Simultaneous blockade of a photon phonon, and magnon induced by a two-level atom
Chengsong Zhao, Xun Li, Shilei Chao, Rui Peng, Chong Li, Ling Zhou
aa r X i v : . [ qu a n t - ph ] S e p Simultaneous blockade of a photon phonon, and magnon induced by a two-level atom
Chengsong Zhao, Xun Li,
1, 2
Shilei Chao, Rui Peng, Chong Li, and Ling Zhou ∗ School of Physics, Dalian University of Technology, Dalian 116024,China National Key Laboratory of Shock Wave and Detonation Physics,Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, China
The hybrid microwave optomechanical-magnetic system has recently emerged as a promisingcandidate for coherent information processing because of the ultrastrong microwave photon-magnoncoupling and the longlife of the magnon and phonon. As a quantum information processing device,the realization of single excitation holds special meaning for the hybrid system. In this paper,we introduce a single two-level atom into the optomechanical-magnetic system and show that anunconventional blockade due to destructive interference cannot offer a blockade of both the photonand magnon. Meanwhile under the condition of single excitation resonance, the blockade of photon,phonon, and magnon can be achieved simultaneously even in a weak optomechanical region, but thephonon blockade still requires the cryogenic temperature condition.
I. INTRODUCTION
The effect of one photon preventing the second photonentrance is called a photon blockade [1, 2], which is thepivotal effect to achieve photons at the quantum level. Itis believed that photon blockade can be used as a singlephoton source and to process quantum information [3].The photon blockade in the cavity-QED systems [4–6]were thoroughly investigated and have been achieved inexperiments [7, 8]. Recently, the optomechanical systemhas attracted significant attention, such as working as asensor to detect tiny mass and force [9–12] , a platform toinvestigate the fundamental physics [13] and a device toprocessing quantum information [14–20]. The most at-tractive characteristic of an optomechanical system is thenonlinearity resulting from the radiation pressure, whichcan induce Kerr nonlinearity [21] and produce the pho-ton blockade [22]. However, currently, the single-photonoptomechanical coupling is still within a weak couplingregion, which only induces only fainter Kerr nonlinear-ity. Therefore, some strategies were put forward to en-hance the nonlinearity [23, 24]. To avoid the weakness ofthe delicate single-photon nonlinear coupling, the photonblockade resulting from destructive interference calledunconventional blockade (UB) was proposed and thor-oughly investigated [25, 26].Most recently, the photon-magnon coupling systemin the microwave [27–29] and optical frequency [30–32] regime has aroused attention. Different from theweak optomechanical coupling, the ultrastrong couplingbetween microwave photons and magnons [the collec-tive spin excitation in yttrium iron garnet (YIG)] wasrealized[28, 33], and the magnons possess a very lowdamping rate. Meanwhile, the magnon excitation in-teracting with phonons (vibrational modes of the YIGsphere) is similar to the optomechanical interaction, [34],so, both kinds of interactions magnetic-mechanical [35]and optical-mechanical are nonlinear. The phonons and ∗ [email protected] magnons posse coupling mediated by cavity fields [36],and the entanglement of a magnon, photon, and phononin cavity magnomechanics has been investigated wherephoton-magnon and magnomechanical interactions wereconsidered [34]. In Ref. [37], the supermode of a photonexhibits blockade under the Kerr effect in optomagnonicmicrocavities system.The photon blockade can be generated from the de-structive interference [25, 38] as well as the single ex-citation resonance [6, 21, 39–41]. Usually, the destruc-tive interference and the single excitation resonance re-sulting from dressed states can supply a better block-ade than the Kerr effect because of the weak couplingstrength of the Kerr interaction. The photon blockadein an optomechanical system [25] as well as in an op-tomagnonic system [37] were separately thoroughly in-vestigated. A magnon blockade via qubit-magnon cou-pling has been studied in Ref. [42]. However, in thehybrid optomechanical-magnetic system, the simultane-ous blockade of the photon, phonon, and magnon has notbeen studied. Meanwhile, the hybrid system has specialsignificance for the realization of quantum informationprocessing, like the quantum internet [43]. If the hybridoptomechanical-magnetic system was used