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

Characterization of PT-symmetric quantum interference based on the coupled mode theory

Junhe Zhou

Open Access Open Access

Abstract

In this paper, we propose a comprehensive quantum theoretical framework to formulate the quantum interference inside the parity-time (PT) symmetric waveguide system which is formed by two coupled optical waveguides with unequal losses. Based on the theory, the expression for the well-known Hong-Ou-Mandel (HOM) dip is derived, which is in an exact agreement with the published results. What’s more, a novel one-photon quantum interference phenomenon is predicted according to the model, which suggests a quantum interference process similar to the HOM effect can be observed for the one-photon state, while the other photon is lost due to the waveguide attenuation. Such phenomenon cannot occur in a Hermitian system or in the system formed by the waveguides with equal losses.

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

1. Introduction

Optical waveguides have been regarded as one of the most commonly used structures to realize various photonics passive and active functional devices, particularly to manipulate the quantum states of photons [1–7]. Theories to formulate the wave propagation inside the optical waveguides have been a mature research topic which has been built upon the Maxwell equations [8] and the quantum optics framework [9–13]. Most of the quantum theories have focused on the single mode waveguides, which treat the loss as the coupling between the propagation mode and the modes in the reservoir. A. Luks et. al. [10], J. Liñares et. al. [11–12] and D. Barral [13] proposed the multimode waveguide quantum wave propagation theory, which includes the constant coupling between the different modes and the linear and nonlinear losses [10,13]. In [14], two coupled single mode waveguides with equal losses are considered for the quantum wave propagation, within which the loss does not affect the Hermitian property of the coupling matrix despite a common attenuation term along the propagation distance.

Recently, quantum wave propagation inside the coupled waveguides with unequal losses has raised significant attentions. Such coupled waveguides have remarkable features including the parity time (PT) symmetric property [5]. PT symmetric systems have the real eigen values despite the non-Hermitian nature of the operator [15], which has triggered tremendous research efforts in the field, both in quantum optics [5–6] and in classical optics [16–22]. While abundant mathematical tools have presented for classical optics [16–22], the related theory in quantum optics remains scarce. In [5], a Lie algebra approach was suggested to tackle the problem, it is, however, relatively complex and difficult to be scaled for the waveguide array with more coupled waveguides. The complex nature of the PT theory in [5] also hinders the efforts to gain a deep physical insight into the entangled photon pairs. For example, the research on the entangled photon pairs focusses on the interference between the two indistinguishable photons, particularly the existence of the Hong-Ou-Mandel (HOM) effect [5]. Up to date, no studies have been reported on what happens after one of the photons is lost due to the attenuation in the PT symmetric system.

In this work, a coupled mode theory approach with the introduction of the Langevin noise is proposed to offer a flexible tool to formulate the quantum interference inside the coupled optical waveguides with unequal losses. The two-photon interference inside the coupled waveguides will result in the well-known Hong-Ou-Mandel (HOM) effect, which is caused by the interference between two indistinguishable photons. A close form expression for the HOM dip can be derived from the theory, which is in an exact agreement with the published results [5], and this demonstrates the validity of the theory.

What’s more, a new phenomenon is predicted by the theory, which indicates that interference can also occur for the one photon quantum state with the other photon lost due to the waveguide attenuation in the PT symmetric systems. Although only one photon retains, it is contributed by the two photons injected into the system before one of them is lost. The interference between the different portions of the one-photon state is caused by the correlated Langevin noise terms, which suggests a dip/peak similar to the HOM dip/peak [23–24] can be expected while conducting the measurements. Such a phenomenon cannot be observed in the Hermitian systems or in the systems formed by the waveguides with equal losses, because the correlation between the Langevin noise terms is zero and the interaction term vanishes accordingly.

2. Coupled mode theory for the coupled waveguides with unequal losses

The Hamiltonian operator in two coupled optical waveguides under the interaction picture can be formulated by [5]:

$$M = \hbar ({\kappa {{\hat{a}}_1}^\dagger {{\hat{a}}_2} + \kappa {{\hat{a}}_2}^\dagger {{\hat{a}}_1}} ),$$
where $\hbar$ is the plank constant, κ is the coupling coefficient between the two waveguides, ${\hat{a}_m}$ and ${\hat{a}_m}^\dagger$ are the annihilation and creation operators of the two waveguides, with m indicating the waveguide number. Under the Heisenberg picture and in the absence of waveguide loss, we have the following coupled mode equation [5],
$$\frac{\partial }{{\partial z}}{\hat{a}_m} = \frac{1}{{i\hbar }}[{{{\hat{a}}_m},M} ].$$

