# Nonreciprocal phonon blockade

Xiao-Yu Yao  Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Hamad Ali  Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Peng-Bo Li<EMAIL_ADDRESS>Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China

###### Abstract

Quantum nonreciprocal devices have received extensive attention in recent
years because they can be used to realize unidirectional quantum routing and
noise isolation. In this work, we show that the shift of resonance frequencies
of propagating phonons induced by spin-orbit interactions (SOI) of phonons in
a rotating acoustic ring cavity can be used to realize nonreciprocal phonon
blockade. When driving the cavity from different directions, nonreciprocal
single-, two-phonon blockade and phonon-induced tunneling can take place by
varying the parameters of the system to an appropriate value. To realize
phonon blockade, the sublevels of the lower orbit branch of the ground state
of silicon-vacancy (SiV) color centers in the diamond membrane are employed to
induce self-interactions of phonons in the cavity. This work provides a way to
achieve acoustic nonreciprocal devices, such as directional acoustic switches
and quantum noise isolation, which may help acoustic information network
processing.

††preprint: APS/123-QED

##  I introduction

In quantum information processing, reliable methods for creation and
manipulation of phonons have become the subject of interests on account of
their potential applications [1, 2, 3, 4, 5]. Phononic devices work in the GHz
frequency range that corresponds to wavelengths on the order of $\mu$m, which
makes them a bridge between optical and artificial circuits [6, 7]. Thus it
has the potential to be a good candidate for solid-state quantum devices [8,
9, 10, 11, 12]. Various methods and architectures have been investigated to
study quantum phononic effects, such as phonon blockade (PB).

Phonon blockade is a phenomenon when one phonon is present in a nonlinear
acoustic cavity, the subsequent phonons cannot be excited, which is one of the
important mechanisms to realize the single phonon source [13]. In close
analogy to Coulomb blockade [14] and photon blockade [15, 16, 17, 18, 19, 20,
21, 22], PB is a pure quantum effect, which plays an important role in
unveiling the quantum behavior of a device. The initial single-phonon blockade
(1PB) was achieved in a nanomechanical resonator, which is coupled to a
superconducting quantum two-level system for inducing self-interactions of
phonons. On this basis, great effort has been made to study other effects such
as two-phonon blockade (2PB), equivalent phonon-induced tunneling (PIT), and
detecting of PB [23, 24, 25, 26, 27, 28]. Optomechanical systems with second
order non-linearities are considered for photon induced phonon blockade [23,
29, 30], with potential applications in hybrid photon-phonon quantum networks.

Recently, nonreciprocal devices have received widespread attention [31, 32,
33, 34, 35]. Reciprocity is a basic theorem in a network, which states that
when the source and observation points of the network change the positions,
the response of the transmission channel is symmetric [31, 36]. Nonreciprocal
devices, allowing the flow of signal from one side but blocking it from the
other, have become fundamental devices for building information processing
networks, such as unidirectional transmission, cloaking and noise-free
information processing and amplifiers [31, 37, 38, 39, 40, 41, 42, 43]. In
classical electromagnetic networks, one usually uses magnetic materials or
strong nonlinearities to destroy reciprocity [44, 45, 46, 47, 48]. But in the
quantum networks, magnetic materials are bulky and hard to integrate with
other devices. Furthermore, by using nonlinear materials, to achieve
nonreciprocity requires higher input strength from only one port at a time. So
in recent years, using temporal modulation to break reciprocity has received
much attention, such as traveling wave modulation [49, 50, 51, 52, 53, 54,
55], Sagnac effect of light [56, 57, 58, 59], and direct photon transition
[60, 61, 62, 63, 64], which have great potential for integrated nonreciprocal
devices. For acoustic networks, nonreciprocal devices have also made great
progress [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. It is reported that
the use circulating fluids and synthetic angular momentum in acoustic meta-
atom or meta-materials can realize acoustic circulators [65]. On this basis,
many other works have been performed, such as using acoustic metamaterials and
sonic crystal structures to achieve unidirectional acoustic transmission [67,
68, 69], and nonreciprocal phonon lasers based on optomechanical systems [72].
But so far, the nonreciprocal realization of phonon blockade has yet to
realize.

