⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 084203 Photon Coalescence in a Lossy Non-Hermitian Beam Splitter Zhiqiang Ren (任志强), Rong Wen (温荣), and J. F. Chen (陈洁菲)* Affiliations State Key Laboratory of Precision Spectroscopy, School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China Received 17 April 2020; accepted 4 June 2020; published online 28 July 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11674100 and 11654005), the Natural Science Foundation of Shanghai (Grant No. 16ZR1448200), and the Shanghai Rising-Star Program (Grant No. 17QA1401300).
*Corresponding author. Email: jfchen@phy.ecnu.edu.cn Citation Text: Ren Z Q, Wen R and Chen J F 2020 Chin. Phys. Lett. 37 084203    Abstract We investigate photon coalescence in a lossy non-Hermitian system and study a dynamic device modeled by a beam splitter with an extra intrinsic phase term added in the transformation matrix, with which the device is a lossy non-Hermitian linear system. The two-photon interference behavior is altered accordingly since this extra intrinsic phase affects the unitary of transformation and the coalescence of the incoming photons. We calculate the coincidence between two single-photon pulses, considering the interferometric phase between two pulses and the extra intrinsic phase as the tunable parameters. The extra phase turns the famous Hong–Ou–Mandel dip into a bump, with the visibility dependent on both the interferometric phase and the extra phase. DOI:10.1088/0256-307X/37/8/084203 PACS:42.50.Ar, 42.25.Hz, 42.50.-p, 42.79.-e © 2020 Chinese Physics Society Article Text Photon coalescence can be revealed by the two-photon interference, i.e., the interference phenomenon observed in the joint probability measurement of two photons meeting at a beam splitter (BS).[1] The interference between two independent photons reveals their intensity correlation, or in other words, the second-order correlation function. The nonclassical nature of the photon source can be verified in this interference.[2–4] Bosonic particles exhibit coalescence, i.e., coincidence between two output ports of BS is perfectly cancelled due to quantum interference. The interference visibility is 50% for coherent pulses, which sets the classical limit.[5–7] The quantum nature of light is demonstrated with the results exceeding this limit and approaching 100%.[8–10] The visibility reveals the purity and indistinguishability of independent light sources,[11,12] and therefore the two-photon interference plays an important role in quantum information processing and communication.[13–20] A typical lossless BS gives a unitary transformation, and the phase angles for the reflection and transmission coefficients satisfy $\phi_{r}-\phi_{t}=\pm\pi/2$.[21,22] This intrinsic phase differs from $\pi/2$ and approaches $\pi$ when the BS cannot be considered to be unitary. In these cases, the system is non-Hermitian, and open to loss or gain from the environment. Our recent work by Wen et al.[23] demonstrated that a cold atomic ensemble functions as a controllable coupler device, whose Hermitian property is tunable through laser excitation detuning, thus this extra intrinsic phase term. This phase term significantly changes the two-photon interference, and hence varies the multi-photon statistics.[24] The boson interferes in the Hermitian BS system, and a bunching effect occurs. However, anti-bunching effects can occur in non-Hermitic systems with energy loss.[25] The non-Hermitian system with PT symmetry due to the loss or gain of system energy has attracted attention. Due to the progress of optical technology and quantum control technology, new optics behavior is found in PT symmetrical non-Hermitian systems, e.g., single-direction lasing and unidirectional perfect absorption.[26,27] In the meantime, the two-photon interference is also heavily affected by the phase difference between two incident pulses. For the Fock-state source, e.g., the single photons, their phase would be naturally considered to be random. However, for single-photon source obtained from attenuating a laser beam, the second-order correlation function strongly depends on the relative phase difference between the incident pulses. We call this phase the interferometric phase, since this phase is able to be drawn out from outputs of an interferometry setup. In this letter, we investigate the photon coalescence in a lossy non-Hermitian BS. To match the practical experimental setups, e.g., Wen et al.,[23] we consider interference of single-photon pulses attenuated from a coherent state. The extra intrinsic phase turns the famous Hong–Ou–Mandel dip into a bump, with the interference visibility dependent on the interferometric phases. Our results here demonstrate that how Hermiticity affects a linear optical device, and therefore change the photon interference. It implies that non-Hermitian physics will find potential new phenomena, which are reported most recently,[28,29] in the playground of quantum science.
cpl-37-8-084203-fig1.png
Fig. 1. The schematics of two-photon interference measurement with general splitting and coupling devices.
