Exceptional points of any order in a single, lossy, waveguide beamsplitter by photon-number-resolved detection

TL;DR

This paper demonstrates that a single lossy waveguide beamsplitter, when excited by N indistinguishable photons and post-selected, can exhibit an exceptional point of order N+1, enabling the realization of high-order EPs in a simple device.

Contribution

It analytically shows how to generate arbitrary-order exceptional points in a single, lossy waveguide beamsplitter using photon-number-resolved detection, overcoming previous symmetry constraints.

Findings

EP of order N+1 observed in N-photon dynamics

Analytical model mapping beamsplitter to perfect state transfer

Numerical simulations confirm high-order EPs in post-selected states

Abstract

Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e. the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remains an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here we analytically show that the number-resolved dynamics of a single, lossy, waveguide beamsplitter, excited by $N$ indistinguishable photons and post-selected to the $N$-photon subspace, will exhibit an EP of order $N+1$. By using the well-established mapping between a beamsplitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general $N$-photon state, and numerically simulate the system's evolution in the post-selected manifold. Our results pave the way towards realizing robust, arbitrary-order EPs on demand in a single device.

Equations

$$\hat{H}={\omega }_0({\hat{a}}^{†}\hat{a}+{\hat{b}}^{†}\hat{b})+\kappa (\hat{a}{\hat{b}}^{†}+{\hat{a}}^{†}\hat{b})-i\Gamma {\hat{b}}^{†}\hat{b}$$

$${\hat{H}}_N=({\omega }_0-i\Gamma /2)\hat{N}+2\kappa {\hat{J}}_x-i\Gamma {\hat{J}}_z$$

$${\hat{J}}_z$$

$${\hat{J}}_x$$

$${\lambda }_r=({\omega }_0-i\Gamma /2)N+r\sqrt{4{\kappa }^2-{\Gamma }^2}$$

$$I(z)=⟨\psi (0)|G^{†}(z)G(z)|\psi (0)⟩,$$

$$G(z)=e^{-i({\omega }_0-i\Gamma /2)\hat{N}z}{e}^{-i{f}_{+}(z){\hat{J}}_{+}}{e}^{-i{f}_z(z){\hat{J}}_z}{e}^{-i{f}_{-}(z){\hat{J}}_{-}}$$

$${\partial }_z{f}_{+}(z)$$

$${\partial }_z{f}_z(z)$$

$${\partial }_z{f}_{-}(z)$$

$$f_{\pm }(z)$$

$$f_z(z)$$

$$P(\left|m\right);z)=\frac{|(m|\varphi (z)⟩{|}^2}{⟨\varphi (z)|\varphi (z)⟩},$$

Full text

I Introduction

A fundamental postulate of quantum theory is that the Hamiltonian of a (closed) system is Hermitian, which guarantees real energy eigenvalues and a unitary time evolution Landau and Lifshitz (1977). This conventional wisdom was upended when Bender and coworkers discovered families of non-Hermitian Hamiltonians with real spectra Bender and Boettcher (1998). The common feature of all such Hamiltonians was that they were invariant under the combined operations of space- and time-reflection, i.e. they were parity and time-reversal ($\mathcal{PT}$) symmetric Hamiltonians. In the past two decades, it has become clear that non-Hermitian, $\mathcal{PT}$-symmetric Hamiltonians represent classical systems with spatially or temporally separated gain and loss Joglekar et al. (2013); Feng et al. (2017); El-Ganainy et al. (2018). The spectrum of a $\mathcal{PT}$ symmetric Hamiltonian changes from purely real to complex conjugate pairs when the strength of its anti-Hermitian part matches the Hermitian energy scale. This $\mathcal{PT}$-symmetry breaking transition occurs at an exceptional point Mueller and Rotter (2008); Heiss (2012); Kato (2013). The phenomenology of $\mathcal{PT}$-symmetric Hamiltonians with second and third order EPs has been extensively explored in optical, mechanical, electrical, and acoustic experimental realizations El-Ganainy et al. (2018). Exceptional point degeneracies also occur in mode-selective lossy Hamiltonians, and this has enabled investigations of EP-related phenomena in dissipative systems in the classical Guo et al. (2009); Ding et al. (2016) and quantum Naghiloo et al. (2019); Li et al. (2019) domains, including the realization of a fourth-order EP with single photons Bian et al. (2019).