In this work we propose a spin-phononic system to study nonreciprocal PB and
PIT in non-stationary ring resonators, having counter prorogating travelling
acoustic modes coupled to color centers in diamond. The spin-phononic system
[77], which can be regarded as a phonon cavity quantum electrodynamics(QED)
system, may provide a promising setup for realizing nonreciprocal PB. The
mechanical vibrations can be controlled through their couplings with spins
[78, 79, 80, 81]. And NV and SiV color centers in diamond have long become
promising qubit platforms [82, 83, 84, 85, 86]. Especially, the SiV has
relatively strong strain coupling, which can be used to induce phonon-phonon
interactions [87], and has good potential for achieving phonon blockade.

To be specific, we use the sublevels of the lower orbit branch of the ground
state of the SiV color centers as a two-level system to induce phonon self-
interactions in the rotating ring cavity [88, 89]. We find that, nonreciprocal
1PB and PIT can be induced in the spinning ring resonator when driving it from
the left- or right-side of a coupled phononic waveguide, because the cavity
phonons satisfy the sub-Poisson distribution or the super-Poisson
distribution. Furthermore, by varying other parameters of the system, we can
get other nonreciprocal phenomena such as 2PB and multi-PB. The physical
origins behind the nonreciprocity phenomena are the SOI of phonons in the
spinning ring resonator. It cause a shift in the resonance frequency and phase
of the propagating phonon mode in the cavity, which has been proved in recent
experiments [90]. Such nonreciprocal PB can be used for many acoustic devices,
such as acoustic diodes [91, 92, 93, 94, 95], which can play a key role in a
phonon-basis information network at the few-phonon levels [96].

The remainder of this paper is organized as follows. In Sec. II, we describe
the physical model of a acoustic ring resonator coupled to a two-level SiV
center in diamond. In Sec. III, we present phonon bloackade in case of the
stationary ring resonator coupled to two level system. In Sec. IV, we present
our result in case of the rotating system. Experimental feasibility of our
scheme is given in Sec. V. Finally, the conclusion is given in Sec. VI.

##  II Model

The system we considered consists of an acoustic ring cavity coupled to a
phononic waveguide on one side and to SiV centers in diamond membrane on the
other side, as illustrated in Fig. 1. The cavity and the diamond membrane are
contact-coupled [97, 98, 99], and the sublevels of the lower orbital branch of
ground-state of SiV centers are coupled to the phonon mode in the cavity.
Under the rotating wave approximation and neglecting two-phonon terms, the
Hamiltonian of such system is given by

$\displaystyle H^{\text{(0)}}$ $\displaystyle=$ $\displaystyle\hbar\omega
a^{\dagger}a+\frac{1}{2}\hbar\omega_{0}\sigma_{z}+\hbar
g\left(a\sigma_{+}+a^{\dagger}\sigma_{-}\right))$ $\displaystyle+$
$\displaystyle\hbar\Omega_{s}\left(\sigma_{+}e^{-i\omega_{1}t}+\sigma_{-}e^{i\omega_{1}t}\right),$
$\displaystyle H^{\text{(d)}}$ $\displaystyle=$
$\displaystyle\hbar\xi\left(a^{\dagger}e^{-i\omega_{L}t}+ae^{i\omega_{L}t}\right),$
(1)

here, $\omega$ is the resonance frequency of the cavity, $a$$(a^{\dagger})$
denote the annihilation(creation) operator of the cavity field,
$\hbar\omega_{0}$ is the level splitting of the two-level system, and SiV
centers is descried by the spin operator $\sigma_{z}=|e\rangle\langle
e|-|g\rangle\langle g|$, with $|e\rangle$ and $|g\rangle$ represent the
excited and ground state of SiV centers, respectively. The ladder operators
$\sigma^{\pm}=\frac{1}{2}\left(\sigma^{x}\pm i\sigma^{y}\right)$ describe the
interaction between the SiV centers and the phononic cavity, with $\sigma^{x}$
and $\sigma^{y}$ as Pauli matrices, where $g$ is the coupling strength between
the cavity mode and SiV center. The parameter $\xi$ is the driving field
amplitude with driving frequency $\omega_{L}$.

Figure 1: (Color online) Schematic of the composite spin-phononic system. One
side of the ring cavity is coupled with the phononic waveguide to the input
and output of the driving field, and the other side is coupled with the SiV
centers in the diamond membrane to realize phonon blockade. In addition, we
add a rotator to the cavity to make the cavity rotate counterclockwise at an
angular frequency $\Omega$. The phonons which input from the left- or right-
side of the waveguide rotates counterclockwise or clockwise in the cavity
corresponding to (a) or (b), respectively.

