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Abstract
We theoretically demonstrate quantum interference in an anti-parity-time (anti-PT) symmetric system based on coupled waveguides. We calculate the coincidence probability in an input polarization-entangled two-photon state, which can be used to simulate different statistical particles. When the birefringence of the waveguides is negligible, our results indicate that the coincidence probabilities of the bosons and fermions decrease exponentially with the propagation distance in both the unbroken and broken anti-PT symmetry regions owing to the dissipation. Particularly, loss-induced transparency is observed for the bosons. Simultaneously, the statistical rule valid in the Hermitian system is violated and the antibunching of bosons is observed. When the birefringence of the waveguides is considered, the coincidence probability of the bosons and fermions is equalized at the exceptional point (EP), whereas that of the bosons is less(greater) than that of the fermions in the broken(unbroken) anti-PT symmetry region. Additionally, we observe the Hong-Ou-Mandel dip for bosons in the broken anti-PT phase. Our research provides a complementary technique for the manipulation of quantum interference compared with the PT symmetric system and may be applied in building quantum devices with anti-PT symmetric quantum mechanics.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that the Hermiticity of a Hamiltonian operator guarantees that the system has a real eigenvalue. As a complex generalization of the conventional quantum theories with the relaxing requirement for the Hermiticity of observables, parity-time (PT) symmetry quantum mechanics, put forward by Bender and Boettcher in $1998$ [1], has attracted considerable interest among scientific researchers [2–4]. For PT symmetry, the relationship between the non-Hermitian Hamiltonian $H$ and $PT$ operator meets $[H, PT]=0$, i.e., $HPT=PTH$. The functions of the parity operator $P$ and time-reversal operator $T$ are defined as $P$: $\hat {x}\rightarrow -\hat {x}, \hat {p}\rightarrow -\hat {p}$, $i\rightarrow i$ and $T$: $\hat {x}\rightarrow \hat {x}, \hat {p}\rightarrow -\hat {p}, i\rightarrow -i$, respectively. Regarding the equivalence between the time-dependent Schrödinger equation and paraxial wave equation, optical systems could be excellent platforms for studying non-Hermitian physics. Presently, the notion of PT symmetry has increasingly widespread applications in optical systems [5–20].
As the counterpart of PT symmetry, anti-PT symmetry has attracted much consideration [21–26] in the last several years. The anti-PT symmetric Hamiltonian satisfies the anticommutation relation $\{H,PT\}=0$, i.e., $HPT=-PTH$. Like the PT symmetry, the anti-PT symmetry has also experienced a spontaneous phase transition from the unbroken anti-PT symmetry region to the broken one. This characteristic has a noticeable influence on the transport properties and could facilitate complementary optical manipulation technology [27]. In contrast with the PT symmetric system realized by engineering gain and loss, the anti-PT symmetric system can be realized without optical gain. The imaginary coupling plays a significant role in the realization of anti-PT symmetry. The earliest literature [22] discussing anti-PT symmetry was proposed with balanced positive- and negative-index materials. It was then shown in [27,28] that indirect dissipative coupling in optical waveguides or microcavity systems can be used to realize anti-PT symmetry. The first anti-PT symmetry experiment was demonstrated in 2016 for two coupled atomic spin waves, where the rapid coherence transport via flying atoms provided the needed dissipative coupling [29]. Recently, anti-PT symmetry has also been observed in the four-wave mixing process for two distinct parametric signals [30].
Presently, quantum interference between two or more photons is a central resource in quantum computation and quantum optics. To reveal the quantum nature of the non-Hermitian system, the quantum interference in the PT symmetric system is demonstrated by several groups. For example, in 2018, S. Longhi theoretically demonstrated quantum interference in a passive PT symmetry optical system near an exceptional point (EP) [31]. In 2019, Javid Naikoo et al. theoretically studied the quantum Zeno and anti-Zeno effects in a PT symmetric system of coupled cavities with gain and loss [32]. By employing the coupled two-waveguide, F. Klauck et al. experimentally demonstrated two-particle quantum interferences in a passive PT symmetric system and observed the counterintuitive quantum interference phenomena [33]. Most recently, Lucas Teuber et al. theoretically studied quantum interference with a passive PT symmetric coupled waveguide system by solving the quantum master equation [34].
