Higher-order exceptional point in a blue-detuned non-Hermitian cavity optomechanical system

Cavity optomechanical (COM) systems, emerged as a promising platform in quantum information science, have been paid considerable attention both theoretically and experimentally Aspelmeyer . The simplest COM system is made up of a mechanical resonator (MR) nonlinearly coupled to a cavity via radiation pressure, which can be well controlled by strong driving fields. In such mystical systems, abundant effects including sensing Schreppler-2014 ; Wu-2017 ; Gil-Santos-2020 ; Fischer-2019 , ground-state cooling Chan-2011 ; Teufel-2011 , squeezed light generation Purdy-2014 ; Safavi-Naeini-2013 ; Aggarwal-2020 , nonreciprocal transport Xu-2019 ; Shen-2016 , optomechanically induced transparency Kronwald-2013 ; Weis-2010 ; Liuy-2013 , coupling enhancement Xiong-2021 ; Lu-2013 ; Xiong2-2021 , and nonlinear behaviors (e.g., bi- and tristability and chaos ) Lu-2015 ; Xiong3-2016 have been investigated.

In addition, COM systems have shown huge potential in studying exceptional points (EPs) of non-Hermitian systems Xiongwei-2021 ; Jing-2014 ; Xu-2016 ; LYL-2017 ; ZhangJQ-2021 ; XuH-2021 ; Xuxw-2015 , at which both eigenvalues and eigenvectors coalesce. This is due to the fact that practical COM systems can be characterized by effective non-Hermitian Hamiltonians when decoherence arising from surrounding environment is considered. Moreover, the driven COM systems can provide fine-tuning parameters for requirement of realizing EPs, assisted by strong driving fields. Owing to these, EPs have been intensively studied in COM systems, especially for the second-order EPs (EP2s) Jing-2014 ; Xu-2016 ; LYL-2017 ; ZhangJQ-2021 ; XuH-2021 ; Xuxw-2015 where two eigenvalues and the corresponding eigenvectors coalesce Minganti-2019 ; Zhang-2021 ; Ozdemir-2019 ; Mostafazadeh1-2002 ; Mostafazadeh2-2002 ; Konotop-2016 ; Bender-2013 ; Parto-2021 ; Bergholtz-2021 ; Wiersig-2020 ; Feng-2017 ; El-Ganainy-2018 . Besides, EP2s are also studied in other systems  Doppler-2016 ; Zhang-2017 ; Harder-2017 ; Quijandria-2018 ; Zhang-2019-2 ; Naghiloo-2019 ; ZhangGQ . Around EP2s, lots of fascinating phenomena like unidirectional invisibility Peng-2014 ; Lin-2011 ; Chang-2014 , single-mode lasing Feng-2014 ; Hodaei-2014 , sensitivity enhancement Chen-2017 ; Hokmabadi-2019 , energy harvesting Fern-2021 , protecting the classification of exceptional nodal topologies Marcus-2021 , electromagnetically induced transparency Guo-2009 ; Wang-2019 ; Wang1-2020 ; Lu-2021 , and quantum squeezing Miranowicz-2019 ; Mukherjee-2019 ; Luo-2022 can be studied.

Instead of EP2s, non-Hermitian systems can also host higher-order EPs, where more than two eigenmodes coalesce into one Graefe-2008 ; Heiss-2008 ; Demange-2012 ; Heiss-2016 ; Jing-2017 ; Ge-2015 ; Lin-2016 ; Quiroz-2019 ; Bian-2020 . It has been shown that higher-order EPs can exhibit greater advantages than EP2s in spontaneous emission enhancement Lin-2016 , sensitive detection Hodaei-2017 ; Zeng-2021 ; Wang-2021 ; Zeng-2019 , topological characteristics Ding-2016 ; Delplace-2021 ; Mandal-2021 . With these superiorities, higher-order EPs are being intensively studied in various systems Roy-2021 ; Zhong-2020 ; Zhang-2020 ; Pan-2019 ; Zhang-2019 ; Kullig-2019 ; Kullig-2018 ; Schnabel-2017 ; Nada-2017 ; Wang-2020 but attract less attention in non-Hermitian COM systems. For this, how to construct higher-order EPs in non-Hermitian COM systems is strongly demanded.