as a quantumdevice, the single excitation level is important, and thesimultaneous blockade of photon, phonon, and magnonshould be pivotal and deserves further investigation.In this paper, we consider a hybrid microwaveoptomechanical-magnetic system aiming to generate thesimultaneous photon-phonon-magnon blockade. Consid-ering the achievement of ultrastrong microwave optical-magnetic coupling in experiments [27, 44], we derivethree-partite interaction among photon, phonon, andmagnon. By introducing a single two-level atom, un-der the condition of single excitation resonance, we showthat the simultaneous blockade of photon, phonon, andmagnon can be achieved with the assistance of the three-partite interaction on the condition of cryogenic temper-ature of the mechanical mode, while the unconventionaldestructive interference can not offer the simultaneouslymulti-modes antibunching. In our scheme the single-photon strong optomechanical coupling is not required, FIG. 1. (a) Sketch of the system. A two-level atom is placedinside a microwave cavity with a movable mirror. A YIGsphere is placed near the maximum magnetic field of the cav-ity mode, and in a uniform bias magnetic field, which estab-lishes the magnon-photon coupling. (b) Energy-level diagramunder the Hamiltonian Eq. (3), where | g ( e ) , n + , n − , n b i de-notes lower-level g (upper level e ), and n j ( j = + , − , b ) is thenumber of the mode ( a + , a − , b ) . The eigenstates are drawnon the right-hand side. therefore, it can be feasible in experiment. Our schemeis a guideline for hybrid optomechanical-magnetic exper-iments nearing the regime of single-photon nonlinearity,and for potential quantum information processing appli-cations with photons, magnons, and phonons. II. THE MODEL AND THE ANALYTICALANALYSIS
We consider a hybrid optomechanical-magnetic sys-tem, where a two-level atom and a YIG microsphere arecontained in the microwave cavity, and one of the mir-rors is movable, shown in Fig. 1(a). The magnons aresourced from a collective spins in a ferrimagnet. Here,we ignore the interaction between magnons and phononsdue to deformation of the YIG sphere, because the single-magnon magnomechanical coupling rate is typically small[34, 36]. The magnetic dipole mediates the coupling be-tween magnons and cavity photons. The Hamiltonian ofthe system reads H = H om + H op + H ao + H d , (1) where H om = ω c a † a + ω m m † m + G m ( a † m + am † ) ,H op = ω b b † b + ga † a ( b † + b ) ,H ao = ω a σ † σ + g a ( σa † + σ † a ) , (2) H d = Ω e ( σe iω L t + σ † e − iω L t ) ,j † ( j, j = a, m, b ) is the creation (annihilation) operatorof the related mode (photon, magnon, and phonon) withfrequency ω c , ω m and ω b , respectively. σ stands for thepseudo-spin of the two-level atom. H om consists of theenergy of the photon and magnon, as well as the photon-magnon interaction with the effective strength G m , whichis called the cavity magnon polaritons [45]. H op is com-posed of the energy of the phonon and the optomechan-ical interaction with coupling strength g . The first termin H ao is the energy of the atom, and the second termdescribes the atom interacting with the cavity field. H d denotes an atom pumped with a classical field with fre-quency ω L .In the frame rotating with H = ω L ( a † a + σ † σ + m † m ),the Hamiltonian can be changed into time-independent.For simplicity, we assume ω m = ω c , then δ = ω c ( m ) − ω L .We diagonalize the Hamiltonian H ′ = δ ( a † a + m † m ) + G m ( a † m + am † ) by introducing supermodes a ± = √ ( a ± m ). Considering photon-magnon interaction larger thanthe optomechanical and atom-photon interaction, i.e., G m ≫ { g, g a } and choosing ω b = 2 G m , we rewrite theHamiltonian as H eff = ∆ a † + a + + (∆ − G m ) a †− a − + ω b b † b + ∆ a σ † σ − η ( a † + a − b + a + a †− b † ) + η a ( a † + σ + a + σ † )+Ω e ( σ + σ † ) , (3)where ∆ = δ + G m , η = g/ η a = g a / √
