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Nonreciprocal phonon blockade

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.


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 µm, 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 phononinduced tunneling (PIT), and detecting of PB [23][24][25][26][27][28]. Optomechanical systems with second order nonlinearities are considered for photon induced phonon blockade [23,29,30], with potential applications in hybrid photon-phonon quantum networks.
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 here, ω is the resonance frequency of the cavity, a(a † ) denote the annihilation(creation) operator of the cavity field, ω 0 is the level splitting of the two-level system, and SiV centers is descried by the spin operator σ z = |e e| − |g g|, with |e and |g represent the excited and ground state of SiV centers, respectively. The ladder operators σ ± = 1 2 (σ x ± iσ y ) describe the interaction between the (Color online) Schematic of the composite spinphononic 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 Ω. 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.
SiV centers and the phononic cavity, with σ x and σ y as Pauli matrices, where g is the coupling strength between the cavity mode and SiV center. The parameter ξ is the driving field amplitude with driving frequency ω L .
In the regime where SiV centers and phonon modes are far detuned to the coupling strength ∆ = (ω − ω 0 ) ≫ g and the Rabi frequency Ω s , by which SiV centers are driven with the field of frequency ω 1 obeys the condition Ω s ≫ g 2 /∆ . Here we assume that ω 1 = ω 0 , and then the Hamiltonian H (0) in Eq.(1) in the rotating frame with V = exp(−iω 0 σ z t/2) can be written as [13,100] with the effective phonon-phonon interaction term (nonlinearity strength) Here, it is worth mentioning that after straightforward calculations we have ignored ρ z = |+ +| − |− −|, with |± = (|g ± |e )/ √ 2, by preparing the SiV in the ground state |− .
In the ring cavity, the phonon modes with two different propagation directions (clockwise (CW) or counterclockwise (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 Ω to the cavity, the resonant frequency of phonons will have a change, i.e. ω → ω + ∆ F , where here χ = ±0.12 is the chirality of the phonon. For the rotator, rotating in the CCW direction, when the cavity phonon rotates CW, ∆ F > 0, otherwise, ∆ F < 0. Now in the rotating reference frame at the driving frequency ω L , the effective Hamiltonian of the driven system can be written as: here, ∆ L = ω L − ω 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 For a Fock state |n , we have Thus we obtain the eigenenergy where n is the number of cavity phonons. The eigenenergy is also effective for the weak driving condition. In the region of weak drive ξ ≪ γ, we can truncate the Hilbert space of the system to n=2. Here, γ = ω/Q is the cavity dissipation rate and Q is the quality factor. The state of the system can be written as |ϕ(t) = 2 n=0 C n (t)|n , 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 eff − (i γ/2)â †â . Then the equation of motion of the dissipative system is given bẏ 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 Now we proceed to study PB and PIT. For this purpose, we will exploit the µ th -order correlation function of zero-time delay g (µ) (0) ≡ â †µâµ / n µ , withn =â †â . 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 For n=3, we can get the third-order correlation function The minimum and the maximum of g (µ) (0) are corresponding to 1PB, multi-PB, and PIT, respectively. For example, when g (2) min (0) = 1/[4 (U/γ) 2 + 1]<1 for ∆ L = ∆ F , the phonons satisfy the super-Poisson distribution (phonon bunching state) which corresponds to 1PB; when g (2) max (0) = 4 (U/γ) 2 + 1>1 for ∆ L = ∆ F + U , the phonons satisfy the sub-Poisson distribution (phonon anti-bunching state), which corresponds to PIT. For µ>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, where H eff is the hamiltonian given in Eq. 5. We can get the phonon number probability P (n) = n |ρ| n from the steady-state solutionρ of the system. We now focus on the properties of the cavity phonons described by the second order correlation function for various deriving field strengths ξ. In Fig. 2 we plot the results given by Eq. (12) as a function of the nonlinear strength U and detuning ∆ L in the case of a stationary cavity i.e. ∆ F = 0. In the weak drive condition (ξ = 0.33γ), 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 ξ = 1γ, ξ = 3γ and ξ = 5γ 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 for PIT, there is Here, m 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≤4). In this case, m ≪ 1 , so Eq. (14) can be rewritten as For nPB, we also have an equivalent standard for comparing the phonon number distribution with the Poisson distribution: with P(n) = n n n! exp(− n ) is the Poissonian distribution. To show a relative deviation of a given phononnumber distribution from the corresponding Poissonian distribution, we use the formula [106] [P (n) − P(n)]/P(n).
To further elaborate the effect of weak drive on PB in the non-rotating system (∆ F = 0), in Fig. 3 we have plotted the n-order correlation function for constant nonlinear strength (U). Figure 3(a) shows the results for the n-order correlation function with respect to the detuning (∆ L ) for U = 20γ. Here, for g (2) (0), the minimum and maximum values are at ∆ L = 0 and ∆ L = 20γ , which corresponds to 1PB and PIT respectively.
According to Eq. (13), 1PB should meet the following conditions for n = 1: (1) . (18) 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 ∆ 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) > P(1), P (2) < P(2), and P (3) < 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) > P(2) and P (3) > P(3) in Fig. 3(e). In addition, we can find that g (2) 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 n-order correlation function versus the detuning (∆ L ) for U = 20γ and ξ = 3γ. One can see the system still shows 1PB at ∆ 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 ∆ L = 20γ and ∆ L = 40γ, respectively.