Anti-PT symmetry, as a subclass of non-Hermitian physics gives rise to intriguing optical phenomena. Furthermore, anti-PT symmetry has clear advantages over PT symmetry, which relies on optical gain. The anti-PT symmetry can therefore provide a practical way for manipulating quantum interference. Comparatively, there are few researches on the quantum interference in an anti-PT symmetric system. Recently, the features of quantum correlation near the EP was experimentally observed in the platform of flying atoms [35]. In this study, we examine the quantum interference in an anti-PT symmetric system based on coupled waveguides. Our results indicate that the coincidence probabilities of different statistical particles decrease exponentially with the propagation distance in both the unbroken and broken anti-PT symmetry regions when the birefringence of waveguides is negligible. The coincidence probability of bosons is always greater than that of fermions, which indicates that the anti-bunching effect can occur for bosons in an anti-PT symmetric system. What is even more interesting is that the bosons exhibit an obvious loss-induced transparency phenomenon. When the two coupled waveguides possess birefringence, owing to their modulation on the polarization of photons, the coincidence probability for bosons equals that of fermions at the EP, whereas the coincidence probability for bosons is less(greater) than that of fermions in the broken(unbroken) anti-PT symmetry region. Furthermore, the Hong-Ou-Mandel(HOM) dip can also be observed in the broken anti-PT symmetry region. These findings imply that the anti-PT symmetry could be a potential way to control and tune quantum interference.
2. Anti-PT symmetry model
The schematic of the three-channel coupled waveguides system is shown in Fig. 1(a). With the coupled-mode approach, the dynamics equations of the three-channel coupled waveguides model are:
where $b_{1}$, $a$, and $b_{2}$ are the guided modes in channels 1, 2 and 3, respectively. The modes $b_{1}$ and $b_{2}$ are coupled to the auxiliary lossy mode $a$ through near-field tunnelling, and are not directly coupled. Parameter $\kappa _{1}$($\kappa _{2}$) represents the coupling strength between modes $b_{1}$($b_{2}$) and $a$, whereas $\kappa _{1}^{\ast }$($\kappa _{2}^{\ast }$) is the complex conjugate of $\kappa _{1}$($\kappa _{2}$). In this work, we assume that both $\kappa _{1}$, $\kappa _{2}$ are real, and we make $\kappa _{1}=\kappa _{2}=\kappa$. $\mu$ is the loss constant of the mode $a$. $\beta _{1}$, $\alpha$ and $\beta _{2}$ are the propagation constants for the modes of $b_{1}$, $a$ and $b_{2}$, respectively.
Fig. 1. Schematic coupled waveguides for implementing anti-PT symmetry. (a) Three-channel coupled waveguides. Modes $b_1$ and $b_2$ are coupled to the auxiliary lossy mode $a$ through near-field tunnelling. (b) The anti-PT symmetric system resulted from the three-channel coupled waveguides by adiabatically eliminating the mode $a$. The birefringence of waveguides 1 and 3 is negligible. (c) The anti-PT symmetric system considering the birefringence of the waveguides. The first and third segments (labelled by $d_1$ and $d_3$) in waveguides 1 and 3 are lossless and provide unitary rotations, and the middle segment (labelled by $d_2$) is dissipation.
To realize anti-PT symmetry, the dissipation in waveguide 2 must satisfy the condition $\mu \gg \left \vert \kappa \right \vert$ [27]. By adiabatically eliminating mode $a$, the dynamics equations for the coupled three-channel waveguides system can be reduced as follows:
where $\delta =\frac {\beta _{1}-\beta _{2}}{2}$ is the effective detuning of modes $b_{1}$ and $b_{2}$, and $\gamma =\frac {\left \vert \kappa \right \vert ^{2}}{\mu }$ is the effective dissipative coupling strength between modes $b_{1}$ and $b_{2}$. From the Eqs. (2a)–(2b), we can deduce that the system satisfies anti-PT symmetry. The simplified scheme is shown in Fig. 1(b).To study the quantum interference, we determine the Hamiltonian of the coupled three-channel waveguides system:
Naturally, the dynamics equations for quantized fields are expressed as follows:
The solutions for Eqs. (4a)–(4d) can be written in the following forms:
Under the Markovian approximation, the transfer matrix elements $T_{i,j}(z)(i,j=1,2,3)$ and $T_{o}(r,z)(o=1,2,3)$ can be obtained from the classical equation of motion. By solving Eqs. (2a)–(2b), the required matrix elements for studying quantum interference can be directly worked out.