2, ∆ a = ω a − ω L .The detailed deduction of Hamiltonian (3) is given inAppendix A. For simplicity, hereafter we will assume∆ = ∆ a . We see that the effective Hamiltonian containsthree-partite interaction, which is similar to in Ref. [25].Differently from their scheme, we introduce a pumpedtwo-level atom aiming to achieve a blockade of the pho-ton, magnon, and phonon. We also would like to com-pare the different effect of a blockade between the de-structive interference mechanism and the single excita-tion resonance mechanism. Observe the last two brack-ets in Eq. (3); the pumped two-level atom interacts withmode a + , which results in the blockade of mode a + . Al-though the three-partite nonlinear interaction means theparametric-down conversion form between a − and b me-diated by absorption or emission of mode a + , the block-ade of the mode a + can not result in the amplification inmode a − and b . Instead, if there is only one excitation inthe mode a + , the transfer of the single excitation createsonly one excitation in every mode of a − and b , that is tosay, the blockade in mode a + will lead to the blockade inmode a − and mode b ; therefore it is possible to generatea blockade in supermodes a + , a − and mode b . We will FIG. 2. The evolution of probabilities P g (a) and P g (b) with original Hamiltonian H (red line) and effectiveHamiltonian H eff (blue squares), respectively, where P g = | C g | , P g = | C g | . The parameters are η = 5 κ , η a = 6 / √ κ , G m = 200 κ , and Ω e = 0 . κ . show that the bare modes a , b , and m can also be block-aded simultaneously.To check the validity of the approximation from Hamil-tonian (1) to Hamiltonian (3), we choose | g i as theinitial state and plot the evolution of the probabilitiesof states | g i and | g i governed by the Hamiltoni-ans H and H eff respectively, shown in Fig. 2, where | e ( g ) , n + , n − , n b i represents a state with atom in | e i ( | g i ),and | n + i , | n i , and | n b i are the number state for the a + , a − , and b modes, respectively. From Fig. 2, we see clearlythat the results of original Hamiltonian agree very wellwith that of effective Hamiltonian H eff , which meansthat the effective Hamiltonian H eff is reliable.Due to the limit of the weak driving field, for under-standing the blockade mechanism of the photon (phonon,magnon), we temporarily ignore the pumping of the atomand derive the eigenstates and eigenvalues of H eff (3) inthe few-photon subspace, yielding | i : λ = 0 , | i : λ = ∆ , | ± i : λ ± = ∆ ± β , | i : λ = 2∆ , | ± i : λ ± = 2∆ ± β , | ± i : λ ± = 2∆ ± β , (4)where β = p η a + η , β = q η a +7 η − D , β = q η a +7 η + D , D = p η a + 26 η a η + 25 η . The ex-pression of the dressed states | s c i ( s = 0 , , c =0, ± , ± , ± ) is given in Appendix A, and the energy-levels are shown on the right side of Fig. 1(b).In the weak driving limit, to analytically derive theequal-time second-order correction function, the state ofthe system can be truncated in few excitation subspaceand approximately expressed as | ψ i = C g | g i + C g | g i + C g | g i + C e | e i + C g | g i + C g | g i + C e | e i + C g | g i + C e | e i . (5) Under the action of the non-Hermite Hamiltonian e H = H eff − i ( κ + a † + a + + κ − a †− a − + κ a σ † σ ) with the decay rate κ j ( j = + , − , a ), the probability amplitude in | ψ i can beobtained by solving the Schr¨odinger equation i∂ | ψ i /∂t = e H | ψ i . The detail of the deduction and the steady-statesolution can be found in Appendix B.To characterize nonclassical photon (magnon, phonon)statistics, we employ and equal-time second-order corre-lation function defined by g i (0) = Tr(c † i c † i c i c i ρ )[Tr(c † i c i ρ )] , (6)where i = a, m, b, a + , a − . The steady-state correlationfunctions of our system can be analytically obtained viathe steady-state wave function (5) as g a + (0) = 2 | C g | ( | C g | + u ) ≈ | C g | | C g | , (7) g a − (0) = 2 | C g | ( | C g | + u ) ≈ | C g | | C g | , with u = 2 | C g | + | C g | + | C e | , and u =2 | C g | + | C g | + | C e | where the second approx-imate equals in Eq. (7) are obtained under the condi-tions | C g | ≫ {| C g | , | C g | , | C e |} ≫ {| C g | , | C g | , | C g | , | C e | , | C e |} . For mode b , it is notreasonable to obtain g b (0) with the analytical solution(5) because its decay has been ignored. We will directlycalculate it from the master equation. The correlationfunction g i (0) ≥ g i (0) < g i (0) → g i (0) < . a + ( a − ) is possible only if the pop-ulation C g ( C g ) ≈
