# Hybrid coupling optomechanically assisted nonreciprocal photon blockade

Yong-Pan Gao<EMAIL_ADDRESS>School of Electronic Engineering and the
State Key Laboratory of Information Photonics and Optical Communications,
Beijing University of Posts and Telecommunications, Beijing 100876, China
Chuan Wang<EMAIL_ADDRESS>School of Artificial Intelligence, Beijing
Normal University, Beijing 100875, China

###### Abstract

The properties of open quantum system in quantum information science is now
extensively investigated more generally by the research community as a
fundamental issue for a variety of applications. Usually, the states of the
open quantum system might be disturbed by the decoherence which will reduce
the fidelity in the quantum information processing. So it is better to
eliminate the influence of the environment. However, as part of the composite
system, rational use of the environment system could be beneficial to quantum
information processing. Here we theoretically studied the environment induced
quantum nonlinearity and energy spectrum tuning method in the optomechanical
system. And we found that the dissipation coupling of the hybrid dissipation
and dispersion optomechanical system can induce the coupling between the
environment and system in the cross-Kerr interaction form. When the symmetry
is broken with a directional pumping environment, the system exhibits the non-
reciprocal behavior during the photon excitation and photon blockade for the
clockwise and counterclockwise modes of the whispering-gallery mode
microcavity. Furthermore, we believe that the cross-Kerr coupling can also be
used in a more widely region in quantum information processing and quantum
simulation.

††preprint: APS/123-QED

##  I Introduction

Quantum information processing and quantum metrology rely on the efficient
manipulation of the qubits which have attracted intense interest during the
past decades. It has been found that the system for quantum information
processing are not isolated and may be characterized by the open quantum
system. Furthermore, the decoherence of the environment may induce noise and
reduce the fidelity of the quantum states. On the other hand, the rational use
of the environment system may be beneficial to the quantum system. Here we are
intending to investigate the dynamics of the dissipation process based on the
cavity optomechanical system. Recently, cavity optomechanical system is
introduced into the study of quantum information science. Cavity
optomechanical system [1, 2, 3, 4, 5, 6] is an artificial microstructure that
combines the interaction between the mechanical mode and the optical mode with
various effect could be observed on it. For example, the optomechanical
induced transparency [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], optomechanical
assisted nonlinearity [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], phonon lasers
[27, 28, 29, 30, 31, 32], in the semi-classical region; the photon blockade
[33, 34, 35, 36, 37, 38, 39], mechanical resonator cooling [40, 41, 42, 43,
44, 45], nonclassical state preparation [46, 47, 48, 49, 50, 51, 52, 53, 54] ,
and the fundamental quantum properties [55, 56, 57, 58, 59] in the full
quantum region.

The geometric deformation characteristics of the optomechanics and the
geometric deformation correlation of its optical properties endow the optical
machine with unique and excellent physical properties. Different from other
systems, such as the cavity-atom coupled system [60, 61, 62, 63, 64] and
cavity-magnetic coupled system [65, 66, 67, 68, 69], optomechanics provides us
a platform to investigate the properties of the non-static coupling between
the system and the environment. The dissipative optomechanics describes the
coupling of the optical mode to the environment of a cavity [70, 71, 72, 73,
74, 75, 76, 77], when it is combined with the dispersion optomechanics [7, 9,
10, 11], the hybridization of the dissipative optomechanics with the
dispersion optomechanics shows potential applications in pure optical
manipulation. The hybrid systems have become a key strategy in microwave
electromechanical system [78, 79, 80, 81, 82]. Recently, various studies have
shown that such kind of optomechanical system with both dispersive coupling
and dissipative coupling can be realized in the whispering-gallery cavities
[83, 84, 85, 86, 87, 88].

In this study, we present a full quantum approach that describes the
dissipation and dispersion hybrid coupled optomechanical system. First, we
developed the self-Kerr effect based on the dispersive coupling, and
illustrated the cross-Kerr effect of the environment caused by dissipative
coupling on the system. In order to show the potential applications of the
cross Kerr effect, relying on the directional coherent manipulation of the
cross-Kerr interaction, this approach is further applied to achieve the
tunable non-reciprocity between the intrinsic optical clockwise and counter-
clockwise modes, and also the non-reciprocal photon blockade.

