Nonreciprocal Photon Blockade
Ran Huang Adam Miranowicz Jie-Qiao Liao Franco Nori Hui Jing




Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum  Effects and Applications, Hunan Normal University, Changsha 410081, China

Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum  Effects and Applications, Hunan Normal University, Changsha 410081, China

Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

jinghui73@foxmail.com
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum  Effects and Applications, Hunan Normal University, Changsha 410081, China



We propose how to create and manipulate one-way nonclassical light
via photon blockade in rotating nonlinear devices. We refer to
this effect as nonreciprocal photon blockade (PB). Specifically, we
show that in a spinning Kerr resonator, PB happens
when the resonator is driven in one direction but not the other.
This occurs because of the Fizeau drag, leading to a full
split of the resonance frequencies of the countercirculating
modes. Different types of purely quantum correlations, such as
single- and two-photon blockades, can emerge in different
directions in a well-controlled manner, and the transition from PB to
photon-induced tunneling is revealed as well. Our work opens up a new
route to achieve quantum nonreciprocal devices, which are crucial
elements in chiral quantum technologies or topological photonics.




Nonreciprocal Photon Blockade
    Hui Jing
    December 30, 2023
=============================



Nonreciprocal devices, allowing the flow of light from one side
but blocking it from the other, are indispensable in a wide range
of practical applications, such as invisible sensing or cloaking,
and noise-free information processing <cit.>. To
avoid the difficulties of conventional magnet-based devices (e.g.,
bulky and quite lossy at optical frequencies), nonreciprocal
optical devices have been demonstrated in recent experiments based
on nonlinear optics <cit.>,
optomechanics <cit.>,
atomic gases <cit.>, and
non-Hermitian
optics <cit.>.
Similar advances have also been achieved in making acoustic and
electronic one-way
devices <cit.>. However, previous
studies have mainly focused on the classical regimes, i.e.,
one-way control of transmission rates instead of quantum noises.
Nonreciprocal quantum devices have been explored very recently,
including one-way quantum
amplifiers <cit.>
and routers of thermal noises <cit.>. Such
devices can find applications for quantum control of light in
chiral and topological quantum
technologies <cit.>.

Here we propose how to induce and control nonreciprocal
quantum effects with rotating nonlinear devices.
Specifically, we show that photon blockade (PB), which is a purely
quantum effect, can emerge nonreciprocally in a spinning Kerr
resonator. We note that single-photon blockade (1PB), i.e.,
blockade of the subsequent photons by absorbing the first
one <cit.>, has been demonstrated experimentally
in diverse systems from cavity or circuit
QED <cit.> to
cavity-free devices <cit.>. In view of its
important role in achieving single-photon devices, optomechanical
PB  <cit.> have also
been explored, offering a way to test, e.g., the quantumness of
massive
objects <cit.>. In a
very recent experiment <cit.>, two-photon blockade
(2PB) <cit.>
has also been observed, opening a route for creating two-photon
devices. Thus, nonreciprocal PB devices, as studied here, together
with other nonreciprocal quantum
devices <cit.>,
are expected to play a key role in quantum
engineering <cit.>,
metrology <cit.>,
and quantum information
processing <cit.> at the
single- or few-photon levels.

In a very recent experiment <cit.>, an optical
diode with 99.6% isolation has been demonstrated by using a
spinning resonator. Inspired by this
experiment <cit.>, here we study nonreciprocal
PB in a spinning Kerr resonator. We find that light with
sub- or super-Poissonian photon-number statistics
can emerge when driving the resonator from its left or right side.
Also, by varying the parameters of the system, different quantum
correlations (i.e., 1PB or 2PB) can be achieved for the clockwise
(CW) or counterclockwise (CCW) modes, for a resonator spinning
along the CCW direction. We note that the main idea of
nonreciprocal PB is analogous to the classical nonreciprocity
induced by the Doppler effect, which has been studied extensively
in various areas of physics (see, e.g.,
Refs. <cit.>).
Here we focus on quantum nonreciprocity induced by the
Fizeau light-dragging effect. This opens up the prospect of
engineering nonreciprocal PB devices for applications in, e.g.,
unidirectional quantum sensing and quantum optical
communications <cit.>.

Model.—We consider a spinning optical Kerr resonator as
shown in Fig. <ref>. As a generic PB
model <cit.>,
Kerr interactions can also be experimentally achieved in
cavity-atom systems <cit.>, or
magnon devices <cit.>, and theoretically in
optomechanical systems <cit.>.
For a resonator spinning at an angular velocity Ω, the
light circulating in the resonator experiences a Fizeau shift,
i.e., ω_0→ω_0+Δ_F,
with <cit.>

    Δ_F=±nrΩω_0/c(1-1/n^2-λ/ndn/dλ),

where ω_0 is the resonance frequency of a nonspinning
resonator, n is the refractive index, r is the resonator
radius, and c (λ) is the speed (wavelength) of light in
vacuum. Usually, the dispersion term dn/dλ, characterizing
the relativistic origin of the Sagnac effect, is relatively small
(up to ∼ 1%) <cit.>. We
fix the CCW rotation of the resonator; hence Δ_F>0
(Δ_F<0) corresponds to the situation of driving the
resonator from its left (right) side, i.e., the CW and CCW mode
frequencies are
ω_↻,↺≡ω_0±|Δ_F|,
respectively.



In a frame rotating at driving frequency ω_L, the effective
Hamiltonian of the system can be written at the simplest level
as <cit.>

    Ĥ=ħ(Δ_k+Δ_F)â^†â+ħ
    Uâ^†â(â^†â-k)+ħξ(â^†+â),

where Δ_k=Δ_L+U(k-1), Δ_L=ω_0-ω_L,
the tuning parameter k is simply k=1-Δ_L/U for
Δ_k=0, â (â^†) is the annihilation
(creation) operator of the cavity field, and ξ=√(γ
P_in/(ħω_L)), with the cavity loss rate γ
and the driving power P_in. The Kerr parameter
is <cit.>
U=ħω_0^2cn_2/(n_0^2V_eff), where n_0
(n_2) is the linear (nonlinear) refraction index, and
V_eff is the effective mode volume. The Kerr coupling
is also attainable by using other kinds of
devices <cit.>.
Note that the term Δ_F makes Eq. (<ref>)
fundamentally different from that used for studying conventional
PB <cit.>.

The energy eigenstates of this system are the Fock states
|n⟩ (n= 0,1,2,...) with eigenenergies

    E_n=nħΔ_L+(n^2-n)ħ U± nħ|Δ_F|,

where n is the cavity photon number. The second term, with U,
leads to an anharmonic energy-level structure. The last term, with
±|Δ_F|, describing upper or lower shifts of energy
levels with an amount being proportional to Ω, is the
origin of nonreciprocal implementations of PB. When
|Δ_F|=U/2 and the probe with frequency
ω_0+|Δ_F| (k=1.5) comes from the left side, the
light is resonantly coupled to the transition
|0⟩→|1⟩. As shown in Fig. <ref>(a), the
transition |1⟩→|2⟩ is detuned by 2ħ U and,
thus, suppressed for U>γ; i.e., once a photon is coupled
into the resonator, it suppresses the probability of the second
photon with the same frequency going into the resonator. In
contrast, by driving from the right side, there is a two-photon
resonance with the transition |0⟩→|2⟩; hence the
absorption of the first photon favors also that of the second or
subsequent photons, i.e., resulting in photon-induced tunneling
(PIT), as defined below and shown in Fig. <ref>(b). This is a
clear signature of nonreciprocal 1PB; i.e., sub-Poissonian
light emerges by driving the system from one side, while
super-Poissonian light emerges by driving from the other
side.

Analytical results.—To confirm this intuitive picture, we
study the μth-order (μ=2,3) correlation function with
zero-time delay, i.e., g^(μ)(0) ≡⟨â^†μâ^μ⟩/⟨n̂⟩^μ,
with n̂=â^†â. The condition
g^(2)(0)>1 [g^(2)(0)<1] characterizes
PIT <cit.> (1PB) via super-Poissonian
(sub-Poissonian) photon-number statistics or photon bunching
(antibunching) <cit.>. The latter terms
can also refer to different (i.e., two-time) optical correlation
effects <cit.>, which are, however, not studied here.
We stress that, although PIT has a classical-like property of
super-Poissonian photon-number
statistics <cit.>,
it is a purely quantum effect <cit.>. The analysis of
higher-order correlation functions g^(μ)(0)>1 with μ>2
can reveal the relation of a particular PIT and
multi-PB <cit.>. Thus, more refined criteria for PIT are
sometimes applied <cit.>, and
we refer here to PIT if the conditions g^(μ)(0)>1 for
μ=2,3,4 are satisfied <cit.>. We also note that partially
coherent mixtures of the vacuum, and single- and multiphoton
states, as generated here, can be described by μth-order
super-Poissonian correlations, i.e., g^(μ)(0)>1, for
specific values of μ <cit.>. Particularly,
g^(3)(0)<1 [g^(3)(0)>1] is a signature of third-order
sub-Poissonian (super-Poissonian) statistics, which is also
interpreted as three-photon antibunching (bunching) in recent
experiments on multi-PB <cit.> and
PIT <cit.>. Thus, g^(3)(0), which is usually
measured with extended Hanbury Brown and Twiss interferometers,
provides a more refined test and classification of the
nonclassical character of light, including 2PB (as studied below)
or unconventional PB <cit.>.

According to the quantum-trajectory
method <cit.>, the optical decay can be included
in the effective Hamiltonian
Ĥ_s=Ĥ-(iħγ/2)â^†â,
where γ=ω_0/Q is the cavity dissipation rate and Q
is the quality factor. In the weak-driving regime
(ξ≪γ), by truncating the Hilbert space to n=2, the
state of this system is written as
|φ(t)⟩=∑_n=0^2C_n(t)|n⟩, with probability
amplitudes C_n. Then we have the following equations of motion

    Ċ_0(t)     =-iν_0C_0(t)-iξ C_1(t),
    Ċ_1(t)     =-i(ν_1-iγ/2)C_1(t)-iξ C_0(t)-iξ√(2)C_2(t),
    Ċ_2(t)    
    =-i(ν_2-iγ)C_2(t)-iξ√(2)C_1(t),

with ħν_n=E_n, C_0(0)=1, C_1(0)=C_2(0)=0. Solving
these equations (and dropping higher-order terms) leads to the
steady-state solutions

    C_1(∞)
    =-ξ/(ν_1-ν_0-iγ/2),  C_2(∞)
    =-√(2)ξ C_1(∞)/(ν_2-ν_0-iγ).



Denoting the probability of finding m photons in the resonator
by P(m)=|C_m|^2, we have

    g^(2)(0) =2P_2/(P_1+2P_2)^2≃(Δ_L+Δ_F)^2+γ^2/4/(Δ_L+Δ_F+U)^2+γ^2/4.

1PB and PIT correspond to the minimum and the maximum of
g^(2)(0), respectively, i.e., when U>γ,
g^(2)_min(0)=1/[4(U/γ)^2+1]<1 for
Δ_L=-Δ_F, and
g^(2)_max(0)=4(U/γ)^2+1>1 for
Δ_L=-Δ_F-U.

Numerical results.—In order to confirm our analytical
results, now we numerically study the full quantum dynamics of the
system. We introduce the density operator ρ̂(t) and then
solve the master
equation <cit.>:

    ρ̇̂̇=i/ħ[ρ̂,Ĥ]+γ/2(2âρ̂â^†-â^†âρ̂-ρ̂â^†â).

