Processing light with an optically tunable mechanical memory

Mechanical systems are one of the promising platforms for classical and quantum information processing and are already widely-used in electronics and photonics. Cavity optomechanics offers many new possibilities for information processing using mechanical degrees of freedom; one of them is storing optical signals in long-lived mechanical vibrations by means of optomechanically induced transparency. However, the memory storage time is limited by intrinsic mechanical dissipation. More over, in-situ control and manipulation of the stored signals processing has not been demonstrated. Here, we address both of these limitations using a multi-mode cavity optomechanical memory. An additional optical field coupled to the memory modifies its dynamics through time-varying parametric feedback. We demonstrate that this can extend the memory decay time by an order of magnitude, decrease its effective mechanical dissipation rate by two orders of magnitude, and deterministically shift the phase of a stored field by over 2π. This further expands the information processing toolkit provided by cavity optomechanics.

whereê in is the input from the environment. ω b and Γ b are intrinsic properties of the device, however cavity optomechanics offers a way to manipulate the mechanical parameters of the cavity by optical means 3 . As we will show here, this can be viewed as coupling the mechanical mode to an optical reservoir mode,r. Not only is this interaction easily controllable, but the reservoir can be arranged to have negligible thermal occupation through use of optical laser light. We consider a reservoir mode with frequency ω r connected to an input port at a rate κ ex r with a total decay rate κ r . This is dissipatively coupled to the mechanical mode with a strength g r . In the presence of a strong control laser with amplitude α r , the optomechanical interaction can be linearized, and we can write expressions for the cavity fluctuation operatorr, which is coupled the mechanics at a rate g r = G r α r x o , where G r = dω dx is the shift in cavity frequency due mechanical displacement, and x o are the zero point fluctuations of the mechanics 3 . This leads to the coupled equations of motion where we define the relevant response functions as The input modes at time t, are given in terms of the time t 0 in the far past as 4ê where E 0 and R 0 are the state of the input modes at time t 0 . For the sake of simplicity, in what follows, we will assume κ r = κ ex r . Using Supplementary Eq. 2, we can solve for the reservoir dynamics aŝ It is interesting to note the role of the optical cavity as a filter. The exponential terms are in the form of a retarded Green's function, and specify a sensitivity to frequencies near ±∆ r to a history on the timescale 1/κ r . Inserting this into the expression for the mechanics given in Supplementary Eq. 2, we find an equation of motion for the mechanics under the influence of both the environment and the reservoir where in the above we used the fact the t − t 0 1/κ r and applied the rotating wave approximation. The right side of the equation can be interpreted as the sum of damping and dissipation terms due to coupling to the environment, and damping and dissipation terms due to coupling to the reservoir. With the assumption that κ r Γ b , we can make further simplifications. First we note that the integral associated with the dissipation term becomes Next we simplify the fluctuation term as where once again we used the assumption that κ r Γ b to simplify. Combining Supplementary Eqs. 7-9 we arrive at the solution This can be rearranged to the simple expression reminiscent of Supplementary Eq. 1 In the above we have absorbed a factor of i into the definition of R 0 and R † 0 , and defined effective frequency and damping terms Comparing Supplementary Eq. 1 and Supplementary Eq. 11, we see that coupling the reservoir mode induces both fluctuation and dissipation. By varying the strength or detuning of the control laser, the coupling to the reservoir is modified. In the sideband resolved regime (ω b κ) we note two special cases. For ∆ r = −ω b the effective interaction Hamiltonian is H eff = −g r b †r +br † , and the mechanics has the equation of motioṅ On the other hand, for ∆ r = ω b the interaction Hamiltonian takes the form H eff = −g r br +b †r † , and the equation of motion isḃ

SUPPLEMENTARY NOTE 2. ENHANCED OMIT
The amplitude in cavity a, as a function of probe-control field detuning, δ a , under the influence of the reservoir mode may be expressed as, From inspection with the OMIT lineshape for a conventional optomechanical system 3 , when our probe is on-resonance, such that δ a = ω b , we can write our effective cooperativity as: where C a = 4|g a | 2 /Γ b κ a is the cooperativity of the device in absence of the reservoir.

A. Group delay
The group delay imparted on the pulse in transmission and reflection can be calculated about a central signal frequency, ω s with the spectrum confined to a small window (< Γ eff b ) following Safavi-Naeini et al. 5 by computing and for the transmission and reflection group delay, respectively. These quantities are shown in Fig. 2

SUPPLEMENTARY NOTE 3. COOLING AND HEATING
As a test of the reservoir engineering expressions, and as a step towards calculating the thermal occupations required for the memory calculations, we calculate full expressions for optomechanical heating and cooling here. Ignoring initial transients, the formal solution of Supplementary Eq. 11 iŝ We quantify the thermal statistics of the reservoir and environment with the correlators where n th r is the number of thermal photons occupying the reservoir, and n th e is the number of thermal phonons occupying the environment. Using these expressions, we can calculate the thermal occupancy of the cavity as In the experiment considered in this work, our reservoir does not have thermal occupation. Setting n th r = 0 we recover the usual limit of optomechanical cooling where, n min = |g r | 2 κχ r † (ω b )/Γ opt r .

SUPPLEMENTARY NOTE 4. STORAGE ENHANCEMENT
Solving the equations of motion explicitly, we can divide the phonon population in the cavity during the storage time into signal phonons, which are proportional toâ in , and undesired thermal phonons, which are a consequence of e in . These each evolve as, Here the presence of Γ opt a terms are due to optomechanical cooling by the OMIT control laser during the write step on the initial thermal population of the resonator mode. Examples of the competing growth and decay of noise and signal phonons is shown in Fig. 3(a). Defining the storage time as the moment the signal level decays to the level of the thermal phonons, we find, This expression is used to generate the plots in Fig. 3(b-d), which analyze the performance of the memory as a function of optomechanical damping of modes a and r. Note that quantum optical noise, e.g. Stokes scattering, is not included in this analysis.

SUPPLEMENTARY NOTE 5. PHASE SHIFTING
Reservoir engineering also allows us to dynamically change the frequency of the mechanical mode. If the frequency is changed over a time interval δt, the change in phase may be expressed as, For simplicity, we assume we change our mechanical frequency as a ramp function, with maximum frequency shift δ b . Under the adibaticity requirement 1 This yields the simple expression for the phase shift,  In the phase shifting experiment in the main text, we operate with the reservoir laser detuning ∆ r ≈ −ω b , so we may approximate the frequency shift as, SUPPLEMENTARY NOTE 6. TIME LENS The reservoir mode also permits the mechanical damping rate to be dynamically adjusted. For example, at ∆ r ≈ −ω b , the damping is approximately, If we ramp the mechanical damping according the expression Γ opt r (t) = ηt, we recover the expression for a time lens 6