Equation (2) is in accordance with the coupled mode equation derived from the coupled mode theory [8]

$$\begin{array}{l} \frac{{d{{\hat{a}}_1}(z )}}{{dz}} ={-} i\kappa {{\hat{a}}_2}(z ),\\ \frac{{d{{\hat{a}}_2}(z )}}{{dz}} ={-} i\kappa {{\hat{a}}_1}(z ). \end{array}$$

By introducing the loss terms in the coupled mode equation, one has [9,25]

$$\begin{array}{l} \frac{{d{{\hat{a}}_1}(z )}}{{dz}} ={-} i\kappa {{\hat{a}}_2}(z )- {\gamma _1}{{\hat{a}}_1}(z )+ {{\hat{f}}_1}(z ),\\ \frac{{d{{\hat{a}}_2}(z )}}{{dz}} ={-} i\kappa {{\hat{a}}_1}(z )- {\gamma _2}{{\hat{a}}_1}(z )+ {{\hat{f}}_2}(z ),\\{[{{{\hat{f}}_m}(z ),{{\hat{f}}_{m^{\prime}}}^\dagger ({z^{\prime}} )} ]}= 2{\delta _{mm^{\prime}}}{\gamma _m}\delta ({z - z^{\prime}} ),\\ \left\langle {{f_m}^\dagger (z ){f_m}({z^{\prime}} )} \right\rangle = 2{{\bar{n}}_\sigma }{\gamma _m}\delta ({z - z^{\prime}} ),\\ \left\langle {{f_m}(z ){f_m}^\dagger ({z^{\prime}} )} \right\rangle = 2({1 + {{\bar{n}}_\sigma }} ){\gamma _m}\delta ({z - z^{\prime}} ), \end{array}$$
where γm is the loss term on the mth waveguide, fm is Langevin noise term respectively, ${\bar{n}_\sigma }$ is the mode density [9,25]:
$${\bar{n}_\sigma } = \frac{1}{{\exp \left( {\frac{{h\nu }}{{{k_B}T}}} \right) - 1}} \approx 0.$$
which is close to zero at the room temperature [25]. The noise term is introduced to ensure the commutation relation to hold along the propagation distance [9,25]. Equation (4) can be analytically solved as:
$$\left( {\begin{array}{{c}} {{{\hat{a}}_1}(z )}\\ {{{\hat{a}}_2}(z )} \end{array}} \right) = \exp \left( { - \left( {\frac{{{\gamma_1} + {\gamma_2}}}{2}} \right)z} \right)\left( {\begin{array}{{cc}} {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )}&{ - \frac{{i\kappa }}{\lambda }\sin ({\lambda z} )}\\ { - \frac{{i\kappa }}{\lambda }\sin ({\lambda z} )}&{\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \end{array}} \right)\left( {\begin{array}{{c}} {{{\hat{a}}_1}(0 )}\\ {{{\hat{a}}_2}(0 )} \end{array}} \right) + {\mathbf F}(z ),$$
with:
$$\begin{array}{l} \gamma = \frac{{{\gamma _2} - {\gamma _1}}}{2},\\ \lambda = \sqrt {{\kappa ^2} - {\gamma ^2}} ,\\ {\mathbf F}(z )= \left( {\begin{array}{{c}} {{F_1}(z )}\\ {{F_2}(z )} \end{array}} \right) = \exp \left( { - \left( {\frac{{{\gamma_1} + {\gamma_2}}}{2}} \right)z} \right)\\ \times \int_0^z {\exp \left( {\left( {\frac{{{\gamma_1} + {\gamma_2}}}{2}} \right)s} \right)\left( {\begin{array}{@{}cc@{}} {\cos ({\lambda ({z - s} )} )+ \frac{\gamma }{\lambda }\sin ({\lambda ({z - s} )} )}&{ - \frac{{i\kappa }}{\lambda }\sin ({\lambda ({z - s} )} )}\\ { - \frac{{i\kappa }}{\lambda }\sin ({\lambda ({z - s} )} )}&{\cos ({\lambda ({z - s} )} )- \frac{\gamma }{\lambda }\sin ({\lambda ({z - s} )} )} \end{array}} \right)\left( {\begin{array}{@{}c@{}} {{{\hat{f}}_1}(s )}\\ {{{\hat{f}}_2}(s )} \end{array}} \right)ds} . \end{array}$$