The photon-number probability P(n)=⟨ n
|ρ̂_ss|n⟩ can be obtained for the
steady-state solutions ρ̂_ss of the master
equation. The experimentally accessible parameters are chosen
as <cit.>:
V_eff=150 μm^3, Q=5×10^9,
n_2=3×10^-14 m^2/W, n_0=1.4,
P_in=2 fW, r=30 μm, and
λ=1550 nm. V_eff is typically
10^2–10^4 μm^3 <cit.>,
Q is typically
10^9–10^12 <cit.>, and
g^(2)(0) as low as ∼0.13 was achieved
experimentally <cit.>. Moreover, in Fig. <ref>,
we set Ω=29 kHz; a similar property of quantum
nonreciprocity is also confirmed for Ω=6.6 kHz
(see the Supplemental Material <cit.>). These values of
Ω are experimentally feasible <cit.>.
Very recently, spinning objects have reached much higher
velocities, reaching the GHz
regime <cit.>; such systems could also be
applied to study the nonreciprocal PB via Kerr-like optomechanical
interactions <cit.>. We note that the Kerr
coefficient can be n_2∼10^-14 m^2/W for
materials with potassium titanyl
phosphate <cit.>, and n_2 can be further
enhanced with various
techniques <cit.>,
e.g., feedback control <cit.> or
quadrature
squeezing <cit.>.




An excellent agreement between our analytical results and the
exact numerical results is seen in Fig. <ref>. Here we use
g^(2)_0(0), g^(2)_↻(0), and
g^(2)_↺(0) to denote the cases with
Δ_F=0, Δ_F>0, and Δ_F<0, respectively. For a
nonspinning resonator, regardless of the driving direction,
g^(2)_0(0) always has a dip at k=1 (i.e., Δ_L=0) or a
peak at k=2 (i.e., Δ_L=-U), corresponding to 1PB or PIT,
respectively. In contrast, for a spinning device, by driving from
the left (right) side, we have Δ_F>0 (Δ_F<0)
and, thus, a redshift (blueshift) for g^(2)(0), leads to 1PB
(PIT) at k=1.5, i.e., g^(2)_↻(0)∼0.001,
g^(2)_↺(0)∼673. This quantum nonreciprocity,
with up to 6 orders of magnitude difference of g^(2)(0) for
opposite directions, is fundamentally different from the classical
transmission-rate nonreciprocity.

Nonreciprocal 2PB.—The absorption of 2 photons can also
suppress the absorption of additional photons <cit.>.
This 2PB effect, featuring three-photon antibunching, but with
two-photon bunching, satisfies <cit.>:

    g^(3)(0)<f≡   e^-⟨n̂⟩,
    
    g^(2)(0)≥
    f^(2)≡   e^-⟨n̂⟩+⟨n̂⟩·
    g^(3)(0).

The third-order correlation function can be obtained analytically
as <cit.>:

    g^(3)(0)=6P_3/(P_1+2P_2+3P_3)^3≃(Δ^2+γ^2/4)g^(2)(0)/(Δ+2U)^2+γ^2/4,

with Δ=Δ_L+Δ_F, also agreeing well with the
numerical results. Figures <ref>(a) and <ref>(b) show
that 2PB emerges around k=2.5 by driving from the left side,
while we have PIT by driving from the right side, i.e.,
g^(2)_↺(0)∼36,
g^(3)_↺(0)∼1003. By tuning the driving
frequency to the three-photon resonance [see Fig. <ref>(d)],
it is indeed possible to observe that g^(3)(0)/g^(2)(0)∼
100, as shown in Fig. <ref>(a) for max⟨
n⟩=0.0185. This means that the probability of
simultaneously measuring three photons can be much larger than
that of two photons in this situation. Similar values of
g^(3)(0)∼ 10^3, g^(2)(0)∼ 10 were also predicted in
the PIT analysis in Ref. <cit.>.




Our results can be further confirmed by comparing the
photon-number distribution P(n) with the Poisson distribution
𝒫(n). Figure <ref>(c) shows that P(2) is
enhanced while P(n>2) are suppressed by driving from the left
side, which is in sharp contrast to the case when driving
from the right side. This unidirectional 2PB effect can be
intuitively understood by considering the energy-level structure
of the system, as shown in Fig. <ref>(d). By choosing
Δ_L=-3U/2 or k=2.5, the transition
|0⟩→|2⟩ is resonantly driven by the left input
laser, but the transition |2⟩→|3⟩ is detuned by
4ħ U, which features the 2PB effect; in contrast, by driving
from the right side, three-photon resonance happens for the
transition |0⟩→|3⟩, leading to PIT. Hence with
such a device, sub-Poissonian light can be achieved by driving
it from the left side, while super-Poissonian light is observed
by driving it from the right side.




Nonreciprocity of 1PB and 2PB.—Figure <ref> shows
that at k=1.5, 1PB emerges by driving from the left side, due to
g^(2)_↻(0)∼0.045, while 2PB occurs by
driving from the right side since the criteria given in
Eq. (<ref>) are fulfilled for Δ_F<0. This indicates a
purely quantum device with direction-dependent counting
statistics, a new nonreciprocal feature, which has not been
revealed previously. This 1PB-2PB nonreciprocity, as also clearly
seen in Fig. <ref>(c) for the populations of different Fock
states, provides a route for creating or processing different
quantum states in a single node of quantum
networks <cit.>.
Figures <ref>-<ref> present our solutions of the
standard master equation, given in Eq. (<ref>), which
describes both a slow continuous nonunitary evolution and quantum
jumps occurring with a small probability <cit.>. By
contrast, our approximate analytical solutions, based on the
complex Hamiltonian H_s and the Schrödinger equation,
were obtained by ignoring these quantum jumps following the
standard approach of Ref. <cit.>.


Conclusions.—We have studied nonreciprocal PB effects in
a spinning Kerr resonator. By fixing the CCW rotation of the
resonator, we find the following: (i) for
P_in=2 fW, Δ_sag=± U/2 and
k=1.5, we have 1PB and PIT for the CW and CCW modes,
respectively. (ii) For P_in=0.3 pW,
Δ_sag=± U/2 and k=2.5, we have 2PB and PIT for
the CW and CCW modes, respectively. More interestingly, (iii) for
P_in=0.3 pW, Δ_sag=± U/2 and
k=1.5, we have 1 and 2PB for the CW and CCW modes, respectively
(for more examples, see the Supplemental Material <cit.>).
These results can be useful in achieving, e.g., nonreciprocal
few-photon sources and quantum one-way devices.

The basic mechanism of this work can be generalized to a wide
range of systems, such as acoustic and electronic
devices <cit.>, to achieve, e.g.,
nonreciprocal phonon blockade <cit.> as a
test of the quantumness of mechanical devices <cit.>. Our
work can also be extended to study, e.g., nonreciprocal photon
turnstiles <cit.>, nonreciprocal photon
routers <cit.>, and
nonreciprocal extraction of a single photon from a laser
pulse <cit.>, by considering a hybrid
device with atoms <cit.>,
quantum dots <cit.>, or nitrogen-vacancy
centers <cit.>.


R.H. and H.J. are supported by the National Natural Science
Foundation of China (NSFC, 11474087 and 11774086). F.N. is
supported by the MURI Center for Dynamic Magneto-Optics via the
Air Force Office of Scientific Research (AFOSR)
(FA9550-14-1-0040), Army Research Office (ARO) (Grant
No. 73315PH), Asian Office of Aerospace Research and Development
(AOARD) (Grant No. FA2386-18-1-4045), Japan Science and
Technology Agency (JST) (the ImPACT program and CREST Grant
No. JPMJCR1676), Japan Society for the Promotion of Science
(JSPS) (JSPS-RFBR Grant No. 17-52-50023, and JSPS-FWO Grant No.
VS.059.18N), and the RIKEN-AIST Challenge Research Fund. A.M. and
F.N. are also supported by a grant from the John Templeton
Foundation. J.Q.L. is supported by the NSFC (11822501 and
11774087).















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Supplementary Material for “Nonreciprocal Photon
Blockade”





Ran Huang^1, Adam Miranowicz^2,3, Jie-Qiao Liao^1,
Franco Nori^2,4, and Hui Jing^1,^*





[]18cm
^1Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,  

Department of Physics and Synergetic Innovation Center for Quantum  Effects and Applications, 

Hunan Normal University, Changsha 410081, China 

^2Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan 

^3Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland 

^4Physics Department, The University of Michigan,
Ann Arbor, Michigan 48109-1040, USA 









Here, we present technical details on nonreciprocal photon
blockade (PB) in a driven Kerr-type model with a Fizeau drag.
Our discussion includes: (1) single- (1PB) and two-photon blockade
(2PB) effects; (2) our analytical solutions for the steady-state
optical-intensity correlation functions; and (3) rotation-induced
quantum nonreciprocity.




§ KERR-TYPE INTERACTION WITH THE FIZEAU DRAG


To realize nonreciprocal photon blockade, we consider a rotating
optical resonator with a nonlinear Kerr medium which can be
described by a Kerr-type interaction with a Fizeau drag term,

    Ĥ_R=ħ(ω_0+Δ_F)â^†â+ħ
    Uâ^†â^†ââ.



Here, Uâ^†â^†ââ is the
standard Kerr interaction term
 <cit.>,
â (â^†) is the annihilation (creation) operator
for the cavity field, while
U=ħω_0^2cn_2/(n_0^2V_eff) is the strength of
the nonlinear interaction with the nonlinear (linear) refraction
index n_2 (n_0), an effective cavity-mode volume
V_eff, and the speed of light in vacuum c . Moreover,
ω_0 is the resonance frequency of the nonspinning
resonator, and the rotation leads to a Fizeau
shift <cit.>:

    ω_0→ω_±=ω_0+Δ_F,

with

    Δ_F=±nrΩω_0/c(1-1/n^2-λ/ndn/dλ)=±ηΩ,

where Δ_F>0 (Δ_F<0) denotes the light
propagating against (along) the direction of the spinning
resonator, λ is the optical wavelength, n is the
refractive index of the resonator, and r is the radius of the
cavity. The dispersion term dn/dλ, characterizing the
relativistic origin of the Sagnac effect, is relatively small
(∼1%) <cit.>.

When the resonator is not spinning, the Fizeau drag is equal to
zero, owing to the same resonance frequency of light coming from
the left or right side. As implied by Eq. (<ref>),
increasing the rotation frequency Ω results in an opposing
frequency linear shift of ηΩ (see Fig. <ref>) for
light coming from opposite directions <cit.>. 




§ PHOTON BLOCKADE EFFECTS




 §.§ Origin of photon blockade


In order to study conventional photon blockade (PB), we consider
the Hamiltonian (<ref>) including the driving term

    Ĥ=Ĥ_R+ħξ(â^†e^-iω_Lt+âe^iω_Lt),

where ξ=√(γ P_in/(ħω_L)) is the
driving amplitude with the cavity loss rate γ, the driving
power P_in, and the driving frequency ω_L
 <cit.>. In a frame rotating with the driving
frequency ω_L, the Hamiltonian is transformed to

    Ĥ_eff=iħdD̂^†/dtD̂+D̂^†ĤD̂,

with
D̂=exp(-iω_Lâ^†ât),
which leads to

    Ĥ_eff    =-ħω_Lâ^†â+ħω_±â^†â+ħ Uâ^†â^†ââ+ħξ(â^†+â)
        =ħ(ω_0+Δ_F-ω_L)â^†â+ħ Uâ^†â^†ââ+ħξ(â^†+â).




Thus, the effective Hamiltonian of this system becomes

    Ĥ_eff=ħ(Δ_L+Δ_F)â^†â+ħ
    Uâ^†â^†ââ+ħξ(â^†+â),

where Δ_L=ω_0-ω_L is the detuning between
the driving field and the cavity field for the nonspinning
resonator. The Hamiltonian of the isolated spinning system, i.e.,

    H_0=ħ(Δ_L+Δ_F)â^†â+ħ
    Uâ^†â^†ââ,

can be expressed as

    Ĥ_0|n⟩    =[ħΔ_Lâ^†â+ħΔ_Fâ^†â+ħ Ua^†(ââ^†-1)â]|n⟩
        =[ħΔ_Lâ^†â+ħΔ_Fâ^†â+ħ Uâ^†ââ^†â-ħ Uâ^†â]|n⟩
        =[ħΔ_Lâ^†â+ħ(Δ_F-U)â^†â+ħ U(â^†â)^2]|n⟩
        =[nħΔ_L+nħ(Δ_F-U)+n^2ħ U]|n⟩
        =E_n|n⟩.

