$${\gamma _1} = \frac{{{{({{n_1} - {n_2}} )}^2}}}{{{n_0}^2{\tau _1}}}$$
In photon transmission experiments, several decoherence mechanisms have been discovered. In addition to the decoherence caused by the birefringent environment, there is decoherence caused by the thermal state of the reservoir. The difference between the two decoherence mechanisms is attributable to the environmental noise: not only the spectral density but also the coupling strength is different. The coupling coefficient between the system and the thermal environment mostly depends on the temperature T, whereas the coupling coefficient between the polarization and the frequency modes is affected by both the correlation time of photons and the refractive index of the optical fiber.

There is a simple but imprecise way to understand the physical roots of the PMD effect: compare the magnitude of ${\gamma _1}$ and ${\gamma _2}$. Since ${\gamma _1}$ represents the strength of the birefringence effect and is independent of temperature, whereas ${\gamma _2}$ reflects the strength of the thermal noise effect and is mainly influenced by temperature. Therefore, it is possible to explore the magnitude of DGD error incurred by temperature drift to comprehend the main causes of the PMD effects. If temperature variations have little effect on DGD, the birefringence effect, rather than thermal noise, is the main source of PMD effects.

5. PMD effects on entangled two-photon states

In this section, we focus on the decoherence of entangled two-photon states induced by the coupling between the different degrees of freedom of photons and the thermal state of the bath at temperature T.

In experiments involving polarization-entangled states, the most common way to produce entangled photon pairs is spontaneous parametric down-conversion (SPDC), in which a pump light acts on a non-linear BBO crystal to produce a pair of polarization-entangled and frequency-entangled photons.

Conservation of energy imposes frequency anti-correlation for the entangled photons produced in a SPDC effect. If the frequency of the pump beam is $2\omega$, the frequencies of the two photons are given by:

$${\omega _L} = \omega + \varepsilon ,{\omega _R} = \omega - \varepsilon$$

The variable $\varepsilon$ makes the polarization-entangled photons distinguishable, and for the convenience of calculation, let us denote by ${\omega _L}$ the frequency of the photon emitted to the left of the fiber and by ${\omega _R}$ the frequency of the other one. A schematic diagram can be used to describe the interaction between the entangled state, “internal” environments (bath 1 and bath 2) and the “external” environments (thermal bath) (see Fig. 1).

 figure: Fig. 1.

Fig. 1. Schematic picture of the interaction between entangled states and the environment. Bath 1 and Bath 2 refer to the non-local frequency modes of entangled photons, and the thermal bath denotes the external environment with the similar temperature. $\sigma _z^L$, $\sigma _z^R$ denote the polarization modes of entangled photons transmitted to each end of the fiber.

Download Full Size | PDF

Next, the initial polarization state of photon pairs can be represented by a 4×4 density matrix:

$${\rho _s}(0 )= \left( {\begin{array}{{cccc}} {{\rho_{0000}}}&{{\rho_{0010}}}&{{\rho_{0001}}}&{{\rho_{0011}}}\\ {{\rho_{1000}}}&{{\rho_{1010}}}&{{\rho_{1001}}}&{{\rho_{1011}}}\\ {{\rho_{0100}}}&{{\rho_{0110}}}&{{\rho_{0101}}}&{{\rho_{0111}}}\\ {{\rho_{1100}}}&{{\rho_{1110}}}&{{\rho_{1101}}}&{{\rho_{1111}}} \end{array}} \right)$$
where the matrix elements are written as:
$${\rho _{klmn}} = \left\langle {kl} \right|{\rho _s}|{mn} \rangle ,k,l,m,n = 0,1$$

If we assume that the initial entangled state is one of the Bell states:

$${\phi _ \pm }\textrm{ = }{\textstyle{1 \over {\sqrt 2 }}}({|{00} \rangle \pm |{11} \rangle } ),{\varphi _ \pm } = {\textstyle{1 \over {\sqrt 2 }}}({|{01} \rangle \pm |{10} \rangle } )$$
the density matrix can be simplified as:
$${\rho _s}(0 )= \frac{1}{2}\left( {\begin{array}{{cccc}} 0&0&0&0\\ 0&1&{ \pm 1}&0\\ 0&{ \pm 1}&1&0\\ 0&0&0&0 \end{array}} \right)or\frac{1}{2}\left( {\begin{array}{{cccc}} 1&0&0&{ \pm 1}\\ 0&0&0&0\\ 0&0&0&0\\ { \pm 1}&0&0&1 \end{array}} \right)$$