In the regime where SiV centers and phonon modes are far detuned to the
coupling strength $\Delta=\left(\omega-\omega_{0}\right)\gg g$ and the Rabi
frequency $\Omega_{s}$, by which SiV centers are driven with the field of
frequency $\omega_{1}$ obeys the condition
$\Omega_{s}\gg\left(g^{2}/\Delta\right)$. Here we assume that
$\omega_{1}=\omega_{0}$, and then the Hamiltonian $H^{\text{(0)}}$ in Eq.(II)
in the rotating frame with $V=\exp(-i\omega_{0}\sigma_{z}t/2)$ can be written
as [13, 100]

$\displaystyle H^{\text{(0)}}_{\text{eff}}=\hbar\omega a^{\dagger}a+\hbar
Ua^{\dagger}a^{\dagger}aa,$ (2)

with the effective phonon-phonon interaction term (nonlinearity strength)

$\displaystyle U=\frac{g^{4}}{\Omega_{s}\Delta^{2}}.$ (3)

Here, it is worth mentioning that after straightforward calculations we have
ignored $\rho_{z}=|+\rangle\langle+|-|-\rangle\langle-|$, with
$|\pm\rangle=(|g\rangle\pm|e\rangle)/\sqrt{2}$, by preparing the SiV in the
ground state $|-\rangle$.

In the ring cavity, the phonon modes with two different propagation directions
(clockwise (CW) or counter-clockwise (CCW), corresponding to driving of the
cavity from the left- or right-hand, respectively) carry different orbital
angular momentum (OAM) [90, 101]. The phonon in different propagation
directions will possess spins of opposite direction because of SOI. Therefore,
phonons propagating CW or CCW experience the centripetal or centrifugal
Coriolis force, respectively [102]. When we add a rotator that rotates CCW at
the angular frequency $\Omega$ to the cavity, the resonant frequency of
phonons will have a change, i.e. $\omega\rightarrow\omega+\Delta_{F}$, where

$\displaystyle\Delta_{F}=\chi\Omega,$ (4)

here $\chi=\pm 0.12$ is the chirality of the phonon. For the rotator, rotating
in the CCW direction, when the cavity phonon rotates CW, $\Delta_{F}>0$,
otherwise, $\Delta_{F}<0$. Now in the rotating reference frame at the driving
frequency $\omega_{L}$, the effective Hamiltonian of the driven system can be
written as:

$\displaystyle H_{\text{eff}}=$ $\displaystyle\hbar$
$\displaystyle\left(-\Delta_{L}+\Delta_{F}\right)a^{\dagger}a+\hbar
Ua^{\dagger}a^{\dagger}aa$ (5) $\displaystyle+$
$\displaystyle\hbar\xi\left(a^{\dagger}+a\right),$

here, $\Delta_{L}=\omega_{L}-\omega$ is the detuning between the field of the
non-rotation resonant cavity and the driving field.

In order to obtain the analytical solution of the system, we first consider
the isolated phononic system. We have

$\displaystyle
H_{\text{eff}}=\hbar\left(-\Delta_{L}+\Delta_{F}\right)a^{\dagger}a+\hbar
Ua^{\dagger}a^{\dagger}aa.$ (6)

For a Fock state $|n\rangle$, we have

$\displaystyle H_{\text{eff}}|n\rangle$ $\displaystyle=$
$\displaystyle\left[-\hbar\Delta_{L}\hat{a}^{\dagger}\hat{a}+\hbar\Delta_{F}\hat{a}^{\dagger}\hat{a}+\hbar
Ua^{\dagger}a^{\dagger}aa\right]|n\rangle$ $\displaystyle=$
$\displaystyle\left[-\hbar\Delta_{L}\hat{a}^{\dagger}\hat{a}+\hbar\left(\Delta_{F}-U\right)\hat{a}^{\dagger}\hat{a}+\hbar
U\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}\right]|n\rangle$
$\displaystyle=$
$\displaystyle\left[-n\hbar\Delta_{L}+n\hbar\left(\Delta_{F}-U\right)+n^{2}\hbar
U\right]|n\rangle$ $\displaystyle=$ $\displaystyle E_{n}|n\rangle.$