Thus, we obtain the eigensystem for the weak-driving case,

    Ĥ_0|n⟩    =E_n|n⟩,

with eigenvalues

    E_n=nħΔ_L+nħΔ_F+(n^2-n)ħ
    U=nħΔ_L+(n^2-n)ħ U± nħ|Δ_F|,

where +nħ|Δ_F| and -nħ|Δ_F| denote
the light propagating against (Δ_F>0) and along
(Δ_F<0) the direction of the spinning resonator,
respectively.





The origin of conventional n-photon blockade can be understood
from the fact that due to the anharmonicity of the energy
structure, i.e., the energy difference between consecutive
manifolds is not constant, the Hilbert space of the system is
restricted to the states containing at most n quanta. For
example, when the optical resonator is nonspinning
(|Δ_F|=0), single-photon blockade (1PB) is illustrated
in Fig. <ref>(a). If a coherent probe beam, tuned to
ω_0 (Δ_L=0), is coupled to the system, the probe is
on resonance with the |0⟩→|1⟩ transition, but the
|1⟩→|2⟩ transition is detuned by 2ħ U and is
suppressed for U>γ (where γ denotes the optical loss
of the resonator). Consequently, once a photon is coupled to the
system, it suppresses the probability of coupling a second photon
with the same frequency. Similarly, two-photon blockade (2PB)
corresponds to a two-photon resonance (2PR) for a nonspinning
case, as shown in Fig. <ref>(b). Moreover, multi-PB
corresponds to a multi-photon resonance <cit.>.
In addition to multi-PB, the energy-level diagrams of multi-photon
resonances in a Kerr-type system <cit.> also
correspond to photon-induced tunneling
(PIT) <cit.>.
This indicates that the absorption of the first photon enhances
the absorption of subsequent photons <cit.>. The
distinction of 1PB, multi-PB, and PIT can be found by analysing
higher-order correlation functions g^(μ)(0) with μ≥2,
as discussed below.

Due to the rotation of the resonator, different cases of
nonreciprocal PB effects can be achieved. For example,
Table <ref> and Fig. <ref> summarize the
main results for P_in=0.3 pW, and these are
elaborated in detail later on in this Supplementary Material.

We observe that the Hamiltonian, given in Eq. (<ref>), can be
rewritten as follows

    Ĥ_k=ħ(Δ_k+Δ_F)â^†â+ħ
    Uâ^†â(â^†â-k)+ħξ(â^†+â),

where Δ_k=Δ_L+U(k-1) is the frequency mismatch for the
nonspinning resonator. For convenience, we refer to k as a
tuning parameter, as in Ref. <cit.>. Hereafter, we
analyze the resonant case of Δ_k=0, which is related to the
resonant k-photon transitions in the nonspinning resonator, as
shown in Fig. <ref>. This condition implies that the
tuning parameter k is related to the Kerr nonlinearity and the
driving-field and cavity frequencies as follows

    k=-Δ_L/U+1.







 §.§ Criteria of photon blockade

We have studied the origin of conventional PB via the anharmonic
energy-level structure. In order to describe this picture
quantitatively, we apply two approaches. One is based on studying
the photon-number distribution of the system
 <cit.>, and the other is based on
investigating the optical intensity correlations
 <cit.>. Both can be
experimentally measured
 <cit.>.

Concerning the first method, in the case of an ideal n-photon
blockade, the cavity field shows the following photon-number
distribution <cit.>:


    (i)     P(m)=0    for m>n, 
    
    (ii)     P(n)≠0.


with normalization ∑_m=0^∞ P(m)=1. While the first n
photons are resonantly absorbed in the system, the generation of
more photons is blockaded in the cavity. However, these
photon-number distribution conditions are hard to achieve in an
experiment, where P(m)≠0 even for m>n. Thus, a comparison
with the Poissonian distribution was proposed by Hamsen et
al. <cit.>:


    (i)     P(m)<𝒫(m)    for m>n, 
    
    (ii)     P(n)≥𝒫(n).


where 𝒫(m) is the Poissonian distribution

    𝒫(m)=⟨m̂⟩^m/m!exp(-⟨m̂⟩),

with the same average photon number ⟨m̂⟩ as the
cavity field. The condition, given in Eq. (<ref>), indicates
that the first n photons are effectively impenetrable to the
following photons; while the condition, given in
Eq. (<ref>), indicates that the coupling of an initial
photon to the system favors the coupling of the subsequent photons
within the first n photons. This leads to the sub-Poissonian
photon-number statistics for (n+1) photons with the simultaneous
super-Poissonian statistics of the first n photons. To show a
relative deviation of a given photon-number distribution from the
corresponding Poissonian distribution, we use the
formula <cit.>:

    [P(n)-𝒫(n)]/𝒫(n).


For the second approach, correlation function
G^(n)(t_1,t_2,⋯,t_n) is the quantity measured at moments
t_1,t_2,⋯,t_n in extended Hanbury Brown-Twiss experiments
with n detectors. Note that g^(n) is G^(n) normalized by
the nth power of the mean photon number. Thus,
g^(n)(0)≡lim_t→∞(t,t,⋯,t) is related to
the probability of simultaneously measuring n photons in their
steady state assuming photon detections at the same time
t=t_1=t_2=⋯=t_n. The larger value of g^(n)(0)>1, the
higher probability of n-photon bunching (photon coalescence).
And the smaller value of g^(n)(0)<1, the lower probability of
n-photon bunching, which corresponds to the higher probability
of n-photon antibunching (photon anticoalescence). The case of
g^(n)(0)=1 is called photon unbunching, which is a typical
feature of coherent light for any n. These correlation functions
G^(n) and g^(n) are basic elements of the quantum
coherence theory of Glauber <cit.>.

The normalized equal-time μth-order photon correlation is
given by

    g^(μ)(0)=∑_m=μ^∞m!/(m-μ)!P(m)/⟨m̂⟩^μ=⟨m̂⟩^-μ∑_m=μ^∞m(m-1)⋯(m-μ+1)P(m)=⟨â^†μâ^μ⟩/⟨â^†â⟩^μ.

In particular, the second-order photon correlation function is

    g^(2)(0)=∑_m=2^∞m(m-1)P(m)/⟨m̂⟩^2=⟨m̂(m̂-1)⟩/⟨m̂⟩^2=⟨â^†2â^2⟩/⟨â^†â⟩^2,

and the third-order photon correlation function is

    g^(3)(0)=∑_m=3^∞m(m-1)(m-2)P(m)/⟨m̂⟩^3=⟨m̂(m̂-1)(m̂-2)⟩/⟨m̂⟩^3=⟨â^†3â^3⟩/⟨â^†â⟩^3.


The photon-number distribution conditions for n-photon blockade,
given in Eqs. (<ref>) and (<ref>), can be translated into
the following conditions:


    (i)     g^(n+1)(0)=0, 
    
    (ii)     g^(n)(0)≠0.


As aforementioned, these strict conditions can only be fulfilled
for an ideal case. The experimentally-realizable conditions can be
obtained based on Eqs. (<ref>) and (<ref>). Since in
the weak-driving regime, the photon-number distribution fulfills
the condition P(m) ≫ P(m+1), it is sufficient to satisfy
P(n+1)<𝒫(n+1) according to the condition in
Eq. (<ref>). Meanwhile, we can approximately express
P(n+1) with g^(n+1)(0) as follows:

    g^(n+1)(0)   =∑_m=n+1^∞m!/(m-n-1)!P(m)/⟨m̂⟩^n+1≈(n+1)!/⟨m̂⟩^n+1P(n+1), 
    
    P(n+1)   ≈⟨m̂⟩^n+1/(n+1)!·
    g^(n+1),

as the P(m) have been neglected for all m>(n+1). Thus, the
condition, given in Eq. (<ref>),
reads <cit.>:

    P(n+1)   <𝒫(n+1), 
    ⟨m̂⟩^n+1/(n+1)!· g^(n+1)(0)   <⟨m̂⟩^n+1/(n+1)!exp(-⟨m̂⟩), 
    
    g^(n+1)(0)   <exp(-⟨m̂⟩).

We can also obtain an approximate P(n) using a similar method as
follows:

    g^(n)(0)   =∑_m=n^∞m!/(m-n)!P(m)/⟨m̂⟩^n≈n!/⟨m̂⟩^nP(n)+(n+1)!/⟨m̂⟩^nP(n+1), 
    
    P(n)   ≈⟨m̂⟩^n/n!· g^(n)(0)-(n+1)P(n+1), 
    
    P(n)   ≈⟨m̂⟩^n/n!·
    g^(n)(0)-⟨m̂⟩^n+1/n!·
    g^(n+1)(0).

Moreover, the condition, given in Eq. (<ref>), then reads:

    P(n)   ≥𝒫(n), 
    ⟨m̂⟩^n/n!· g^(n)(0)-⟨m̂⟩^n+1/n!· g^(n+1)(0)   ≥⟨m̂⟩^n/n!exp(-⟨m̂⟩), 
    
    g^(n)(0)-⟨m̂⟩· g^(n+1)(0)   ≥exp(-⟨m̂⟩), 
    
    g^(n)(0)   ≥exp(-⟨m̂⟩)+⟨m̂⟩·
    g^(n+1)(0),

i.e., the experimentally-realizable conditions, given in
Eqs. (<ref>) and (<ref>), can be translated into the
following conditions <cit.>:


    (i)     g^(n+1)(0)<exp(-⟨m̂⟩),  
    
    (ii)     g^(n)(0)≥exp(-⟨m̂⟩)+⟨m̂⟩·
    g^(n+1)(0),


indicating a higher-order sub-Poissonian photon-number
statistics.

Moreover, PIT can be quantified by photon-number correlation
functions. Table <ref> shows that more refined
criteria for PIT are sometimes applied based on higher-order
correlation functions g^(μ)(0) with
μ>2 <cit.>. Here, we refer to PIT if
the following conditions are satisfied for μ≥2:

    g^(μ)(0)>exp(-⟨m̂⟩).

For simplicity, in this work, we consider these conditions only
for 2≤μ≤4. This indicates light with higher-order
super-Poissonian photon-number statistics, i.e., once, a photon
is coupled in a resonator, it enhances the probabilities of more
photons entering the resonator. In the few-photon regime
(⟨m̂⟩≪1), these criteria become

    g^(μ)(0)>1   for   μ=2,3,4.

We provide a more basic criteria to identify multi-PB and PIT by
using μth-order correlation functions g^(μ)(0). These
criteria lead to the same conclusions as those based on
Eq. (<ref>).








 §.§ Single- and Multi-photon blockade

In this section, we only consider the nonspinning case
(Δ_F=0), while the spinning case is discussed in
Sec. <ref>. According to criteria, given in Eqs. (<ref>)
and (<ref>), 1PB has to fulfill the following conditions for
n=1:


    (i)     g^(2)(0)<exp(-⟨m̂⟩)≡ f, 
    
    (ii)     g^(1)(0)≥exp(-⟨m̂⟩)+⟨m̂⟩·
    g^(2)(0)≡ f^(1).


As expected from the intuitive picture discussed in
Sec. <ref>, the strongest 1PB occurs at Δ_L=0 (k=1),
since the correlation functions fulfill the criteria of 1PB given
in Eqs. (<ref>) and (<ref>) [see Fig. <ref>(a)].
In the weak-driving regime, ⟨m̂⟩≪1 implies
that f→1 and f^(1)→1. Then we obtain g^(2)(0)<1,
which corresponds to the usual criterion of 1PB, as known in the
published literature.

As aforementioned in 1PB, the first photon blocks the entrance of
a second photon, which indicates the enhancement of the
single-photon probability, and also the suppression of the two- or
more-photon probabilities. We can clearly see that
P(1)>𝒫(1), while P(2)<𝒫(2) and
P(3)<𝒫(3) at k=1 in Fig. <ref>(b). Moreover, 1PB
can be recognized from the deviations of the photon distribution
from the standard Poissonian distribution with the same mean
photon number [i.e., Eq. (<ref>)], as shown in
Fig. <ref>(c-i).

