Enhanced quantum nonlinearities in a two mode optomechanical system
In cavity optomechanics, nanomechanical motion couples to a localized optical mode. The regime of single-photon strong coupling is reached when the optical shift induced by a single phonon becomes comparable to the cavity linewidth. We consider a setup in this regime comprising two optical modes and one mechanical mode. For mechanical frequencies nearly resonant to the optical level splitting, we find the photon-phonon and the photon-photon interactions to be significantly enhanced. In addition to dispersive phonon detection in a novel regime, this offers the prospect of optomechanical photon measurement. We study these QND detection processes using both analytical and numerical approaches.
Introduction. - By coupling mechanical resonators to the light of optical cavities the emerging field of optomechanics  aims at observing quantum mechanical behavior of macroscopic systems. The ultimate goal is the regime where single phonons and photons interact strongly. New architectures and progress in design and fabrication pave the way towards realizing strong coupling even at the single-photon level in optomechanical systems ; ; ; ; ; . This development has stimulated several theoretical works that analyze the generic optomechanical system, i.e. a single optical mode coupled to a single mechanical mode, in the regime of strong coupling. Non-classical effects are found in the dynamics of the mechanical resonator ; ;  and the statistics of the light field ; ;  if the photon-phonon coupling rate becomes comparable to both the decay rate of the cavity and the mechanical oscillation frequency .
In this paper, we show how an optomechanical setup consisting of two optical modes coupled to a mechanical resonator ; ;  can be brought into a novel regime that significantly enhances the size of the quantum nonlinearity. We derive an effective Hamiltonian of the system that captures the regime of strong single-photon optomechanical coupling and large mechanical frequencies. In our analysis the difference between optical level splitting and mechanical frequency, , appears as a crucial parameter. It enters the coupling rate that characterizes the coherent interaction among photons and between photons and phonons. If this dispersive optical frequency shift exceeds the cavity decay rate, one enters what we will call the strong dispersive coupling regime: . Since can be made much smaller than , this condition is easier to achieve than the corresponding one for the generic optomechanical system, . This is relevant in particular because optomechanical systems have by now reached the regime of large mechanical frequencies, see for example ; ; , where they are less susceptible to thermal fluctuations and optomechanical cooling is more efficient.
As a first application of the enhanced phonon-photon interaction we investigate the possibility of a quantum non-demolition (QND) detection of the phonon number. A measurement of this kind has been proposed in a pioneering work by Thompson et al.  for a setup where a dielectric membrane is placed inside an optical cavity. Subsequently, this QND scheme ; ; ;  and other features of such a two mode system ; ; ; ; ;  have been studied in detail. An increase of the nonlinear coupling by making use of the full spectrum of cavity modes has been demonstrated in ; ; . However, the analysis has so far been restricted to cases, where the influence of individual photons is weak. Furthermore, it was assumed that the mechanical and optical timescales separate. Hence the previous analysis did not capture the enhancement of the optomechanical nonlinearity, which, as we show below, results in an increased read-out rate.
As a completely new feature of optomechanical systems, our effective description reveals strong photon-photon interaction for mechanical frequencies comparable to the optical mode splitting. As we show below, this interaction opens up the possibility of a QND measurement of the photon number. The two mode optomechanical system can therefore be assigned to a larger class of optical systems whose ultimate goal is the realization of QND photon detection on the level of single quanta .
In our analysis of the phonon and photon Fock state measurements we discuss the limitations due to quantum noise and confirm our predictions by numerical simulations of the dissipative quantum dynamics.
Model. - We consider an optomechanical setup consisting of two optical modes (, frequencies ) and one mechanical mode (, frequency ) that is described by a Hamiltonian
The optomechanical coupling rate is denoted by , and both optical modes are pumped by laser sources at rates . The optical cavities are characterized by the photon decay rates into the reflection channel () and into the transmission channel () with . We assume that the transmitted signal from each of the modes can be filtered and measured independently using a photodetector (), see Fig. 1(c). The mechanical resonator couples to a thermal bath at a rate with a bath occupation given by . In the following, we assume the mechanical frequency to be high enough and the bath temperature to be low enough such that the oscillator is sufficiently close to the ground state.
A Hamiltonian of the form of Eq. (1) is found both in the “membrane in the middle”-setup , in coupled microtoroid resonators  and in optomechanical crystals . The optical modes constitute normal modes , where denotes geometrically distinct modes with an original Hamiltonian , where
and . The frequency splitting of the normal modes is thus given by the photon tunnel coupling rate , .