We also note that EPs in non-Hermitian COM systems, including EP2s and EP3s, are mainly focused in the red-sideband regime Xiongwei-2021 ; Jing-2014 . In this regime, fast oscillating terms related to mode squeezing are neglected. This limits nonclassical quantum effects such as quantum squeezing investigation around EPs. For this, we here theoretically propose a paradigmatic COM system consisting of a MR coupled to two cavities via radiation pressure for predicting EP3s, where two cavities are respectively passive (loss) and active (gain), and driven by two blue-detuned classical fields. First, we derive an effective non-Hermitian Hamiltonian for the proposed COM system and analytically give the pseudo-Hermitian condition of the proposed COM system in the general case. Then, three scenarios are specifically considered in the pseudo-Hermitian condition: (i) the neutral MR; (ii) the passive MR; (iii) the active MR. In case (i), the proposed non-Hermitian COM system with symmetric coupling strength can host both two degenerate EP3s and two non-degenerate EP3s in the parameter space. When we tune the system parameters deviation from the critical paraters at EP3s, four non-degenerate EP2s can be predicted, which is different from the situation in the red-sideband non-Hermitian COM system. For the cases (ii) and (iii), the proposed non-Hermitian COM system is required to have asymmetric coupling strength for satisfying pseudo-Hermitian condition. We find only two degenerate EP3s or two degenerate EP2s can be predicted. By investigating the effects of system paramters on EP3s or EP2s, we find large coupling strength or large frequency detuning is benefit to observe EPs more clearly. Our proposal provides a promising path to study nonclassical quantum effects around EP2s and EP3s in non-Hermitian COM systems, and it is the first scheme to study higher-order EPs in the blue-detuned COM system, although two-mode quantum squeezing has been investigated in a system with pseudo-anti-parity-time symmetry Luo-2022 .

This paper is organized as follows. In Sec. II, the model is described and the system effective Hamiltonian is given. Then we derive the pesudo-Hermitian condition for the considered non-Hermitian COM system in Sec. III. In Sec. IV, critical parameters of the proposed COM system at EP3 are anatically derived. In Sec. V, phase diagram of the descriminant for the characteristic equation is studied to predict EP3 and EP2. In Sec. VI, EP3 and EP2 in three cases are specifically studied. Finally, a conclusion is given in Sec. VII.

We consider an experimental three-mode optomechanical system Dong-2012 ; Hill-2012 ; Andrews-2014 consisting of two driven cavities (labeled as cavity $a$ and cavity $c$) coupled to a MR with frequency $\omega_{b}$ via radiation pressure (see Fig. 1). At the rotating frame respect to two laser fields, the Hamiltonian of the total system can be written as (setting $\hbar=1$) Zhangk-2015

	$\displaystyle H_{\rm total}=$	$\displaystyle\delta_{a}a^{\dagger}a+\omega_{b}b^{\dagger}b+\delta_{c}c^{\dagger}c$
		$\displaystyle+g_{a}a^{\dagger}a(b^{\dagger}+b)+g_{c}c^{\dagger}c(b^{\dagger}+b)+H_{D},$		(1)

where $\delta_{a(c)}=\omega_{a(c)}-\nu_{a(c)}$, with $\omega_{a(c)}$ being the frequency of the cavity $a~{}(c)$ and $\nu_{a(c)}$ the frequency of the laser field acting on the cavity $a~{}(c)$, is the frequency detuning of the cavity $a~{}(c)$ from the laser field acting on the cavity $a~{}(c)$. $g_{a}$ and $g_{c}$ are the single-photon optomechanical coupling strengths of the MR coupled to the cavity $a$ and cavity $c$, respectively. The operators $a~{}(c)$ and $a^{\dagger}~{}(c^{\dagger})$ are the annihilation and creation operators of the cavity $a~{}(c)$. The last term $H_{D}=i(\Omega_{a}a^{\dagger}+\Omega_{c}c^{\dagger})+{\rm H.c.}$ in Eq. (1) represents the coupling between two cavities and two laser fields with Rabi frequencies $\Omega_{a}$ and $\Omega_{c}$. As we are interested in the blue-sideband regime of the proposed COM system, thus $\delta_{a},~{}\delta_{c}<0$ is assumed below.