At k=2, we find the correlation functions fulfill
g^(2)(0)>g^(3)(0)>g^(4)(0)>1, as shown in the inset in
Fig. <ref>(a). This shows that PIT corresponding to
super-Poissonian photon-number behavior of light, which occurs at
k=2, since the correlation functions satisfy the conditions
given in Eq. (<ref>). PIT can also be recognized from the
photon-number distributions and the deviations given in
Eq. (<ref>). As shown in Figs. <ref>(b) and
<ref>(c-ii), we find that P(1)<𝒫(1),
P(2)>𝒫(2), P(3)>𝒫(3), and
P(4)>𝒫(4) at k=2. This is a clear signature of PIT.
Since the case for k=2 corresponds to a two-photon resonance, we
refer to this PIT as two-photon resonance-induced PIT.








Similarly, the 2PB has to fulfill the criteria in Eqs. (<ref>)
and (<ref>) for n=2:


    (i)     g^(3)(0)<exp(-⟨m̂⟩)≡ f, 
    
    (ii)     g^(2)(0)≥exp(-⟨m̂⟩)+⟨m̂⟩·
    g^(3)(0)≡ f^(2).


As expected from the intuitive picture discussed in
Sec. <ref>, 2PB occurs at Δ_L=-U (k=2), since the
correlation functions fulfill the conditions of 2PB given in
Eqs. (<ref>) and (<ref>) [see Fig. <ref>(a)]. We
find that, at k=2, g^(3)(0) is smaller than f defined in
the criterion given in Eq. (<ref>), while g^(2)(0) is
greater than f^(2) defined in the criterion given in
Eq. (<ref>). Here, 2PB indicates that the two-photon
probability is enhanced as P(2)>𝒫(2), while the other
photon-number probabilities are suppressed, as shown in
Figs. <ref>(b) and <ref>(c-ii). In Fig. <ref>, there
is PIT at k=2. However, in Fig. <ref>, there is 2PB at k=2
with an enhanced input power. We note that it is necessary to
properly increase the driving power to obtain a good-quality 2PB,
since we need a larger average photon number. Thus, we enhance the
input power from P_in=2 fW (Fig. <ref>)
to P_in=0.3 pW (Fig. <ref>). Also, the
1PB still emerges at k=1, since the second-order correlation
function fulfills g^(2)(0)<1 [see Fig. <ref>(a)], or only
the single-photon probability is enhanced at k=1 [see
Figs. <ref>(b) and <ref>(c-i)].

At k=3, we find the correlation functions fulfill
g^(4)(0)>g^(3)(0)>g^(2)(0)>1, as shown in the inset in
Fig. <ref>(a). It shows PIT occurs at k=3, since the
correlation functions satisfy the conditions given in
Eq. (<ref>). PIT can also be recognized from the
photon-number distributions and the deviations given in
Eq. (<ref>). As shown in Figs. <ref>(b) and
<ref>(c-iii), we find that P(1)<𝒫(1),
P(2)>𝒫(2), P(3)>𝒫(3), and
P(4)>𝒫(4) at k=3. This is a clear signature of PIT.
Since the case for k=3 corresponds to a three-photon resonance,
we refer to this PIT as three-photon resonance-induced PIT.

 In a sense, light with g^(3)(0)∼1000 has three-photon
correlations 1000 stronger than those for coherent light. We note
that the ratio of g^(3)(0)/g^(2)(0) can be quite large. For
example, g^(3)(0)/g^(2)(0)∼ 100 can be seen in
Fig. <ref>(a). A similar prediction g^(3)(0)/g^(2)(0)∼
100 has been reported in <cit.>. This is possible
since the mean photon number is ⟨n̂⟩≪ 1. For
example, if additionally ⟨â^† 3â^3⟩≈⟨â^† 2â^2⟩, then
g^(3)(0)/g^(2)(0)≈ 1/⟨n̂⟩≫ 1. 



§ ANALYTIC SOLUTION OF THE OPTICAL INTENSITY CORRELATION FUNCTIONS



 §.§ Second-order correlation function

According to the quantum trajectory method
 <cit.>, we introduce an anti-Hermitian term to
the Hamiltonian in Eq. (<ref>) to describe the dissipation of
the cavity photons. The effective non-Hermitian Hamiltonian is,
thus, given by

    Ĥ_t=ħ(Δ_L+Δ_F)â^†â+ħ
    Uâ^†â^†ââ+ħξ(â^†+â)-iħγ/2â^†â,

where γ is the rate of the cavity dissipation. Then the
Hamiltonian (<ref>) can be expressed in a spectral
representation as

    Ĥ_t    =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n=0^∞|n⟩⟨ n|(â^†+â)∑_n'=0^∞|n'⟩⟨ n'|
        =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n=0^∞∑_n'=0^∞|n⟩(⟨ n|â^†|n'⟩+⟨ n|â|n'⟩)⟨ n'|
        =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n=0^∞∑_n'=0^∞|n⟩(√(n'+1)⟨ n|n'+1⟩+√(n')⟨ n|n'-1⟩)⟨ n'|
        =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n=0^∞∑_n'=0^∞|n⟩(√(n'+1)δ_n,n'+1+√(n')δ_n,n'-1)⟨ n'| 
       =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n=0^∞∑_n'=0^∞|n⟩(√(n'+1)δ_n,n'+1+√(n')δ_n',n+1)⟨ n'| i
        =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨ n|+ħξ∑_n'=0^∞√(n'+1)|n'+1⟩⟨ n'|+ħξ∑_n=0^∞√(n+1)|n⟩⟨ n+1| 
       
    =∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨
    n|+ħξ∑_n=0^∞√(n+1)|n+1⟩⟨
    n|+ħξ∑_n=0^∞√(n+1)|n⟩⟨
    n+1|,ii

(i) To avoid negative n, we changed the subscript of the second
δ; Also, (ii) we substituted n for n', for convenience.
Therefore, we obtain the Hamiltonian of the whole system as

    Ĥ_t=∑_n=0^∞(E_n-iħγ/2n)|n⟩⟨
    n|+ħξ∑_n=0^∞√(n+1)|n+1⟩⟨
    n|+ħξ∑_n=0^∞√(n+1)|n⟩⟨
    n+1|,

with eigenenergies

    E_n=nħΔ_L+ nħΔ_F+(n^2-n)ħ
    U,

where Δ_F>0 (Δ_F<0) denotes the light
propagating against (along) the direction of the spinning
resonator.

For the weak-driving case, we restrict to a subspace spanned by
the basis states {|0⟩,|1⟩,|2⟩}. Then, the
Hamiltonian in Eq. (<ref>) becomes

    Ĥ_t    =E_0|0⟩⟨ 0|+(E_1-iħγ/2)|1⟩⟨ 1|+(E_2-iħγ)|2⟩⟨ 2|
         +ħξ√(1)|1⟩⟨ 0|+ħξ√(2)|2⟩⟨ 1|+ħξ√(3)|3⟩⟨ 2|
         +ħξ√(1)|0⟩⟨ 1|+ħξ√(2)|1⟩⟨ 2|+ħξ√(3)|2⟩⟨ 3|.

Due to the limits of the basis states, the terms including
|3⟩ can be neglected. Then we have

    Ĥ_t=E_0|0⟩⟨
    0|+(E_1-iħγ/2)|1⟩⟨
    1|+(E_2-iħγ)|2⟩⟨
    2|+ħξ|1⟩⟨
    0|+ħξ√(2)|2⟩⟨
    1|+ħξ|0⟩⟨
    1|+ħξ√(2)|1⟩⟨
    2|,

where:

    E_0    =0,
    
    E_1    =ħΔ_L+ħΔ_F,
    
    E_2    =2ħΔ_L+2ħΔ_F+2ħ U.

In this subspace, a general state can be written as

    |φ(t)⟩=∑_n=0^2C_n(t)|n⟩=C_0(t)|0⟩+C_1(t)|1⟩+C_2(t)|2⟩.

where C_n are probability amplitudes. We substitute the
Hamiltonian (<ref>) and the general state (<ref>) into the
Schrödinger equation

    iħ|φ̇(t)⟩=Ĥ_t|φ(t)⟩.

Then we have

    iħ|φ̇(t)⟩
    =iħĊ_0(t)|0⟩+iħĊ_1(t)|1⟩+iħĊ_2(t)|2⟩,

and

    Ĥ_t|φ(t)⟩
    =Ĥ_tC_0(t)|0⟩+Ĥ_tC_1(t)|1⟩+Ĥ_tC_2(t)|2⟩,

where:

    Ĥ_tC_0(t)|0⟩=(E_0|0⟩⟨
    0|+ħξ|1⟩⟨
    0|)C_0(t)|0⟩=E_0C_0(t)|0⟩+ħξ
    C_0(t)|1⟩,


    Ĥ_tC_1(t)|1⟩    =[(E_1-iħγ/2)|1⟩⟨ 1|+ħξ√(2)|2⟩⟨ 1|+ħξ|0⟩⟨ 1|]C_1(t)|1⟩
        =ħξ C_1(t)|0⟩+(E_1-iħγ/2)C_1(t)|1⟩+ħξ√(2)C_1(t)|2⟩,


    Ĥ_tC_2(t)|2⟩
    =[(E_2-iħγ)|2⟩⟨
    2|+ħξ√(2)|1⟩⟨
    2|]C_2(t)|2⟩=ħξ√(2)C_2(t)|1⟩+(E_2-iħγ)C_2(t)|2⟩,

i.e.,

    Ĥ_t|φ(t)⟩    =[E_0C_0(t)+ħξ C_1(t)]|0⟩+[(E_1-iħγ/2)C_1(t)+ħξ C_0(t)+ħξ√(2)C_2(t)]|1⟩
         +[(E_2-iħγ)C_2(t)+ħξ√(2)C_1(t)]|2⟩.

By comparing the coefficients of the same basis states in
Eqs. (<ref>) and (<ref>), we have:

    iħĊ_0(t)|0⟩    =[E_0C_0(t)+ħξ C_1(t)]|0⟩,
    
    iħĊ_1(t)|1⟩    =[(E_1-iħγ/2)C_1(t)+ħξ C_0(t)+ħξ√(2)C_2(t)]|1⟩,
    
    iħĊ_2(t)|2⟩    =[(E_2-iħγ)C_2(t)+ħξ√(2)C_1(t)]|2⟩,

with ν_n=E_n/ħ. Then we obtain the following equations
of motion for the probability amplitudes C_n(t):

    Ċ_0(t)     =-iν_0C_0(t)-iξ C_1(t),
    Ċ_1(t)     =-i(ν_1-iγ/2)C_1(t)-iξ C_0(t)-iξ√(2)C_2(t),
    Ċ_2(t)    
    =-i(ν_2-iγ)C_2(t)-iξ√(2)C_1(t),

where ν_n=E_n/ħ.

Weak driving means the driving strength is smaller than the cavity
damping rate ξ<γ. If there is no driving field, the
cavity field remains in the vacuum. When a weak-driving field is
applied to the cavity, it may excite a single photon or two
photons in the cavity. Thus, we have the following approximate
expressions: C_0∼1, C_1∼ξ/γ, and
C_2∼ξ^2/γ^2. Then we can approximately solve
the equations in Eq. (<ref>) using a perturbation method by
discarding higher-order terms in each equation for lower-order
variables. Thus, the Eq. (<ref>) becomes:

    Ċ_0(t)     =-iν_0C_0(t),
    Ċ_1(t)     =-i(ν_1-iγ/2)C_1(t)-iξ C_0(t),
    Ċ_2(t)    
    =-i(ν_2-iγ)C_2(t)-iξ√(2)C_1(t),

where ν_n=E_n/ħ.

For the initially empty cavity, the initial conditions read as:
C_0(0)=C_0(0), and C_1(0)=C_2(0)=0. Accordingly, the
solution of the zero-photon amplitude can be obtained as

    C_0(t)=C_0(0)exp(-iν_0t).

