Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems

We investigate a hybrid electro-optomechanical system that allows us to realize controllable strong Kerr nonlinearities even in the weak-coupling regime. We show that when the controllable electromechanical subsystem is close to its quantum critical point, strong photon-photon interactions can be generated by adjusting the intensity (or frequency) of the microwave driving field. Nonlinear optical phenomena, such as the appearance of the photon blockade and the generation of nonclassical states (e.g., Schrödinger cat states), are demonstrated in the weak-coupling regime, making the observation of strong Kerr nonlinearities feasible with currently available optomechanical technology.


Results
Hybrid electro-optomechanical system. In the hybrid electrooptomechanical system of Fig. 1, the mechanical oscillator is parametrically coupled to both the optical cavity and the microwave resonator. The microwave resonator is driven by a strong field with amplitude e c and frequency v ci , where e c is related to the input microwave power P and microwave decay rate k c by e c j j~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Pk c = v ci p . In a frame rotating with frequency v ci , the Hamiltonian for this hybrid systems reads 48 where the detuning d c 5 v c 2 v ci and the microwave frequency v c~1 ffiffiffiffiffiffi LC p , g a (g c ) denotes the optomechanical (electromechanical) coupling strength at the single-photon level, andâ (b orĉ) is the annihilation operator of the optical cavity (the mechanical oscillator or the microwave resonator). Under a strong microwave driving field, following the standard linearization procedure [49][50][51][52] (shiftingĉ andb with their steady-state mean values a and b, i.e., c?ĉza,b?bzb), the Hamiltonian can be transformed intô where G is the linearized electromechanical coupling strength; D c andṽ a are, respectively, the effective microwave detuning and optical frequency including the radiation-pressure-induced optical resonance shift. Their explicit expressions are given by where the angle h is defined by Then, the HamiltonianĤ' OMS becomeŝ where v 6 are the normal mode frequencies of the electromechanical subsystem, and g +~+ g a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v b 1+cos2h ð Þ =2v + p are the effective coupling strengths between the optical photon and the normal modes. Equation (5) shows that v 2 { becomes zero (negative) when as shown in Fig. 2(a). This corresponds to a critical property 53 , namely, the normal mode v 2 will change from a standard harmonic oscillator (G , G cp ) to a free particle, and further becomes dynamically unstable (G . G cp ) as G crosses its critical value G cp . The above critical property can become more transparent with the following canonical relationships: Here x b , x c are the dimensionless displacements of the mechanical and microwave resonators from their stable points, and p b , p c are the corresponding dimensionless momentums. The Hamiltonian of the electromechanical system can then be written in terms of the usual canonical x-p variables, H e-m 5 H 0 1 H int with denoting the free Hamiltonian of the microwave and the mechanical resonators, and the coupling between them. The potential of the free  Hamiltonian (6a) can be further expressed as It shows that the intrinsic potential of the electro and mechanical resonators is characterized by ffiffiffiffiffiffiffiffiffiffi ffi D c v b p . Comparing Eq. (7) with the coupling Hamiltonian (6b), one can see that there is an interplay between the intrinsic potential and the coupling interaction between them. This interplay leads to the above critical property. In other words, when G approaches (or exceeds) ffiffiffiffiffiffiffiffiffiffi ffi D c v b p 2, the normal mode v 2 is dragged out from its effective potential, and becomes increasingly flat (or inverted) [see the Fig. 3].
Quantum-criticality-induced strong Kerr nonlinearities. As one can see, the last two terms in the Hamiltonian (4) show that optical photons can interact with each other through the exchange of the normal modesB + , very similar to electrons interacting with each other through the exchange of photons in the QED Hamiltonian. In particular, when the electromechanical subsystem approaches its quantum critical point, the optical cavity shows a strong effective Kerr nonlinearity. This quantum-criticality-induced strong Kerr nonlinearity becomes clear after taking a displacement transformation,Ĥ'' OMSṼ  The result iŝ and g is the photon-photon interaction strength, Notice that the photon-photon interaction strength g remains unchanged when the system-environment interaction is explicitly included (see the detailed derivation in Methods). On the other hand, it also shows in Figs. 2(c,d) that even in the weak-coupling regime g m =k m m~a,c ð Þ , a strong photon-photon interaction g (g . k a ) can still be obtained when G (or D c ) approaches the quantum critical point. In particular, Fig. 2 shows that when the coupling strength G (or the detuning D c ) is close to its quantum critical point, a very small normal mode frequency v 2 is obtained, which induces a large photon-photon interaction with g / 1/v 2 . The system parameters G and D c , determined by the power P and the frequency detuning d c of the input microwave driving field, can be directly tuned in experiments. Figure 4 shows explicitly the practical parameter range of P and d c for obtaining the strong Kerr nonlinear parameter g (g . k a ).