0. We will plot second-ordercorrelation function and discuss it further in the nextsection.Although the polariton modes [45–47] consisting of op-tical mode and magnetic mode can be indirectly derivedby directly detecting the output spectrum of photons,the blockade of the photon, phonon, and magnon stilldeserve our investigation. Due to the combination of theoptical mode and magnetic mode, the statistical prop-erties of supermodes a ± and bare modes a and m aredifferent. In order to see clearly the difference, we derivethe relations between the two bases | i d = | i , | i d = 1 √ | i + | i ) , | i d = 1 √ | i − | i ) , | i d = 12 ( | i − √ | i + | i ) , | i d = 12 ( | i + √ | i + | i ) , | i d = 1 √ | i − | i ) , (8)where the left side states are labeled by | n + , n − i d ( n + and n − correspond to the Fock state of mode a + and a − ) while right-side state are labeled with | n m , n a i ( n m and n a denote the Fock state of mode m and a ). Thederivation of Eq. (8) is given in Appendix C. See thelast line in Eq. (8), where the state | i d means only oneexcitation in mode a + and a − , however for the modes m and a , they might be populated in two excitations. Thatis to say, the blockade of supermodes a + and a − does notmean the blockade of bare modes a and m . Therefore,we need to calculate the second-order correlation of themode m and a : g a (0) ≈ | C g | + | C g | + 2 | C g | )( | C g | + | C g | ) ,g m (0) = g a (0) . (9)We can see that the correlation functions for the opti-cal and magnetic mode are the same. The blockades inthe modes m and a require that C g , C g and C g reach zero simultaneously. Fortunately, as one can ob-serve from Eq. (B2), when C g equals zero, C g isequal to zero too. That is to say, when both a + and a − modes are a blockade, the photon and magnon modes a and m are both a blockade too. III. THE STATISTICAL PROPERTIES OF THEMULTIMODE FIELD
In the above analytical calculation of g i (0) ( i = a ± , b, a, m ), we have made some approximations. Wenow show the correction of the approximations and in-vestigate the statistical properties of the multimode field.For simplicity, we assume that the decay rates of the opti-cal mode, magnetic mode, and atom are equal, and thenwe can derive the master equation as˙ ρ = − i [ H eff , ρ ] + κ ( D [ a + ] + D [ a − ] + D [ σ ]) ρ + ( n th + 1) κ b D [ b ] ρ + n th κ b D [ b † ] ρ, (10)where ρ is the density matrix of the hybrid system, D [ o ] ρ = 2 oρo † − o † oρ − ρo † o , and n th is the thermalphonon population. We assume that the average particle FIG. 3. (a) Equal-time second-order correlation function formodes a + (red solid, red square), a − (blue dashed, blue dot),and b (black asterisk) versus detuning ∆, where lines andmarks represent analytical and numerical solutions respec-tively. (b) The average particle number. The relative proba-bility population function y (c)-(f) corresponding to the markpoint A-D in (a), where red (light gray) and blue (dark gray)bars represent supermode a + and a − , respectively. The pa-rameters are η a = 40 / √ κ , η = 15 κ , κ b = 0 . κ , Ω = 0 . κ , G m = 800 κ , and n th = 0. numbers of photons (magnons) in thermal equilibriumare zero because of their high frequencies.We truncate the Fock space up to | i for modes a ± and b . Based on the subspace consisting of the two-level atom and the modes a ± and b , we numerically solveEq. (10) and calculate the second-order correlation func-tion of mode a ± and b . In Fig. 3(a), we plot g a ± (0)with an analytical solution of Eqs. (7) and numerical re-sults of Eq. (10), respectively. We see that they agreewell, which means that we can understand the secondorder correlation with the analytic analysis. In order tomake clear the relation between the mechanism of block-ade and the probability distribution, we define the func-tion y ( N ) = log P ( N ) P p ( N ) where P ( N ) is the probability in | N i , and P p ( N ) is Poissonian distribution; thus the valueof y reveals the relative difference between the popula-tion and Poissonian distribution. In Fig. 3(c) to 3(f), weplot y ( N ) corresponding to point A to D respectively. If y is positive, population at N excitation is higher thanPoissonian distribution, or otherwise it is lower than thePoissonian distribution.For the mark point A in Fig. 3(a), ∆ = β , λ − = 0,which means the single excitation resonance. Then | − i can be easily populated (for the symmetry point of A,∆ = − β , λ = 0, then | + i is easy to be populated). Notice the expression | − i in Eq. (A6), where thereis only one excitation in modes a ± and b , so we can seestrong blockade in a + , a − , and b modes under the samecondition. Meanwhile the average numbers n a + , n a − ,and n b reach their local maximum of n a ± ( b ) [see Fig. 3(b)].All of the probability at N > g a + (0) achieves alocal minimum value where the real part of numeratorof C g is zero. By observing Fig. 1(b), the two jumps | e i → | g i and | g i → | g i destructively in-terfere each other, such that the population in | g i islow, so the mode a + is blockade. That is the so-calledUB. However, under this condition, the a mode is super-Poissonian because there is a population in | g i , result-ing in population | g i . By observing Fig. 3 (d), thedestructive interference only decreases the probability in N = 2 for the mode a + , while for the mode a − the prob-ability for N > a + and a − .For the point C in Fig. 3(a), g a (0) achieves a local min-imum value. As one can observe from Eq. (B2), the re-quirement for C g ≈ C g ≈ { η, η a } 6 = 0. As seen in Fig. 1(b), there are two jumps | g i → | g i and | e i → | g i . Their destructiveinterference results in blockade in mode a − . Meanwhile,there is a population in the state | g i , which meansthe super-Poissonian in mode a + . Correspondingly, inFig. 