##  II The dissipation-dispersion hybrid optomechanics

Figure 1: The hybrid coupling optomecanics and its energy level scheme. The
whisper gallery mode (WGM) has an unstable frequency
$\omega_{R}+g_{\text{s}}(X)$ and dissipation $\kappa+g_{\text{d}}(X)$. Both
the clockwise (CW) and counterclockwise (CCW) of thie WGM can be excited. An
auxiliary field is weakly coupled with the CCW mode and tuning it resonant
features through the dissipation induced cross-kerr effect. Here we studied
the quantum feature of the weak drive field which is strong coupled with the
cavity with the frequency $\omega_{\text{L}}$.

The system is shown in FIG.1, the whispering-gallery mode (WGM) microcavity
supports both the optical mode and the mechanical breath mode. Due to the
optomechanical interaction, the mechanical breach mode will change both the
curvature [89] and effective length of the microcavity, in which the
dissipation and dispersion hybrid coupling optomechanics appears. The
mechanical mode has the resonant frequency of $\omega_{M}$ and negligible
damping rate, while the optical mode has the position dependent frequency
$\omega_{R}+g_{s}(X)$ and dissipation rate $\kappa+g_{d}(X)$. The
optomechanical microcavity is coupled with the add-drop fiber tapers. For the
two fiber waveguides, the add fiber is strongly coupled with the cavity, and
the weak driving field is input and output with the frequency $\omega_{L}$
through this fiber; the drop fiber is weakly coupled with the cavity (The feed
back of the cavity is negligible), the input field plays the role of
controllable environment which is defined as the auxiliary field. The
Hamiltonian of the dissipation-dispersion hybrid optomechanical system in
FIG.1 could be expressed as (with $\hbar=1$)

$\hat{H}=\omega_{\text{R}}\hat{a}^{\dagger}\hat{a}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c}+\sum_{q}\omega_{q}\hat{b}_{q}^{\dagger}\hat{b}_{q}+\hat{H}_{\text{d}}+\hat{H}_{\text{s}}+\hat{H}_{\gamma}$
(1)

Here the optical mode has the frequency of $\omega_{R}$ with the annihilation
operator denoted as $\hat{a}$. $\hat{c}$ denotes the annihilation operator of
the mechanical mode whose frequency is $\omega_{M}$. $\hat{b}_{q}$ is the
annihilation operator of the environment with the frequency $\omega_{q}$. And
$H_{d}$ corresponds to the dissipation coupling Hamiltonian, $H_{s}$
represents the dispersion coupling Hamiltonian, and $H_{\gamma}$ shows the
dissipation part of the system. The present system could be achieved both in
the microwave electro-mechanical system [78] and optical WGM system. The
microwave system offer easy manipulation and processing, yet have limited
thermal noise, while the present system enlarge the hybrid optomechics from
the microwave to optical wavelength. It can have much less thermal noise than
the microwave case, but is usually hard to approach the strong coupling regime
of the system because the co-existence of the dissipation and the dispersion
coupling.

The dissipative optomechanical coupling part could be described as

$\hat{H}_{\text{d}}=g_{\text{d}}(X)\equiv
X\frac{\text{d}\,\kappa}{\text{d}x}(\hat{b}_{q}\hat{a}^{\dagger}-\hat{b}_{q}^{\dagger}\hat{a}).$
(2)

here $X=\hat{c}^{\dagger}+\hat{c}$, and the dispersion coupling term is

$\hat{H}_{\text{s}}=g_{\text{s}}(X)\hat{a}^{\dagger}\hat{a}=g_{\text{s}}\hat{X}\hat{a}^{\dagger}\hat{a}.$
(3)

For simplicity, we set
$\hat{A}^{\dagger}=\sum_{q}\hat{b}_{q}\hat{a}^{\dagger}$. And the Hamilton
could be written as

$\hat{H}=\frac{\omega_{\text{R}}}{\hat{n}_{q}}\hat{a}^{\dagger}\hat{a}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c}+X[ig_{\text{d}}(\hat{A^{\dagger}-\hat{A}})+\frac{g_{\text{s}}}{\hat{n}_{q}}\hat{A}^{\dagger}\hat{A}].$
(4)

As $\hat{n}_{q}=\sum_{q}\hat{b}_{q}^{\dagger}\hat{b}_{q}$, we assignment the
above Hamilton as

$\displaystyle\hat{H}_{0}$
$\displaystyle=\Omega_{\text{R}}\hat{A}^{\dagger}\hat{A}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c},$
(5a) $\displaystyle\hat{W}$
$\displaystyle=X\left[ig_{\text{d}}(\hat{A}^{\dagger}-\hat{A})+\frac{g_{\text{s}}}{\hat{n}_{q}}\hat{A}^{\dagger}\hat{A}\right],$
(5b) $\displaystyle\hat{H}$ $\displaystyle=\hat{H}_{0}+\hat{W}.$ (5c)