Many remarkable properties of non-Hermitian systems, such as asymmetric mode switching Doppler et al. (2016), topological energy transfer Xu et al. (2016), robust wireless power transfer Assawaworrarit et al. (2017), and enhanced classical sensitivity Wiersig (2014, 2016); Hodaei et al. (2017); Chen et al. (2017) are due to their EP degeneracies. In a sharp contrast with Hermitian Hamiltonians whose eigenmodes continue to span the space irrespective of eigenvalue degeneracies, in the non-Hermitian case, the eigenmodes of the Hamiltonian at an EP do not span the space and the deficit grows proportional to the order of the EP. This key difference is instrumental to the system sensitivity that scales with the order of the EP Hodaei et al. (2017); Chen et al. (2017) and has led to the tremendous interest developing systems with higher order EPs Zhao et al. (2018); Zhong et al. (2018); Wang et al. (2019), and understanding their fundamental quantum limits Lau and Clerk (2018). However, experimentally realizing classical or quantum systems with higher order EPs has proven extremely challenging and EPs beyond the fourth order have not been realized. In particular, integrated platforms where EPs of different orders can be realized are absent.

In this paper, we propose and theoretically investigate such a platform in a single, lossy waveguide beamsplitter in the quantum domain. When excited by a state with $N$ indistinguishable photons and confined to the $N$-photon subspace, we show that the dynamics of such beamsplitter has an EP of order $N+1$, which is observable with currently available and near-term number-resolving single-photon detectors Harder et al. (2016); Banchi et al. (2018); Magaña-Loaiza et al. (2019). In contrast to the past proposals with multiple waveguides or resonators, where precise parameter tuning is needed to ensure that the higher-order EP does not split into lower-order ones, we show that these EPs are robust due to the bosonic nature of photons and linear nature of the loss at low intensities.

The paper is structured as follows. In section II we present the formal treatment of the model in the photon-number basis and show analytical results for the time evolution of an arbitrary state. Results from numerical simulations based on NOON states are presented in Sec. III. The paper is concluded in Sec. IV with a discussion.

II Lossy beamsplitter in the photon-number basis

The general beamsplitter Hamiltonian in second-quantized notation is given by Lai et al. (1991)

[TABLE]

where $\hat{a}^{\dagger}$ ($\hat{a}$) and $\hat{b}^{\dagger}$ ($\hat{b}$) represent bosonic creation (annihilation) operators for photonic modes in the two waveguides, $\omega_{0}$ is their common propagation constant, the coupling between the two waveguides is given by $\kappa$, and $\Gamma$ is the dissipation coefficient of the lossy waveguide.

To unveil the link between the waveguide beamsplitter and arbitrary-order exceptional points, we represent the Hamiltonian Eq.  (1) in the two-mode, $N$-photon subspace. This subspace is spanned by $N+1$ orthonormal states $\left.|m\right):=\left|N-m,m\right>=\left|N-m\right>_{a}\left|m\right>_{b}$ ($0\leq m\leq N$) corresponding $(N-m)$ photons in the neutral waveguide and $m$ photons in the lossy waveguide. We emphasize that photon-number-resolving detection is necessary to access different basis states in this subspace [Fig. (1-a)]. In this basis, Eq.  (1) becomes Tschernig et al. (2018); Graefe et al. (2008).

[TABLE]

where $\hat{N}=\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}$ is the total photon-number operator. The Hermitian generators

[TABLE]

satisfy the angular momentum algebra $[\hat{J}_{z},\hat{J}_{x}]=i\hat{J}_{y}$, with $\hat{J}_{y}=i(\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger})/2$, and its cyclic permutations. Thus, in the $N$-photon subspace, $\hat{J}_{x}$ and $\hat{J}_{z}$ are spin $S=N/2$ representations of the angular momentum operators. In another, equivalent language, Hamiltonian (2) describes an $(N+1)$-mode tight-binding lattice with nearest-neighbor tunnelings given by the nonzero matrix elements of $\hat{J}_{x}$, i.e $f(n)=2\kappa\sqrt{n(N+1-n)}$ ($1\leq n\leq N$), and a linearly varying, on-site, loss potential given by the diagonal matrix elements of $\hat{J}_{z}$ [see Fig. (1-b)]. In the absence of loss, $\Gamma=0$, Eq.  (2) reduces to the perfect-state transfer model Christandl et al. (2004); Perez-Leija et al. (2013a, b); Joglekar and Saxena (2011); Joglekar et al. (2011); Chapman et al. (2016).