The correlation matrix of the noise can be evaluated as

$$\begin{array}{l} \left\langle {{\mathbf F}(z ){{\mathbf F}^H}(z )} \right\rangle = \left( {\begin{array}{{cc}} {\left\langle {{{|{{F_1}(z )} |}^2}} \right\rangle }&{\left\langle {{F_1}(z ){F_2}^\ast (z )} \right\rangle }\\ {\left\langle {{F_1}{{(z )}^\ast }{F_2}(z )} \right\rangle }&{\left\langle {{{|{{F_2}(z )} |}^2}} \right\rangle } \end{array}} \right)\\ = 2\int_0^z {\exp ({ - ({{\gamma_1} + {\gamma_2}} )s} ){\mathbf A}(s )} \left( {\begin{array}{{cc}} {{\gamma_1}} & {}\\{}& {{\gamma_2}} \end{array}} \right){{\mathbf A}^H}(s )ds,\\ {\mathbf A}(s )= \left( {\begin{array}{{cc}} {\cos ({\lambda s} )+ \frac{\gamma }{\lambda }\sin ({\lambda s} )}&{ - \frac{{i\kappa }}{\lambda }\sin ({\lambda s} )}\\ { - \frac{{i\kappa }}{\lambda }\sin ({\lambda s} )}&{\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda s} )} \end{array}} \right). \end{array}$$

It can be seen from the equation that when the losses are equal, the correlation matrix becomes a diagonal matrix. When the losses are not equal, the off-diagonal terms are non-zero.

3. Analysis of PT symmetric quantum entanglement

Considering the propagation of state $|{1,1} \rangle$ in the waveguides, which suggests a two-photon pair is injected into the coupled waveguides,

$$|{1,1} \rangle = \hat{a}_1^\dagger \hat{a}_2^\dagger |{0,0} \rangle .$$

According to Eq. (6), with the evolution of the creation operators along the propagation distance, the state becomes:

$$\hat{a}_1^\dagger (z )\hat{a}_2^\dagger (z )|{0,0} \rangle = {\beta _{|{2,0} \rangle }}|{2,0} \rangle + {\beta _{|{0,2} \rangle }}|{0,2} \rangle + {\beta _{|{1,1} \rangle }}|{1,1} \rangle + {\beta _{|{1,0} \rangle }}|{1,0} \rangle + {\beta _{|{0,1} \rangle }}|{0,1} \rangle + {\beta _{|{0,0} \rangle }}|{0,0} \rangle ,$$
with the coefficients as:
$$\begin{array}{l} {\beta _{|{2,0} \rangle }} = \sqrt 2 \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)\frac{{i\kappa }}{\lambda }\sin ({\lambda z} ),\\ {\beta _{|{0,2} \rangle }} = \sqrt 2 \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)\frac{{i\kappa }}{\lambda }\sin ({\lambda z} ),\\ {\beta _{|{1,1} \rangle }} = \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\left( {\left( {\cos {{({\lambda z} )}^2} - \frac{{{\gamma^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right) - \frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right),\\ {\beta _{|{1,0} \rangle }} = \exp \left( { - \left( {\frac{{{\gamma_1} + {\gamma_2}}}{2}} \right)z} \right)\left( {\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right){F_2}^\dagger + \frac{{i\kappa }}{\lambda }\sin ({\lambda z} ){F_1}^\dagger } \right),\\ {\beta _{|{0,1} \rangle }} = \exp \left( { - \left( {\frac{{{\gamma_1} + {\gamma_2}}}{2}} \right)z} \right)\left( {\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right){F_1}^\dagger + \frac{{i\kappa }}{\lambda }\sin ({\lambda z} ){F_2}^\dagger } \right),\\ {\beta _{|{0,0} \rangle }} = {F_1}^\dagger {F_2}^\dagger . \end{array}$$

The probabilities to detect the related quantum states are:

$$\begin{array}{l} {p_{|{2,0} \rangle }} = 2\exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)^2}\frac{{{\kappa ^2}}}{{{\lambda ^2}}}\sin {({\lambda z} )^2},\\ {p_{|{0,2} \rangle }} = 2\exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)^2}\frac{{{\kappa ^2}}}{{{\lambda ^2}}}\sin {({\lambda z} )^2},\\ {p_{|{1,1} \rangle }} = \exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\frac{{{\kappa^2}\cos ({2\lambda z} )- {\gamma^2}}}{{{\lambda^2}}}} \right)^2},\\ {p_{|{1,0} \rangle }} = \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\\ \times \left( {{\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)}^2}\left\langle {{F_2}{F_2}^\dagger } \right\rangle + \frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}\left\langle {{F_1}{F_1}^\dagger } \right\rangle \right.\\ \left.+ 2Re \left( { - \left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)\frac{{i\kappa }}{\lambda }\sin ({\lambda z} )\left\langle {{F_1}{F_2}^\dagger } \right\rangle } \right) \right),\\ {p_{|{0,1} \rangle }} = \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\\ \times \left( \frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}\left\langle {{F_2}{F_2}^\dagger } \right\rangle + {{\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)}^2}\left\langle {{F_1}{F_1}^\dagger } \right\rangle \right.\\ \left.+ 2Re \left( { - \frac{{i\kappa }}{\lambda }\sin ({\lambda z} )\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)\left\langle {{F_2}{F_1}^\dagger } \right\rangle } \right) \right),\\ {p_{|{0,0} \rangle }} = \left( {\left\langle {{F_1}{F_1}^\dagger } \right\rangle \left\langle {{F_2}{F_2}^\dagger } \right\rangle + \left\langle {{F_2}{F_1}^\dagger } \right\rangle \left\langle {{F_1}{F_2}^\dagger } \right\rangle } \right). \end{array}$$