By choosing the quadrature amplitudes as
$\hat{P}=i(\hat{A}^{\dagger}-\hat{A})$ and
$\hat{Q}=\hat{A}^{\dagger}+\hat{A}$, then we have

$\hat{W}=X[g_{\text{d}}\hat{P}+\frac{g_{\text{s}}}{4|B|^{2}}(\hat{Q}^{2}+\hat{P^{2}})].$
(6)

Figure 2: The energy scheme and shift of the hybrid optomechanical system.
(a). How the dispersion and dissipation coupling modulation the energy scheme.
(b). The energy shift $\Delta_{d}$ various with the auxiliary field strength
$\mathcal{E}$ under different dissipation coupling strength
$g_{\text{d}}/\omega_{\text{M}}=0.1,0.5,1$.

With the definition as $B=\langle\sum_{q}\hat{b}_{q}\rangle$, we perform the
transformation by using the operator $U_{1}=\text{e}^{S_{1}}$,where
$S_{1}=i\frac{g_{\text{d}}|B|^{2}}{g_{\text{s}}}(\hat{A}^{\dagger}+\hat{A})$.
Then we can get

$\begin{split}\hat{H}=&\frac{\Omega_{R}}{\hat{n}_{p}}(A-i\frac{g_{\text{d}}\hat{n}_{q}}{g_{\text{s}}})(A^{\dagger}+i\frac{g_{\text{d}}\hat{n}_{q}}{g_{\text{s}}})+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c}\\\
+&X\left[\frac{g_{\text{s}}}{\hat{n}}\hat{A}^{\dagger}\hat{A}+\frac{g_{\text{d}}^{2}\hat{n}_{q}}{g_{\text{s}}}\right],\end{split}$
(7)

here the term $\hat{A}-\hat{A}^{\dagger}$ is the environment driven term that
can be eliminated by the out-phase drive. Then, the Hamiltonian of the system
could be changed to

$\hat{H}=\frac{\Omega_{R}}{\hat{n}}\hat{A}^{\dagger}\hat{A}+\omega_{M}\hat{c}^{\dagger}\hat{c}+X\left[\frac{g_{\text{s}}}{\hat{n}}\hat{A}^{\dagger}\hat{A}+\frac{g_{\text{d}}^{2}\hat{n}_{q}}{g_{\text{s}}}\right].$
(8)

The environment is single mode $\hat{n}_{q}=\hat{b}^{\dagger}\hat{b}$. Then by
using the transformation operator
$S_{2}=\exp{(\hat{c}^{\dagger}-\hat{c})(\frac{g_{\text{s}}}{\hat{n}}\hat{A}^{\dagger}\hat{A}+\frac{g_{\text{d}}^{2}\hat{n}}{g_{\text{s}}})}$,
the system could be changed to

$\hat{H}=\left(\omega_{\text{R}}-\frac{g_{\text{s}}^{2}}{\omega_{\text{M}}}\hat{a}^{\dagger}\hat{a}-\frac{2g_{\text{d}}^{2}}{\omega_{\text{M}}}\hat{b}^{\dagger}\hat{b}\right)\hat{a}^{\dagger}\hat{a}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c}.$
(9)

Previously, the cross-Kerr effect is usually discussed in the semi-classical
region which induced directly [90, 91, 92, 93]. For example, the cross-Kerr
effect between the mechanical mode and optical mode [94, 95]. Here we consider
the cross-Kerr nonlinearity as a port which is open to the environment, and
provides us a tool for the control of the cavity.

##  III Energy structure and its modulation with the auxiliary field

As the environment plays the role of the cross-Kerr term, in order to study
the dynamics of the system, here an auxiliary filed is introduced to simulate
the environment which is stronger than the driving field. The environment for
the directional features of the CW and CCW fields are different for the field
in the waveguide.