3(e), the population of mode a + is still higher thanthe Poissonian distribution, while for the mode a − , thedestructive interference only decrease the probability inonly N = 2. This result is similar to what we havepointed in the analysis of point B, i.e., the destructiveinterference can not offer us a simultaneous blockade insupermode a + and a − . For the point D, in Fig. 3(a), ∆ = 0, λ = λ = 0,which means that the single excitation resonance | i anddouble resonant excitation | i are both satisfied. Ob-serving Eq. (A6), the resonance between state | i andstate | i can lead to the populations in the states | g i and | e i . Likewise, the population in | i means thatthe states | g i , and | e i are easily populated too,while the state | g i is not so easily populated becauseof the mutual cancellation between η and η a [the factor η a − η √ η A is smaller than η a β , see Eq. (A6)]. Therefore, themode a + will be strong super-Poissonian, and the mode a − is sub-Poissonian. The results are corresponding toFig. 3(f), where the population for mode a + is higherthan the Poissonian distribution, and the probabilitiesdistribution for mode a − are less than Poissonian.As we have mentioned before, for mode b , g b (0) shouldnot be calculated from an analytical solution Eq. (5). Wedirectly calculate g b (0) with the master equation (10),shown in Fig. 3(a). We see that around point A, we canalso achieve blockade in mode b . Therefore, under singleexcitation resonance, all of the modes a + , a , and b ex-hibit the blockade phenomenon. In addition, the param- FIG. 4. (a): Equal time second correlation function for modes a (solid line), m (dots). (b): average number for optical mode a (solid line) and magnetic mode m (dots) as functions ofdetuning ∆. The other parameters are the same as in Fig. 3. eters, in Fig. 3, g ω b κ = 9 / < g is still larger than the damping rate κ . We will show that the single excitation resonant doesnot require g > κ ; that is to say, even under the condition g < κ , we still can obtain the simultaneous blockade forthe three modes.In Fig. 4, we plot g a (0) (solid) and g m (0) (dots), where,obviously, they are the same and agree well with Eq. (9).As we have analyzed before, the blockade of a + modemeans | C g | ≈
0, and the a − mode blockade corre-sponds to | C g | ≈ | C g | ≈ a + and a − modes are a blockade, from the expression Eq(9), the photon a and the magnon m are both blockade;therefore at point A [see Fig. 4(a)], the optical mode andmagnetic mode are both a blockade. However, around∆ = 0 (point D), the statistical property of g a + (0) isdifferent from that of g a − (0), g a ( m ) (0) still showing sub-Poissonian. From Eqs. (8), (9), and (B2), we obtain g a (0) ≈ F g a + (0) + ( 2 F − F ) g a − (0) , (11)where F = | ˜∆ η | + 1, F = | η ˜∆ | + 1, and ˜∆ = ∆ − iκ . So,when ∆ is extremely small, F → F → ∞ , then, g a (0) is dominated by g a − (0). Therefore, we can observea sub-Poissonian around ∆ = 0 regime. Comparing thevalue of g a ( m ) (0) around point B with that around pointA, we see that the sub-Poissonian resulting from destruc-tive interference (point B) does not exist, but the block-ade resulting from single excitation resonance (point A)still exists.To further characterize the blockade of modes a ± , b , a , and m , choosing a single excitation resonance con-dition ∆ = β , we plot a second-order delay correla-tion function defined by g (2) i ( τ ) = h c † i (0) c † i ( τ ) c i ( τ ) c i (0) ih c † i (0) c i (0) i in Fig. 5. g i ( τ ) ≤ g i (0) is called bunching, and g i ( τ ) > g i (0) is called antibunching which is also thequantum signature [48]. Meanwhile, g ( τ ) is proportional FIG. 5. Time-delay second-order correlation function for su-permode a + (red solid line) and a − (blue dashed line) in (a),and for optical mode (blue solid line), magnetic mode (orangedashed line), and mechanical mode (black dotted line) in (b).We set ∆ = β , and other parameters are the same as inFig. 