Thus we obtain the eigenenergy

$\displaystyle
E_{n}=-n\hbar\Delta_{L}+n\hbar\Delta_{F}+\left(n^{2}-n\right)\hbar U,$ (7)

where n is the number of cavity phonons. The eigenenergy is also effective for
the weak driving condition. In the region of weak drive $\xi\ll\gamma$, we can
truncate the Hilbert space of the system to n=2. Here, $\gamma=\omega/Q$ is
the cavity dissipation rate and $Q$ is the quality factor. The state of the
system can be written as $|\varphi(t)\rangle=\sum_{n=0}^{2}C_{n}(t)|n\rangle$,
with $C_{n}$ is the probability amplitude. For the effective Hamiltonian
(Eq.5), we can introduce a dissipation term according to the quantum-
trajectory method [103]:
$H=H_{\text{eff}}-(i\hbar\gamma/2)\hat{a}^{\dagger}\hat{a}$. Then the equation
of motion of the dissipative system is given by

$\displaystyle\dot{C}_{0}(t)$
$\displaystyle=-i\frac{E_{0}}{\hbar}C_{0}(t)-i\xi C_{1}(t),$ (8)
$\displaystyle\dot{C}_{1}(t)$
$\displaystyle=-i\left(\frac{E_{1}}{\hbar}-i\frac{\gamma}{2}\right)C_{1}(t)-i\xi
C_{0}(t)-i\xi\sqrt{2}C_{2}(t),$ $\displaystyle\dot{C}_{2}(t)$
$\displaystyle=-i\left(\frac{E_{2}}{\hbar}-i\gamma\right)C_{2}(t)-i\xi\sqrt{2}C_{1}(t).$

With the initial conditions: $C_{0}(0)=1$, and $C_{1}(0)=C_{2}(0)=0$, the
Steady-state solutions of the system can be derived as

$\displaystyle C_{1}(\infty)$
$\displaystyle=\frac{-\xi}{\left(\frac{E_{1}}{\hbar}-\frac{E_{0}}{\hbar}-i\frac{\gamma}{2}\right)},$
(9) $\displaystyle C_{2}(\infty)$ $\displaystyle=\frac{-\sqrt{2}\xi
C_{1}(\infty)}{\left(\frac{E_{2}}{\hbar}-\frac{E_{0}}{\hbar}-i\gamma\right)}.$

Now we proceed to study PB and PIT. For this purpose, we will exploit the
$\mu^{th}$-order correlation function of zero-time delay
$g^{(\mu)}(0)\equiv\left\langle\hat{a}^{\dagger\mu}\hat{a}^{\mu}\right\rangle/\langle\hat{n}\rangle^{\mu}$,
with $\hat{n}=\hat{a}^{\dagger}\hat{a}$. Describing the probability of finding
n phonons in the acoustic cavity by $P(n)=|C(n)|^{2}$, we can define the
correlation function as follows

$\displaystyle g^{(2)}(0)$ $\displaystyle=$
$\displaystyle\frac{2P_{2}}{\left(P_{1}+2P_{2}\right)^{2}}$ (10)
$\displaystyle\simeq$
$\displaystyle\frac{\left(-\Delta_{L}+\Delta_{F}\right)^{2}+\gamma^{2}/4}{\left(-\Delta_{L}+\Delta_{F}+U\right)^{2}+\gamma^{2}/4}.$

For n=3, we can get the third-order correlation function

$\displaystyle g^{(3)}(0)=\frac{6P_{3}}{\left(P_{1}+2P_{2}+3P_{3}\right)^{3}}$
$\displaystyle\simeq$
$\displaystyle\frac{\left[\left(\Delta_{L}+\Delta_{F}\right)^{2}+\frac{\gamma^{2}}{4}\right]^{2}}{\left[\left(\Delta_{L}+\Delta_{F}+U\right)^{2}+\frac{\gamma^{2}}{4}\right]\left[\left(\Delta_{L}+\Delta_{F}+2U\right)^{2}+\frac{\gamma^{2}}{4}\right]}.$

The minimum and the maximum of $g^{(\mu)}(0)$ are corresponding to 1PB, multi-
PB, and PIT, respectively. For example, when
$g_{min}^{(2)}(0)=1/[4\left(U/\gamma\right)^{2}+1]\textless 1$ for
$\Delta_{L}=\Delta_{F}$, the phonons satisfy the super-Poisson distribution
(phonon bunching state) which corresponds to 1PB; when
$g_{max}^{(2)}(0)=4\left(U/\gamma\right)^{2}+1\textgreater 1$ for
$\Delta_{L}=\Delta_{F}+U$, the phonons satisfy the sub-Poisson distribution
(phonon anti-bunching state), which corresponds to PIT. For $\mu\textgreater
2$, we have similar results for multi-PB and PIT.