Hence, the equation for the single-photon amplitude in
Eq. (<ref>) becomes

    Ċ_1(t)=-i(ν_1-iγ/2)C_1(t)-iξ
    C_0(t)exp(-iν_0t).

To solve this equation, we introduce a slowly-varying amplitude:

    C_1(t)     =c_1(t)exp[-i(ν_1-iγ/2)t],
    
    C_1(0)     =c_1(0).

Then we obtain

    Ċ_1(t)=ċ_1(t)exp[-i(ν_1-iγ/2)t]-i(ν_1-iγ/2)c_1(t)exp[-i(ν_1-iγ/2)t],

and Eq. (<ref>) becomes:

    ċ_1(t)e^-i(ν_1-iγ/2)t-i(ν_1-iγ/2)c_1(t)e^-i(ν_1-iγ/2)t =    -i(ν_1-iγ/2)c_1(t)e^-i(ν_1-iγ/2)t-iξ C_0(t)e^-iν_0t,
    ċ_1(t) =    -iξ
    C_0(t)exp[i(ν_1-ν_0-iγ/2)t].

The solution can be obtained by integrating both sides of
Eq. (<ref>), as follows:

    c_1(t)-c_1(0)     =-iξ C_0(t)∫_0^texp[i(ν_1-ν_0-iγ/2)t'] dt',
    
    c_1(t)-c_1(0)     =-iξC_0(t)/i(ν_1-ν_0-iγ/2){exp[i(ν_1-ν_0-iγ/2)t]-1},
    
    c_1(t)exp[-i(ν_1-iγ/2)t]     =c_1(0)exp[-i(ν_1-iγ/2)t]-iξC_0(t)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]},
    
    C_1(t)    
    =C_1(0)exp[-i(ν_1-iγ/2)t]-iξC_0(t)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.

With the initial condition C_1(0)=0, we have the solution for
the single-photon amplitude given by

    C_1(t)=-iξC_0(t)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.


Consider the solution of the single-photon amplitude in
Eq. (<ref>), the equation for the two-photon amplitude in
Eq. (<ref>) becomes

    Ċ_2(t)=-i(ν_2-iγ)C_2(t)-√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.

To solve this equation, we introduce another slowly-varying
amplitude:

    C_2(t)     =c_2(t)exp[-i(ν_2-iγ)t],
    
    C_2(0)     =c_2(0),

and obtain

    Ċ_2(t)=ċ_2(t)exp[-i(ν_2-iγ)t]-i(ν_2-iγ)c_2(t)exp[-i(ν_2-iγ)t],

then Eq. (<ref>) becomes:

    ċ_2(t)e^-i(ν_2-iγ)t-i(ν_2-iγ)c_2(t)e^-i(ν_2-iγ)t=-i(ν_2-iγ)c_2(t)e^-i(ν_2-iγ)t-√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)[e^-iν_0t-e^-i(ν_1-iγ/2)t],
       ċ_2(t)=-√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2){exp[i(ν_2-ν_0-iγ)t]-exp[i(ν_2-ν_1-iγ/2)t]}.

The solution can also be obtained by integrating both sides of
Eq. (<ref>), as follows:

    c_2(t)-c_2(0) =    -√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)∫_0^t{exp[i(ν_2-ν_0-iγ)t']-exp[i(ν_2-ν_1-iγ/2)t']} dt',
    
    c_2(t)-c_2(0) =    -√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2){exp[i(ν_2-ν_0-iγ)t]-1/i(ν_2-ν_0-iγ)-exp[i(ν_2-ν_1-iγ/2)t]-1/i(ν_2-ν_1-iγ/2)},
    
    c_2(t)exp[-i(ν_2-iγ)t] =    c_2(0)exp[-i(ν_2-iγ)t]-√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)·exp(-iν_0t)-exp[-i(ν_2-iγ)t]/i(ν_2-ν_0-iγ)
        +√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)·exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/i(ν_2-ν_1-iγ/2),
    
    C_2(t) =     C_2(0)exp[-i(ν_2-iγ)t]-√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)·exp(-iν_0t)-exp[-i(ν_2-iγ)t]/i(ν_2-ν_0-iγ)
       +√(2)ξ^2C_0(t)/i(ν_1-ν_0-iγ/2)·exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/i(ν_2-ν_1-iγ/2).

With the initial condition C_2(0)=0, we have the following
solution of the two-photon amplitude

    C_2(t)=√(2)ξ^2C_0(t)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_2-iγ)t]/(ν_2-ν_0-iγ)-exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/(ν_2-ν_1-iγ/2)}.


Thus, for the initially empty resonator, the solutions of the
equations of motion for the probability amplitudes in the
equations in Eq. (<ref>) can be obtained as:

    C_0(t)     =C_0(0)exp(-iν_0t),
    
    C_1(t)     =-ξC_0(t)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]},
    
    C_2(t)    
    =√(2)ξ^2C_0(t)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_2-iγ)t]/(ν_2-ν_0-iγ)-exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/(ν_2-ν_1-iγ/2)},

where

    ν_0 =0,   ν_1 =Δ_L+Δ_F,   ν_2
    =2Δ_L+2Δ_F+2U.

When the initial state of the system is the vacuum state
|0⟩, i.e., the initial condition C_0(0)=1,
then the solutions in Eq. (<ref>) are reduced to:

    C_0(t)=     1,
    
    C_1(t)=     -iξ1/i(Δ_L+Δ_F-iγ/2){1-exp[-i(Δ_L+Δ_F-iγ/2)t]},
    
    C_2(t)=    √(2)ξ^2/(Δ_L+Δ_F-iγ/2){1-exp[-i(2Δ_L+2Δ_F+2U-iγ)t]/(2Δ_L+2Δ_F+2U-iγ)-exp[-i(Δ_L+Δ_F-iγ/2)t]/(Δ_L+Δ_F+2U-iγ/2)}
       +√(2)ξ^2·-exp[i(2Δ_L+2Δ_F+2U-iγ)t]/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+2U-iγ/2),

and for the infinite-time limit exp(-At)→0 (t→∞),
we have:

    C_0(∞)    ≡ C_0=1,
    
    C_1(∞)    ≡ C_1=-ξ/(Δ_L+Δ_F-iγ/2),
    
    C_2(∞)    ≡
    C_2=-√(2)ξC_1/(2Δ_L+2Δ_F+2U-iγ).



For the state given in Eq. (<ref>), the infinite-time state
(steady state) of the system reads as

    |φ(t→∞)⟩=|0⟩+-ξ/(Δ_L+Δ_F-iγ/2)|1⟩+√(2)ξ^2/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)|2⟩,

and the normalization coefficient of the state is given by

    N=1+|C_1|^2+|C_2|^2,

where:

    |C_1|^2   
    =|ξ/(Δ_L+Δ_F-iγ/2)|^2=ξ^2/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+iγ/2)=ξ^2/[(Δ_L+Δ_F)^2+γ^2/4],


    |C_2|^2    =|√(2)ξ^2/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)|^2
       =2ξ^4/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+iγ/2)(2Δ_L+2Δ_F+2U-iγ)(2Δ_L+2Δ_F+2U+iγ)
       
    =2ξ^4/[(Δ_L+Δ_F)^2+γ^2/4][4(Δ_L+Δ_F+U)^2+γ^2].

The probabilities of finding single and two photons in the cavity
are, respectively, given by:

    P_1   =|C_1|^2/N,
    
    P_2   =|C_2|^2/N.


As mentioned in Sec. <ref>, the equal-time (namely
zero-time-delay) second-order correlation function can be written
as

    g^(2)(0)≡⟨â^†2â^2⟩/⟨â^†â⟩ ^2=⟨â^†ââ^†â⟩
    -⟨â^†â⟩/⟨â^†â⟩ ^2.

When the cavity field is in the state given in (<ref>), we
have

    g^(2)(0)     =∑_n,n'=0^2C_n^*C_n⟨ n|â^†ââ^†â|n'⟩-∑_n,n'=0^2C_n^*C_n⟨ n|â^†â|n'⟩/(∑_n,n'=0^2C_n^*C_n⟨ n|â^†â|n'⟩)^2
        =0+|C_1|^2+4|C_2|^2-(0+|C_1|^2+2|C_2|^2)/(0+|C_1|^2+2|C_2|^2)^2
        =N(P_1+4P_2-P_1-2P_2)/N^2(P_1+2P_2)^2
        =2P_2/N(P_1+2P_2)^2.

In the weak-driving regime, we have the following approximate
formulas: C_0∼1, C_1∼ξ/γ, and
C_2∼ξ^2/γ^2, i.e., N∼1 with
|C_2|^2≪|C_1|^2≪1. Hence, the
second-order correlation function can be written as

    g^(2)(0)≈2P_2/(P_1+2P_2)^2.

Because P_1≫ P_2, we have

    g^(2)(0)≈2P_2/P_1^2.

Substituting Eqs. (<ref>) and (<ref>) into
Eq. (<ref>), we can easily obtain

    g^(2)(0)    ≈4ξ^4/[(Δ_L+Δ_F)^2+γ^2/4][4(Δ_L+Δ_F+U)^2+γ^2]·[(Δ_L+Δ_F)^2+γ^2/4]^2/ξ^4
        =(Δ_L+Δ_F)^2+γ^2/4/(Δ_L+Δ_F+U)^2+γ^2/4,

where Δ_F>0 (Δ_F<0) denotes the light
propagating against (along) the direction of the spinning
resonator.

Here, we focus on the nonspinning case (Δ_F=0), the
rotating case is discussed in Sec. <ref>. Then, the
second-order correlation function becomes

    g^(2)_0(0) =
    Δ_L^2+γ^2/4/(Δ_L+U)^2+γ^2/4.

When the driving laser tuned to a single-photon resonance,
Δ_L=0 (k=1), the minimum of g^(2)_0(0) is
g_0min^(2)=(γ^2/4)/(U^2+γ^2/4)=[4(U/γ)^2+1]^-1.
We have g_0min^(2)<1, when U≠0. The larger
U/γ, the smaller is the correlation function
g_0min^(2). This indicates that 1PB can be
achieved. On the other hand, for the driving laser tuning to the
two-photon resonance, Δ_L=-U (k=2), there is
g_0max^(2)=(U^2+γ^2/4)/(γ^2/4)=4(U/γ)^2+1.
We have g_0max^(2)>1 when U≠0. The larger
U/γ, the larger is the correlation function
g_0max^(2), which indicates a strong photon-induced
tunneling caused by two-photon resonance. In Sec. <ref>, we
find that this conclusion is completely confirmed by our numerical
results.




 §.§ Third-order correlation function

Using a method similar to that in Sec. <ref>, we calculate the
third-order photon-number correlation function. For the
weak-driving case, we restrict to a subspace spanned by the basis
states {|0⟩,|1⟩,|2⟩,|3⟩}. Then, the
Hamiltonian in Eq. (<ref>) becomes

    Ĥ_t    =E_0|0⟩⟨ 0|+(E_1-iħγ/2)|1⟩⟨ 1|+(E_2-iħγ)|2⟩⟨ 2|+(E_3-iħ3γ/2)|3⟩⟨ 3|
         +ħξ√(1)|1⟩⟨ 0|+ħξ√(2)|2⟩⟨ 1|+ħξ√(3)|3⟩⟨ 2|+ħξ√(4)|4⟩⟨ 3|
         +ħξ√(1)|0⟩⟨ 1|+ħξ√(2)|1⟩⟨ 2|+ħξ√(3)|2⟩⟨ 3|+ħξ√(4)|3⟩⟨ 4|.