3. The panel in (b) shows the partial enlarged detail. to the condition probability for detecting a second pho-ton (magnon, phonon) at t = τ , given that a photon(magnon, phonon) has been detected earlier at t = 0[49]. Observing Figs. 5(a) and (b), because of the singleexcitation resonance, the time-delay correction functionsfor supermodes a ± and optical, magnetic, or mechani-cal mode are all antibunching even in the weak photonnonlinear region. g (2) m ( τ ) agrees well with g (2) a ( τ ) which isjust like the equal-time second-order correlation function.Comparing Figs. 5(a) and (b), the time-delay correctionfunction of supermodes a ± has no quick oscillations, butthat of the optical and magnetic mode exhibits quick os-cillations. The quick local oscillations in the time-delaysecond-order function for optical and magnetic mode re-sults from the interference between supermodes a + and a − and the frequency of mechanical mode b [39].We now investigate the second-order correlation func-tion g a ( m ) affected by the coupling strength η a shown inFig. 6. From Figs. 6(a) and (b), we observe that with theincreasing of η a , the low value log g (2) a (0) points (singleexcitation resonance) in terms of ∆ are increased, whichis because the resonant condition ∆ = β is increasedwith η a . In Fig. 6(b), interestingly, the minimum valueof g a ( m ) is not monotonous decreasing with increasing η a . When η a ≈ . κ , g a ( m ) is abnormal where the ef-fect of the single excitation resonance does not result ina blockade as in the other case. See the mark point P inFig. 6(a), where there is a cross where the ∆ = β (thesingle excitation resonance ) and ∆ = β / g a ( m ) can not showa blockade. Except for the cross point, the larger valueof η a , the better the blockade.We now show that it is possible to generate a photon,magnon, and phonon blockade without a strong optome-chanical coupling coefficient. In Fig. 7, both g ω b κ ≪ g < κ are satisfied, and we plot the equal-time second-order correlation function for modes b , a , and m . InFig. 7(a), due to single excitation resonance, the strongsub-Poissonian for modes a , m , and b can be observed, FIG. 6. (a): Contour plot log g a (0) as function of η a and ∆.(b): g a (0) change with ∆ for several values of η a = 40 / √ κ (black solid line), 17 . κ (orange dashed line), 0 . κ (green dot-ted line). The other parameters are the same as in Fig. 3 while the destructive interference resulting in a block-ade is not observed in the weak coupling regime. Here,although the single-photon optomechanical coupling issmall, the large atom-photon interaction g a makes β larger than κ , which ensures the blockade of the pho-ton, magnon, and phonon. We can understand it fromEq.(B2). In order to keep single excitation, the denomi-nator of C g ( C g , C e ) should be as low as possible,i.e., min | e ∆ − η − η a | , then we deduce the condition∆ = p η + η a − κ . Therefore, even η < κ , the rela-tive large value of η a still can make η + η a > κ , andthen the single excitation will dominate the wave func-tion, and the blockade can be obtained. We can concludethat the single excitation resonance can result in a mul-timode blockade even in a weak optomechanical couplingregion while the destructive interference can not offer usmultimode antibunching. FIG. 7. (a) Equal-time second-order correlation function formode a (blue solid line), m (orange dots), and b (black dottedline). We set η = 0 . κ , Ω e = 0 . κ , η a = 20 / √ κ , and κ b = κ in (a). The other parameters are same as in Fig. 3. (b)Equal-time second-order correlation function versus thermalphonon population. We set ∆ = β , and other parametersare the same as in panel (a). In Fig. 7(b), we plot the equal-time second-order cor-relation functions of a photon, magnon, and phonon af-fected by thermal phonon number. As we can observe theblockade of a photon and magnon under a weak couplingregime still exists after considering the thermal environ-ment of a phonon, but the phonon blockade disappearsand the correlation function approaches 2 with increas-ing n th . When the thermal phonon population is takeninto account, the state of the system truncated the in fewexcitation subspace can be expressed as mixed a state of | ψ n i [25] where | ψ n i = C g n | g n i + C g n | g n i + C g n +1 | g n + 1 i + C e n | e n i + C g n | g n i + C g n +1 | g n + 1 i + C e n | e n i + C g n +2 | g n + 2 i + C e n +1 | e n + 1 i . (12)Because of the three-partite interaction a + a †− b † + h.c. , thethermal phonon cannot be converted into a photon andmagnon. From Eq. (12), although the thermal phononcan be in the state | n i , the states of photon and magnonstill can be in | i or | i , which means the blockade ofmodes a and m still exists, but phonon blockade willbe destroyed ( n >