In the approach of ; ; ;  the optical resonances are calculated as (see Fig. 2(a)), where is the mechanical displacement in units of the mechanical ground state width and where it is assumed that . Note that is treated as a quasi-static variable (in the sense of the Born-Oppenheimer approximation, with photons playing the role of electrons). This approach therefore has to fail if the optical frequency splitting and the mechanical excitation energy become comparable.
Effective Description. - The effect of the optomechanical interaction to first order in can be readily described in the following picture. A photon initially placed in the left (or right) cavity mode starts oscillating between the left and right part of the cavity at a frequency : . Accordingly, the radiation pressure force varies sinusoidally in time. This force drives mechanical oscillations and , where .
To take these elementary dynamics into account, we shift the oscillator by and via a unitary transformation , with . This procedure exactly eliminates the interaction to first order in and results in an effective Hamiltonian
where and where we disregard terms of order . In the limit of vanishing tunnel coupling, , the unitary transformation reduces to a shift of the mechanical position due to a static radiation pressure force. In this case the effective Hamiltonian is given by in correspondence to the “polaron transformation” for the generic single-mode setup ; ; ; . The most interesting regime is entered if the mechanical frequency becomes comparable to the optical splitting, i.e. :
where and where we neglect terms of the order and rapidly rotating terms like , .
Phonon detection. - The effective Hamiltonian of Eq. (6) enables us to discuss optomechanical QND phonon detection in its most general form, going beyond previous discussions ; ; ; . The optical frequencies are shifted by . We note that in the limit the result of  is recovered. However, for mechanical frequencies comparable to the optical splitting, i.e. , the frequency shift per phonon is greatly enhanced. We stress that the enhancement of the frequency shift is observable even in the weak coupling regime , where the cavity modes have to be strongly driven in order to detect the transmission phase shift in a homodyne measurement ; . In the following, however, we focus on the regime where both and and where single quanta affect the optical and mechanical modes strongly.
The experimental protocol for detecting the phonon number is to pump one of the optical modes (here ) with a laser at frequency and measure the transmitted signal using a photodetector (). The second mode () is undriven, playing the role of an idle spectator (though it will become important for dissipative processes, see below). We first study the spectrum of the detection mode , i.e. the photon number as a function of detuning . In steady state, the spectrum consists of several resonances with spacing corresponding to different phonon number states. In a situation where the optical frequency shift per phonon is smaller than the cavity linewidth , the resonances overlap, see Fig. 3(a). In the following section, we will discuss this weak dispersive coupling regime (even though will still be taken on the order of one). Note that the strong dispersive regime is also relevant, both for phonon and photon detection, and we will come back to it when discussing photon measurements. The time evolution of the mechanical state can be monitored by pumping the detection mode at fixed detuning and recording the photon counts at the detector during an interval . A quantum jump in the phonon number changes the number of intracavity photons by and, accordingly, the number of detected photons by . The shift in photon number can be estimated as , where we disregard a prefactor that depends on the detuning. The measurement time has to be chosen large enough, such that the measured signal exceeds the photon number uncertainty, i.e.  or equivalently:
On the other hand, the measurement time has to be smaller than the lifetime of a phonon Fock state which is governed by thermal fluctuations at rate and by decoherence induced via the optical modes at rate :
The thermalization rate of the phonon state is given by in the uncoupled system. The major contribution to stems from the process where a phonon is annihilated while a photon tunnels from the to the mode and decays. A calculation according to Fermi’s golden rule yields . It follows that single-photon strong coupling, i.e. , is required to obtain a signal to noise ratio bigger than one, as has already been shown by  for the limiting case of small mechanical frequencies . We note that a phonon measurement using the mode for detection can be described analogously, the main qualitative difference being that the cavity-induced decoherence processes excite phonons and potentially cause an instability.
To simulate the envisaged QND phonon measurement, we employ the Lindblad master equation for the system’s density matrix ,
where . The unobserved channels are the photon decay into the reflection channels and the coupling between the mechanical resonator and the thermal environment with and , while the transmission channels are under observation. We unravel the time evolution into quantum jumps  that occur with probability , and into the deterministic part plus subsequent normalization. A quantum jump with is interpreted as a detection event at the photodetector or , respectively. Figure 3 (b)-(d) shows trajectories from such a simulation. The phonon number jumps between the Fock states and , driven by thermal fluctuations (Fig. 3d). The photon number in the detection mode follows the time evolution of the mechanical mode (Fig. 3b). Thus, by monitoring the photon counts at the photodetector (Fig. 3c) a QND measurement of the phonon number is achieved. In contrast to earlier numerical analysis , our results apply to the general case of a two-sided cavity and thereby confirm the limits imposed by quantum noise . Moreover, they show the strong enhancement of the coupling in the design considered here.