Photon blockade. The strong Kerr nonlinearity in the present system can be further demonstrated by the steady-state secondorder correlation function of the optical field g (2) (0). g (2) (0) R 0 in the weak-coupling regime signals the photon blockade effect, and can be directly detected by a Hanbury-Brown-Twiss Interferometer 3 . Explicitly, by driving the optical cavity with a weak laser field of frequency v ai and amplitude e a , the Hamiltonian of the system becomesĤ where all the similarity transformations used before have been taken into account, and D a~ṽa {v ai . The damping effect arising from the coupling of the optical field to the electromagnetic vacuum modes of the environment can also be taken into account, and the dissipative dynamics of cavity modeâ is described by the quantum Langevin equation, Here k a is the decay rate of cavity modeâ andf in is a vacuum noise With a weak optical driving field, the quantum Langevin equation is solved by truncating them to the lowest relevant order in e a . The resulting two-photon correlation function is given by lim t??â where is the normalized s-photon probability in the cavity (P s ?P sz1 in the weak-driving regime), and The noise operatorl in t ð Þ~1 ffiffiffiffi Þ t , which comes from the environment of the microwave resonator. The environment is initially in the thermal equilibrium state r th with temperature T, andl 0 v ð Þ is the initial environment operators of the microwave resonator. Here, we have safely ignored the dissipation of the mechanical oscillator because the mechanical decay rate k b is extremely small, k b /k a , k b /k c , 10 23 . Thus, the effective decay rates k j is determined by the original decay rate of the microwave resonator k c (see the detailed derivation in Methods).
In Fig. 5, we show the dependences of k 6 on the system parameters G, D c and k c . From Fig. 5 one can see that the effective decay rate k 2 sharply changes from a positive value to a negative value when the system parameter G (or D c ) crosses its quantum critical point G cp (or D cp ). This result demonstrates that the mode v 2 will become a gain mode when G . G cp or D c , D cp . Near the quantum critical points G cp and D cp , the effective decays k 6 almost become constant with G or D c [see the inserts of Fig. 5(a) and 5(c)]. In Fig. 5(b) k 6 is plotted via the microwave field decay rate k c when G (or D c ) is near the quantum critical points. As it is shown, k 6 exhibit a linear increase with the decay rate of the microwave field k c .
When the microwave (mechanical) mode is initially in the coherent state jaae (jbae), and the optical field in the vacuum state, the twopoint correlation function exp(2W 2 ) and the four-point correlation function exp(2W 4 ) are calculated. With numerically integrating Eqs. (13), the dependence of g (2) (0) on k 2 , G, and D c is shown in Fig. 6. As one see, in the quantum critical regime, the photon antibunching effect g (2) (0) , 1 (even the photon blockade g (2) (0) R 0) occurs because the two-photon transition is largely suppressed in comparison with the single-photon transition when k 2 /2p . 60 kHz [see the insert in Fig. 6(a)]. Figures 6(b) and (c) further show that the photon blockade [g (2) (0) R 0] occurs when the tunable parameter G (or D c ) approaches its quantum critical value even if the optomechanical coupling g a is very weak.
Furthermore, we also find that the photon antibunching [g (2) (0) , 1] disappears when k 2 /2p , 60 kHz [see the inserts in Figs. 6(b) and (c)]. Physically, this is because in the hybrid OMS, a relatively large decay rate k 2 (k 2 /2p . 60 kHz) occurs when the electromechanical subsystem approaches the quantum critical point. This decay will significantly suppress the steadystate sideband transition in the electromechanical subsystem. Then, in the quantum critical regime, the hybrid OMS becomes a pure optical nonlinear system, and g . k a is the exclusive condition for achieving the photon blockade. Meanwhile, the very small v 2 (v 2 R 10 kHz) near the quantum critical point effectively enhances the photon-photon interaction to g . k a because g / 1/v 2 , namely the photon blockade can still be reachable even if the effective electromechanical frequency extends beyond the resolved sideband regime, i.e. v 2 , k a . Notice that the original mechanical frequency used here is still in the resolved sideband regime (v b ?k a ) so that there is no problem in cooling the mechanical oscillator at the initial time.