The Hamiltonian describing the interaction of the two photons with the different degrees of freedom and the thermal noise in a birefringent environment may be written as:

$$\begin{array}{c} {H_I}\textrm{ = }\hbar \chi [{\sigma_z^L({a_L^ + \textrm{ + }{a_L}} )\textrm{ + }\sigma_z^R({a_R^ + \textrm{ + }{a_R}} )} ]\\ \textrm{ + }\hbar ({\sigma_z^L + \sigma_z^R} )\sum\limits_k^N {({{g_k}{b_k}^ +{+} {g_k}^\ast {b_k}} )} \end{array}$$
$a_L^ +$, ${a_L}$, $a_R^ +$, ${a_R}$ are the generation and annihilation bosonic operators of the two modes, which represent the different “internal” environments for photons. $\textrm{b}_\textrm{k}^\textrm{ + }$, ${\textrm{b}_\textrm{k}}$ are the bosonic operators of the “external” environment. $\sigma _z^L$, $\sigma _z^R$ are spin-1/2 Pauli z-operators. Considering that the structure of the fiber in the two paths is the same, we may safely assume that Pauli operators $\sigma _z^L$, $\sigma _z^R$ are expressed in the same basis.

Using the Born and Markov approximation, we obtain the following master equation for the reduced density matrix describing the two-photon polarization

$$\begin{aligned} \frac{{d{\rho _{klmn}}(t )}}{{dt}} &={-} i[{({k - m} ){\alpha_L} + ({l - n} ){\alpha_R}} ]{\rho _{klmn}}(t )\\ &- \left[ {{{({k - m} )}^2}\int\limits_0^t {d{t^\prime }{f_L}({t^{\prime}} )+ {{({l - n} )}^2}\int\limits_0^t {d{t^\prime }{f_R}({t^{\prime}} )} } } \right]{\rho _{klmn}}(t )\\ &- [{{{({k - m} )}^2} + {{({l - n} )}^2}} ]\int\limits_0^t {dt^{\prime}{f_2}({t^{\prime}} )} {\rho _{klmn}}(t )\end{aligned}$$

The solution of the master equation is given by

$$\begin{array}{l} {\rho _{klmn}}(t )= {\rho _{klmn}}{e^{ - \Gamma (t )}},\\ \Gamma (t )= i[{({k - m} ){\alpha_L} + ({l - n} ){\alpha_R}} ]t + {({k - m} )^2}{\Gamma _L}(t )+ {({l - n} )^2}{\Gamma _R}(t )\\ + [{{{({k - m} )}^2} + {{({l - n} )}^2}} ]{\Gamma _2}(t )\end{array}$$
where the decay function is given by
$$\begin{array}{l} {\Gamma _L}(t )= {\gamma _L}\left( {t + {\tau_L}{e^{ - \frac{t}{{{\tau_L}}}}} - {\tau_L}} \right),\\ {\Gamma _R}(t )= {\gamma _R}\left( {t + {\tau_R}{e^{ - \frac{t}{{{\tau_R}}}}} - {\tau_R}} \right),\\ {\Gamma _2}(t )\textrm{ = }\frac{1}{2}\ln \left( {1 + \frac{{{t^2}}}{{{\tau_2}^2}}} \right) + \ln \left[ {\frac{{\sinh ({{\gamma_2}t} )}}{{{\gamma_2}t}}} \right] \end{array}$$
where ${\gamma _L}$, ${\gamma _R}$ are the coupling coefficients, respectively, which are related to the optical properties of the fibers and the frequency of photons.

To assess and quantify decoherence in a birefringent environment, we use the concurrence [29], which can be obtained from the density matrix ${\rho _s}(t )$ as [30]

$$C = \max \left\{ {0,\sqrt {{k_1}} - \sqrt {{k_2}} - \sqrt {{k_3}} - \sqrt {{k_4}} } \right\}$$
where ${k_1}$, ${k_2}$, ${k_3}$, ${k_4}$ are the eigenvalues of the matrix:
$$\varsigma \textrm{ = }{\rho _s}(t )({\sigma_y^1 \otimes \sigma_y^2} ){\rho _s}^ \ast (t )({\sigma_y^1 \otimes \sigma_y^2} )$$
${\sigma _y}$ is the Pauli matrix of photons.