Photon detection. - As a novel feature of the system, we identify the dispersive photon-photon interaction in the effective Hamiltonian (7). We note that the interaction term vanishes in the limit of small mechanical frequencies and therefore did not appear in previous works. Here we demonstrate the prospects of a QND measurement of the photon number using the mode for detection. The roles of the two optical modes are chosen as to suppress the influence of unwanted transitions from the mode to the energetically lower-lying mode. Both modes are driven independently by a laser and the data from the photodetector is used to extract the information about the photon number . We assume that the detection mode has a lower finesse than the signal mode, i.e. , such that a sufficiently large number of photons arrives at the detector while the state of is only weakly perturbed by the photons in .
In the weak dispersive coupling regime, , we find a required measurement time of
with a frequency shift per photon of , in analogy to the case of phonon detection discussed above (see also Fig. 3). In order to detect the photon state within its lifetime, it is also required that . Moreover, the measurement would be spoiled if a phonon were to be excited during the measurement time, since actually measures . We therefore demand that both the thermalization rate and the rate for the optically induced heating process, given by , are smaller than the measurement rate . From the latter condition it follows that single-photon strong coupling, i.e. , is also required for an undisturbed photon detection.
In the strong dispersive regime, , a strong projective measurement of the photon number (or analogously the phonon number) can be performed as illustrated in Fig. 4. The spectrum of the detection mode , i.e. the intensity as a function of laser detuning, shows well-resolved resonances with spacing , see Fig.4 (a). The weights of the peaks correspond to the photon number distribution of the signal mode. This is in close analogy to the theoretical and experimental results of ;  where a qubit coupled to a microwave cavity was used to measure the photon distribution. The quantum trajectory simulations (Fig. 4(b),(c)) reveal strong measurement induced back-action leading to (anti-)correlation between signal and detection mode. Whenever the photodetector registers photons from the detection mode, the state of the signal mode is projected into the zero- or one-photon Fock state depending on the detuning of the detection mode. This projection leads to a disruption of the coherent evolution of the signal mode as is clearly visible in Figs. 4(b),(c)). We note that in the regime , this kind of measurement backaction affects the quantum evolution significantly. Indeed, it can be shown that the photons impinging on the signal mode from the coherent laser source tend to be prevented from entering the cavity due to the continuous observation of the photon number inside the cavity. This is a manifestation of the Quantum Zeno effect, as analyzed in .
Experimental prospects. Single-photon strong coupling, i.e. , has been demonstrated in optomechanical systems where the mechanical element is a cloud of cold atoms ; ; . In principle, currently available setups of this kind are extensible to a two-mode design by making use of the spectrum of transverse cavity modes . Reaching would additionally require larger trapping frequencies, .
A number of optomechanical systems exhibit large mechanical frequencies of a few , and has been demonstrated ; ; . Single-photon strong coupling, however, is yet to be reached in solid-state systems. The current record is achieved in optomechanical crystal setups, . Utilizing nanoslots  to enhance the local optical field in such structures offers the prospect of coupling rates above . Advances in design, fabrication and material properties are expected to lead to high-quality optical cavities with ; . These developments, taken together, should make attainable.
Conclusions and Outlook. - The results presented here demonstrate how the design flexibility of photonic crystals and other optomechanical systems can be exploited to significantly enhance nonlinear coupling rates, and how to benefit therefrom in the deep quantum regime. Besides the dispersive QND measurement schemes based on the two-mode structure addressed here, one may think of applying the enhanced photon-photon and photon-phonon coupling for studies of optomechanical quantum many-body effects (e.g. in arrays), or for further applications in quantum information processing (see also the related work by Stannigel et al. ). The coherent Kerr-type interaction introduced here can form the basis for an all-optical switch and moreover directly permits to engineer a quantum phase gate (based on the conditional phase shift) for photonic or phononic qubits. In addition, the mechanical degrees of freedom can also serve as a quantum memory , and optomechanical interactions yield a quantum interface between solid-state, optical and atomic qubits ; . The combination of these ingredients will make optomechanical systems a promising integrated platform for quantum repeaters and general “hybrid quantum networks“.
This work was supported by the DARPA/MTO ORCHID program through a grant from the AFOSR, the DFG Emmy-Noether and an ERC starting grant, and the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation. ML thanks OJP for his hospitality at Caltech.
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