The effect of Landau-Zener dynamics on phonon lasing

Optomechanical systems couple light to the motion of nanomechanical objects. Intriguing new effects are observed in recent experiments that involve the dynamics of more than one optical mode. There, mechanical motion can stimulate strongly driven multi-mode photon dynamics that acts back on the mechanics via radiation forces. We show that even for two optical modes Landau-Zener-Stueckelberg oscillations of the light field drastically change the nonlinear attractor diagram of the resulting phonon lasing oscillations. Our findings illustrate the generic effects of Landau-Zener physics on back-action induced self-oscillations.

Optomechanical systems couple light to the motion of nanomechanical objects. Intriguing new effects are observed in recent experiments that involve the dynamics of more than one optical mode. There, mechanical motion can stimulate strongly driven multi-mode photon dynamics that acts back on the mechanics via radiation forces. We show that even for two optical modes Landau-Zener-Stueckelberg oscillations of the light field drastically change the nonlinear attractor diagram of the resulting phonon lasing oscillations. Our findings illustrate the generic effects of Landau-Zener physics on back-action induced self-oscillations. The exploration of nanomechanical objects and their interaction with light constitutes the rapidly evolving field of optomechanics (see [1] for a recent review). The key element of any optomechanical system is a laserdriven optical mode whose resonance frequency shifts in response to the displacement of a mechanical object. The photon dynamics conversely acts back on the mechanics in terms of a radiation pressure force. These dynamical back-action effects, mediated by the light field, can cool or amplify mechanical motion, and even drive the system into a regime of self-induced mechanical oscillations [2][3][4][5][6][7][8] akin to lasing.
An exciting new development has introduced optomechanical setups with multiple coupled optical and vibrational modes [9][10][11]. These systems allow one to realize sophisticated measurement schemes [12][13][14], to study collective phenomena [15], or to mechanically drive coherent photon dynamics [16]. For applications, they furthermore stimulate prospects of integrated optomechanical circuits [17,18]. Recently, phonon lasing for an optomechanical setup involving a tunable optical two-level system has been demonstrated [19]. Hence, implementing a nanomechanical analog of a laser has finally been achieved [20]. Here, we develop the fully nonlinear theory of phonon lasing (self-induced mechanical oscillations) in such multimode optomechanical setups. In particular, we will point out that the mechanical oscillations may induce Landau-Zener physics with respect to the optical two-level system, and that this has a strong effect on the dynamical back-action. The resulting phenomena drastically change the nonlinear attractor diagram, i.e. the relation between the mechanical lasing amplitudes and the experimentally tunable parameters. We will refer to an existing optomechanical setup [9] where our predictions can be verified experimentally. However, most of our analysis and discussion are applicable to the quite generic situation where self-induced oscillations are pumped by a parametrically coupled, driven two-level system. Our Mechanical oscillations x(t) = A cos(Ωt) + xa periodically sweep the system along the photon branches (red). (c) Cavity resonance frequency ω±(x(t)) depending on time. For non-adiabatic sweeps through the anti-crossing, repeated LZ transitions (highlighted regions) split the photon state. After each passage, the two contributions gather a phase difference that leads to subsequent interference. The resulting LZS oscillations in the light field act back on the mechanics via the radiation pressure force. (d) For sufficiently large backaction-induced anti-damping, the system enters a regime of mechanical self-oscillations (phonon lasing). findings thus are also relevant for nanomechanical structures or microwave modes whose oscillations are amplified by coupling to, e.g., current-driven double quantum dot setups, superconducting single-electron transistors, or Cooper-pair boxes [21]. We consider the system depicted in Fig. 1a. A dielectric membrane is placed in the middle between two highfinesse mirrors [9]. Transmission through the membrane couples the optical modes of the left and right half of the arXiv:1102.1647v1 [cond-mat.mes-hall] 8 Feb 2011 cavity, respectively. Focussing on two nearly degenerate modes, the Hamiltonian of the cavity readŝ Here,â † Lâ L (â † Râ R ) is the photon number operator of the left (right) cavity mode, ω 0 is the modes' frequency for x = 0 (where the two modes are degenerate), and 2l is the length of the full cavity. The membrane's displacement x linearly changes the modes' bare frequencies, while the optical coupling g leads to an avoided crossing for the system's two optical resonances, ω ± = g 2 + (ω 0 x/l) 2 [ Fig. 1b]. Thus, mechanical oscillationsx(t) periodically sweep the system along the hyperbola branches ω ± .