3. Quantum interference in anti-PT symmetric system
3.1 Waveguides without birefringence
Here, we focus on the polarization-entangled two-photon state $\left \vert \psi (0)\right \rangle =\frac {1}{\sqrt {2}}(\hat {b}_{1}^{(H)\dagger }\hat {b}_{2}^{(V)\dagger }+e^{i\phi }\hat {b}_{1}^{(V)\dagger }\hat {b}_{2}^{(H)\dagger })\left \vert 0,0\right \rangle$, in which different $\phi$ values can simulate different statistical particles [31]. Among these, H and V indicate horizontal and vertical polarization, respectively. For the bosons $\phi =0$, whereas $\phi =\pi$ for the fermions. When $\phi$ takes other values, i.e., $0<\phi <\pi$, it represents the non-semi-integer spin particles that are the extension of the bosons and fermions [36–39].
In the following sections, we use the coincidence probability to study the quantum interference of various statistical particles in the anti-PT symmetric system. The terminology of the survival probability [31] is the equivalent of the coincidence probability and also used in some literature. One particle is launched into waveguide 1 and the other is launched into waveguide 3 in Fig. 1(b). In this section, we neglect the birefringence of the waveguides, then the modes in both waveguides are kept linearly polarized. When two particles propagate in the waveguides, the coincidence probability is defined as the probability that none of the two particles have decayed and both particles can be simultaneously detected in their output ports after some propagation distance, which is given by [39]:
According to the definition, the coincidence probability is given by:
For the bosons and fermions, the corresponding coincidence probability $C_{bos}(z)$ and $C_{f\!erm}(z)$ are individually expressed as: $C_{bos}(z)=A(z)[\delta ^{2}-\gamma ^{2}\cosh (2mz)]^{2}/m^{4}$ for $\gamma > \delta$ and $C_{bos}(z)=A(z)[\delta ^{2}-\gamma ^{2}\cos (2nz)]^{2}/n^{4}$ for $\gamma <\delta$, $C_{f\!erm}(z)=A(z)$ in both the unbroken and broken anti-PT symmetry region.
Because the coincidence probabilities for the bosons and fermions all contain an exponential decay factor $A(z)$, we can deduce that $C_{bos}(z)$ and $C_{f\!erm}(z)$ exponentially decay versus $z$ in both the unbroken and broken anti-PT symmetry regions, as shown in Fig. 2. This is the reflection of the dissipative system. For an arbitrary fixed $\gamma$, comparing Fig. 2(a) with Fig. 2(b), we find that the coincidence probability of the bosons in the broken anti-PT symmetry region decays faster than that in the unbroken one. The difference is because the cosine function in the broken region is bound, whereas the hyperbolic cosine function in the unbroken region is infinite. The transition from the unbroken anti-PT symmetry region to the broken one can be achieved by altering the effective detuning $\delta$. This provides us a method to modulate the two-particle quantum interference of the bosons. However, the parameter $\delta$ has no impact on the coincidence probability of the fermions, as shown in Figs. 2(c)‐2(d). Therefore, the coincidence probability of the fermions is insensitive to the fact that the system is in the unbroken or broken anti-PT symmetry region.
Fig. 2. Coincidence probability $C_{bos}(z)$ and $C_{f\!erm}(z)$ vs the propagation distance $z$ and dissipative coupling strength $\gamma$. (a)(c) Unbroken anti-PT symmetry region with $\delta =0.8\gamma$; (b)(d) Broken anti-PT symmetry region with $\delta =2.5\gamma$.