The pump strength of the auxiliary field on the cavity is set as
$\mathcal{E}$, then frequency shift of the CCW mode could be denoted as
$\frac{2g_{\text{d}}^{2}\mathcal{E}^{2}}{\omega_{\text{M}}}\langle\alpha|\hat{b}^{\dagger}\hat{b}|\alpha\rangle$
according to Eq.9. $|\alpha\rangle$ represents the coherent state. Here it is
obvious that the frequency shift is different in the CW and CCW directions. We
assume the intensity of the probe beam is weaker compared with the pumping
beam. And the probe beam is bidirectional, the pumping beam is only coupled
with the CW mode. Then the Hamiltonian of the two modes could be expressed as

$\displaystyle\hat{H}_{\text{CW}}$
$\displaystyle=\left(\omega_{\text{R}}-\frac{g_{\text{s}}^{2}}{\omega_{\text{M}}}\hat{a}_{\text{CW}}^{\dagger}\hat{a}_{\text{CW}}-\frac{2g_{\text{d}}^{2}}{\omega_{\text{M}}}\mathcal{E}^{2}\right)\hat{a}_{\text{CW}}^{\dagger}\hat{a}_{\text{CW}}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c},$
(10a) $\displaystyle\hat{H}_{\text{CCW}}$
$\displaystyle=\left(\omega_{\text{R}}-\frac{g_{\text{s}}^{2}}{\omega_{\text{M}}}\hat{a}_{\text{CCW}}^{\dagger}\hat{a}_{\text{CCW}}\right)\hat{a}_{\text{CCW}}^{\dagger}\hat{a}_{\text{CCW}}+\omega_{\text{M}}\hat{c}^{\dagger}\hat{c}.$
(10b)

From Eq.10, we can find frequency shift of both the CW and CCW modes are
related to the intensity of the auxiliary field. And the eigen-energy of the
two modes could be solved as

$\displaystyle E_{n,\text{CW}}$
$\displaystyle=n\hbar\omega_{R}-\frac{g_{\text{s}}^{2}}{\omega_{\text{M}}}n^{2}-\frac{2g_{\text{d}}^{2}}{\omega_{\text{M}}}\mathcal{E}^{2}+n_{\text{M}}\omega_{M},$
(11a) $\displaystyle E_{n,\text{CCW}}$
$\displaystyle=n\hbar\omega_{R}-\frac{g_{\text{s}}^{2}}{\omega_{\text{M}}}n^{2}+n_{\text{M}}\omega_{\text{M}}.$
(11b)

As shown in FIG.2, the energy-level structure generation of our scheme is
presented. Except that the energy level distribution is no longer uniform, the
mechanical system will generate many mechanical sub-level which is different
from the pure Kerr case. Thus, the dispersion coupling will shift the energy
level with different amplitude and there are several mechanical dressed energy
level generated. Then the energy could be shifted by the dissipation coupling
generation, and the variance is related to the auxiliary field strength. In
FIG.2 (b) we show the frequency shift varies with the auxiliary field strength
$\mathcal{E}$ under the dissipation coupling strength of
$g_{\text{d}}=0.1,0.5,1$.

##  IV The photon excitation and photon blockade of the hybrid coupling system

As the energy gap is no longer equally spaced and auxiliary field is
adjustable, the non-reciprocal photon blockade could be achieved in our hybrid
coupling system. The dynamical equation of the system without environment
noise is

$\dot{\hat{a}}(t)=(i\Delta_{i}-\kappa)\hat{a}_{i}(t)+e^{-iP(t)}\mathcal{E}_{p}+\mathcal{O}\left(\mathcal{E}_{p}^{2}a_{\mathrm{in}}\right).$
(12)

Here, $i$ denotes the mode CW or CCW. The detunings are
$\Delta_{\text{CW}}=\Delta-\frac{g_{der}^{2}}{\omega_{M}}-\frac{2g_{dis}^{2}}{\omega_{M}}\mathcal{E}^{2}$,
$\Delta_{\text{CCW}}=\Delta-\frac{g_{der}^{2}}{\omega_{M}}$, and
$\Delta=\omega_{\text{L}}-\omega_{\text{R}}$ is the detuning of the laser from
the bare cavity frequency. $P(t)=e^{-iH_{m}t}Pe^{iH_{m}t}$,
$H_{m}=\omega_{M}\hat{c}^{\dagger}\hat{c}$ is the free mechanical Hamilton.
Here we assume the pumping beam is the environment which is not affected by
the feedback. By integrating Eq.12 under the weak driven condition, we find
the spectrum function of the system could be solved as [33, 34, 96, 97, 98]:

$S\left(\Delta\right)\simeq\kappa\sum_{n=-\infty}^{\infty}A_{n}\frac{\kappa_{n}}{\kappa_{n}^{2}+\left(\Delta_{i}-n\omega_{m}\right)^{2}},$
(13)

where $A_{n}=e^{-\eta^{2}(2N+1)}I_{n}[2\eta^{2}\sqrt{N(N+1)}]((N+1)/N)^{n/2}$.
$N$ denotes the total photon number, in which
$N(\omega)=1/\left(e^{\hbar\omega/k_{B}T}-1\right)$ for the thermal state,
$k_{B}$ is the Boltzmann constant.