Within the $(N+1)$-dimensional subspace, the equidistant eigenvalues of the Hamiltonian are analytically given by the expression

[TABLE]

where $r=\{-S,-S+1,\ldots,S\}$. It follows from Eq. (5) that the adjacent difference $\Delta\lambda\equiv\lambda_{r}-\lambda_{r-1}=\sqrt{4\kappa^{2}-\Gamma^{2}}$ is purely real when the dissipation coefficient $\Gamma\leq 2\kappa$, and becomes purely imaginary when the dissipation coefficient is larger, i.e. $\Gamma>2\kappa$. At the transition point $\Gamma_{c}=2\kappa$, all eigenvalues become degenerate and all the eigenmodes coalesce, thus giving rise to an exceptional point of order $N+1$ Graefe et al. (2008). Figure (1-c) shows the (analytical) flow of eigenvalues for a lossy beamsplitter excited by $N=4$ photons, a realistic number that has been achieved in recent experiments Harder et al. (2016); Banchi et al. (2018); Magaña-Loaiza et al. (2019). The beamsplitter parameters are set to $\omega_{0}=\kappa=1$ cm*-1*. The top panel in Fig. (1-c) shows that as $\Gamma$ increases, $\Re(\lambda_{r})$ undergo level attraction and become degenerate at $\Gamma_{c}$, remaining constant thereafter. The bottom panel shows that $\Im(\lambda_{r})$ increase linearly with $\Gamma$ and are the same for all eigenmodes for $\Gamma\leq\Gamma_{c}$, whereas past the transition point, slowly (and rapidly) decaying eigenmodes emerge. It is worth to note that in a waveguide beamsplitter, the EP of order $N+1$ appears naturally in the $N$-photon subspace, and it is always located at $\Gamma_{c}=2\kappa$ irrespective of $N$.

To detect the order of the EP in an experimentally friendly manner León-Montiel et al. (2018), we consider the behavior of the intensity $I(z)$ within the $N$-photon subspace as a function of the propagation distance $z$, or equivalently, the time. In general, when the lossy beamsplitter is excited with an $N$-photon input, the number-resolving detectors at the output will register any of the $(N+1)(N+2)/2$ possibilities $|p\rangle_{a}|q\rangle_{b}$ where $0\leq p,q\leq N$ with $p+q\leq N$. Thus, $I(z)$ registers the fraction of trials where the total number of photons detected is exactly $N$, i.e. we post-select on the manifold where no photons are absorbed in the lossy waveguide Naghiloo et al. (2019). For a normalized initial state $|\psi(0)\rangle$, this intensity is given by

[TABLE]

where $G(z)=\exp(-i\hat{H}_{N}z)$ is the decaying time evolution operator. At the exceptional point $\Gamma=2\kappa$, the Hamiltonian $\hat{H}_{N}$ satisfies the characteristic equation $[\lambda_{r}-(\omega_{0}-i\kappa)N]^{N+1}=0$, and therefore the power series expansion for $G(z)$ terminates at $N^{\mathrm{th}}$ order in $z$. This implies the post-selection probability $I(z)\propto z^{2N}\exp(-N\Gamma z)$ at long distance $\kappa z\gg 1$. The numerically obtained post-selection probability $I(z)$, for input states where $N$ photons are injected into the neutral waveguide for $N\in\{5,7,10\}$, is shown in Fig. (1-d). It clearly shows that the order of the exceptional point is reflected in the results.

In order to obtain the photon-number-resolved population dynamics of the lossy beamsplitter, we note that the time-evolution operator satisfies the Schrödinger equation $i\partial_{z}G(z)=\hat{H}_{N}G(z)$. Therefore it can be expressed in terms of the total number operator $\hat{N}$ and the angular momentum operators $\hat{J}_{\alpha}$ (with $\alpha=x,y,z$) by using the Wei-Norman method Wei and Norman (1963); Louisell (1973),

[TABLE]

where $\hat{J}_{\pm}=\hat{J}_{x}\pm i\hat{J}_{y}$ are the angular momentum raising and lowering operator respectively. Note that since the photon-number operator $\hat{N}$ commutes with $\hat{J}_{\alpha}$, it can be treated as a $c$-number. $f_{\pm}(z),f_{z}(z)$ are three complex functions that parameterize the non-unitary time evolution operator, and satisfy the following set of coupled, nonlinear differential equations,

[TABLE]

The solutions of Eqs. (8), subject to the initial condition $G(0)=\mathbb{1}$, or equivalently $f_{\pm}(0)=f_{z}(0)=0$, are given by

[TABLE]