To clearly show the difference between the quantum interference of the bosons and fermions below and above the EP, we calculate the coincidence probability $C_{bos}$ and $C_{f\!erm}$ versus the loss rate $\gamma /\delta$ for a fixed propagation distance in Fig. 3. We deduce that the coincidence probability $C_{f\!erm}$ has a continuous exponential decay, whereas $C_{bos}$ first decreases and then increases near the EP with the increase in the loss rate. Generally, it is intuitively expected that the coincidence probability would drop with the increasing loss rate. However, the coincidence probability of bosons becomes anomalously transparent as the loss rate increases. This counterintuitive phenomenon is often described as loss-induced transparency [11], which is the manifestation of the EP behavior of non-Hermiticity. This behavior occurs because the propagation of one of the eigenstates of the system is lossless when $\gamma \gg \left \vert \delta \right \vert$ in the unbroken region. Moreover, this eigenstate corresponds to the two waveguides sharing an equal probability to propagate the photons [27,40]. The existence of this eigenstate means the coincidence probability will be transparent after an adequate propagation length. For fermions, no evidence of EP transition is found. This is because bosons can bunch together and partially avoid the lossy regions during propagation in the structure, whereas fermions cannot because of the Pauli exclusion. The results show that quantum interference can realize loss-induced transparency for bosons and hide the phase transition for fermions in the anti-PT symmetric system, which can be used to tune the quantum interference.
Fig. 3. Coincidence probability $C_{bos(f\!erm)}$ vs the loss rate $\gamma /\delta$ in an anti-PT symmetric optical waveguides system with $z=3.5$ and $\delta =1$. The blue solid line represents $C_{bos}$ for bosons and the red dotted line is $C_{f\!erm}$ for fermions. The $y$ axis is in the logarithm scale. The pink region marks the unbroken anti-PT symmetry, whereas the purple region marks the broken anti-PT symmetry.
It is worth noting that the coincidence probabilities $C_{bos}$ decay slower than the $C_{f\!erm}$ in Fig. 3. This indicates that $C_{bos}> C_{f\!erm}$ when the other condition is the same. As is well known, in the Hermitian system, the coincidence probability satisfies $C_{H\!bos}\leq C_{H\!f\!erm}$. As aforementioned, in the present scheme, $C_{bos}$ is larger than $C_{f\!erm}$, which violates the statistics rule $C_{H\!bos}\leq C_{H\!f\!erm}$. This is because our system consists of two coupled optical waveguides that can be considered as the integrated analogue of a bulk beam splitter. Compared to bulk beam splitters, the coupled waveguides are lossy beam splitters owing to the dissipative coupling between them. This makes the bosons exhibit antibunching behavior in the lossy anti-PT symmetric system. Similar results have been reported in other kinds of dissipative system [31,41i,42].
Next, we present the dependence of the coincidence probability $C(z)$ on the phase shift $\phi$. As shown in Fig. 4, in both the unbroken (Fig. 4(a)) and broken (Fig. 4(b)) anti-PT symmetry regions, the coincidence probabilities all decay with $z$ but with different decay rates. In the unbroken region the particles (except for $\phi =\pi$) experience slower propagation decay than in the broken region. Simultaneously, the results indicate that the quantum interference for different statistical particles in the same region has different decay rates. The bosons decay slowest, followed by the non-semi-integer spin particles with $0<\phi <\pi$, whereas the fermions decay fastest.
Fig. 4. Coincidence probability $C(z)$ vs phase shift $\phi$ and propagation distance $z$. (a) The case for the unbroken anti-PT symmetry region with $\delta =0.8\gamma$; (b) The case for the broken anti-PT symmetry region with $\delta =2.5\gamma$. The other parameter is $\gamma =2$.
3.2 Waveguides with birefringence
In this section, we examine waveguides with birefringence. As is well known, the polarization state of light is changed when it passes through a birefringence medium. Here, we still focus on the polarization-entangled two-photon state for comparison with the results in the previous section.