Figure 3: The photon number shift with high thermal phonon number and lower
photon number. (a). The photon excitation spectrum various with frequency
detuning $\Delta$ under 1 thermal phonon environment. (b). The photon
excitation spectrum various with frequency detuning $\Delta$ under 0.01
thermal phonon environment. In both figure, we have
$\kappa/\omega_{\text{M}}=0.1$, the quality factor of the mechanical mode is
$10^{4}$, and the dispersion coupling strength $g_{\text{s}}/\omega_{M}=1$.
The shift strength is $\Delta_{\text{d}}/\omega_{\text{M}}=0.5$ for the red
line, while it is 1 for the black line.

To further clarify the properties of the hybrid coupling system, we show the
photon excitation spectrum in FIG.3. Here the black line represents the photon
excitation of CW mode, while the red and blue line show the auxiliary field
shift photon excitation spectrum of the CCW mode. In figure FIG.3(a), the
thermal photon number is set as 1. We can find that there are resonance peaks
at both the positive and negative detuning regions. When the thermal photon
number is chosen as 0.01 in FIG.3 (b), there are only resonance peaks in the
positive detuning region. It is because the thermal noise will bridge
different mechanical sub-levels.

One more important thing is that the auxiliary field will cause energy level
displacement. Here we show the cross-Kerr induced shift $\Delta_{d}=0$,
$\Delta_{d}=0.5\omega_{m}$ and $\Delta_{d}=\omega_{m}$ with different line in
this figure, respectively. There are obviously spectrum shift for both cases.
However, the reciprocity of the excitation spectrum is different in FIG.3(a)
and (b). When the thermal photon number is higher (1 thermal photon in the
environment), the blue line and the blackline is partial reciprocition when
$\Delta/\eta=1$, or $2$, while it is fully reciprocal under the lower thermal
environment condition. Moreover, we can also achieve the fully reciprocity by
tuning the cross-Kerr induced frequency shift. when compare the blue and red
line, we can find their peak points are alternately distributed. Then we can
also achieve the fully reciprocition by choosing suitable $\Delta$.

Beyond the energy levels of the system, we also studied the properties of the
correlation functions. The dynamical equation of the second order term is

$\dot{\hat{a}}^{2}(t)=2(i\Delta_{i}+i\Delta_{g}-\kappa)\hat{a}^{2}(t)+\mathcal{E}_{p}e^{-iP(t)}a(t).$
(14)

By integrating the above equation, $a^{2}$ could be solved. Then, the second
order correction function
$g^{(2)}_{i}(0)=\langle{a^{\dagger}}^{2}a^{2}\rangle/\langle
a^{\dagger}a\rangle^{2}$ can be written as

$\begin{split}g_{i}^{(2)}(0)&=\operatorname{Re}\sum_{n,m,p}\frac{B_{n,m,p}}{\left[\kappa+i\left(\Delta_{i}-n\omega_{M}\right)\right]\left[\kappa-i\left(\Delta_{i}-m\omega_{M}\right)\right]}\\\
&\times\frac{2\kappa^{3}}{\left[2\kappa-i\left(2\Delta_{i}+2\Delta_{g}-p\omega_{m}\right)\right]S^{2}\left(\Delta\right)}.\end{split}$
(15)

For simplify, we set the temperature is near zero, then we have $B_{n,m,p}=$
$e^{-2\eta^{2}}\left(\eta^{2}\right)^{p}W_{n,p}(\eta)W_{m,p}(\eta)/n!m!p!$,
$W_{n,p}(\eta)=(-1)^{n}U\left[-n,1-n+p,\eta^{2}\right]$. The $U[a,b,x]$ is a
confluent hypergeometric function, and $\eta=g_{der}/\omega_{M}$ is a
dimensionless parameter.

Figure 4: The second correlation function and its shift of the hybrid
optomechanics. (a). The second correlation function with 0.001 thermal
phonons. (b). The correlation spectrum shift with the auxiliary field. The
$\kappa/\omega_{M}=0.15$ and thermal phonons number is 0.001. We ignore the
loss of mechanical mode. The shift strength is
$\Delta_{\text{d}}/\omega_{\text{M}}=0.5$ for the red line, while it is 1 for
the black line. (c). The zoom of (b).