##  III Phonon blockade

In this section, we study the realization of PB in the hybrid quantum system
including the dissipation effect. The master equation for the system is given
as,

$\displaystyle\dot{\hat{\rho}}=\frac{i}{\hbar}[\hat{\rho},H_{\text{eff}}]+\frac{\gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right),$
(12)

where $H_{\text{eff}}$ is the hamiltonian given in Eq. 5. We can get the
phonon number probability $P(n)=\left\langle
n\left|\hat{\rho}_{\mathrm{}}\right|n\right\rangle$ from the steady-state
solution $\hat{\rho}_{\mathrm{}}$ of the system.

Figure 2: (Color online) $g^{(2)}(0)$ as a function of the nonlinear strength
$U$ and the detuning $\Delta_{L}$, here $\Delta_{F}=0$. (a), (b), (c), and (d)
corresponds to the situation when $\xi=0.33\gamma$, $\xi=1\gamma$,
$\xi=3\gamma$, and $\xi=5\gamma$ respectively.

We now focus on the properties of the cavity phonons described by the second
order correlation function for various deriving field strengths $\xi$. In Fig.
2 we plot the results given by Eq. (12) as a function of the non-linear
strength U and detuning $\Delta_{L}$ in the case of a stationary cavity i.e.
$\Delta_{F}=0$. In the weak drive condition ($\xi=0.33\gamma$), one can easily
see the maximum and minimum regions of the second-order correlation function
shown in Fig. 2(a). The multiple relative maximum curves in Fig. 2(b), (c) and
(d) corresponding to $\xi=1\gamma$, $\xi=3\gamma$ and $\xi=5\gamma$
respectively, are derived from the contribution of multi-phonon resonances
rather than just two-phonon resonance; this effect does not appear in the weak
drive case. At this time, the entire system is equivalent to a conventional PB
model. Here we introduce a standard criterion for judging PB and PIT [104,
105, 106].

For n-phonon blockade (nPB), there is

$\displaystyle($ $\displaystyle i)\quad
g^{(n+1)}(0)<\exp(-\langle\hat{m}\rangle)=f,\quad$ (13) $\displaystyle($
$\displaystyle ii)\quad
g^{(n)}(0)\geq\exp(-\langle\hat{m}\rangle)+\langle\hat{m}\rangle\cdot
g^{(n+1)}(0)=f^{(n)};$

for PIT, there is

$\displaystyle
g^{(\mu)}(0)>\exp(-\langle\hat{m}\rangle),\quad\mathrm{for}\quad\mu\geq 2.$
(14)

Here, $\langle\hat{m}\rangle$ is the average phonon number of the cavity
field.

In this work, we mainly consider the situation where the number of phonons is
small (n$\leq$4). In this case, $\langle\hat{m}\rangle\ll 1$ , so Eq. (14) can
be rewritten as

$\displaystyle g^{(\mu)}(0)>1,\quad\mathrm{for}\quad 4\geq\mu\geq 2.$ (15)

For nPB, we also have an equivalent standard for comparing the phonon number
distribution with the Poisson distribution:

$\displaystyle($ $\displaystyle i)\quad
P(m)<\mathcal{P}(m),\quad\mathrm{for}\quad m>n$ (16) $\displaystyle($
$\displaystyle ii)\quad P(n)\geq\mathcal{P}(n),$

with
$\mathcal{P}(n)=\frac{\langle\hat{n}\rangle^{n}}{n!}\exp(-\langle\hat{n}\rangle)$
is the Poissonian distribution.

Figure 3: (Color online) (a) Under the weak driving strength region
$\xi=0.33\gamma$, the correlation function versus the detuning $\Delta_{L}$ at
$U=20\gamma$. (b) and (c) show the comparison for 1PB between the correlation
function and the standard value in Eq. (18). (d) and (e) show the deviation
between the phonon population and the Poisson distribution for 1PB and PIT,
respectively at $\Delta_{L}=0$ and $\Delta_{L}=20\gamma$.