We consider the situation that the two input fields $\hat{V}_{01}$ and $\hat{V}_{02}$ are coming from the same source $\hat{V}$, and they have a specific phase difference. The schematics of the system is shown in Fig. 1. We denote the transformation operator for a generalized BS as $\hat{U}$, while $\hat{U}_{\rm BS}$ is for a unitary BS. $$ \begin{pmatrix} \hat{V}_{1}(t) \\ \hat{V}_{2}(t)\end{pmatrix} = \hat{U}\begin{pmatrix} \hat{V}_{01}(t) \\ \hat{V}_{02}(t) \end{pmatrix} + \begin{pmatrix} \hat{F}_{1}(t) \\ \hat{F}_{2}(t) \end{pmatrix}, $$ where $\hat{F}_{i}$ is noise operators necessary in the presence of loss to preserve the commutators of the observable outputs for a generic beam splitter. The matrix form of a general BS is written as, $$\begin{alignat}{1} \hat{U}\doteq \,& \begin{pmatrix} t & r \\ r' & t' \end{pmatrix} \\ = \,&\begin{pmatrix} \sqrt{T}e^{i\phi_{T}} & \sqrt{R}e^{ i\left(\frac{\pi}{2}+\phi_{T}+\phi_{1}\right) } \\ \sqrt{R}e^{ i\left(\frac{\pi}{2}+\phi_{T'}+\phi_{2}\right) } & \sqrt{T}e^{i\phi_{T'}} \end{pmatrix},~~ \tag {1} \end{alignat} $$ where $r$ is the reflection coefficient, and $t$ is the transmission coefficient. $R$ and $T$ are the beam-splitter reflectivity and transmittivity, respectively. Lossless BS requires that the transformation matrix $U$ is unitary,[30] i.e., $r't'^{*}+tr^{*}=0$ in Eq. (1). To consider a general BS which may not necessarily satisfy the unitary relation, we include extra intrinsic phases $\phi_{1}$ and $\phi_{2}$, and $\frac{\pi}{2}+\phi_1+\frac{\pi}{2}+\phi_2=\pi$ indicates unitary relation. For simplicity and without loss of generality, we consider the symmetric BS in this model and set $\phi_{T}=\phi_{T'}=0$ and $\phi_1=\phi_2\equiv\phi$. Now in the case of a lossy beam splitter, $|t|^2 + |r|^2 < 1$, the noise operators account for this loss. Loss is a function of $\phi$ (which is a parameter characterizing loss). Thus, when $\phi$ increases, losses increase and reach maximum at $\phi=\pi/2$. For the first splitting BS, a linear lossless BS, the transformation matrix turns out to be unitary, i.e., $$ \hat{U}_{\rm BS}\doteq \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}. $$ Therefore, followed by a phase shift $\varphi_1$, the incident fields are $\hat{V}_{01}=\frac{1}{\sqrt{2}}\hat{V}(t)$ and $\hat{V}_{02}=\frac{1}{\sqrt{2}}\hat{V}(t)\exp\left(i\varphi_{1}-i\pi/2\right)$. To simplify the following calculation, we rewrite the incident fields as $\hat{V}_{01}=\hat{V}(t)$ and $\hat{V}_{02}=\hat{V}(t)\exp\left(i\varphi_{1}\right)$. For the second coupling device in Fig. 1, a general form of BS is utilized, therefore two output fields $\hat{V}_{1}(t)$ and $\hat{V}_{2}(t)$ after the general BS are expressed as $$\begin{align} \hat{V}_{1}(t) ={}&\sqrt{T}\hat{V}(t-\tau_{1})+\Big\{\sqrt{R}\hat{V}(t-\tau _{1}-\delta\tau) \\ &\cdot\exp\Big[ i\Big(\frac{\pi}{2}+\phi\Big) \Big] \exp(i\varphi_{1})\Big\} +\hat{F}_{1}(t),~~ \tag {2} \end{align} $$ $$\begin{align} \hat{V}_{2}(t) ={}&\sqrt{T}\hat{V}(t-\tau_{1}) \exp(i\varphi _{1}) +\Big\{\sqrt{R}\hat{V}\left(t-\tau_{1}+\delta\tau\right) \\ &\cdot\exp\Big[ i\Big( \frac{\pi}{2}+\phi\Big)\Big]\Big\} + \hat{F}_{2}(t).