Assuming that the initial state is one of the pure entangled states

$${\phi _ \pm }\textrm{ = }{c_1}|{00} \rangle \pm {c_2}|{11} \rangle ,{\varphi _ \pm } = {c_1}|{01} \rangle \pm {c_2}|{10} \rangle ,{|{{c_1}} |^2} + {|{{c_2}} |^2} = 1$$
the concurrence as a function of time may be calculated as:
$$\begin{array}{l} {C_{{\phi _ \pm }}}(t )= \max \{{0,2|{{\rho_{0011}}(t )} |} \},\\ {C_{{\varphi _ \pm }}}(t )= \max \{{0,2|{{\rho_{1001}}(t )} |} \}\end{array}$$

Using the solution of the master equation, we obtain

$${C_{{\phi _ \pm }}}(t )= {C_{{\varphi _ \pm }}}(t )= 2|{{c_1}{c_2}} |{e^{ - ({{\Gamma _L}(t )+ {\Gamma _R}(t )} )- 2{\Gamma _2}(t )}}$$

Note that the concurrence indicates the magnitude of the entanglement degree, and the decoherence rate is independent of the initial state. The concurrence as a function of $|{{c_1}} |$ and of the dimensionless quantity ${\Gamma _L}(t )+ {\Gamma _R}(t )\textrm{ + }2{\Gamma _2}(t )$ is plotted in Fig. 2. The plot makes apparent that concurrence decreases with time (the sum of the Gammas is a monotonically increasing function of time which has been shown in Eq. (18) and Eq. (19)) and the initially entangled state eventually decays into a separable mixture. The quantity, ${\Gamma _L}(t )+ {\Gamma _R}(t )\textrm{ + }2{\Gamma _2}(t )$,which is called as the decoherence function, describes the behavior of the off-diagonals of the density matrix of entangled states. ${\Gamma _L}(t )$ and ${\Gamma _R}(t )$ indicate the destruction of entangled states by birefringent elements in different fiber paths, and ${\Gamma _2}(t )$ represents the decoherence caused by the common thermal environment which the entangled photons are coupled to.

 figure: Fig. 2.

Fig. 2. Concurrence for pure states ${\phi _ \pm }$ and ${\varphi _ \pm }$ as a function of $|{{c_1}} |$ and of the quantity ${\Gamma _L}(t )+ {\Gamma _R}(t )\textrm{ + }2{\Gamma _2}(t )$

Download Full Size | PDF

From the above results, we conclude the PMD causes decoherence due to the birefringence and thermal reservoir, and this effect may contribute to the loss of entanglement over long distances. As a result, entanglement is difficult to maintain in long-distance optical fiber communication, unless some compensation technique is used to correct polarization mode dispersion.

6. Conclusion

In conclusion, we have addressed polarization mode dispersion in optical fibers as a source of decoherence for polarization states. We have employed a master equation to describe the time evolution of the polarization density matrix assuming a coupling between polarization and frequency modes in a birefringent and thermal environment. We have investigated the resulting phase diffusion process and evaluated the corresponding dephasing rates. Eventually, we have analyzed the effects of polarization mode dispersion on the dynamics of polarization-entangled two-photon states and calculated the corresponding concurrence as a function of time.

Our results show that the dephasing mechanism caused by birefringence in optical fiber is similar to that originating from a thermal bath in both the short-time and the long-time regime. Although there are non-local correlations between entangled photons, the interaction of qubits between the environment is local.

Acknowledgments

We wish to express our gratitude to EditSprings for the expert linguistic services provided.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. H.M. Wiseman and G.J. Milburn, “Quantum Measurement and Control, “Cambridge University,(2014).

2. D.F. Walls and G. J. Milburn, “Quantum optics (2nd edition),” Springer-Verlag Berlin Heidelberg, (2008).

3. A Quantum Information Science and Technology Roadmap, Part 1: Quantum Computation, Section 6.9, LA-UR-04-1778(2004).

4. A. Galtarossa and C.R. Menyuk, “Polarization Mode Dispersion,” SpringerNew York, (2005).

5. W. Tittel, J. Brendel, T. Herzog, H. Zbinden, and N. Gisin, “Non-local two-photon correlations using interferometers physically separated by 35 meters,” Epl. 40(6), 595–600 (1997). [CrossRef]  

6. M. Shtaif and M. Boroditsky, “The Effect of the Frequency Dependence of PMD on the Performance of Optical Communications Systems,” IEEE Photonics Technol. Lett. 15(10), 1369–1371 (2003). [CrossRef]  

7. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U. S. A. (2000).