In the strong-field limit, the nonlinear COM system can be linearized by writing each operator as the expectation value ($a_{s},~{}b_{s},~{}c_{s}$) plus the corresponding fluctuation ($\delta a,~{}\delta_{b},~{}\delta_{c}$). Neglecting the higher-order fluctuations, the linearized Hamiltonian including dissipations can be given by (see details in the Appendix)

	$\displaystyle H_{\rm eff}=$	$\displaystyle(\delta_{a}^{\prime}-i\kappa_{a})\delta a^{\dagger}\delta a+(\omega_{b}-i\gamma_{b})\delta b^{\dagger}\delta b+(\delta_{c}^{\prime}-i\kappa_{c})\delta c^{\dagger}\delta c$
		$\displaystyle+G_{a}(\delta a^{\dagger}\delta b^{\dagger}+\delta a\delta b)+G_{c}(\delta b^{\dagger}\delta c^{\dagger}+\delta b\delta c).$		(2)

Here, fast oscillating terms have been discarded with the condition $|\delta_{a(c)}^{\prime}+\omega_{b}|\ll|\delta_{a(c)}^{\prime}-\omega_{b}|$ and $|G_{a(c)}|\ll|\delta_{a(c)}^{\prime}|$, where $\delta_{a(c)}^{\prime}$ is the effective frequency detuning of the cavity $a~{}(c)$ shifted by the displacement of the mechanical resonator, and $G_{a(c)}$ is the effective optomechanical coupling strength enhanced by the photon number in the cavity $a~{}(c)$. This effective Hamiltonian is the typical three-mode squeezing Hamiltonian without dissipations. For convenience, we assume $G_{a}$ and $G_{c}$ to be real, which can be realized by tuning the phase of two laser fields.

The effective Hamiltonian in Eq. (2) can also be equivalently expressed as

$$H_{\rm eff}=\left(\begin{array}[]{ccc}\delta_{a}^{\prime}-i\kappa_{a}&G_{a}&0\\ -G_{a}^{*}&-\omega_{b}-i\gamma_{b}&-G_{c}^{*}\\ 0&G_{c}&\delta_{c}^{\prime}-i\kappa_{c}\end{array}\right)$$

(3)

is just the matrix form of $H_{\rm eff}$. For the non-Hermitian Hamiltonian in Eq. (3), three eigenvalues can be predicted. When these three eigenvalues are all real, or one is real and the other two are a complex-conjugate pair, the considered three-mode optomechanical system characterized by the Hamiltonian in Eq. (2) or Eq. (3) is pseudo-Hermitian Mostafazadeh1-2002 ; Mostafazadeh2-2002 . For the pseudo-Hermitian systems, the characteristic polynomial equation

$\displaystyle|H_{\rm eff}-\Omega\mathbb{I}|=0$

(4)

is the same as

$\displaystyle|H_{\rm eff}^{*}-\Omega\mathbb{I}|=0,$

(5)

where $H_{\rm eff}^{*}$ is the complex conjugate transpose of $H_{\rm eff}$, $\mathbb{I}$ is a $3\times 3$ identity matrix, and $\Omega$ denotes the eigenvalue of the effective Hamiltonian $H_{\rm eff}$. By expanding Eqs. (4) and (5), and comparing the corresponding coefficients, we can obtain

	$\displaystyle\kappa_{a}+\gamma_{b}+\kappa_{c}=$	$\displaystyle 0,$
	$\displaystyle\gamma_{b}(\delta_{a}^{\prime}+\delta_{c}^{\prime})=$	$\displaystyle\kappa_{a}(\omega_{b}-\delta_{c}^{\prime})+\kappa_{c}(\omega_{b}-\delta_{a}^{\prime}),$		(6)
	$\displaystyle(\delta_{a}^{\prime}\omega_{b}+\kappa_{a}\gamma_{b})\kappa_{c}=$	$\displaystyle G_{a}^{2}\kappa_{c}+G_{c}^{2}\kappa_{a}+(\delta_{a}^{\prime}\gamma_{b}-\kappa_{a}\omega_{b})\delta_{c}^{\prime}.$

By setting

$\displaystyle\eta=$

$\displaystyle\kappa_{a}/\kappa_{c},~{}~{}~{}\lambda=G_{c}/G_{a},~{}~{}\Delta_{\rm a(c)}=\delta_{a(c)}^{\prime}+\omega_{b},$

(7)

Eq. (III) can be further simplified as

	$\displaystyle\gamma_{b}+(1+\eta)\kappa_{c}=0,$
	$\displaystyle\Delta_{c}+\Delta_{a}\eta=0,$		(8)
	$\displaystyle(1+\lambda^{2}\eta)G_{a}^{2}+\eta(1+\eta)(\Delta_{a}^{2}+\kappa_{c}^{2})=0.$