We focus on the experimentally accessible, nonadiabatic regime [12,16] where fast periodic sweeping through the avoided crossing results in consecutive Landau-Zener (LZ) transitions [22,23]. For a photon inserted into the left mode, the first transition splits the photon state into a coherent superposition, the two contributions gather different phases and interfere the next time the system traverses the avoided crossing [ Fig. 1c]. For a two-state system, the resulting interference patterns are known as Landau-Zener-Stueckelberg (LZS) oscillations [24]. These have been demonstrated in many setups, ranging from atomic systems [25][26][27] to quantum dots and superconducting qubits [28][29][30][31]. In all of these situations, LZS effects are produced by a fixed external periodic driving. In contrast, here we address the case where LZS oscillations act back on the mechanism that drives them (i.e. the mechanical motion), via the radiation pressure force. We will see that LZS interference strongly influences this back-action force and thereby drastically affects the mechanical self-oscillations that occur when this force overcomes the internal friction [ Fig. 1d]. More generally, the following discussion thus illustrates the effect of LZS dynamics on back-action induced instabilities.
Given the radiation pressure forceF rad = −∂Ĥ cav /∂x, the coupled equations of motion for the displacementx(t) andâ i (t) (i = L, R), read where we used input-output theory for the light fields and set A 0 = ω 0 /lm. The membrane has a mechanical frequency Ω, an intrinsic damping rate Γ and a rest position x 0 . Photons decay at a rate κ out of the cavity. We assume the left modeâ L to be driven by a laser at frequency ω L ; the input fieldsb i in (t) contain this contribution. In the following, we will consider purely classical (largeamplitude) nonlinear dynamics and replace the operators a i (t) by the coherent light amplitudes α i (t). The classical input fields then read β R in = 0, β L in = e −iω L t P in / ω L where P in is the laser input power, and the mechanical Langevin force will be neglected (ξ ≈ 0). For convenience, we define the laser detuning ∆ L = ω L − ω 0 .
The radiation pressure force gives rise to a timeaveraged net mechanical power input F radẋ . In analogy to the intrinsic friction Γ, see Eq. (2), we can define F radẋ = −mΓ opt ẋ 2 such that we obtain an effective optomechanical damping rate For Γ opt > 0 (Γ opt < 0) the light-field interaction damps (anti-damps) the mechanics. For given oscillations x(t) = A cos(Ωt)+x a , Γ opt can be calculated via the periodic light field dynamics α L (t), α R (t) that is found by solving Eq. (3); see also Eq. (6) further below. Note that our Γ opt is amplitude-dependent, and the usual linearized case [13,32] is recovered for A → 0. In the following we will express displacement in terms of frequency, x(t) = (ω 0 /l)x(t) (see Eq. (1)); likewise forĀ,x a . Fig. 2a shows results for Γ opt in this setup, at moderate amplitudes A. Optomechanical damping and heating is largest if the optical modes' frequency difference is in resonance with the mechanical frequency Ω [8,19]. In this case, photon transfer from the laser-driven left mode into the right one involves absorption (or emission) of a phonon, that yields strong mechanical heating (or cooling), see Fig. 2b-c. For finite amplitudes, we observe an Autler-Townes (AT) splitting [33] that scales as 2gĀ/Ω [16]. Given Γ opt , we now turn to discuss back-action driven mechanical self-oscillations (phonon lasing) of the membrane.
For suitable laser input powers, the radiation pressure force only weakly affects the mechanics over one oscillation period and the mechanics approximately performs sinusoidal oscillations at its unperturbed eigenfrequency Ω; x(t) = A cos(Ωt) + x a . The possible attractors of the dynamics (A, x a ) have to meet two conditions [5,6]. First, the time-averaged total force must vanish: ẍ = 0. Second, the overall mechanical power input due to radiation pressure must equal the power loss due to friction, ẍẋ = 0. From Eq. 2, the power balance ẍẋ = 0 is equivalent to The force balance ẍ = 0 yields F rad (t) = mΩ 2 (x a − x 0 ), i.e. the radiation pressure force displaces the membrane's average position x a from its rest position x 0 . In general, one solves the force balance to find x a = x a (A, x 0 ) and uses this to calculate Γ opt (A, x a ) [5,6]. For high quality mechanics (Ω/Γ 1), the power balance (Eq. (5)) is met for weak radiation pressure forces where x a x 0 . For clarity, we will focus on this case. Otherwise, attractor diagrams get deformed slightly [5]. Fig. 3a displays the effective optomechanical damping Γ opt depending on laser-detuning ∆ L and amplitude A. The structure of this diagram is drastically different from the standard case with one optical mode [5,6]. There are "ridges" of high Γ opt which display an oscillatory shape (clarified in the inset).