8. A. J. Berglund, “Quantum coherence and control in one- and two-photon optical systems,” Preprint arxiv: quant-ph/0010001v2 (2000).

9. R. T. Thew, S. Tanzilli, W. Tittel, H. Zbinden, and N. Gisin, “Experimental investigation of the robustness of partially entangled qubits over 11 km,” Phys. Rev. A 66(6), 062304 (2002). [CrossRef]  

10. C. Antonelli, M. Shtaif, and M. Brodsky, “Sudden Death of Entanglement Induced by Polarization Mode Dispersion,” Phys. Rev. Lett. 106(8), 080404 (2011). [CrossRef]  

11. M. Brodsky, E. C. George, C. Antonelli, and M. Shtaif, “Loss of polarization entanglement in a fiber-optic system with polarization mode dispersion in one optical path,” Opt. Lett. 36(1), 43 (2011). [CrossRef]  

12. M. Shtaif, C. Antonelli, and M. Brodsky, “Nonlocal compensation of polarization mode dispersion in the transmission of polarization entangled photons,” Opt. Express 19(3), 1728 (2011). [CrossRef]  

13. D. E. Jones, B. T. Kirby, G. Riccardi, C. Antonelli, and M. Brodsky, “Exploring classical correlations in noise to recover quantum information using local filtering,” New J. Phys. 22(7), 073037 (2020). [CrossRef]  

14. G. Riccardi, C. Antonelli, D. E. Jones, and M. Brodsky, “Simultaneous Decoherence and Mode Filtering in Quantum Channels: Theory and Experiment,” Phys. Rev. App. 15(1), 014060 (2021). [CrossRef]  

15. N. Tang, G. Y. Wang, Z. L. Fan, and H. S. Zeng, “Non-Markovian Dynamics of an Open Two-Level System with Amplitude-Phase Damping,” Journal of Quantum Information Science 03(01), 27–33 (2013). [CrossRef]  

16. P. Zanardi, “Dissipative dynamics in a quantum register,” Phys. Rev. A 56(6), 4445–4451 (1997). [CrossRef]  

17. D. Tamascelli, C. Benedetti, H.P. Breuer, and M.G. A. Paris, “Quantum probing beyond pure dephasing,” New J. Phys. 22(8), 083027 (2020). [CrossRef]  

18. D. Crow and R. Joynt, “Classical simulation of quantum dephasing and depolarizing noise,” Phys. Rev. A 89(4), 042123 (2014). [CrossRef]  

19. H. P. Breuer and F. Petruccione, “The Theory of Open Quantum Systems,” Oxford University, (2006).

20. C. W. Gardiner and P. Zoller, “Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics,” Springer, Berlin, (2004).

21. P. Knight, “Quantum mechanics: Where the weirdness comes from,” Nature 395(6697), 12–13 (1998). [CrossRef]  

22. W. G. Unruh, “Maintaining Coherence in Quantum Computers,” Phys. Rev. A 51(2), 992–997 (1995). [CrossRef]  

23. J. S. Xu, C. F. Li, M. Gong, M. Gong, X. B. Guo, C. H. Shi, G. Chen, and G. C. Guo, “Experimental Demonstration of Photonic Entanglement Collapse and Revival,” Phys. Rev. Lett. 104(10), 100502 (2010). [CrossRef]  

24. A. Shabani and D. A. Lidar, “Completely positive post-Markovian master equation via a measurement approach,” Phys. Rev. A 71(2), 020101 (2005). [CrossRef]  

25. S. Maniscalco and F. Petruccione, “Non-Markovian dynamics of a qubit,” Phys. Rev. A 73(1), 012111 (2006). [CrossRef]  

26. S. Salemian and S. Mohammadnejad, “Analysis of polarization mode dispersion effect on quantum state decoherence in fiber-based optical quantum communication,” International Conference on Telecommunications. IEEE, 2011.