Due to the limits of the basis states, the terms including
|4⟩ can be neglected. Then we have

    Ĥ_t    =E_0|0⟩⟨ 0|+(E_1-iħγ/2)|1⟩⟨ 1|+(E_2-iħγ)|2⟩⟨ 2|+(E_3-iħ3γ/2)|3⟩⟨ 3|
         ++ħξ|1⟩⟨ 0|+ħξ√(2)|2⟩⟨ 1|+ħξ√(3)|3⟩⟨ 2|+ħξ|0⟩⟨ 1|+ħξ√(2)|1⟩⟨ 2|+ħξ√(3)|2⟩⟨ 3|,

where:

    E_0    =0,
    
    E_1    =ħΔ_L+ħΔ_F,
    
    E_2    =2ħΔ_L+2ħΔ_F+2ħ U,
    
    E_3    =3ħΔ_L+3ħΔ_F+6ħ U.


In this subspace, a general state can be written as

    |φ(t)⟩=∑_n=0^3C_n(t)|n⟩=C_0(t)|0⟩+C_1(t)|1⟩+C_2(t)|2⟩+C_3(t)|3⟩.

where C_n are probability amplitudes. We substitute
Hamiltonian (<ref>) and the general state (<ref>) into the
Schrödinger equation (<ref>) to obtain

    iħ|φ̇(t)⟩=iħĊ_0(t)|0⟩+iħĊ_1(t)|1⟩+iħĊ_2(t)|2⟩+iħĊ_3(t)|3⟩;

and

    Ĥ_t|φ(t)⟩=Ĥ_tC_0(t)|0⟩+Ĥ_tC_1(t)|1⟩+Ĥ_tC_2(t)|2⟩+Ĥ_tC_3(t)|3⟩,

where:

    Ĥ_tC_0(t)|0⟩=[E_0|0⟩⟨
    0|+ħξ|1⟩⟨
    0|]C_0(t)|0⟩=E_0C_0(t)|0⟩+ħξ
    C_0(t)|1⟩,


    Ĥ_tC_1(t)|1⟩    =[(E_1-iħγ/2)|1⟩⟨ 1|+ħξ√(2)|2⟩⟨ 1|+ħξ|0⟩⟨ 1|]C_1(t)|1⟩
        =ħξ C_1(t)|0⟩+(E_1-iħγ/2)C_1(t)|1⟩+ħξ√(2)C_1(t)|2⟩,


    Ĥ_tC_2(t)|2⟩    =[(E_2-iħγ)|2⟩⟨ 2|+ħξ√(2)|1⟩⟨ 2|+ħξ√(3)|3⟩⟨ 2|]C_2(t)|2⟩
        =ħξ√(2)C_2(t)|1⟩+(E_2-iħγ)C_2(t)|2⟩+ħξ√(3)C_2(t)|3⟩,


    Ĥ_tC_3(t)|0⟩=[(E_3-iħ3γ/2)|3⟩⟨
    3|+ħξ√(3)|2⟩⟨
    3|]C_3(t)|3⟩=ħξ√(3)C_3(t)|2⟩+(E_3-iħ3γ/2)C_3(t)|3⟩,

i.e.,

    Ĥ_t|φ(t)⟩    =[E_0C_0(t)+ħξ C_1(t)]|0⟩+[ħξ C_0(t)+(E_1-iħγ/2)C_1(t)+ħξ√(2)C_2(t)]|1⟩
         +[ħξ√(2)C_1(t)+(E_2-iħγ)C_2(t)+ħξ√(3)C_3(t)]|2⟩+[ħξ√(3)C_2(t)+(E_3-iħ3γ/2)C_3(t)]|3⟩.

By comparing the coefficients of the same basis states in
Eqs. (<ref>) and (<ref>), we have:

    iħĊ_0(t)|0⟩    =[E_0C_0(t)+ħξ C_1(t)]|0⟩,
    
    iħĊ_1(t)|1⟩    =[ħξ C_0(t)+(E_1-iħγ/2)C_1(t)+ħξ√(2)C_2(t)]|1⟩,
    
    iħĊ_2(t)|2⟩    =[ħξ√(2)C_1(t)+(E_2-iħγ)C_2(t)+ħξ√(3)C_3(t)]|2⟩,
    
    iħĊ_3(t)|3⟩    =[ħξ√(3)C_2(t)+(E_3-iħ3γ/2)C_3(t)]|3⟩,

with ν_n=E_n/ħ. Then we obtain the following equations
of motion for the probability amplitudes C_n(t):

    Ċ_0(t)     =-iν_0C_0(t)-iξ C_1(t),
    Ċ_1(t)     =-iξ C_0(t)-i(ν_1-iγ/2)C_1(t)-iξ√(2)C_2(t),
    Ċ_2(t)     =-iξ√(2)C_1(t)-i(ν_2-iγ)C_2(t)-iξ√(3)C_3(t),
    Ċ_3(t)    
    =-iξ√(3)C_2(t)-i(ν_3-i3γ/2)C_3(t),

where ν_n=E_n/ħ.

Similarly, due to the weak-driving case, we have the following
approximate formulas: C_0∼1, C_1∼ξ/γ,
C_2∼ξ^2/γ^2, and
C_3∼ξ^3/γ^3. Then we can approximately solve
the equations in Eq. (<ref>) using a perturbation method by
discarding higher-order terms in each equation for lower-order
variables. Thus, the Eq. (<ref>) becomes:

    Ċ_0(t)     =-iν_0C_0(t),
    Ċ_1(t)     =-i(ν_1-iγ/2)C_1(t)-iξ C_0(t),
    Ċ_2(t)     =-i(ν_2-iγ)C_2(t)-iξ√(2)C_1(t),
    Ċ_3(t)    
    =-i(ν_3-i3γ/2)C_3(t)-iξ√(3)C_2(t),

where ν_n=E_n/ħ.

For an initially empty cavity, the initial conditions read as:
C_0(0)=C_0(0), and C_1(0)=C_2(0)=C_3(0)=0. Then, the
solution of the zero-photon amplitude can be obtained as

    C_0(t)=C_0(0)exp(-iν_0t).

Hence, the equation for the single-photon amplitude in
Eq. (<ref>) becomes

    Ċ_1(t)=-i(ν_1-iγ/2)C_1(t)-iξ
    C_0(0)exp(-iν_0t).

To solve this equation, we introduce a slowly-varying amplitude:

    C_1(t)     =c_1(t)exp[-i(ν_1-iγ/2)t],
    
    C_1(0)     =c_1(0),

then we obtain

    Ċ_1(t)=ċ_1(t)exp[-i(ν_1-iγ/2)t]-i(ν_1-iγ/2)c_1(t)exp[-i(ν_1-iγ/2)t],

and Eq. (<ref>) becomes:

    ċ_1(t)e^-i(ν_1-iγ/2)t-i(ν_1-iγ/2)c_1(t)e^-i(ν_1-iγ/2)t =    -i(ν_1-iγ/2)c_1(t)e^-i(ν_1-iγ/2)t-iξ C_0(0)e^-iν_0t,
    ċ_1(t) =    -iξ
    C_0(0)exp[i(ν_1-ν_0-iγ/2)t].

The solution can be obtained by integrating both sides of
Eq. (<ref>), as follows:

    c_1(t)-c_1(0)     =-iξ C_0(0)∫_0^texp[i(ν_1-ν_0-iγ/2)t'] dt',
    
    c_1(t)-c_1(0)     =-iξC_0(0)/i(ν_1-ν_0-iγ/2){exp[i(ν_1-ν_0-iγ/2)t]-1},
    
    c_1(t)exp[-i(ν_1-iγ/2)t]     =c_1(0)exp[-i(ν_1-iγ/2)t]-iξC_0(0)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]},
    
    C_1(t)    
    =C_1(0)exp[-i(ν_1-iγ/2)t]-iξC_0(0)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.

With the initial condition C_1(0)=0, we have the solution for
the single-photon amplitude given by

    C_1(t)=-iξC_0(0)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.


Consider the solution of the single-photon amplitude in
Eq. (<ref>), the equation for the two-photon amplitude in
Eq. (<ref>) becomes

    Ċ_2(t)=-i(ν_2-iγ)C_2(t)-√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]}.

To solve this equation, we introduce another slowly-varying
amplitude:

    C_2(t)     =c_2(t)exp[-i(ν_2-iγ)t],
    
    C_2(0)     =c_2(0),

and obtain

    Ċ_2(t)=ċ_2(t)exp[-i(ν_2-iγ)t]-i(ν_2-iγ)c_2(t)exp[-i(ν_2-iγ)t],

then Eq. (<ref>) becomes:

    ċ_2(t)e^-i(ν_2-iγ)t-i(ν_2-iγ)c_2(t)e^-i(ν_2-iγ)t=-i(ν_2-iγ)c_2(t)e^-i(ν_2-iγ)t-√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)[e^-iν_0t-e^-i(ν_1-iγ/2)t],
       ċ_2(t)
    =-√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2){exp[i(ν_2-ν_0-iγ)t]-exp[i(ν_2-ν_1-iγ/2)t]}.

The solution can also be obtained by integrating both sides of
Eq. (<ref>), as follows:

    c_2(t)-c_2(0) =    -√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)∫_0^t{exp[i(ν_2-ν_0-iγ)t']-exp[i(ν_2-ν_1-iγ/2)t']} dt',
    
    c_2(t)-c_2(0) =    -√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2){exp[i(ν_2-ν_0-iγ)t]-1/i(ν_2-ν_0-iγ)-exp[i(ν_2-ν_1-iγ/2)t]-1/i(ν_2-ν_1-iγ/2)},
    
    c_2(t)exp[-i(ν_2-iγ)t] =    c_2(0)exp[-i(ν_2-iγ)t]-√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)·exp(-iν_0t)-exp[-i(ν_2-iγ)t]/i(ν_2-ν_0-iγ)
       +√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)·exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/i(ν_2-ν_1-iγ/2),
    
    C_2(t) =    C_2(0)exp[-i(ν_2-iγ)t]-√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)·exp(-iν_0t)-exp[-i(ν_2-iγ)t]/i(ν_2-ν_0-iγ)
       +√(2)ξ^2C_0(0)/i(ν_1-ν_0-iγ/2)·exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/i(ν_2-ν_1-iγ/2).

With the initial condition C_2(0)=0, we have the following
solution of the two-photon amplitude

    C_2(t)=√(2)ξ^2C_0(0)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_2-iγ)t]/(ν_2-ν_0-iγ)-exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/(ν_2-ν_1-iγ/2)}.


Consider the solution of the two-photon amplitude in
Eq. (<ref>), the equation for the three-photon amplitude in
Eq. (<ref>) becomes

    Ċ_3(t)=   -i(ν_3-i3γ/2)C_3(t)+√(6)ξ^3C_0(0)exp(-iν_0t)-exp[-i(ν_2-iγ)t]/i(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)
       -√(6)ξ^3C_0(0)exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/i(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2).

To solve this equation, we introduce the slowly-varying amplitude:

    C_3(t)     =c_3(t)exp[-i(ν_3-i3γ/2)t],
    
    C_3(0)     =c_3(0),

and obtain

    Ċ_3(t)=ċ_3(t)exp[-i(ν_3-i3γ/2)t]-i(ν_3-i3γ/2)c_3(t)exp[-i(ν_3-i3γ/2)t],

then Eq. (<ref>) becomes:

    ċ_3(t)e^-i(ν_3-i3γ/2)t-i(ν_3-i3γ/2)c_3(t)e^-i(ν_3-i3γ/2)t=   -i(ν_3-i3γ/2)C_3(t)+√(6)ξ^3C_0(0)e^-iν_0t-e^-i(ν_2-iγ)t/i(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)
       -√(6)ξ^3C_0(0)e^-i(ν_1-iγ/2)t-e^-i(ν_2-iγ)t/i(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2),


    ċ_3(t)=   √(6)ξ^3C_0(0)exp[i(ν_3-ν_0-i3γ/2)t]-exp[i(ν_3-ν_2-iγ/2)t]/i(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)
       -√(6)ξ^3C_0(0)exp[i(ν_3-ν_1-iγ)t]-exp[-i(ν_3-ν_2-iγ/2)t]/i(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2).