Nonclassical states. As demonstrated in previous studies [21][22][23] , strong Kerr nonlinearities generally lead to the periodic generation of nonclassical states, (e.g., cat states) of the cavity field. With the help of the Hamiltonian (4), we obtain the time evolution operator in the interaction picture, where the term corresponding to f 1 has been omitted due to its negligible effect on the evolution of the cavity modeâ (f 1 /v b , 10 24 ) near the quantum critical point. If the cavity fieldâ is initially in a coherent state U j i, the cavity field at time t n 5 2np/ v 2 (n 5 1, 2…) will be in the state The state jY a (t n )ae is a multi-component cat state, depending on the value of g/v 2 . Figure 7 shows the different multi-component cat states for different values of the tunable parameters G and D c near the quantum critical point. Figures 7(b,c,d) present the specific realization of two-, three-and four-component cat states, respectively. Here damping effects (given by k a , k c , k b ) have been ignored. In principle, this is valid when the cut-off time t n =1=k a , 1/ k c , 1/k b . The optical field damping is similar to that in a recent cavity-QED experiment 54 . Moreover, inspired by Ref. 54, the Wigner function can be measured (or reconstructed) by detecting the states of the atoms interacting with the optical field. Nevertheless, the above result indicates that the quantum-criticality-induced strong Kerr nonlinearities in this hybrid OMS can generate  nonclassical states by cutting off the optomechanical interaction at the appropriate time, which can be detected via Wigner tomography.

Discussion
We have provided a new mechanism for obtaining strong Kerr nonlinear effects in a hybrid OMS in the weak-coupling regime. We found that the electromechanical subsystem displays a critical property when adjusting the intensity (or frequency) of the microwave driving field, and a strong controllable photon-photon interaction is induced in the quantum critical regime. Usually, the phonon modulation effect influences the photon statistics in the usual OMSs 24 , and in general will also weaken the photon-photon interaction effect, except in the single-photon strong-coupling (g a . k a ) and the resolved sideband (k a =v b ) regime 24 . The essence of the strong photonphoton interaction presented in this paper can be understood as follows. In the quantum critical regime, the electromechanical normal modeB { coupled to the optical field is highly softened (or a very small normal-mode frequency v 2 is obtained). At the same time, the sideband phonon transitions in the electromechanical subsystem are significantly suppressed by the relative large decay rate of the electromechanical normal mode, which makes the hybrid OMS essentially a pure optical nonlinear system. Thus, the quantumcriticality-induced strong self-Kerr nonlinearity is very different from previous investigations in the usual OMSs 24,34,35 .
Experimentally, the strong photon-photon interaction achieved in the present hybrid OMS requires driving the electromechanical subsystem into its quantum critical region (shaded area in Fig. 2). Normal-mode splitting in the driven electromechanical system has been observed 43 . The quantum critical region could be easily reached by increasing the intensity of the microwave driving field. Moreover, as shown in Figs. 2 and 4, the interesting ranges of G and D c are respectively on the order of 0.1 kHz and 1 kHz for the quantum critical region, and this parameter precision is experimentally realizable 55 . We believe that our proposal will provide a new avenue for experimentally realizing strong optical nonlinearities in the weakcoupling regime and largely enrich the scope of implementing quantum information processing and quantum metrology with cavity OMSs.

Methods
Derivation of the photon-photon interaction with system-environment couplings. The total Hamiltonian of the hybrid OMS plus the environment can be written asĤ where the system HamiltonianĤ' OMS is given by Eq. (4) and are the Hamiltonians of the environment and the system-environment interaction, respectively. Notice that the system-environment interaction is invariant to the linearization procedure applied on the electromechanical subsystem. Heref v ð Þ, h v ð Þ,l v ð Þ are the bath operators forâ,b,ĉ, and K j (v) (j 5 a, b, c) are the corresponding coupling constants. For a slowly-varying bath spectrum, we can simply replace K j (v) by the decay rate ffiffiffiffiffiffiffiffiffi ffi k j p q . Here the last term can be safely neglected because the decay rate k b of the mechanical oscillator is extremely small (k b / k a , k b /k c , 10 23 ). By applying a Bogoliubov transformationR~MB to the total HamiltonianĤ tot , the hybrid OMS HamiltonianĤ' OMS and the interaction between the system and the environmentĤ SE can be rewritten in terms of the normal-mode canonical operators Then, the dynamics of the canonical operatorR is given by Equation (25) shows that the imaginary and real parts of the eigenvalues of D correspond to the eigenfrequencies v 6 and the effective decay rates k 6 of the normal modes. For the undamped case (k c 5 0), the eigenvalues of D are purely imaginary and we obtain the expression Eq. (5) for the normal-mode frequencies. For the general k c , we numerically diagonalized the coefficient matrix D and shown the results in Fig. 5.