~~ \tag {3} \end{align} $$ Specifically, these output fields are wave-packets with well-defined pulse shape, detected by single-photon counting modules (SPCM), and $\tau_{1}$ is the propagation time to these SPCMs. A time delay $\delta\tau$ is included to consider the time difference between the pulse center of the incident fields denoted by $\hat{V}_1$ and $\hat{V}_2$. The coincidence measurement by two SPCMs is calculated from the joint probability of photon clicking at SPCM1 at $t$ and SPCM2 at $t+\tau$, which is obtained through the second-order cross correlation: $$ G^{(2)}(t,t+\tau)=\langle \hat{V}_{1}^†(t) \hat{V}_{2}^†(t+\tau) \hat{V}_{2}(t+\tau) \hat{V}_{1}(t)\rangle . $$ The noise operators $\hat{F}_1$ and $\hat{F}_2$ are ignorable in the second-correlation function. After integration of $G^{(2)}\left(t,t+\tau \right) $ with respect to $\tau $ over $\tau _{R}$, we obtain the measured correlation $G _{\rm M}$ as follows: $$\begin{alignat}{1} &G_{\rm M}=\int_{-\frac{1}{2}\tau_{R}}^{\frac{1}{2}\tau_{R}}d\tau G^{(2)}(t,t+\tau)\\ ={}&\langle I\rangle^{2}\tau_{R}(T^{2}+R^{2}+2TR)\\ &+2TR\tau_{R}[g^{(1)}(\delta\tau)]^{2} \cos(\pi+2\phi)\\ &+2TR\tau_{R}[g^{(1)}(\delta\tau)]^{2} \cos(2\varphi_{1}) +\Big[4\sqrt{T}\sqrt{R}(T+R)\\ &\cdot \tau_{R}\langle I \rangle g^{(1)}(\delta\tau) \cos(\varphi _{1})\cos\Big(\frac{\pi}{2}+\phi\Big)\Big],~~ \tag {4} \end{alignat} $$ where we have used the optical intensity $I(t)$ to replace $\hat{V}^{† }(t)\hat{V}(t)$. A perfect second-order interference occurs at $\delta\tau=0$, i.e., the two wave-packets completely overlap at the BS, and in contrast, no interference occurs at $\delta\tau\gg0$. Here $g^{(1)}(\delta\tau)$ is the normalized first-order correlation function for an optical field; $\tau_{R}$ is the time resolution of detection, which is in practice the length of a time bin when measuring photon counts; $\tau_{R}$ is larger than any correlation time of the field, $\tau_{R}>\int_{-\frac{1}{2}\tau_{R}}^{\frac{1}{2}\tau_{R}}d\tau g^{(1)}(\delta\tau) $, which can always be achieved in practice because the electronic resolving time can be designed as long as we wish. When the incident phase is random, with $T=R$ and $\phi=0$, the result is $\frac{G_{\rm M}\left(\delta\tau\right) }{G_{\rm M}(\infty) }=1-0.5[g^{(1)}\left(\delta\tau\right)]^{2}$, reproducing the typical results of Hong–Ou–Mandel dip, as shown in Fig. 2(a). Two points are worth being emphasized here. Firstly, the incident fields are in coherent states, i.e., the incoming photon sources are constituted of classical pulses, in which the average photon number is less than 1. Secondly, the results are obtained when the phase difference $\varphi_1$ of two pulses is averaged from $0$ to $2\pi$. It is straightforward that Fig. 2(a) gives 50% of interference visibility, consistent with the well-known classical limit. On the other hand, Eq. (4) with $\phi= \frac{\pi}{2}$ produces a constructive interference at $\delta\tau=0$, therefore a bump with visibility of 50%, as shown in Fig. 2(b).
cpl-37-8-084203-fig2.png
Fig. 2. The two-photon interference with coherent pulses with a photon number less than 1: (a) $\phi=0$, Hong–Ou–Mandel dip, (b) $\phi=\pi/2$, an interference bump. The results are calculated from Eq. (4), in which the phase shift is taken to be random and $T=R$. The shape of both pulses is taken as a Gaussian function, $g^{(1)}(\delta\tau)=e^{-(\delta\tau/\tau_c)^2}$, where the pulse width is $\tau_c=100$ ns.