The solution can also be obtained by integrating both sides of
Eq. (<ref>), as follows:

    c_3(t)-c_3(0)     =√(6)ξ^3C_0(0)/i(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)∫_0^t{exp[i(ν_3-ν_0-i3γ/2)t']-exp[i(ν_3-ν_2-iγ/2)t']} dt'
         -√(6)ξ^3C_0(0)/i(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)∫_0^t{exp[i(ν_3-ν_1-iγ)t']-exp[-i(ν_3-ν_2-iγ/2)t']} dt'
        =√(6)ξ^3C_0(0)/i(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ){exp[i(ν_3-ν_0-i3γ/2)t]-1/i(ν_3-ν_0-i3γ/2)-exp[i(ν_3-ν_2-iγ/2)t]-1/i(ν_3-ν_2-iγ/2)}
         -√(6)ξ^3C_0(0)/i(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2){exp[i(ν_3-ν_1-iγ)t]-1/i(ν_3-ν_1-iγ)-exp[i(ν_3-ν_2-iγ/2)t]-1/i(ν_3-ν_2-iγ/2)},


    c_3(t)exp[-i(ν_3-i3γ/2)t]     =c_3(0)exp[-i(ν_3-i3γ/2)t]-√(6)ξ^3C_0(0){exp(-iν_0t)-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_0-i3γ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_2-iγ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_1-iγ/2)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_1-iγ)
         -√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_2-iγ/2),
    
    C_3(t)     =C_3(0)exp[-i(ν_3-i3γ/2)t]-√(6)ξ^3C_0(0){exp(-iν_0t)-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_0-i3γ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_2-iγ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_1-iγ/2)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_1-iγ)
         -√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_2-iγ/2).

With the initial condition C_3(0)=0, we have the following
solution of the three-photon amplitude

    C_3(t)     =-√(6)ξ^3C_0(0){exp(-iν_0t)-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_0-i3γ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_2-iγ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_1-iγ/2)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_1-iγ)
         -√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_2-iγ/2).


Thus, for the initially empty resonator, the solutions of the
equations of motion for the probability amplitudes in the
equations in Eq. (<ref>) can be obtained as:

    C_0(t)     =C_0(0)exp(-iν_0t),
    
    C_1(t)     =-ξC_0(0)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_1-iγ/2)t]},
    
    C_2(t)     =√(2)ξ^2C_0(0)/(ν_1-ν_0-iγ/2){exp(-iν_0t)-exp[-i(ν_2-iγ)t]/(ν_2-ν_0-iγ)-exp[-i(ν_1-iγ/2)t]-exp[-i(ν_2-iγ)t]/(ν_2-ν_1-iγ/2)},
    
    C_3(t)     =-√(6)ξ^3C_0(0){exp(-iν_0t)-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_0-i3γ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_0-iγ)(ν_3-ν_2-iγ/2)
         +√(6)ξ^3C_0(0){exp[-i(ν_1-iγ/2)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_1-iγ)
         -√(6)ξ^3C_0(0){exp[-i(ν_2-iγ)t]-exp[-i(ν_3-i3γ/2)t]}/(ν_1-ν_0-iγ/2)(ν_2-ν_1-iγ/2)(ν_3-ν_2-iγ/2),

where

    ν_0 =0,   ν_1 =Δ_L+Δ_F,   ν_2
    =2Δ_L+2Δ_F+2U,   ν_3
    =3Δ_L+3Δ_F+6U.

When the initial state of the system is the vacuum state
|0⟩, i.e., the initial condition C_0(0)=1,
the solutions in Eq. (<ref>) are reduced to:

    C_0(t) =   1,
    
    C_1(t) =   -ξ1/(Δ_L+Δ_F-iγ/2){1-exp[-i(Δ_L+Δ_F-iγ/2)t]},
    
    C_2(t) =   √(2)ξ^2/(Δ_L+Δ_F-iγ/2){1-exp[-i(2Δ_L+2Δ_F+2U-iγ)t]/(2Δ_L+2Δ_F+2U-iγ)-exp[-i(Δ_L+Δ_F-iγ/2)t]/(Δ_L+Δ_F+2U-iγ/2)}
       +√(2)ξ^2exp[-i(2Δ_L+2Δ_F+2U-iγ)t]/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+2U-iγ/2),


    C_3(t)     =-√(6)ξ^3{1-exp[-i(3Δ_L+3Δ_F+6U-i3γ/2)t]}/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)(3Δ_L+3Δ_F+6U-i3γ/2)
         +√(6)ξ^3{exp[-i(2Δ_L+2Δ_F+2U-iγ)t]-exp[-i(3Δ_L+3Δ_F+6U-i3γ/2)t]}/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)(Δ_L+Δ_F+4U-iγ/2)
         +√(6)ξ^3{exp[-i(Δ_L+Δ_F-iγ/2)t]-exp[-i(3Δ_L+3Δ_F+6U-i3γ/2)t]}/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+2U-iγ/2)(2Δ_L+2Δ_F+6U-iγ)
         -√(6)ξ^3{exp[-i(2Δ_L+2Δ_F+2U-iγ)t]-exp[-i(3Δ_L+3Δ_F+6U-i3γ/2)t]}/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+2U-iγ/2)(Δ_L+Δ_F+4U-iγ/2),

and for the infinite-time limit exp(-At)→0 (t→∞),
we have:

    C_0(∞)    ≡ C_0=1,
    
    C_1(∞)    ≡ C_1=-ξ/(Δ_L+Δ_F-iγ/2),
    
    C_2(∞)    ≡ C_2=-√(2)ξC_1/(2Δ_L+2Δ_F+2U-iγ),
    
    C_3(∞)    ≡
    C_3=-√(3)ξC_2/(3Δ_L+3Δ_F+6U-i3γ/2).


For the state given in Eq. (<ref>), the infinite-time state
(steady state) of the system reads as

    |φ(t→∞)⟩    =|0⟩+-ξ/(Δ_L+Δ_F-iγ/2)|1⟩+√(2)ξ^2/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)|2⟩
         +-√(6)ξ^3/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)(3Δ_L+3Δ_F+6U-i3γ/2)|3⟩,

and the normalization constant of the state is given by

    N=1+|C_1|^2+|C_2|^2+|C_3|^2,

where:

    |C_1|^2   
    =|ξ/(Δ_L+Δ_F-iγ/2)|^2=ξ^2/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+iγ/2)=ξ^2/[(Δ_L+Δ_F)^2+γ^2/4],


    |C_2|^2    =|√(2)ξ^2/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)|^2
       =2ξ^4/(Δ_L+Δ_F-iγ/2)(Δ_L+Δ_F+iγ/2)(2Δ_L+2Δ_F+2U-iγ)(2Δ_L+2Δ_F+2U+iγ)
       
    =2ξ^4/[(Δ_L+Δ_F)^2+γ^2/4][4(Δ_L+Δ_F+U)^2+γ^2],


    |C_3|^2    =|-√(6)ξ^3/(Δ_L+Δ_F-iγ/2)(2Δ_L+2Δ_F+2U-iγ)(3Δ_L+3Δ_F+6U-i3γ/2)|^2
        =6ξ^6/|Δ_L+Δ_F-iγ/2|^2|(2Δ_L+2Δ_F+2U-iγ)|^2|3Δ_L+3Δ_F+6U-i3γ/2|^2
        =6ξ^6/[(Δ_L+Δ_F)^2+γ^2/4][4(Δ_L+Δ_F+U)^2+γ^2][9(Δ_L+Δ_F+2U)^2+9γ^2/4].

The probabilities of finding single, two and three photons in the
cavity are, respectively, given by:

    P_1   =|C_1|^2/N,
    
    P_2   =|C_2|^2/N,
    
    P_3   =|C_3|^2/N.


As mentioned in Sec. <ref>, the equal-time third-order
correlation function can be written as

    g^(3)(0,0) ≡ g^(3)(0)≡⟨
    a^†3a^3⟩/⟨â^†â⟩ ^3=⟨â^†â(â^†â-1)(â^†â-2)⟩/⟨â^†â⟩ ^3=⟨
    (â^†â)^3-3(â^†â)^2+2â^†â⟩/⟨â^†â⟩ ^3

When the cavity field is in the state (<ref>), we have

    g^(3)(0)     =∑_n,n'=0^3C_n'^*C_n⟨ n'|(â^†â)^3|n⟩-3∑_n,n'=0^3C_n'^*C_n⟨ n'|(â^†â)^2|n⟩+2∑_n,n'=0^3C_n'^*C_n⟨ n'|â^†â|n⟩/(∑_n,n'=0^3C_n'^*C_n⟨ n'|â^†â|n⟩)^3
        =|C_1|^2+8|C_2|^2+27|C_3|^2-3(|C_1|^2+4|C_2|^2+9|C_3|^2)+2(|C_1|^2+2|C_2|^2+3|C_3|^2)/(|C_1|^2+2|C_2|^2+3|C_3|^2)^3
        =N(P_1+8P_2+27P_3-3P_1-12P_2-27P_3+2P_1+4P_2+6P_3)/N^2(P_1+2P_2+3P_3)^3
        =6P_3/N(P_1+2P_2+3P_3)^3

In the weak-driving regime, we have the following approximate
amplitudes: C_0∼1, C_1∼ξ/γ,
C_2∼ξ^2/γ^2, and
C_3∼ξ^3/γ^3, i.e., N∼1 with
|C_3|^2≪|C_2|^2≪|C_1|^2≪1.
Hence, the third-order correlation function can be written as

    g^(3)(0)≈6P_3/P_1^3.


Substituting Eqs. (<ref>) and (<ref>) into
Eq. (<ref>), we can easily obtain

    g^(3)(0)    ≈36ξ^6/[(Δ_L+Δ_F)^2+γ^2/4][4(Δ_L+Δ_F+U)^2+γ^2][9(Δ_L+Δ_F+2U)^2+9γ^2/4]·[(Δ_L+Δ_F)^2+γ^2/4]^3/ξ^6
        =[(Δ_L+Δ_F)^2+γ^2/4]^2/[(Δ_L+Δ_F+U)^2+γ^2/4][(Δ_L+Δ_F+2U)^2+γ^2/4],

where Δ_F>0 (Δ_F<0) denotes the light
propagating against (along) the direction of the spinning
resonator.

Here, we focus on the nonspinning case (Δ_F=0), the
rotating case is discussed in Sec. <ref>. For this case, the
third-order correlation function becomes

    g_0^(3)(0)=(Δ_L^2+γ^2/4)^2/[(Δ_L+U)^2+γ^2/4][(Δ_L+2U)^2+γ^2/4].

Including the second-order correlation function, we can
quantitatively compare our analytical results with numerical
calculations <cit.>. We
find an excellent agreement between the numerical calculations and
the approximate analytical solutions, as shown in
Fig. <ref>. Here, the solid curves are plotted using the
numerical solution, while the curves with symbols are based on the
analytical solution given in Eqs. (<ref>) and
(<ref>). As for the g_0^(2)(0)≈
2P(2)/P(1)^2, given in Eq. (<ref>), the dip D^(2)
and the peak P^(2) in the light green curves correspond to the
single- and two-photon resonant driving cases, respectively. In
the single-photon resonant driving case (k=1), a single photon
can be resonantly injected into the cavity, while the probability
of finding two photons in the cavity is largely suppressed due to
the energy restriction; this represents 1PB. We find that the
analytical value of g_0^(2)(0)∼0.0008 at this dip
D^(2), which is well-matched with our numerical value
g_0^(2)(0)∼0.0009. In the two-photon resonant
driving case (k=2), the probability for finding two photons
inside the cavity is resonantly enhanced, and this corresponds to
a peak in the curve of g^(2)(0). We find that the analytical
value of g^(2)(0)∼974 at this peak P^(2) is above the
numerical solution g^(2)(0)∼673, since we neglected the
two-photon probability in the denominator of the analytical
formula [this can be seen more clearly in Eqs. (<ref>) and
(<ref>)]. As for the g_0^(3)(0)≈
6P(3)/P(1)^3, given in Eq. (<ref>), the dip D^(3)
and the peaks P^(3)_1 and P^(3)_2 in the dark green
curves correspond to the single-, two-, and three-photon
resonant-driving cases, respectively. In the single-photon
resonant-driving case (k=1), P(1)≫ P(2)≫ P(3), thus, there
is a dip [i.e., D^(3)] in the g_0^(3)(0) curve.
For the two-photon resonant-driving case (k=2), the
single-photon probability is suppressed, which causes the
occurrence of the peak P^(3)_1. However, the peak
P^(3)_1 is lower than the peak P^(3)_2 at k=3, since
the three-photon probability is enhanced at k=3 (i.e.,
three-photon resonant-driving case), but still suppressed at k=2
(i.e., two-photon resonant-driving case).