To show a relative deviation of a given phonon-number distribution from the
corresponding Poissonian distribution, we use the formula [106]

$\displaystyle[P(n)-\mathcal{P}(n)]/\mathcal{P}(n).$ (17)

To further elaborate the effect of weak drive on PB in the non-rotating system
($\Delta_{\mathrm{F}}=0$), in Fig. 3 we have plotted the n-order correlation
function for constant non-linear strength (U). Figure 3(a) shows the results
for the n-order correlation function with respect to the detuning
$(\Delta_{L})$ for $U=20\gamma$. Here, for $g^{(2)}(0)$, the minimum and
maximum values are at $\Delta_{L}=0$ and $\Delta_{L}=20\gamma$ , which
corresponds to 1PB and PIT respectively.

According to Eq. (13), 1PB should meet the following conditions for $n=1$:

$\displaystyle($ $\displaystyle i)\quad
g^{(2)}(0)<\exp(-\langle\hat{m}\rangle)=f,$ (18) $\displaystyle($
$\displaystyle ii)\quad
g^{(1)}(0)\geq\exp(-\langle\hat{m}\rangle)+\langle\hat{m}\rangle\cdot
g^{(2)}(0)=f^{(1)}.$

Next, in Fig. 3(b) and Fig. 3(c) we plot the n-order correlation functions and
standard values given in Eq. 18 as functions of $\Delta_{L}$. We find that the
curves of the correlation function satisfy the conditions in Eq.(18).
Furthermore, in Fig. 3(d), we find that $P(1)>\mathcal{P}(1)$,
$P(2)<\mathcal{P}(2)$, and $P(3)<\mathcal{P}(3)$, which is a clear signature
of 1PB. Due to PIT corresponding to the super-Poisson distribution of phonons,
we can find that $P(2)>\mathcal{P}(2)$ and $P(3)>\mathcal{P}(3)$ in Fig. 3(e).
In addition, we can find that $g^{(2)}(0)>g^{(3)}(0)>g^{(4)}(0)>1$ in Fig.
3(a), which indicates the emergence of PIT at this time.

Now we investigate PB under strong drive conditions depicted in Fig. 4. Figure
4(a), shows the results for the n-order correlation function versus the
detuning $(\Delta_{L})$ for $U=20\gamma$ and $\xi=3\gamma$. One can see the
system still shows 1PB at $\Delta_{L}=0$. Unlike the aforementioned case in
Fig. 3(a), one may notice that there is second minimum in the correlation
curves in Fig. 4(a), corresponding to 2 phonon blockade. The results of 2PB
and PIT can be seen at $\Delta_{L}=20\gamma$ and $\Delta_{L}=40\gamma$,
respectively.

Figure 4: (Color online) (a) Under the strong driving region $\xi=3\gamma$,
the correlation function versus the detuning $\Delta_{L}$ at $U=20\gamma$. (b)
and (c) show the comparison for 2PB between the correlation function and the
standard value in Eq. (19). (d) and (e) show the deviation between the phonon
population and the Poisson distributionfor 2PB amd PIT, respectively at
$\Delta_{L}=20\gamma$ and $\Delta_{L}=40\gamma$.

According to Eq. (13), 2PB should meet the following conditions for $n=2$:

$\displaystyle($ $\displaystyle i)\quad
g^{(3)}(0)<\exp(-\langle\hat{m}\rangle)=f,$ (19) $\displaystyle($
$\displaystyle ii)\quad
g^{(2)}(0)\geq\exp(-\langle\hat{m}\rangle)+\langle\hat{m}\rangle\cdot
g^{(3)}(0)=f^{(2)}.$

In Fig. 4(b) and 4(c) we show the correlation function versus the detuning
$\Delta$, which satisfies the conditions in Eq. (19). Further, in Fig. 4(d),
we can see $P(2)>\mathcal{P}(2)$, $P(3)<\mathcal{P}(3)$, and
$P(4)<\mathcal{P}(4)$, which meets our criterion given in Eq. (16) for $n=2$.
Here, we can see that $P(3)>\mathcal{P}(3)$ and $P(4)>\mathcal{P}(4)$ in Fig.
4(e). Different from Fig. 3(a), here $g^{(4)}(0)>g^{(3)}(0)>g^{(2)}(0)>1$ in
Fig. 4(a), corresponding to 3 phonon resonance, because of the significant
increase in the drive strength. It is worth mentioning that the PIT at
$\xi=0.33\gamma$ (Fig. 3(a)) corresponds to the 2PB at $\xi=3\gamma$, which
can explain the emergence of the hollow window of the maximum area in Fig.
2(b) and 2(c), i.e., the transformation of PIT to PB by enhancing the driving
strength. This can illustrate the equivalence of PB and PIT.