1) [42, 50–52], and the correlationfunction of the phonon will close to the that of thermalfield. But, the blockade of the photon and magnon isaffected slightly by the thermal environment because ofthe change in the single excitation resonance for | ψ n i [25]. Therefore, to generate simultaneous blockade of aphoton, phonon, and magnon, the small thermal phononpopulation is necessary. IV. DISCUSSION AND CONCLUSION
When the single excitation resonance condition is satis-fied, The simultaneous blockade of a photon, phonon andmagnon can offer us some potential applications. Theusual hybrid system mainly contains two different physi-cal systems, but the quantum internet may require morecomplex quantum information processing, like the pro-cessing and storing of information while simultaneouslyupdating the information in a quantum information cir-cuit and network [43]. The simultaneous blockade of amultimode field could be used in this process and be morepowerful than the usual single mode blockade. If we re-alize the single excitation, from Eq. (8), the photon andmagnon will be a in Bell state 1 / √ | i ± | i ), whichis useful in quantum information processing.From Fig. 3 to Fig. 7, the parameter G m is seem-ingly not important in numerical simulation, but we doneed strong magnon-photon coupling, because we re-quire the condition G m ≫ { η, η a } to achieve the ef-fective Hamiltonian, and the three-partite interactionis true only under this condition. Recently, strongand even ultrastrong coupling between photons andmagnons at microwave frequencies, using of a YIGsphere, has been reported [27, 28]. For instance, inRef. [27], the magnon-photon coupling strength wasachieved as high as g = 2 π × . κ a = 2 π × κ m = 2 π × g > κ is still a challenge.Most of the experiments of the optomechanical systemare still within the single-photon weak coupling regime[17, 53, 54]. In our scheme, the three-partite interactionis results from the optomechanical interaction, but thesingle-photon strong coupling is not necessary.In this paper, we put forward a scheme to generate aphoton, phonon and magnon blockade in a hybrid mi-crowave optomechanical-magnetic system. By introduc-ing a two-level atom interacting with the cavity field, wecarefully compare the blockade resulting from destructiveinterference and that resulting from single excitation res-onance. We find that the blockade resulting from singleexcitation resonance is much better than that resultingfrom destructive interference. Most importantly, underthe same detuning condition, the photon, phonon andmagnon can be blockade simultaneously. Furthermore,we find that the phonon blockade is easy to be destroyedby thermal excitation, while the blockade of the photonand magnon are affected slightly by the thermal environ-ment. To generate simultaneous blockade of the photon,phonon and magnon, the small thermal phonon popula-tion is necessary.In our system, the multipartite interaction results fromoptomechanical coupling, which is the key factor to ob-tain the simultaneous blockade of the photon, phonon,magnon. However, the single excitation is the condi-tion of the simultaneous blockade, and the single-photonstrong optomechanical coupling condition is not required.Therefore, the present scheme is feasible in experiment,which is a guideline for hybrid optomechanical-magneticexperiments nearing the regime of single-photon nonlin-earity, and for potential quantum information processingapplications with photons, magnons and phonons. ACKNOWLEDGEMENTS
We are grateful to J.Q. You and Guo-Qiang Zhang forenlightening discussions. This work was supported byNSFC under Grant No. 11874099.
Appendix A: The derivation of an effectiveHamiltonian and its eigenstates