Obviously, only when the conditions in Eq. (III) are simutaneously satisfied, the considered three-mode optomechanical system is pseudo-Hermitian. From the first condition in Eq. (III), we can see that the decay rates from the cavity $a$, the mecahanical resonator and the cavity $c$ are required to be balanced. This means gain effect must be introduced to the considered system. From the third condition, $\eta<0$ is obtained, which shows one loss cavity and the other gain cavity are always needed to satisfy the pseudo-Hermitian condition for the proposed COM system in the blue-sideband regime. This situation is completely different from the previous study of EPs using a COM system in the red-sideband regime. Without loss of generality, the cavity $a$ with gain, and the cavity $c$ with loss are taken, i.e., $\kappa_{a}<0$ and $\kappa_{c}>0$. From the third equation in Eq. (III), it is not difficult to find the fact that $\lambda=1$ when $\eta=-1$, which indicates that the coupling strengths between the MR and two cavities must be uniform, i.e., $G_{a}=G_{c}$. When $\eta\neq-1$,

$\displaystyle(1+\eta)(1+\lambda^{2}\eta)>0~{}~{}{\rm for}~{}~{}\eta\neq-1$

(9)

is directly given by the third equality in Eq. (III), which in turn gives rise to a boundary for the parameter $\lambda$ or equivalently $G_{a}$ and $G_{c}$. Such the boundary can be achieved here owing to the tunable parameters $\Delta_{a},~{}\Delta_{c},~{}G_{a}$ and $G_{c}$.

When the pseudo-Hermitian conditions in Eq. (III) are satisfied and $x=\Omega+\omega_{b}$ is defined, the characteristic equation in Eq. (4) reduces to

$\displaystyle x^{3}+c_{2}x^{2}+c_{1}x+c_{0}=0,$

(10)

where

	$\displaystyle c_{2}=$	$\displaystyle(\eta-1)\Delta_{a},$
	$\displaystyle c_{1}=$	$\displaystyle(1+\lambda^{2})G_{a}^{2}-\eta\Delta_{a}^{2}+(1+\eta+\eta^{2})\kappa_{c}^{2},$		(11)
	$\displaystyle c_{0}=$	$\displaystyle(\eta-\lambda^{2})G_{a}^{2}\Delta_{a}-(1+\eta)^{2}(1-\eta)\kappa_{c}^{2}\Delta_{a}.$

According to Cardano’s formula kORN-1968 , the solutions of this characteristic equation is determined by the discriminant

$\displaystyle D=B^{2}-4AC,$

(12)

where

$\displaystyle A=$

$\displaystyle c_{2}^{2}-3c_{1},~{}~{}B=c_{1}c_{2}-9c_{0},~{}~{}C=c_{1}^{2}-3c_{0}c_{2}.$

(13)

For $D<0$, Eq. (10) has three real roots. But for $D>0$, Eq. (10) only has one real root and the other two are complex roots. Interestingly, three roots coalesce to the same value $\Omega_{\rm EP3}$ at $D=0$ with $A=B=0$, which is so-called EP3. For the case of $D=0$ but $A\neq 0$ and $B\neq 0$, only two roots coalesce to the value $\Omega_{\rm EP2}$, corresponding to EP2.

Below we analytically derive the critical parameters at EP3. When EP3 appears at $\Omega=\Omega_{\rm EP3}$, we have

$\displaystyle(\Omega-\Omega_{\rm EP3})^{3}=0$

(14)

Comparing coefficients of this equation with Eq. (10),

$\displaystyle-3x_{\rm EP3}=c_{2},~{}~{}3x_{\rm EP3}^{2}=c_{1},~{}~{}x_{\rm EP3}^{3}=-c_{0}$

(15)

are obtained. The first equation directly leads to

$\displaystyle x_{\rm EP3}=\frac{1}{3}(1-\eta)\Delta_{a}.$

(16)

Substituting this solution back into the second equation in Eq. (15), the critical coupling strength at EP3 is given by

$\displaystyle G_{\rm a,EP3}=$

$\displaystyle 2\kappa_{c}\left[-\frac{3(1+\lambda^{2})}{1+\eta+\eta^{2}}-\frac{1+\lambda^{2}\eta}{\eta(1+\eta)}\right]^{-1/2},$

(17)

where

$\displaystyle\Delta_{\rm a,EP3}=\pm\left[-\frac{1+\lambda^{2}\eta}{\eta(1+\eta)}G_{\rm a,EP3}^{2}-\kappa_{c}^{2}\right]^{1/2}$