##  IV Nonreciprocal effect

Now, let’s consider the effect of rotating the cavity. In Fig. 5 we plot the
correlation function $g^{(2)}(0)$ as a function of the detuning $\Delta_{L}$
when the angular velocity $\Omega$ takes various values in the weak driving
regime. In Fig. 5(a) (Fig. 5(b)) we can see the correlation curves shift to
the left (right) after the angular velocity $\Omega$ increases.

Figure 5: (Color online) The correlation function $g^{(2)}(0)$ versus the
detuning $\Delta_{L}$ under weak driving $\xi=0.33\gamma$ and nonlinear
constant $U=20\gamma$. (a) and (b) is corresponding to the results of driving
the cavity from the right side and the left side of the waveguide,
respectively. Figure 6: (Color online) (a) The correlation function
$g^{(2)}(0)$ versus the detuning $\Delta_{L}$ with $\xi=0.33\gamma$,
$\Delta_{F}=10\gamma$, and $U=20\gamma$. The chain (solid) line is the
numerical solution result of driving the system from the left- (right-) side,
and the triangle (square) scatter is the analytical solution result of driving
the system from the left- (right-) side. (b) The energy levels of driving the
system from the left-side and driving the system from the right-side become
non-equidistant at $\Delta_{L}=10\gamma$.

Thus, the rotation of the cavity is the source of reciprocity, which causes
the input driving from two different directions to have different resonance
frequencies in the cavity.

By choosing the appropriate rotational angular velocity $\Omega$, we can make
different quantum effects appear at the same time. As shown in Fig. 6(a), we
have 1PB that appears when driving from one side, and PIT, which indicates
that the absorption of the first phonon enhances the absorption of subsequent
phonons appears when driving from the other side at
$\Delta_{F}=\Delta_{L}=U/2=10\gamma$. Furthermore, Fig. 6(a) shows that our
analytical solution and the numerical solution are consistent; here we use
$g_{\circlearrowright}^{(2)}(0)$ and $g_{\circlearrowleft}^{(2)}(0)$ to
indicate that $\Delta_{F}>0$ and $\Delta_{F}<0$, respectively. We can clearly
see that at $\Delta_{L}=10\gamma$, there is 1PB when driving the system from
the left side, and PIT when driving the system from the right side, which is
conformance to our expectation. For the difference of $g^{(2)}(0)$ for
opposite directions, there is nonreciprocity with up to 6 orders of magnitude.
This magnitude depends on the intensity of nonlinearity, which can be proved
in Fig. 2. In Fig. 6(b), we can intuitively understand the reason why we have
1PB and PIT by considering the change of energy-level structure of the system.

Figure 7: (Color online) The correlation function versus the detuning
$\Delta_{L}$ with $\xi=3\gamma$, $\Delta_{F}=10\gamma$, and $U=20\gamma$.

In Fig. 7, We show the correlation function versus the detuning $\Delta_{L}$
in the strong driving regime $(\xi=3\gamma)$ for $\Delta_{F}=10\gamma$. The
solid curves represent the result of $g_{\circlearrowleft}^{(n)}(0)$, and the
dashed line represents the result of $g_{\circlearrowright}^{(n)}(0)$. Here,
at $\Delta_{L}=10\gamma$, we have 1PB and 2PB for the different driving
directions. Furthermore, at $\Delta_{L}=30\gamma$, we obtain 2PB and PIT
caused by the three-phonon resonance for different driving directions. These
results are discussed in detail in Fig. 8 and Fig. 9.

Figure 8: (Color online) (a) The energy levels of driving the system from the
left- and driving the system from the right-side become non-equidistant for
$\Delta_{L}=10\gamma$. (b) and (c) show the comparison for 2PB between the
correlation function and the standard value in (19), (d) and (e) show the
deviation of the phonon population and the Poisson distribution for 1PB and
2PB, respectively.

In Fig. 8(a), we can intuitively understand the reason why we have 1PB and 2PB
by considering the energy-level structure of the system. The plots in Fig.
8(b) and 8(c) show the comparison of correlation functions
$(g^{2}(0)~{}\text{and}~{}g^{3}(0))$ with the standard values given in Eq.19
for strong driving strengths. We find that the curves of correlation functions
satisfy the 2PB equation given in Eq. (19). Moreover, in Fig. 8(d), we can
find that $P(2)>\mathcal{P}(2)$, $P(3)<\mathcal{P}(3)$ and
$P(4)<\mathcal{P}(4)$, which meets the criterion given in Eq. (16) for $n=2$.
In Fig. 8(e), we find that $P(1)>\mathcal{P}(1)$, $P(2)<\mathcal{P}(2)$ and
$P(3)<\mathcal{P}(3)$, which meets the criterion in Eq.16 for $n=1$.