27. J. P. Gordon, “Statistical properties of polarization mode dispersion,” J. Opt. Fiber. Commun. Rep 1(3), 210–217 (2004). [CrossRef]  

28. L. E. Nelson and R. M. Jopson, “Introduction to polarization mode dispersion in optical systems,” J. Opt. Fiber. Commun. Rep 1(4), 312–344 (2004). [CrossRef]  

29. B. Bellomo, R. L. Franco, and G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99(16), 160502 (2007). [CrossRef]  

30. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998). [CrossRef]  

Data availability

No data were generated or analyzed in the presented research.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)
Fig. 1.
Fig. 1. Schematic picture of the interaction between entangled states and the environment. Bath 1 and Bath 2 refer to the non-local frequency modes of entangled photons, and the thermal bath denotes the external environment with the similar temperature. $\sigma _z^L$, $\sigma _z^R$ denote the polarization modes of entangled photons transmitted to each end of the fiber.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Concurrence for pure states ${\phi _ \pm }$ and ${\varphi _ \pm }$ as a function of $|{{c_1}} |$ and of the quantity ${\Gamma _L}(t )+ {\Gamma _R}(t )\textrm{ + }2{\Gamma _2}(t )$

View in Article | Download Full Size | PDF

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

H b  =  ∑ i ∞ ℏ ω i b i + b i
H = H 0 + H I = H s + H i n + H e x + H I 1  +  H I 2
H s = Δ σ z , H i n = ℏ ω ( a + a + 1 2 ) , H e x  =  ∑ k N ℏ ω k ( b k + b k + 1 2 ) , H I 1  =  ℏ χ ( a +  +  a ) σ z , H I 2  =  ℏ σ z ∑ k N ( g k b k + + g k ∗ b k )
σ z = | 0 ⟩ ⟨ 0 | − | 1 ⟩ ⟨ 1 |
d ρ s ( t ) d t = − i ℏ T r i n [ H I 1 ( t ) , ρ i n ( 0 ) ⊗ ρ s ( t ) ] − i ℏ T r e x [ H I 2 ( t ) , ρ e x ( 0 ) ⊗ ρ s ( t ) ] − 1 ℏ 2 ∫ 0 t d t ′ T r i n [ H I 1 ( t ) , [ H I 1 ( t ′ ) , ρ i n ( 0 ) ⊗ ρ s ( t ′ ) ] ] − 1 ℏ 2 ∫ 0 t d t ′ T r e x [ H I 2 ( t ) , [ H I 2 ( t ′ ) , ρ e x ( 0 ) ⊗ ρ s ( t ′ ) ] ]
d ρ s ( t ) d t = − i α [ σ z 2 , ρ s ( t ) ] − ∫ 0 t d t ′ f 1 ( t ′ ) [ σ z 2 , [ σ z 2 , ρ s ( t ) ] ] − ∫ 0 t d t ′ f 2 ( t ′ ) [ σ z 2 , [ σ z 2 , ρ s ( t ) ] ]
J 1 ( ω ) = δ ( ω − ω 0 ) ∼ γ 1 π ω c 2 ( ω − ω 0 ) 2 + ω c 2
f 1 ( t ′ ) = ∫ d ω J 1 ( ω ) exp [ i ( ω − ω 0 ) t ′ ] = γ 1 τ 1 e − t ′ τ 1
J 2 ( ω ) = 4 ∑ k | g k | 2 δ ( ω − ω k ) ∼ A ω e − ω Ω