The above two-photon interference occurring at a specific type of BS indicates the coalescence of the bosonic particles, e.g., photons, according to the coincidence counts at the completely overlapping point $\delta\tau=0$ over that of the no interference point, i.e., $\delta\tau=\pm\infty$. However, this result is strongly dependent on the phase shift $\varphi_1$. For the unitary BS, $\phi=0$, from Eq. (4), $$\begin{align} \frac{G_{\rm M}(0) }{G_{\rm M}(\infty) }={}&1+\frac{2TR}{(T+R) ^{2}}\cos(\pi) +\frac{2TR}{(T+R) ^{2}}\cos(2\varphi _{1}) \\ &+\frac{4\sqrt{T}\sqrt{R}}{T+R}\cos(\varphi _{1})\cos\left(\frac{\pi}{2}\right),~~ \tag {5} \end{align} $$ where $g^{(1)}(0) =1$ at $\delta\tau=0$ and $g^{(1)}(\infty) =0$ for a coherent optical field. The ratio therefore varies from 0 and 1 by changing the interferometric phase $\varphi_1$. The envelop of the red regime as shown in Fig. 3 is given by Eq. (5) with $T=R=\frac{1}{\sqrt{2}}$, therefore $\frac{G_{\rm M}(0) }{G_{\rm M}(\infty)}=0.5+0.5\cos(2\varphi_{1})$. The area within the regime is obtained by scanning $T$ and $R$ from 0 to 1. Averaging the phase gives a classical limit of the two-photon interference with random phase, $\frac{G_{\rm M}(0) }{G_{\rm M}(\infty)}=1+\frac{2TR}{(T+R) ^{2}}\cos(\pi)$. The blue area in Fig. 3(a) shows the covering value. For a non-Hermitian BS model, $\phi=\pi/2$, the coincidence at $\delta\tau=0$ can be ranges from 0 to 4, with $G_M(\infty)$ normalized to 1. That is, $$\begin{align} \frac{G_{\rm M}(0) }{G_{\rm M}(\infty) }={}&1+\frac{2TR}{(T+R) ^{2}}\cos\left( 2\pi\right) +\frac{2TR}{(T+R) ^{2}}\cos\left( 2\varphi_{1}\right)\\ &+\frac{4\sqrt{T}\sqrt{R}}{T+R}\cos \varphi_{1} \cos\pi .~~ \tag {6} \end{align} $$ Substituting $T=R$ into Eq. (6) (when $T=R$, the value of $T$ does not affect this result), we have $\frac{G_{\rm M}(0) }{G_{\rm M}(\infty)}=1.5+0.5\cos(2\varphi_{1}) -2\cos\varphi _{1}$. This is depicted as the envelope of the red regime in Fig. 3(b), and the whole area is obtained from varying $T$ and $R$ in Eq. (6). When the phase is averaged, we obtain $\frac{G_{\rm M}(0) }{G_{\rm M}(\infty)}=1+\frac{2TR}{(T+R) ^{2}}\cos(2\pi)$, ranging from 1 to 1.5. To mimic the interference of two independent photons, these two coherent pulses can be processed with randomized phase modulation across the whole $2\pi$, or simply across a small span around the equivalent phases of $\varphi_1$. These equivalent values are obtained by averaging the results in Eqs. (5) and (6) when varying the interferometric phase around $2\pi$. In other words, the equivalent phases are obtained as $\{\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\}$ when Eq. (5) equals $0.5$, and $\{1.80, 4.48\}$ when Eq. (6) equals $1.5$.
cpl-37-8-084203-fig3.png
Fig. 3. (a) For $\phi=0$. The red area is the theoretical prediction given by Eq. (5). The blue area is the theoretical prediction given in the following. (b) For $\phi=\frac{\pi}{2}$. The red area is the theoretical prediction given by Eq. (6). The blue area is the theoretical prediction given in the following.