The schematic of the anti-PT symmetric system based on coupled waveguides with birefringence is shown in Fig. 1(c). In this case, each waveguide is divided into three segments. The middle segment $d_{2}$ is lossy, whereas the first one $d_{1}$ and third one $d_{3}$ are lossless. $d_{1}$ and $d_{3}$ are used to realize an appropriate polarization rotation of photons, and the corresponding operations can be represented by $X$ and $X^{'}$. Thus, the transfer matrix for all the waveguides can be written as $T_{R}(z) =X^{'}TX$, where $X^{'}$=$X^{-1}$. $T$ is the transfer matrix in the previous case of waveguides without birefringence. In this section, we assume that the roles of segments $d_{1}$ and $d_{3}$ are played by a quarter-wave plate and its optical axis is set to ${\pi }/{4}$ and ${3\pi }/{4}$ radians, respectively.
Therefore, the coincidence probability in this case is denoted as $C_{R}(z)$ and can be expressed by:
The coincidence probability $C_{Rbos}(z)$ of bosons can be expressed as: $C_{Rbos}(z)=A(z)\cosh ^{2} (2mz)$ for $\gamma > \delta$ and $C_{Rbos}(z)=A(z)\cos ^{2}(2nz)$ for $\gamma <\delta$. Furthermore, the coincidence probability $C_{Rferm}(z)=A(z)$ for both $\gamma > \delta$ and $\gamma <\delta$, is in accordance with the results of the waveguides without birefringence.
First, we evaluate the quantum interference of the bosons and fermions. Here, we determine the behavior of $C_{Rbos}$ and $C_{Rferm}$ versus the loss rate $\gamma /\delta$. As shown in Fig. 5, at $\gamma =0$, the coincidence probability is 0 for bosons and 1 for fermions, inrdicating $C_{Rbos}<C_{Rferm}$. These results are consistent with the Hermitian case. The figure also shows that the coincidence probability $C_{Rbos}$ increases with the increase of the loss rate $\gamma /\delta$, whereas the coincidence probability $C_{Rferm}$ decreases with increasing the loss rate. For the coupled waveguides system considering birefringence, $C_{Rbos}<C_{Rferm}$ in the broken anti-PT symmetry region and $C_{Rbos}>C_{Rferm}$ in the unbroken anti-PT symmetry region. $C_{Rbos}$ equals $C_{Rferm}$ at the EP. This behavior is different from the case shown in Fig. 3, in which the statistics rule for bosons is antibunching and has no EP crossing behavior. This could be caused by the additional relative phase from the birefringence between the two guided photons of the input polarization-entangled two-photon state. Similar results have been shown in a passive PT symmetric waveguides system [43]. Here, our system provides complementary techniques for the manipulation of quantum interference compared with the PT symmetric system.
Fig. 5. Coincidence probability $C_{Rbos(Rferm)}$ vs loss rate $\gamma /\delta$ in anti-PT symmetric optical waveguides with birefringence. The blue solid line is for bosons and the red dotted line is for fermions. The $y$ axis is in the logarithm scale. Other parameters are $\delta =1$ and $z=0.75$, which represent the waveguide length of the lossy segment as $d_{2}=0.75$. The pink region marks the unbroken anti-PT symmetry, whereas the purple one marks the broken anti-PT symmetry.
Figure 6 shows the coincidence probability $C_{Rbos}(z)$ and $C_{Rferm}(z)$ versus the propagation distance $z$ and dissipative coupling strength $\gamma$. For fermions, in both the unbroken (Fig. 6(c)) and broken (Fig. 6(d)) anti-PT symmetry regions, the coincidence probability $C_{Rferm}(z)$ versus $z$ decays exponentially. For bosons, it can be observed from Fig. 6(a) that the coincidence probability decays with $z$ exponentially and continuously in the unbroken region. However, in the broken anti-PT symmetry region shown in Fig. 6(b), an intriguing feature occurs where the coincidence probability $C_{Rbos}(z)$ shows the HOM dip, which refers to the position where two photons appear at the same output. This result indicates that the bosons exhibit bunching behavior in the broken anti-PT symmetry region, which is consistent with the results in Fig. 5, where $C_{Rbos}<C_{Rferm}$ in the broken region.