f 2 ( t ′ ) = ∫ d ω J 2 ( ω ) cos ⁡ ( ω t ′ ) coth ⁡ ( ℏ ω 2 k B T )
ρ s ( 0 ) = | a | 0 ⟩ + b | 1 ⟩ | 2 = ( | a | 2 a b ∗ a ∗ b | b | 2 )
ρ s ( t ) = ( | a | 2 a b ∗ e i α t − Γ 1 ( t ) − Γ 2 ( t ) a ∗ b e − i α t − Γ 1 ( t ) − Γ 2 ( t ) | b | 2 )
Γ 1 ( t ) = γ 1 ( t + τ 1 e − t τ 1 − τ 1 )
Γ 2 ( t ) = ∫ 0 ∞ d ω J 2 ( ω ) coth ⁡ ( ω 2 k B T ) 1 − cos ⁡ ω t ω 2 = 1 2 ln ⁡ ( 1 + t 2 τ 2 2 ) + ln ⁡ [ sinh ⁡ ( γ 2 t ) γ 2 t ]
ρ s ′  =  ( | a | 2 0 0 | b | 2 )
τ 1 ≈ ω c − 1 , τ 2 ≈ Ω − 1
τ R 1 ≈ γ 1 − 1 , τ R 2 ≈ γ 2 − 1
Γ 1 ( t ) ≈ γ 1 2 τ 1 t 2 , Γ 2 ( t ) ≈ 1 2 τ 2 2 t 2
Γ 1 ( t ) ≈ γ 1 t , Γ 2 ( t ) ≈ γ 2 t
γ 2 = 8 A k B T
⟨ τ 2 ⟩ ∝ Γ 1 ( t )
⟨ τ 2 ⟩  =  2 ( Δ n c L c ) 2 ( L L c + e − L L c − 1 ) , Δ n = n 1 − n 2
L = c n 0 t , L c = c n 0 τ 1
γ 1 = ( n 1 − n 2 ) 2 n 0 2 τ 1
ω L = ω + ε , ω R = ω − ε
ρ s ( 0 ) = ( ρ 0000 ρ 0010 ρ 0001 ρ 0011 ρ 1000 ρ 1010 ρ 1001 ρ 1011 ρ 0100 ρ 0110 ρ 0101 ρ 0111 ρ 1100 ρ 1110 ρ 1101 ρ 1111 )
ρ k l m n = ⟨ k l | ρ s | m n ⟩ , k , l , m , n = 0 , 1
ϕ ±  =  1 2 ( | 00 ⟩ ± | 11 ⟩ ) , φ ± = 1 2 ( | 01 ⟩ ± | 10 ⟩ )
ρ s ( 0 ) = 1 2 ( 0 0 0 0 0 1 ± 1 0 0 ± 1 1 0 0 0 0 0 ) o r 1 2 ( 1 0 0 ± 1 0 0 0 0 0 0 0 0 ± 1 0 0 1 )
H I  =  ℏ χ [ σ z L ( a L +  +  a L )  +  σ z R ( a R +  +  a R ) ]  +  ℏ ( σ z L + σ z R ) ∑ k N ( g k b k + + g k ∗ b k )
d ρ k l m n ( t ) d t = − i [ ( k − m ) α L + ( l − n ) α R ] ρ k l m n ( t ) − [ ( k − m ) 2 ∫ 0 t d t ′ f L ( t ′ ) + ( l − n ) 2 ∫ 0 t d t ′ f R ( t ′ ) ] ρ k l m n ( t ) − [ ( k − m ) 2 + ( l − n ) 2 ] ∫ 0 t d t ′ f 2 ( t ′ ) ρ k l m n ( t )
ρ k l m n ( t ) = ρ k l m n e − Γ ( t ) , Γ ( t ) = i [ ( k − m ) α L + ( l − n ) α R ] t + ( k − m ) 2 Γ L ( t ) + ( l − n ) 2 Γ R ( t ) + [ ( k − m ) 2 + ( l − n ) 2 ] Γ 2 ( t )
Γ L ( t ) = γ L ( t + τ L e − t τ L − τ L ) , Γ R ( t ) = γ R ( t + τ R e − t τ R − τ R ) , Γ 2 ( t )  =  1 2 ln ⁡ ( 1 + t 2 τ 2 2 ) + ln ⁡ [ sinh ⁡ ( γ 2 t ) γ 2 t ]
C = max { 0 , k 1 − k 2 − k 3 − k 4 }
ς  =  ρ s ( t ) ( σ y 1 ⊗ σ y 2 ) ρ s ∗ ( t ) ( σ y 1 ⊗ σ y 2 )
ϕ ±  =  c 1 | 00 ⟩ ± c 2 | 11 ⟩ , φ ± = c 1 | 01 ⟩ ± c 2 | 10 ⟩ , | c 1 | 2 + | c 2 | 2 = 1
C ϕ ± ( t ) = max { 0 , 2 | ρ 0011 ( t ) | } , C φ ± ( t ) = max { 0 , 2 | ρ 1001 ( t ) | }
C ϕ ± ( t ) = C φ ± ( t ) = 2 | c 1 c 2 | e − ( Γ L ( t ) + Γ R ( t ) ) − 2 Γ 2 ( t )
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use