(18)

is derived from the third equation in Eq. (III). As $\Delta_{\rm a,EP3}^{2}\geq 0$, so the minimal value of $G_{a}$ for predicting EP3 is

$\displaystyle G_{\rm a,EP3}^{\rm min}=\left[-\frac{\eta(1+\eta)}{1+\lambda^{2}\eta}\kappa_{c}\right]^{1/2}.$

(19)

At EP3, the parameter $\lambda$ is required to meet

$\displaystyle\lambda_{\rm EP3}=$

$\displaystyle\left[\frac{2\eta+1}{\eta(\eta+2)}\right]^{3/2},$

(20)

which is obtained by substituting the solution in Eq. (16) back into the third equality in Eq. (15). To see the dependent relationship between $\lambda$ and $\eta$ at EP3 more clearly, we plot $\lambda$ as a function of $\eta$ in Fig. 2. Obviously, $\lambda$ monotonously decreases with the absolute value of $\eta$. Eq. (20) also requires $\eta$ to satisfy

$\displaystyle(\eta+2)(2\eta+1)<0.$

(21)

Combine Eqs. (9) and (21) together, the parameter $\eta$ for predicting EP3 can take

$$\left\{\begin{aligned} -2<&\eta<-1,\\ &\eta=-1,\\ -1<&\eta<-\frac{1}{2},\end{aligned}\right.$$

(22)

leading to

$$\left\{\begin{aligned} &\gamma_{b}>0,~{}~{}~{}~{}~{}{\rm loss~{}mechanical~{}resonator},\\ &\gamma_{b}=0,~{}~{}~{}~{}~{}{\rm neural~{}mechanical~{}resonator},\\ &\gamma_{b}<0,~{}~{}~{}~{}~{}{\rm gain~{}mechanical~{}resonator}.\end{aligned}\right.$$

(23)

The corresponding value of $\lambda_{\rm EP3}$ is given by Eq. (20).

Next, we numerically predict EP3 and EP2 via phase diagram of the discriminant [see Eq. (12)] with in the three cases given by Eq. (22).

Note that our studies are constrainted in the pseudo-Hermitian condition, which can ensure the emergence of EPs in our proposed non-Hermitian COM system. But actually, the strict pseudo-Hermitian condition in general can not be fully satisfied, which means that the pseudo-Hermitian condition is broken [see Eq. (III) or Eq. (III)]. For example, the gain and loss in Eq. (III) are not balanced, i.e., $\kappa_{a}+\kappa_{b}+\kappa_{c}\neq 0$. In this situation, we find that both EP3 and EP2 can also be predicted in our setup, as shown in Fig. 8 where $\eta=-1$ and $\kappa_{a}+\kappa_{b}+\kappa_{c}=0.1\kappa_{c}$ is taken. This shows that EPs in our proposal are robust against the slightly unbalanced gain and loss, which also reveals that the pseudo-Hermitian condition is neither sufficient nor necessary condition for predicting EPs. For the case of $\eta\neq-1$, we also numerically check it, and the same result is obtained.

In summary, we have proposed a blue-detuned non-Hermitian cavity optomechanical system consisting of a MR coupled to both a passive (loss) and an active (gain) cavities via radiation pressure for predicting EP3s. Under the pseudo-Hermitian condition, the cases of the neural, loss and gain MRs are considered. By investigating the phase diagram of the discriminant, we find that both two degenerate or two non-degenerate EP3s can be predicted by tuning system parameters in the parameter space for the neutral MR. Also, four non-degenerate EP2s can be observed when system parameters deviate from EP3s, which is distinguished from the previous study in the red-detuned optomechanical system. For the gain (loss) MR, we find only two degenerate EP3s or EP2s can be predicted by tuning enhanced coupling strength. By studying the effect of parameters on EP3s or EP2s, we show that large parameters, such as frequency detuning and enhanced optomechanical coupling strength, can be employed to observe EPs more clearly. Our proposal is the first scheme to study higher-order EPs in the blue-detuned COM system, and it provides a potential way to investigate multimode quantum squeezing effects around higher-order EPs

This paper is supported by the key program of the Natural Science Foundation of Anhui (Grant No. KJ2021A1301), the National Natural Science Foundation of China (Grants No. 12205069, No. 11904201 and No. 12104214), and the Natural Science Foundation of Hunan Province of China (Grant No. 2020JJ5466).