According to the second-order correlation function given in Eq.15, the photon
blockade characteristic of the hybrid optomechanical system is also studied.
In FIG.4 (a), we report the theoretical calculations of the CW (black line)
and CCW (red and blue line) correlation spectrum under different driving
strength and cavity dissipation. We find that the second-order correlation
function is almost zero when the detuning $\Delta/\omega_{M}$ is $0,1,2,3,4$
which displays the photon blockade, and it decreases along with the decrement
of the parameter $\kappa$. So, we choose $\kappa=0.15$ as a key parameter in
our study. Then the coherent environment can shift the $g^{(2)}(0)$ of the CCW
mode in FIG.4 (b). Thus, the non-reciprocity could be achieved by choosing the
parameter for the $g^{(2)}(0)$, we can easily achieve non-reciprocity. In
order to show the non-reciprocity more obviously, we show its zoom in FIG.4
(c). However, the degree of non-reciprocity under different pumping strength
is different. When the shift is $0.5\omega_{M}$, the $g^{(2)}(0)$ is ranging
between 1 and 2. When the shift is $\omega_{M}$, the $g^{(2)}(0)$ is above 2.
So, when we give a frequency as $\omega_{M}$, the system shows better
performance of the non-reciprocity.

The non-reciprocal photon blockade is a part of the non-reciprocity feature of
the second-order correlation function. To quantitatively study the
nonreciprocal feature of the this system, we define the photon nonreciprocity
as

$\mathcal{R}(\Delta)=\frac{\left(\log(g_{CW}^{(2)})-\log(g_{CCW}^{(2)})\right)^{2}}{|\log(g_{CW}^{(2)})|+|\log(g_{CCW}^{(2)})|}.$
(16)

There, the denominator represents the sum of the correlation function of the
CW and CCW mode, while the numerator is the size of their difference. We
choose the logarithm operation of the correlation as the 1 is the decision
point of photon correlation.

Figure 5: The Non-Reciprocity under different environment pumping. (a). The
shift is $\Delta_{\text{d}}=0.1$. (b). The shift is $\Delta_{\text{d}}=0.5$.
(c). The shift is $\Delta_{\text{d}}=1$. In all figures, we have
$\kappa/\omega_{\text{M}}=0.15$, the dispersion coupling strength
$g_{\text{s}}/\omega_{M}=0.5$, the thermal phonon numbers is 0.001,and the
mechanical dissipation is neglected.

The performance of Eq.16 is shown in FIG.5. The coherent induced frequency
shift is 0.1, 0.5, and 1 for the CCW mode, respectively. When the detuning is
small as 0.1 as shown in FIG.5(a), the main peak appears in the near-zero
region, while the maximal value is smaller than $\log_{10}(4)$. So the non-
reciprocal photon blockade will not occur. While when we set the environment
induced shift as 0.5, as shown in FIG.5(b), there are transmission peaks in
the resonance position ($\Delta/\omega_{\text{M}}=0,1,2$). It is because the
resonance dip in the transmission spectrum of the CCW mode is shifted to the
peak of the CW mode, and the non-reciprocal photon blockade occur. When we
continuously increasing the shift to 1, as shown in FIG.5(c), the maximal
values of the non-reciprocity decreases, for the resonant peaks of the CW and
CCW modes meet again. When we return to FIG.4(b)(c) again, we can find the
non-reciprocity is not large as the Eq.16(b), but the non-reciprocal photon
blockade can still occur.

##  V Summary

In summary, we present a full quantum approach to study the dynamical behavior
of the dissipation and dispersion coupled optomechanical system. The coupling
would introduce the Kerr type nonlinearity, in particular, dissipative
coupling will cause cross Kerr coupling between the cavity mode and the
environment. On the other hand, by exploiting the cross-Kerr nonlinearity in
the system, the non-reciprocity of the second-order correlation function on
the CW and CCW propagation optical modes is illustrated. Moreover, we find the
photon blockade effect on both the CW and the CCW propagation directions. Our
research give an insight view of the environment of the whispering gallery
mode in the full quantum region. When we simply apply this scheme to an
coherent environment, is shows excellent photon non-reciprocal blocking
performance which will be the powerful tool in the realization of quantum
networking, nonlinear quantum gate, quantum information processing.

##  VI Acknowledgements

The authors gratefully acknowledge the support from the Fundamental Research
Funds for the Central Universities(BNU) and the National Natural Science
Foundation of China through Grants Nos. 61622103.

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