In this appendix, we give the detailed derivation forHamiltonian (3). In the frame rotating with H = ω L ( a † a + σ † σ + m † m ), the Hamiltonian (1) can be writ-ten as H = δ c a † a + δ m m † m + G m ( a † m + am † )+ ω b b † b + ga † a ( b † + b )+∆ a σ † σ + g a ( σa † + σ † a ) (A1)+Ω e ( σ + σ † ) , with δ c ( m ) = ω c ( m ) − ω L . For simplicity, we assume ω m = ω c , then δ c = δ m = δ . We diagonalize the Hamil-tonian H ′ = δ ( a † a + m † m ) + G m ( a † m + am † ) by intro-ducing a ± = √ ( a ± m ), then H ′ = ( δ + G m ) a † + a + + ( δ − G m ) a †− a − . Choosing H f = ∆ a † + a + + (∆ − G m ) a †− a − + ω b b † b + ∆ a σ † σ and assuming ω b = 2 G m , ∆ = ∆ a , weswitch into the interaction picture and obtain H I = η ( a † + a + + a †− a − − a † + a − e i G m t − a †− a + e − i G m t ) × ( be − iω b t + b † e iω b t ) + Ω e ( σe − i ∆ t + σ † e i ∆ t )+ η a ( a † + σ + a + σ † − a †− σe − i G m t − a − σ † e i G m t ) , (A2)where η a = g a √ , η = g . The detuning ∆ can be arbitraryvalue. Considering G m ≫ { η, η a } , we take rotating waveapproximation and ignore high frequency terms, then theHamiltonian could be written as H I = − η ( a † + a − b + a †− a + b † ) + η a ( a † + σ + a + σ † )+ Ω e ( σe i ∆ t + σ † e − i ∆ t ) . (A3)We would like to rewrite the Hamiltonian into time-independent form by switching back into original picture,then we have H eff =∆ a † + a + + (∆ − G m ) a †− a − + ω b b † b + ∆ a σ † σ − g a / a † + a − b + a †− a + b † )+ g/ √ a † + σ + a + σ † ) + Ω e ( σ + σ † ) . (A4)It is exactly the effective Hamiltonian (3).In the limit of a weak driving field, we temporary forgetthe pumping of the atom and derive the eigenstates andeigenvalues of H eff in the few-photon subspace, yielding | i : λ = 0 , | i : λ = ∆ , | ± i : λ ± = ∆ ± β , | i : λ = 2∆ , | ± i : λ ± = 2∆ ± β , | ± i : λ ± = 2∆ ± β , (A5)where β = p η a + η , β = q η a +7 η − D , β = q η a +7 η + D , D = p η a + 26 η a η + 25 η . The corre- sponding eigenstates are | i = | g i , | i = 1 β ( η a | g i + η | e i ) , | − i = 1 √ | g i + ηβ | g i − η a β | e i ) , | + i = 1 √ | g i − ηβ | g i + η a β | e i ) , | i = 1 A ( | g i + η a − η √ η | g i + √ η a η | e i ) , | − i = 1 A ( d | g i + d | g i + d | e i + d | g i + | e i ) , | i = 1 A ( d | g i − d | g i − d | e i + d | g i + | e i ) , | − i = 1 A ( d | g i + d | g i + d | e i + d | g i + | e i ) , | i = 1 A ( d | g i − d | g i − d | e i + d | g i + | e i ) , (A6)with the coefficients: A = √ β +2 η √ η , d = β ( D − η − η a ) √ η a ηM , d = β ( − β + D )2 η a M , d = β ( β − D )2 ηM , d = η ( D − β ) η a M , d = − β ( D +5 η + η a ) √ η a ηM , d = − β (5 β + D )2 η a M , d = β ( β + D )2 ηM , d = − η ( D +5 β ) η a M , M = 3 β − D , M = 3 β + D , and A = p | d | + | d | + | d | + | d | + 1. Appendix B: The dynamic equation and steadystates solution
In this appendix, we derive probability amplitude fora steady state. Substitute the | ψ i expressed by Eq. (5)into the Schr¨odinger equation: i ∂∂t | ψ i = H eff | ψ i , and we obtain the differential equations as i ˙ C g = 0 , (B1) i ˙ C g = ˜∆ C g − ηC g + η a C e ,i ˙ C g = − ηC g + ˜∆ C g ,i ˙ C e = η a C g + ˜∆ C e + Ω e C g ,i ˙ C g = 2 ˜∆ C g − √ ηC g + √ η a C e ,i ˙ C g = −√ ηC g + 2 ˜∆ C g − ηC g + η a C e ,i ˙ C e = Ω e C g + √ η a C g + 2 ˜∆ C e − ηC e ,i ˙ C g = − ηC g + 2 ˜∆ C g ,i ˙ C e = Ω e C g + η a C g − ηC e + 2 ˜∆ C e , where for simplicity, we set κ + = κ − = κ e = κ , ˜∆ =∆ − iκ and temporarily ignore the small mechanical decayrate κ b ≪ κ , and the jumping from high level to low levelis ignored as it is done in Ref. [25].The steady-state solution of Eq. (B1) is derived as C g = 1 , (B2) C g = η a Ω e ˜∆ − η a − η ,C g = ηη a Ω e ˜∆( ˜∆ − η a − η ) ,C e = − ( ˜∆ − η )Ω e ˜∆( ˜∆ − η a − η ) ,C g = η a η (5 ˜∆ − η a + η )Ω e ˜∆ B ,C g = η a (4 ˜∆ + ˜∆ ( η − η a ) − η )Ω e √ B ,C e = η a (4 ˜∆ − ˜∆ ( η a + 4 η ) + η η a − η )Ω e ˜∆ B ,C g = η η a (5 ˜∆ − η a + η )Ω e ˜∆ B ,C e = − η a η (6 ˜∆ − ˜∆ ( η a + 9 η ) + 2 η η a )Ω e ˜∆ B , where B = ( ˜∆ − β )(4 ˜∆ − β )(4 ˜∆ − β ). Appendix C: The deduction of the relations betweentwo bases
In this Appendix, we provide the certification ofEq. (8). We define the Fock basis of the supermode a ± as | n + n − i d and the bare modes of a and m as | nm i . Forthe supermodes, we have a † + | n + n − i d = p n + + 1 | n + + 1 n − i d ,a + | n + n − i d = √ n + | n + − n − i d ,a †− | n + n − i d = p n − + 1 | n + n − + 1 i d ,a − | n + n − i d = √ n − | n + n − − i d , (C1)Specifically, for n + , n − = 0, we have the relation of theannihilation operator a ± | i d = 0 . Since a ± = √ ( a ± m ), we have a | i d = 0, m | i d = 0.We expand the state | i d by using the bare basis | n, m i of mode a and m as | i d = X n,m C nm | nm i = C | i + C | i + C | i + . . . (C2)Thus C nm = h nm | i d , for example C = h | i d = h | a | i d = 0. Finally,we have | i d = | i . (C3)In addition, we can write a † + | i d = | i d , i.e., 1 / √ a † + m † ) | i = 1 / √ | i + | i ). Then, we can obtain | i d = 1 √ | i + | i ) . (C4)Similarly, we can have | i d = 1 √ | i − | i ) . 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