Fig. 6. Coincidence probability $C_{Rbos}(z)$ and $C_{Rferm}(z)$ vs propagation distance $z$ and dissipative coupling strength $\gamma$. (a) and (b) are the results for the bosons, whereas (c) and (d) are those for the fermions. (a) and (c) show the results in the unbroken anti-PT symmetry region with $\delta =0.8\gamma$. (b) and (d) show the results in the broken anti-PT symmetry region with $\delta =2.5\gamma$.
To obtain a clear physical image of the HOM dip, we show that the coincidence probability $C_{R}$ varies with the propagation distance $z$ in Fig. 7, where the dotted lines represent the Hermitian case and the solid lines are for the case of the broken anti-PT symmetry. In the following, we use $z_{i}(i=0,1,2)$ to denote the position of the HOM dip for various examples. In the first instance, we use the Hermitian instance for bosons, from Eq. (8) we can find $z_{0}=\frac {\pi }{4\delta }$. Next, under conditions of the broken anti-PT symmetry, namely, $\gamma \neq 0$, $z_{1}=\frac {\pi }{4n }$ is obtained for the bosons. Owing to $n< \delta$, $z_{1}>z_{0}$. This indicates that the position of the HOM dip in the broken anti-PT symmetry region moves to a longer $z$ compared to that in the Hermitian example. In the broken anti-PT symmetry region, when $\phi$ takes other values, $z_{2}=\frac {\arccos (-\tan ^{2}\frac {\phi }{2})}{2n}$. Clearly, with the increase of the phase shift $\phi$, the value of $z_{2}$ increases until the HOM dip vanishes.
Fig. 7. Coincidence probability $C_{R}(z)$ vs propagation distance $z$ for the Hermitian case (dotted line) and broken anti-PT symmetry case (solid line) with $\phi =0, \frac {\pi }{4}, \frac {\pi }{2}$. $\gamma =0$ for the Hermitian case and $\gamma =1$ for the broken anti-PT symmetry case. The other parameter is $\delta =4$.
In the subsequent part, we present the coincidence probability $C_{R}(z)$, which varies with $\phi$ in Fig. 8(a) for the anti-PT unbroken symmetry region and in Fig. 8(b) for the broken anti-PT symmetry region. Comparing Fig. 8(a) with Fig. 8(b), we deduce that no matter what value the $\phi$ takes in the unbroken anti-PT symmetry region, the coincidence probability $C_{R}(z)$ decays exponentially versus $z$. However, for some statistical particles in the broken anti-PT symmetry region, the HOM dips appear as shown in Fig. 8(b).
Fig. 8. Coincidence probability $C_{R}(z)$ vs phase shift $\phi$ and propagation distance z for the polarization-entangled two-photon state. (a) Unbroken anti-PT symmetry region with $\delta =0.8\gamma$; (b) Broken anti-PT symmetry region with $\delta =2.5\gamma$. The other parameter is $\gamma =2$.
4. Conclusion
In this work, we studied quantum interference through the evolution of coincidence probability in an anti-PT symmetric coupled waveguides system. The polarization-entangled two-photon state is used as input to simulate different statistical particles. When the birefringence of waveguides is negligible, the coincidence probabilities for bosons and fermions decrease exponentially with the propagation distance $z$ in both unbroken and broken anti-PT symmetry regions. Particularly, under anti-PT symmetry, bosons can exhibit loss-induced transparency and possess the antibunching effect. When the birefringence of waveguides is considered, the results reveal that, $C_{Rbos}$ equals $C_{Rferm}$ at the EP; $C_{Rbos}<C_{Rferm}$ in the broken anti-PT symmetry region and $C_{Rbos}>C_{Rferm}$ in the unbroken anti-PT symmetry region. Significant EP crossing of the statistical rule is observed. Furthermore, the HOM dip can also be observed for bosons in the broken anti-PT symmetry region. Our results show that the EP phase transition is hidden for fermions in two of the proposed schemes. Our results may be applied in fundamental quantum technologies based on anti-PT symmetric quantum mechanics and provide a new way to tune the quantum interference.
Funding
National Natural Science Foundation of China (11874296, 11574247); Natural Science Foundation of Shaanxi Province (2019JM-279).
Acknowledgement
Y. Qin is grateful to F. Yang from Tsinghua University for the helpful discussion.
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.
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