The HOM dip is

$${p_{|{1,1} \rangle }} = \exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\frac{{{\kappa^2}\cos ({2\lambda z} )- {\gamma^2}}}{{{\lambda^2}}}} \right)^2},$$
which is in an exact agreement with the results in [5], despite some definition differences. The visibility is defined as $\frac{{{p_{|{1,1} \rangle }}}}{{{p_{|{1,1} \rangle d}}}} - 1$, where ${p_{|{1,1} \rangle d}}$ stands for the probability in case the two photons are distinguishable. For the sake of clarity and completeness, the expressions for the cases of distinguishable photons are listed below:
$$\begin{array}{l} {p_{|{2,0} \rangle d}} = 2\exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)^2}\frac{{{\kappa ^2}}}{{{\lambda ^2}}}\sin {({\lambda z} )^2},\\ {p_{|{0,2} \rangle d}} = 2\exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} ){\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)^2}\frac{{{\kappa ^2}}}{{{\lambda ^2}}}\sin {({\lambda z} )^2},\\ {p_{|{1,1} \rangle d}} = \exp ({ - 2({{\gamma_1} + {\gamma_2}} )z} )\left( {{{\left( {\cos {{({\lambda z} )}^2} - \frac{{{\gamma^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)}^2} + {{\left( {\frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)}^2}} \right),\\ {p_{|{1,0} \rangle d}} = \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\left( {{{\left( {\cos ({\lambda z} )+ \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)}^2}\left\langle {{F_2}{F_2}^\dagger } \right\rangle + \frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}\left\langle {{F_1}{F_1}^\dagger } \right\rangle } \right),\\ {p_{|{0,1} \rangle d}} = \exp ({ - ({{\gamma_1} + {\gamma_2}} )z} )\left( {\frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}\left\langle {{F_2}{F_2}^\dagger } \right\rangle + {{\left( {\cos ({\lambda z} )- \frac{\gamma }{\lambda }\sin ({\lambda z} )} \right)}^2}\left\langle {{F_1}{F_1}^\dagger } \right\rangle } \right),\\ {p_{|{0,0} \rangle d}} = \left\langle {{F_1}{F_1}^\dagger } \right\rangle \left\langle {{F_2}{F_2}^\dagger } \right\rangle . \end{array}$$

The visibility $\frac{{{p_{|{1,1} \rangle }}}}{{{p_{|{1,1} \rangle d}}}} - 1$ can be calculated as

$$\frac{{ - 2\left( {\cos {{({\lambda z} )}^2} - \frac{{{\gamma^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)\left( {\frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)}}{{{{\left( {\cos {{({\lambda z} )}^2} - \frac{{{\gamma^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)}^2} + {{\left( {\frac{{{\kappa^2}}}{{{\lambda^2}}}\sin {{({\lambda z} )}^2}} \right)}^2}}}.$$

Considering the evolution of the state $|{1,1} \rangle$ in the two coupled waveguides with the coupling coefficient κ=0.25 cm-1, the losses on waveguides γ1 = 0 cm-1 and γ2 = 0.35 cm-1[5], the coincidence to observe the two photons simultaneously at the output of the two waveguides is shown as the curves in Fig. 1(a). When the two photons are indistinguishable, the curve for the PT symmetric waveguides system is plotted as the blue solid line, and curve for the Hermitian system (with the losses as 0) is the plotted as red dashed line. When the two photons are distinguishable, the curves are plotted as the cyan dash-dotted line and the green dotted line respectively. It can be observed from the figure that the interference (the difference between the distinguishable and indistinguishable photons) does exist, which leads to the visibility of the HOM effect varying along the propagation distance as is shown in Fig. 1(b). In Fig. 1(b), the deepest HOM dip shifts slightly from 3cm in the Hermitian system to about 2.8cm in the PT symmetric system, which is in accordance with the results in [5].

 figure: Fig. 1.

Fig. 1. the coincidence and the visibility for the two-photon state varying with respect to propagation distance. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs.