§ ROTATION-INDUCED QUANTUM NONRECIPROCITY



 §.§ Rotation-induced shifts


For the optical microtoroid resonator, an input-laser light
applied from the left or right side of the cavity causes a
clockwise (CW) circulating mode or a counterclockwise (CCW)
circulating mode. When the microresonator is rotating,
Δ_F>0 and Δ_F<0 denote the cases with the
light propagating against and along the spinning direction of the
resonator, respectively, i.e., for the CCW spinning resonator,
Δ_F>0 (Δ_F<0) indicates an input-laser
applied from the left (right) side; for the CW spinning resonator,
Δ_F>0 (Δ_F<0) indicates an input-laser used
from the right (left) side.

When the resonator is rotating, the second-order correlation
function in Eq. (<ref>) can be written as

    g_±^(2)(0)
    =(Δ_L±|Δ_F|)^2+γ^2/4/(Δ_L±|Δ_F|+U)^2+γ^2/4,

where g_-^(2)(0) [g_+^(2)(0)] denotes the equal-time
second-order correlation function for Δ_F<0
(Δ_F>0).



For the Δ_F<0 case, 1PB emerges at
Δ_L=|Δ_F| with
g^(2)_-(0)=(γ^2/4)/(U^2+γ^2/4)=[4(U/γ)^2+1]^-1.
This minimum value of g_-^(2)(0) is independent of the
angular speed Ω; thus, the minimum value of
g_-^(2)(0) is a constant. Since |Δ_F|
is an amount proportional to the angular speed Ω, the dip
D^(2) experiences linearly shifts with Ω. Also,
D^(2) experiences linearly shifts to the opposite direction
for the Δ_F<0 case, since now 1PB emerges at
Δ_L=-|Δ_F|. The shifts of the curve
can also be understood from an energy-level structure, where the
rotation of the resonator causes upper or lower shifts of energy
levels, as shown in Fig. <ref>.

Here, we plot the correlation function g^(2)(0) as a function
of k when the angular speed Ω takes various values, as
shown in Fig. <ref>. For the Δ_F<0 case, a blue
shift of the g^(2)(0) curve can be clearly seen in
Fig. <ref>(a). For the Δ_F>0 case, a red shift can
be seen in Fig. <ref>(b). This indicates a highly-tunable
nonreciprocal PB device, i.e., sub-Poissonian light can be
achieved by driving from one side; super-Poissonian light
emerges by driving from the opposite side (see Fig. 2 in the main
article).




For example, let us now fix the CCW rotation of the resonator;
hence Δ_F>0 (Δ_F<0) corresponds to the situation
of driving the resonator from its left (right) side, i.e., the CW
(CCW) mode frequency is
ω_↻≡ω_0+|Δ_F|
(ω_↺≡ω_0-|Δ_F|), as
aforementioned. When the optical resonator rotates with an angular
velocity Ω=6.6 kHz <cit.>, we
find g^(2)_↻(0)∼0.39 and
g^(2)_↺(0)∼2.53, i.e., sub-Poissonian
light can be achieved by driving the device from its left side,
while super-Poissonian light emerges by driving from the
right side, as shown in Fig. <ref>.

The third-order correlation function Eq. (<ref>) in
the rotating resonator becomes

    g_±^(3)(0)    
    =[(Δ_L±|Δ_F|)^2+γ^2/4]^2/[(Δ_L±|Δ_F|+U)^2+γ^2/4][(Δ_L±|Δ_F|+2U)^2+γ^2/4],

where g_-^(3)(0) (g_+^(3)(0)) denotes the third-order
optical intensity correlation for the Δ_F<0
(Δ_F>0) case. Similarly, the curve of g^(3)(0) also
experiences opposite shifts for different driving directions.






 §.§ Nonreciprocal photon blockade


We have investigated PB effects (witnessing sub-Poissonian light)
and photon-induced tunneling (PIT, corresponding to
super-Poissonian light) for the nonspinning case in the former
Sections. Note that PB and PIT always emerge at fixed locations of
the tuning parameter k, no matter if the input-laser comes from
the left or right side (see Figs. <ref> and <ref>).
However, the rotation of the resonator can lead to upper or lower
shifts of energy levels for different driving directions, as
discussed in Sec. <ref>. Therefore, using a spinning
nonlinear optical resonator, under the same driving frequencies,
PIT can emerge by driving from one side and 1PB/2PB can emerge by
driving from the other direction, i.e., unidirectional
1PB/2PB. Furthermore, 1PB for driving from one side and 2PB for
driving from the opposite direction can also be realized with this
spinning device.

As shown in Figs. <ref>(a) and <ref>(b), when the
angular speed of the resonator is Ω=58 kHz, we
find (i) 1PB for Δ_F>0 and PIT for Δ_F<0, at
k=2.0; (ii) 2PB for Δ_F>0 and PIT for
Δ_F<0, at k=3.0. These nonreciprocal 1PB and 2PB can
also be confirmed by comparing the photon-number distribution
P(n) with the Poissonian distribution 𝒫(n).
Figure <ref>(b) shows that: (i) single-photon probability
P(1) is enhanced while two- and more-photon probabilities
P(m>1) are suppressed for the Δ_F>0 case, leading to
1PB; in contrast, P(1) is suppressed while P(m>1) are enhanced
for the Δ_F<0 case, leading to PIT. (ii) only
two-photon probability P(2) is enhanced for Δ_F>0,
which corresponds to 2PB; in contrast, PIT emerges for
Δ_F<0. The unidirectional 2PB can also be achieved at
k=2.5 when Ω=29 kHz, as shown in
Figs. <ref>(c) and <ref>(d). Such quantum
nonreciprocities indicate one-way quantum devices at the
few-photon level, and open up exciting prospects for applications
in nonreciprocal quantum technologies, such as nonreciprocal
quantum information processing or few-photon topological
devices <cit.>.





More interestingly, when the angular speed of the nonlinear
optical resonator is Ω=29 kHz, 2PB emerges at k=1.5 for
Δ_F<0, while 1PB emerges with the same driving strength
for Δ_F>0, as shown in Figs. <ref>(c)
and <ref>(d). In contrast to the nonreciprocities of the
former cases between the sub- and super-Poissonian states of
light, this is a new kind of nonreciprocal PB between two
sub-Poissonian states of light, indicating possible applications
for few-photon nonreciprocal devices with direction-dependent
counting-statics.

All of the cases of nonreciprocal PB can be intuitively understood
by considering the energy-level structure of the system. As shown
in Fig. <ref>(a), for the Δ_F>0 case,
when angular speed fulfills |Δ_F|=U and the probe light
with frequency ω_0+|Δ_F| (k=2.0), the light is
resonantly coupled to the transition |0⟩→|1⟩. The
transition |1⟩→|2⟩ is detuned by 2ħ U and,
thus, suppressed for U>γ, i.e., once, a photon is coupled
into the resonator, it suppresses the probability of the second
photon with the same frequency going into the resonator. In
contrast, for the Δ_F<0 case, there is a three-photon
resonance with the transition |0⟩→|3⟩, hence the
absorption of the first photon favors also that of the second or
subsequent photons, i.e., resulting in PIT. This is a clear
signature of nonreciprocal 1PB, i.e., sub-Poissonian light
emerges for Δ_F>0, while super-Poissonian light
can be observed for Δ_F<0.

As shown in Figs. <ref>(c) [<ref>(e)],
for the Δ_F>0 case, by choosing |Δ_F|=U
(|Δ_F|=U/2) and Δ_L=-2U (Δ_L=-3U/2), the
transition |0⟩→|2⟩ is resonantly driven by the
input laser, but the transition |2⟩→|3⟩ is detuned
by 4ħ U, which features the 2PB effect; in contrast, for the
Δ_F<0 case, four-photon resonance (three-photon
resonance) happens for the transition |0⟩→|4⟩
(|0⟩→|3⟩), leading to PIT. This is also a
nonreciprocal PB.

As shown in Fig. <ref>(g), for the Δ_F>0
case, when |Δ_F|=U/2 and Δ_L=-U/2 (k=1.5), the
input light is resonantly coupled to the transition
|0⟩→|1⟩, and the transition
|1⟩→|2⟩ is detuned by 2ħ U, leading to 1PB.
More interestingly, for the Δ_F<0 case, the input light
is just resonantly coupled to the transition
|0⟩→|2⟩, and the transition
|2⟩→|3⟩ is detuned by 4ħ U, i.e., resulting
in 2PB. This 1PB-2PB nonreciprocity can suggest an application for
a purely quantum device with direction-dependent counting
statistics. This new nonreciprocal feature, which (to our
knowledge) has not been revealed previously.



Table <ref> shows different cases of nonreciprocal PB.
Interestingly, both PB-PIT and 1PB-2PB nonreciprocities can only
occur in an irreversible way. Unidirectional 1PB for
Δ_F>0, i.e., 1PB emerges for Δ_F>0 and PIT
emerges for Δ_F<0, can occur with the same angular
speeds (Δ_F=± U), and the same driving frequencies
(Δ_L=-U). However, the case of PIT for Δ_F>0 and
1PB for Δ_F<0 cannot be observed with the same angular
speeds and driving frequencies, i.e., one-way 1PB is an
irreversible quantum nonreciprocal effect. Also, 1PB-2PB
nonreciprocity can only happen in the case of 1PB for
Δ_F>0 and 2PB for Δ_F<0, but not vice
versa.

Note that 1PB and 2PB correspond to the single- and two-photon
resonances, respectively. PIT is also caused by a multi-photon
resonance. The multi-photon resonance can be clearly seen in
energy-level diagrams, thus, the origin of this irreversible
feature can be understood from the energy-level diagrams for
Δ_F>0 and Δ_F<0. Without the rotation, the
energy-level diagrams for the Δ_F>0 and
Δ_F<0 cases are symmetric. Due to the rotation, energy
levels experience shifts to different directions for
Δ_F>0 and Δ_F<0, leading to asymmetries of
energy-level diagrams, as shown in Fig. <ref>. From
Sec. <ref>, the energy levels of this spinning system are
E_n=nħΔ_L+nħΔ_F+(n^2-n)ħ U. Thus,
we have

    E_n/n=ħ(Δ_L+Δ_F)+(n-1)ħ U.

Then the driving frequency of an n-photon resonance for the
Δ_F>0 case is

    ω_L=ω_0+|Δ_F|+nU-U,

and the driving frequency of an m-photon resonance for the
Δ_F<0 case is

    ω_L'=ω_0-|Δ_F|+mU-U.

Under the same driving frequency, we have

    ω_0+|Δ_F|+nU-U   =ω_0-|Δ_F|+mU-U 
    
    |Δ_F|+nU   =-|Δ_F|+mU 
    
    2|Δ_F|   =(m-n)U.

Because |Δ_F|>0 (i.e., Ω≠0) and U>0, we have
the following condition for the allowed cases of nonreciprocal PB

    n<m.

When the driving frequencies for Δ_F>0 and
Δ_F<0 are the same, an n-photon resonance for
Δ_F>0 and an m-photon resonance for Δ_F<0
can only happen under the condition n<m. In contrast to this,
the cases of n>m are prohibited, as shown in
Figs. <ref>(b), <ref>(d),
<ref>(f), and <ref>(h).


 ^*Corresponding author.

jinghui73@foxmail.com













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