We note that the functions $f_{+}(z)=f_{-}(z)$ are real irrespective of whether $\Delta\lambda$ is real or purely imaginary, while $f_{z}(z)$ is, in general, complex. It is straightforward to check that as the system approaches the exceptional point, i.e. $\Delta\lambda\rightarrow 0$, the functions $f_{\pm}(z)\approx\kappa z/(1+\kappa z)$ approach unity at $\kappa z\gg 1$. On the other hand, the function $f_{z}(z)\approx-2i\ln(1+\kappa z)$, in conjunction with the diagonal operator $\hat{J}_{z}=\mathrm{diag}(-N/2,\ldots,N/2)$, gives rise to an algebraically growing time evolution operator $G(z)\propto z^{N}\exp(-N\Gamma z/2)$. Thus, our exact solution, Eq.  (7), encodes the order of the exceptional point.

III Post-selected dynamics: numerical results

Motivated by the realization of high-order multiphoton entangled states Afek et al. (2010); Zhang et al. (2018), we explore the dynamics of the lossy beamsplitter excited with NOON-state initial conditions, i.e. $|\phi(0)\rangle=\left[|N\rangle_{a}|0\rangle_{b}+|0\rangle_{a}|N\rangle_{b}\right]/\sqrt{2}=\left[\left|0\right)+\left|N\right)\right]/\sqrt{2}$. Although the post-selection probability $I(z)=\langle\phi(z)|\phi(z)\rangle$ decreases exponentially with the propagation distance, we will see that within the post-selected $N$-photon manifold, signatures of the $\mathcal{PT}$-symmetry breaking transition and the order of the exceptional point are clearly visible. To that end, we consider the normalized, $z$-dependent occupation function

[TABLE]

which satisfies $\sum_{m=0}^{N}P(\left|m\right);z)=1$. Figure (2) shows the results for this occupation function for two different input states. The top row in Fig. (2) shows the normalized mode occupations as a function of $z$ for an $N=5$ state. When $\Gamma=\Gamma_{c}/4$, panel (a), we see an asymmetric, oscillatory motion across the six modes with an energy flow from a low-loss region to the high-loss region. At the EP, $\Gamma=\Gamma_{c}$, the system reaches a steady state with a weight distributed largely in the low-loss region [panel (b)] . Past the transition, $\Gamma=1.2\Gamma_{c}$, the steady-state is reached slower [panel (c)], indicating the emergence of slowly decaying eigenmodes for the Hamiltonian Eq.  (2). The bottom row in Fig. (2) shows corresponding results for an $N=8$ NOON state input. Comparing the two rows, it is clear that the period of asymmetric oscillations does not depend on $N$ and the order of the EP is reflected in the post-selected, $N$-photon manifold results.

IV Discussion

Despite tremendous interest due to the classical sensitivity enhancement they offer Wiersig (2014, 2016); Hodaei et al. (2017); Chen et al. (2017), experimental realizations of exceptional points of higher order has remained elusive. The primary obstacle for such realizations in coupled waveguides, resonators, or other traditional platforms is the fine tuning of system parameters that is required by a higher-order symmetry necessary for eigenmode degeneracy Teimourpour et al. (2018). When such stringent constraints regarding the ratio of losses or nearest-neighboring coupling amplitudes are not satisfied, a higher order EP splits into EPs of lower order.

Here, we have shown that a single, lossy waveguide beamsplitter can be used to realize robust EPs of arbitrary order without any fine tuning required. In our proposal, the stringent symmetries required for higher-order EPs are guaranteed by the bosonic nature of input photons, and the linear nature of loss. We have shown that the dynamics observed within the post-selected $N$-photon subspace has an EP of order $N+1$. Thus, our analysis passes the burden of fine-tuning the Hermitian and lossy parts of the Hamiltonian on to the dual tasks of creating higher-order NOON states and number-resolving photon-detectors, a rapidly maturing technology found in quantum optics laboratories across the globe Lita et al. (2008); Harder et al. (2016); Banchi et al. (2018); Magaña-Loaiza et al. (2019). Our results, therefore, offer a realistic pathway for realizing EPs of arbitrary order on demand in a single platform.

Acknowledgements

This work was supported by CONACyT under the project CB-2016-01/284372 (MAQJ and RJLM) and by the NSF grant DMR-1054020 (YNJ). BMRL acknowledges financial support from the Marcos Moshinsky Foundation through the 2018 Marcos Moshinsky Young Researcher Chair. OSML acknowledges startup funding from Louisiana State University.