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 ∆ L when the angular velocity Ω 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 Ω increases. 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 Ω, 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 ∆ F = ∆ L = U/2 = 10γ. Furthermore, Fig. 6(a) shows that our analytical solution and the numerical solution are consistent; here we use g (2) (0) and g (2) (0) to indicate that ∆ F > 0 and ∆ F < 0, respectively. We can clearly see that at ∆ L = 10γ, 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.
In Fig. 7, We show the correlation function versus the detuning ∆ L in the strong driving regime (ξ = 3γ) for ∆ F = 10γ. The solid curves represent the result of g (n) (0), and the dashed line represents the result of g (n) (0). Here, at ∆ L = 10γ, we have 1PB and 2PB for the different driving directions. Furthermore, at ∆ L = 30γ, we obtain 2PB and PIT caused by the three-phonon resonance for different driving directions. These results are  Correlation g (2) (0) D L /g g (2) 0 Q (Num) g (2) 0 P (Num) g (2) 0 Q (Ana) g (2) 0 P ( discussed in detail in Fig. 8 and Fig. 9. In Fig. 8(a), we can intuitively understand the reason why we have 1PB and 2PB by considering the energylevel structure of the system. The plots in Fig. 8(b) and 8(c) show the comparison of correlation functions (g 2 (0) 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) > P(2), P (3) < P(3) and P (4) < P(4), which meets the criterion given in Eq. (16) for n = 2. In    8(e), we find that P (1) > P(1), P (2) < P(2) and P (3) < 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) > P(3) and P (4) > P(4) in Fig. 9(d). In Fig. 9(e), one can clearly see that P (2) > P(2), P (3) < P(3) and P (4) < 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 ∆ F = 20γ and ∆ L = 20γ, nonreciprocity of 2PB and 4PR at ∆ F = 20γ and ∆ L = 40γ for U = 20γ.

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 be- tween the phononic waveguide and ring cavity by using wrap-around couplers. The dimensions of the cavity are (r, w, h) = (50 µm, 5 µm, 0.5 µm), and the mode volume is V ef f = 180 µm 3 [107]. In our calculations, the resonance frequency of the cavity is ω/2π = 3 GHz, and the quality factor is Q = 10 7 . The frequency of phonons, which is in the 1 ∼ 10 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 kHz is easy to obtain. In our calculations, U is in the 0 ∼ 12 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.