A physical understanding of Fig. 3 can be found from the general structure of the light field dynamics that enters the optomechanical damping, Eq. (4). For given mechanical oscillations x(t) = A cos(Ωt) + x a , the formal solution to Eq. (3) can be expressed as where the Green's function G i (t, t ) describes the amplitude for a photon entering the left mode at time t and to be found in the left or right one (i = L, R) at time t. From Eq.
with t ≥ t and initial conditionã R (t , t ) = 0,ã L (t , t ) = 1. Thus, the internal photon dynamics between the two modesã i (t, t ) is expressed in terms of a two-level system with a time-dependent coupling ge 2iφ(t) . With ψ = (ã R ,ã L ) T , Eq. (7) is the Schrödinger equation including a time-periodic Hamiltonian, H(t + T ) = H(T ). In this case it is appropriate to consider the time-evolution operator for one period, ψ(t + T ) = U (T )ψ(t ), and its two eigenvalues, the so-called Floquet eigenvalues ± : U (T )χ ± = exp(−i ± T )χ ± . U (T ) is obtained by integrating Eq. (7).
Using Floquet theory [34], we find the general structure of the Green's function G i (t, t ) = j,n,n C n,n ,j i e −iΩ(nt−n t ) e −i j (t−t ) , where C n,n ,j i are time-independent coefficients. Then, via Eq. (6) we obtain pronounced resonances in Γ opt located at ∆ L = mΩ + ± (Ā), corresponding to the ridges in Fig. 3. The interference between consecutive LZ transitions renormalizes the coupling between modes in terms of Bessel functions J n : ge 2iφ(t) = g n J n (2Ā/Ω)e inΩt (Eq. 7). This results in an oscillatory modulation of the Floquet eigenvalues ± (Ā). At certain amplitudes, these vanish due to total destructive interference, see Fig. 3a. The oscillatory shape of the ridges in Γ opt then directly determines the attractor diagram for the self-induced oscillations, via the power balance equation (5), see Fig. 3b.
Regarding the global structure of Fig. 3a, Γ opt tends to be large near ∆ L = ±Ā (dashed lines). This is because then the left mode gets into resonance with the laser at the motion's turning point. For larger amplitudes, we recover the predictions for the standard optomechanical setup [5] (checkerboard in Fig. 3a).
So far, we discussed dynamical back-action effects for parameters where the mechanical frequency is larger than the optical splitting, Ω > 2g [ Fig. 2,3]. In general, the parameter space can be subdivided as shown in Fig. 4a. Multimode dynamics that goes beyond the standard scenario [5,6] can only be observed if the photon lifetime inside the cavity is larger than the timescale for photons to tunnel between modes, 2g > κ (colored region, Fig. 4a). Otherwise, photons inserted into the left mode decay before the second mode affects the dynamics and we recover the standard results [5,6]. Within the new region (colored in Fig. 4a), the most interesting regime is where mechanical sidebands can in fact be resolved, i.e. κ < Ω. Above, we had focussed on the sector 2g < Ω within this regime. Now Fig. 4b displays Γ opt in the opposite sector where 2g > Ω. Here, several mechanical sidebands lie within the avoided crossing. With respect to self-induced mechanical oscillations, these sidebands and their interaction yield an intricate web of multistable attractors, see Fig. 4c. The global asymptotics of these structures (green lines) can be found from the quasistatic approximation, i.e. from the time-averaged transition frequency: ∆ L = 2 ω + (t) = 4 g 2 +Ā 2 E(π/2, k)/π, where k = Ā2 / g 2 +Ā 2 and E( π 2 , k) is the complete elliptic integral of the second kind.
To conclude, we have investigated self-induced mechanical oscillations (phonon lasing) in a multimode optomechanical system. The mechanical motion drives Stueckelberg oscillations in the light field of two coupled optical modes, and this drastically modifies the attractor diagram. The additional influence of quantum (and thermal) noise could be analyzed along the lines of [5,6]. Our example, which can be realized in present optomechanical setups, illustrates the potential of Landau-Zener physics to appreciably alter lasing behavior.