Download Full Size | PDF

A new phenomenon can be predicted by analyzing ${p_{|{1,0} \rangle }}$ and ${p_{|{0,1} \rangle }}$. The third term in the 4th and 5th expressions of Eq. (12) suggests that interference can occur in case the two photons are indistinguishable. The curves are shown in Fig. 2. Figure 2(a) shows the probability evolution along the propagation distance, with the blue solid line and the red dashed line indicating the probabilities of the two different single-photon states formed by the indistinguishable two-photon pair with one of the photons lost, i.e., $|{1,0} \rangle$ and $|{0,1} \rangle$. The solid line is higher because the loss in the first waveguide is 0 and the second waveguide has the non-zero loss. The distinguishable photon pair results are plotted as the cyan dash-dotted and green dotted lines. The difference between the probabilities for the distinguishable and indistinguishable input photon pairs gives rise to the visibilities in Fig. 2(b), which is similar to the HOM effect for the two-photon pairs. As is shown in Fig. 2(b), state $|{1,0} \rangle$ will have the interference dip, while state $|{0,1} \rangle$ will have the interference peak. This suggests that the quantum interference can be measured by observing only one photon counter at the output of waveguide 1 or waveguide 2. The existence of a dip or a peak will suggest a quantum interference has occurred.

 figure: Fig. 2.

Fig. 2. Interference of the one-photon state with the other photon lost due to the waveguide attenuation. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs. State $|{1,0} \rangle$ has the interference dip, while state $|{0,1} \rangle$ has the interference peak.

Download Full Size | PDF

The one photon interference phenomenon can be explained by the fact that the one-photon state is actually formed by the two-photon pair at the different input of the waveguides. Due to the existence of the waveguide loss in the PT symmetric system, one photon is lost and the Langevin noise term replaces it. However, it still remains unknown which photon is lost, i.e., whether it is from waveguide 1 or waveguide 2. The combination of the two contributions could be in phase or out of phase and thus leads to the quantum interference. Such a phenomenon cannot be observed in the Hermitian system or in the waveguides with equal losses, because the correlation between the Langevin noise term is zero in these cases and the interaction term (third term in the 4th and 5th expressions of Eq. (12)) vanishes accordingly.

The evolutionary behaviors of the probabilities ${p_{|{2,0} \rangle }}$, ${p_{|{0,2} \rangle }}$ and ${p_{|{0,0} \rangle }}$ along the propagation distance are shown in Fig. 3. As the propagation distance increases, the photons tend to be lost and the probability for the state of $|{0,0} \rangle$ increases. The summation of all the probabilities (including the probabilities shown in Figs. 1–3) equals 1, which is shown as the black dot line, i.e,

$${p_{|{2,0} \rangle }} + {p_{|{0,2} \rangle }} + {p_{|{1,1} \rangle }} + {p_{|{1,0} \rangle }} + {p_{|{0,1} \rangle }} + {p_{|{0,0} \rangle }} = 1.$$

 figure: Fig. 3.

Fig. 3. The evolution behaviors of the probabilities ${p_{|{2,0} \rangle }}$, ${p_{|{0,2} \rangle }}$ and ${p_{|{0,0} \rangle }}$ along the propagation distance.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. The coincidence and the visibility for the two-photon state varying with respect to propagation distance in two lossy waveguides with γ1 = 0.15 cm-1 and γ2 = 0.5 cm-1. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs.

Download Full Size | PDF

The results suggest that the proposed theory is self-consistent.

Finally, the validity of the theory is further checked by analyzing two lossy waveguides with γ1 = 0.15 cm-1 and γ2 = 0.5 cm-1. According to the previously presented theory, the coincidence of the photon pair should decrease in comparison to the one in Fig. 1, however, the visibility should maintain unchanged. This is because Eq. (15) suggests that the visibility is dependent on the loss difference between the two waveguides, which remains 0.35 cm-1. The above mentioned facts are shown in Fig. 4.

4. Summary

In summary, we have derived a comprehensive theory for the quantum propagation of the light inside two coupled waveguides with unequal radiation losses, which can be referred to as the PT symmetric system. The derived results for the HOM effect are in an exact agreement with the published results. In addition to that, a quantum phenomenon is predicted, which indicates that the one-photon state can also have the interference phenomenon. The interference comes from the contribution of the two-photon pair before one of them is lost due to the unequal waveguide losses in the PT symmetric system.