Next in Fig. 9(a), we show the energy-level structure of the system which
enables us to understand the nonreciprocity of 2PB and PIT. Similarly in Fig.
9(b) and (c), we show that the curves of the correlation function satisfy the
results given equation 19. Furthermore, we can see that $P(3)>\mathcal{P}(3)$
and $P(4)>\mathcal{P}(4)$ in Fig. 9(d). In Fig. 9(e), one can clearly see that
$P(2)>\mathcal{P}(2)$, $P(3)<\mathcal{P}(3)$ and $P(4)<\mathcal{P}(4)$, which
meets the criterion given in Eq. (16) for $n=2$. In addition to the above
results, we can also obtain nonreciprocity of 1PB and 3PR at
$\Delta_{F}=20\gamma$ and $\Delta_{L}=20\gamma$, nonreciprocity of 2PB and 4PR
at $\Delta_{F}=20\gamma$ and $\Delta_{L}=40\gamma$ for $U=20\gamma$.

Figure 9: (Color online) (a) Under strong driving strength region
$\xi=3\gamma$, The energy levels of driving the system from the left- and
driving the system from the right-side become non-equidistant for
$\Delta_{L}=30\gamma$. (b) and (c) show the comparison for 2PB between the
correlation function and the standard value in (19), (d) and (e) show the
deviation of the phonon population and the Poisson distribution for 2PB and
PIT, respectively.

##  V Experimental feasibility

For examining the feasibility of this scheme in the experiment, we now discuss
the relevant parameters. The acoustic ring cavity can be mounted on a rotating
stage in the experiment [90]. We can realize the coupling between the phononic
waveguide and ring cavity by using wrap-around couplers. The dimensions of the
cavity are $(r,w,h)=(50\ \mu\mathrm{m},5\ \mu\mathrm{m},0.5\ \mu\mathrm{m})$,
and the mode volume is $V_{eff}=180\ \mu\mathrm{m}^{3}$[107]. In our
calculations, the resonance frequency of the cavity is $\omega/2\pi=3\
\mathrm{GHz}$, and the quality factor is $Q=10^{7}$. The frequency of phonons,
which is in the $1\sim 10\ \mathrm{GHz}$ range, can be easily implemented with
the standard fabrication process.

For the SiV centers, the splitting of the sublevels of the lower orbital
branch in the ground-state can be tuned by a few GHz by simply controlling
local strain [89]. The production scheme of the diamond membrane doped with
SiV color centers is quite mature. The diamond membrane can not only be
coupled with the cavity in contact on one side, and can also be placed above
the cavity for coupling together [99]. If placed above the cavity, the diamond
membrane can also be moved to change the coupling strength between the cavity
mode and SiV center. The coupling strength of SiV centers and cavity is
generally on the order of MHz. Thus the nonlinearity strength $U$ of
$\mathrm{kHz}$ is easy to obtain. In our calculations, $U$ is in the $0\sim
12\ \mathrm{kHz}$ range.

##  VI CONCLUSION

In conclusion, we study the nonreciprocal phonon blockade in a spinning
acoustic ring resonator coupled to SiV centers in diamond. We show that in
this hybrid system, the SOI of phonons causes a nonreciprocal response. The
statistical properties of phonons are exploited through calculating the
higher-order correlation functions. We investigate the PB for various system
parameters like the nonlinearity strength, driving strength, and drive
frequency in a cavity rotating in the CCW direction. We find that in the weak
driving regime, nonreciprocity of 1PB and PIT for CW and CCW modes occurs,
respectively. For strong driving we obtain nonreciprocity of 1PB and 2PB with
the same system parameters. Furthermore, in the strong driving regime we have
nonreciprocity of 2PB and PIT for CW and CCW modes, respectively. These
results combined with the current quantum technologies can be used to
implement nonreciprocal few-phonon sources, nonreciprocal phonon routers and
other quantum unidirectional devices.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
under Grant No. 92065105 and Natural Science Basic Research Program of Shaanxi
(Program No. 2020JC-02).

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