To directly reveal the photon coalescence while varying the extra phase $\phi$, which can be easily manipulated in some particular systems, e.g., an atomic ensemble,[23] we define a parameter $\gamma\equiv\frac{G_{\rm M}(0,\phi) }{G_{\rm M}(\infty,\phi)}$. Again when $\varphi_1$ is randomized from 0 to $2\pi$, we straightforwardly obtain the visibility as a function of the extra phase $\phi$, $$\begin{align} \gamma=1+\frac{2TR}{(T+R) ^{2}}\cos \left(\pi+2 \phi\right),~~ \tag {7} \end{align} $$ where the $\gamma$ factor is indeed the visibility of the two-photon interference. The visibility therefore varies from $1/2$, which gives a Hong–Ou–Mandel dip, to $3/2$, which gives a bump, when the extra phase $\phi$ increases from 0 to $\pi/2$, as shown in Fig. 4. While $\gamma < 1$, the system exhibits a certain degree of coincidence cancellation and hence photon coalescence. On the other hand, for $\gamma>1$, the system exhibits increased coincidence and hence photon anti-coalescence.
cpl-37-8-084203-fig4.png
Fig. 4. Results of changing the extra phase from 0 to $\pi/2$. The red area is the theoretical prediction given by Eq. (7).
Based on a lossy beam splitter model, with an extra intrinsic phase term added in the transformation matrix, the two-photon interference of two coherent pulses is investigated. When the two incident pulses are with random relative phase, the interference obtained from coincidence counts of the two different output ports of BS has 50% of visibility. When the extra intrinsic phase of BS takes zero, the two-photon interference reveals a Hong–Ou–Mandel dip; when the extra intrinsic phase approaches $\pi/2$, the interference reveals a bump, in which it indicates that photons tend to choose different ports to exit, i.e., photon anti-coalescence, and therefore form $\vert 1,1\rangle $. Further, we vary the interferometric phase between two coherent pulses. When their phase difference differs, the dip of interference changes from 0 to 1 for a unitary BS. On the other hand, when the intrinsic phase takes $\pi$, the interference changes from a dip of 0 to a bump of 4 times when scanning the interferometric phase. For weak attenuated laser pulses, their phase difference should take designated values to make the results equivalent to those of two independent photons. References Measurement of subpicosecond time intervals between two photons by interferenceDense Coding in Experimental Quantum CommunicationObservation of Spatial Quantum Beating with Separated PhotodetectorsFourth-order interference technique for determining the coherence time of a light beamInterference and correlation of two independent photonsSpatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometerHong–Ou–Mandel interference of two phonons in trapped ionsTwo-plasmon quantum interferenceQuantum interference in plasmonic circuitsQuantum interference between two single photons emitted by independently trapped atomsTemporal Purity and Quantum Interference of Single Photons from Two Independent Cold Atomic EnsemblesTwo-photon interference with true thermal lightTwo-photon interference from a bright single-photon source at telecom wavelengthsObservation of Genuine Three-Photon InterferenceCavity-enhanced two-photon interference using remote quantum dot sourcesTwo-Photon Quantum Interference from Separate Nitrogen Vacancy Centers in DiamondMeasurement of the single-photon tunneling timeSecond-order interference of two independent photons with different spectraSecond-order interference of two independent and tunable single-mode continuous-wave lasersInterference and the lossless lossy beam splitterOptical Polarization Möbius Strips and Points of Purely Transverse Spin DensityNon-Hermitian Magnon-Photon Interference in an Atomic EnsembleQuantum optics of lossy beam splittersAnti-coalescence of bosons on a lossy beam splitterIncident Direction Independent Wave Propagation and Unidirectional LasingReciprocal and unidirectional scattering of parity-time symmetric structuresPT symmetry dips into two-photon interferenceObservation of PT-symmetric quantum interference
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