Funding

National Natural Science Foundation of China (61775168).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. G. Moody, V. Sorger, P. Juodawlkis, W. Loh, C. Sorace-Agaskar, A. E. Jones, K. Balram, J. Matthews, A. Laing, M. Davanco, L. Chang, J. Bowers, N. Quack, C. Galland, I. Aharonovich, M. Wolff, C. Schuck, N. Sinclair, M. Loncar, T. Komljenovic, D. M. Weld, S. Mookherjea, S. Buckley, M. Radulaski, S. Reitzenstein, G. S. Agarwal, B. Pingault, B. Machielse, D. Mukhopadhyay, A. V. Akimov, A. Zheltikov, K. Srinivasan, W. Jiang, T. McKenna, J. Lu, H. Tang, A. H. Safavi-Naeini, S. Steinhauer, A. Elshaari, V. Zwiller, P. Davids, N. Martinez, M. Gehl, J. Chiaverini, K. Mehta, J. Romero, N. Lingaraju, A. M. Weiner, D. Peace, R. Cernansky, M. Lobino, E. Diamanti, R. Camacho, and L. Trigo Vidarte, “Roadmap on integrated quantum photonics,” JPhys Photonics 4(1), 012501 (2022). [CrossRef]  

2. E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature 469(7331), 512–515 (2011). [CrossRef]  

3. A. Javadi, I. Söllner, M. Arcari, S. Lindskov Hansen, L. Midolo, S. Mahmoodian, G. Kiršanskė, T. Pregnolato, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Lodahl “Single-photon non-linear optics with a quantum dot in a waveguide,” Nat. Commun. 6(1), 8655 (2015). [CrossRef]  

4. T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, “Frequency-domain Hong-Ou-Mandel interference,” Nat. Photonics 10(7), 441–444 (2016). [CrossRef]  

5. F. Klauck, L. Teuber, M. Ornigotti, M. Heinrich, S. Scheel, and A. Szameit, “Observation of PT-symmetric quantum interference,” Nat. Photonics 13(12), 883–887 (2019). [CrossRef]  

6. Y. Qin, H. Chen, D. Luo, C. Pan, H. Hu, Y. Zhang, and D. Wei, “Quantum interference in anti-parity-time symmetric coupled waveguide system,” Opt. Express 29(18), 29175–29185 (2021). [CrossRef]  

7. M. Ehrhardt, M. Heinrich, and A. Szameit, “Observation-dependent suppression and enhancement of two-photon coincidences by tailored losses,” Nat. Photonics 16(3), 191–195 (2022). [CrossRef]  

8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

9. M. O. Scully and M. S. Zubairy,. Quantum Optics (Cambridge University, Cambridge, 2001).

10. A. Luks and V. Perinova, “Canonical quantum description of light propagation in dielectric media,” Progress in Optics 43, Chap. 4, 295–431(Elsevier Science, New York, 2002).

11. J. Liñares and M. C. Nistal, “Quantization of coupled modes propagation in integrated optical waveguides,” J. Mod. Opt. 50(5), 781–790 (2003). [CrossRef]  

12. J. Liñares, M. C. Nistal, and D. Barral, “Quantization of coupled 1D vector modes in integrated photonic waveguides,” New J. Phys. 10(6), 063023 (2008). [CrossRef]  

13. D. Barral, Spatial Propagation and Characterization of Quantum States of Light in Integrated Photonic Devices, (Universidade De Santiago De Compostela, 2015).

14. A. Rai, S. Das, and G. S. Agarwal, “Quantum entanglement in coupled lossy waveguides,” Opt. Express 18(6), 6241–6254 (2010). [CrossRef]  

15. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]  

16. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]  

17. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef]  

18. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]  

19. Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Technol. 31(15), 2477–2481 (2013). [CrossRef]  

20. A. Cerjan and S. Fan, “Eigenvalue dynamics in the presence of nonuniform gain and loss,” Phys. Rev. A 94(3), 033857 (2016). [CrossRef]  

21. Z. Liu, Q. Zhang, Y. Chen, and J. Xiao, “General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss,” Photonics Res. 5(2), 57–63 (2017). [CrossRef]  

22. X. Zhang, V. A. Vysloukh, Y. V. Kartashov, X. Chen, F. Ye, and M. R. Belić, “PT symmetry in nonlinear twisted multicore fibers,” Opt. Lett. 42(15), 2972–2975 (2017). [CrossRef]  

23. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef]  

24. B. Vest, M. C. Dheur, É Devaux, A. Baron, E. Rousseau, J. P. Hugonin, J. J. Greffet, G. Messin, and F. Marquier, “Anti-coalescence of bosons on a lossy beam splitter,” Science 356(6345), 1373–1376 (2017). [CrossRef]  

25. S. Jiang, Contributions a l'etude Theorique des Bruits Quantiques et Classiques dans les Amplificateurs Raman distribues. (Telecom ParisTech, 2008).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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 (4)
Fig. 1.
Fig. 1. the coincidence and the visibility for the two-photon state varying with respect to propagation distance. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Interference of the one-photon state with the other photon lost due to the waveguide attenuation. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs. State $|{1,0} \rangle$ has the interference dip, while state $|{0,1} \rangle$ has the interference peak.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. The evolution behaviors of the probabilities ${p_{|{2,0} \rangle }}$, ${p_{|{0,2} \rangle }}$ and ${p_{|{0,0} \rangle }}$ along the propagation distance.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. The coincidence and the visibility for the two-photon state varying with respect to propagation distance in two lossy waveguides with γ1 = 0.15 cm-1 and γ2 = 0.5 cm-1. Dis. Stands for distinguishable photon pairs and indis. stands for the indistinguishable photon pairs.

View in Article | Download Full Size | PDF

Equations (16)

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

M = ℏ ( κ a ^ 1 † a ^ 2 + κ a ^ 2 † a ^ 1 ) ,
∂ ∂ z a ^ m = 1 i ℏ [ a ^ m , M ] .
d a ^ 1 ( z ) d z = − i κ a ^ 2 ( z ) , d a ^ 2 ( z ) d z = − i κ a ^ 1 ( z ) .
d a ^ 1 ( z ) d z = − i κ a ^ 2 ( z ) − γ 1 a ^ 1 ( z ) + f ^ 1 ( z ) , d a ^ 2 ( z ) d z = − i κ a ^ 1 ( z ) − γ 2 a ^ 1 ( z ) + f ^ 2 ( z ) , [ f ^ m ( z ) , f ^ m ′ † ( z ′ ) ] = 2 δ m m ′ γ m δ ( z − z ′ ) , ⟨ f m † ( z ) f m ( z ′ ) ⟩ = 2 n ¯ σ γ m δ ( z − z ′ ) , ⟨ f m ( z ) f m † ( z ′ ) ⟩ = 2 ( 1 + n ¯ σ ) γ m δ ( z − z ′ ) ,
n ¯ σ = 1 exp ⁡ ( h ν k B T ) − 1 ≈ 0.
( a ^ 1 ( z ) a ^ 2 ( z ) ) = exp ⁡ ( − ( γ 1 + γ 2 2 ) z ) ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) − i κ λ sin ⁡ ( λ z ) − i κ λ sin ⁡ ( λ z ) cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) ( a ^ 1 ( 0 ) a ^ 2 ( 0 ) ) + F ( z ) ,
γ = γ 2 − γ 1 2 , λ = κ 2 − γ 2 , F ( z ) = ( F 1 ( z ) F 2 ( z ) ) = exp ⁡ ( − ( γ 1 + γ 2 2 ) z ) × ∫ 0 z exp ⁡ ( ( γ 1 + γ 2 2 ) s ) ( cos ⁡ ( λ ( z − s ) ) + γ λ sin ⁡ ( λ ( z − s ) ) − i κ λ sin ⁡ ( λ ( z − s ) ) − i κ λ sin ⁡ ( λ ( z − s ) ) cos ⁡ ( λ ( z − s ) ) − γ λ sin ⁡ ( λ ( z − s ) ) ) ( f ^ 1 ( s ) f ^ 2 ( s ) ) d s .
⟨ F ( z ) F H ( z ) ⟩ = ( ⟨ | F 1 ( z ) | 2 ⟩ ⟨ F 1 ( z ) F 2 ∗ ( z ) ⟩ ⟨ F 1 ( z ) ∗ F 2 ( z ) ⟩ ⟨ | F 2 ( z ) | 2 ⟩ ) = 2 ∫ 0 z exp ⁡ ( − ( γ 1 + γ 2 ) s ) A ( s ) ( γ 1 γ 2 ) A H ( s ) d s , A ( s ) = ( cos ⁡ ( λ s ) + γ λ sin ⁡ ( λ s ) − i κ λ sin ⁡ ( λ s ) − i κ λ sin ⁡ ( λ s ) cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ s ) ) .
| 1 , 1 ⟩ = a ^ 1 † a ^ 2 † | 0 , 0 ⟩ .
a ^ 1 † ( z ) a ^ 2 † ( z ) | 0 , 0 ⟩ = β | 2 , 0 ⟩ | 2 , 0 ⟩ + β | 0 , 2 ⟩ | 0 , 2 ⟩ + β | 1 , 1 ⟩ | 1 , 1 ⟩ + β | 1 , 0 ⟩ | 1 , 0 ⟩ + β | 0 , 1 ⟩ | 0 , 1 ⟩ + β | 0 , 0 ⟩ | 0 , 0 ⟩ ,
β | 2 , 0 ⟩ = 2 exp ⁡ ( − ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) i κ λ sin ⁡ ( λ z ) , β | 0 , 2 ⟩ = 2 exp ⁡ ( − ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) i κ λ sin ⁡ ( λ z ) , β | 1 , 1 ⟩ = exp ⁡ ( − ( γ 1 + γ 2 ) z ) ( ( cos ⁡ ( λ z ) 2 − γ 2 λ 2 sin ⁡ ( λ z ) 2 ) − κ 2 λ 2 sin ⁡ ( λ z ) 2 ) , β | 1 , 0 ⟩ = exp ⁡ ( − ( γ 1 + γ 2 2 ) z ) ( ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) F 2 † + i κ λ sin ⁡ ( λ z ) F 1 † ) , β | 0 , 1 ⟩ = exp ⁡ ( − ( γ 1 + γ 2 2 ) z ) ( ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) F 1 † + i κ λ sin ⁡ ( λ z ) F 2 † ) , β | 0 , 0 ⟩ = F 1 † F 2 † .
p | 2 , 0 ⟩ = 2 exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) 2 κ 2 λ 2 sin ⁡ ( λ z ) 2 , p | 0 , 2 ⟩ = 2 exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) 2 κ 2 λ 2 sin ⁡ ( λ z ) 2 , p | 1 , 1 ⟩ = exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( κ 2 cos ⁡ ( 2 λ z ) − γ 2 λ 2 ) 2 , p | 1 , 0 ⟩ = exp ⁡ ( − ( γ 1 + γ 2 ) z ) × ( ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) 2 ⟨ F 2 F 2 † ⟩ + κ 2 λ 2 sin ⁡ ( λ z ) 2 ⟨ F 1 F 1 † ⟩ + 2 R e ( − ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) i κ λ sin ⁡ ( λ z ) ⟨ F 1 F 2 † ⟩ ) ) , p | 0 , 1 ⟩ = exp ⁡ ( − ( γ 1 + γ 2 ) z ) × ( κ 2 λ 2 sin ⁡ ( λ z ) 2 ⟨ F 2 F 2 † ⟩ + ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) 2 ⟨ F 1 F 1 † ⟩ + 2 R e ( − i κ λ sin ⁡ ( λ z ) ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) ⟨ F 2 F 1 † ⟩ ) ) , p | 0 , 0 ⟩ = ( ⟨ F 1 F 1 † ⟩ ⟨ F 2 F 2 † ⟩ + ⟨ F 2 F 1 † ⟩ ⟨ F 1 F 2 † ⟩ ) .
p | 1 , 1 ⟩ = exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( κ 2 cos ⁡ ( 2 λ z ) − γ 2 λ 2 ) 2 ,
p | 2 , 0 ⟩ d = 2 exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) 2 κ 2 λ 2 sin ⁡ ( λ z ) 2 , p | 0 , 2 ⟩ d = 2 exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) 2 κ 2 λ 2 sin ⁡ ( λ z ) 2 , p | 1 , 1 ⟩ d = exp ⁡ ( − 2 ( γ 1 + γ 2 ) z ) ( ( cos ⁡ ( λ z ) 2 − γ 2 λ 2 sin ⁡ ( λ z ) 2 ) 2 + ( κ 2 λ 2 sin ⁡ ( λ z ) 2 ) 2 ) , p | 1 , 0 ⟩ d = exp ⁡ ( − ( γ 1 + γ 2 ) z ) ( ( cos ⁡ ( λ z ) + γ λ sin ⁡ ( λ z ) ) 2 ⟨ F 2 F 2 † ⟩ + κ 2 λ 2 sin ⁡ ( λ z ) 2 ⟨ F 1 F 1 † ⟩ ) , p | 0 , 1 ⟩ d = exp ⁡ ( − ( γ 1 + γ 2 ) z ) ( κ 2 λ 2 sin ⁡ ( λ z ) 2 ⟨ F 2 F 2 † ⟩ + ( cos ⁡ ( λ z ) − γ λ sin ⁡ ( λ z ) ) 2 ⟨ F 1 F 1 † ⟩ ) , p | 0 , 0 ⟩ d = ⟨ F 1 F 1 † ⟩ ⟨ F 2 F 2 † ⟩ .
− 2 ( cos ⁡ ( λ z ) 2 − γ 2 λ 2 sin ⁡ ( λ z ) 2 ) ( κ 2 λ 2 sin ⁡ ( λ z ) 2 ) ( cos ⁡ ( λ z ) 2 − γ 2 λ 2 sin ⁡ ( λ z ) 2 ) 2 + ( κ 2 λ 2 sin ⁡ ( λ z ) 2 ) 2 .
p | 2 , 0 ⟩ + p | 0 , 2 ⟩ + p | 1 , 1 ⟩ + p | 1 , 0 ⟩ + p | 0 , 1 ⟩ + p | 0 , 0 ⟩ = 1.
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