1.2 Reduction to a set of dimensionless and independent parame-ters
We now identify the dimensionless parameters the system dynamics depends on. Expressed in terms of the mechanical oscillator frequencyωM, the parameters describing the classical system are
mechanical damping : ΓM/ωM cavity decay : κ/ωM
detuning : ∆/ωM
driving strength : P = 8|αL|2gM2 /ωM4 =ωcavκ2Emaxcav/(ωM5 mML2). (1.2) HereEmaxcav is the light energy circulating inside the cavity when the laser is in resonance with the optical mode.
The quantum mechanical nature of the system is described by the “quantum parameter” ζ, comparing the magnitude of the cantilever’s zero-point fluctuations, xZPF, with the full width at half maximum (FWHM) of the cavity (translated into a cantilever displacement xFWHM)
quantum parameter : ζ = xZPF xFWHM
= gM
κ . (1.3)
The resonance width of the cavity can be expressed as xF W HM =κL/ωcav, where L is the cavity’s length. The quantum parameter ζ vanishes in the classical limit~→0, as the zero-point fluctuations xZPF of the cantilever go to zero. The magnitude of ζ determines the effect of quantum fluctuations on the dynamics of the coupled cavity-cantilever system.
We note that there is an alternative way to introduce the quantum parameter (1.3). Here we made use of the two characteristic length scales of the system. Alternatively, we could compare the zero-point momentum fluctuations of the cantilever to the impulse a single intracavity photon transfers to the cantilever. When the photon is reflected at the cantilever, it transfers an impulse of2~k.This process is repeated after one cavity round-trip time 2Lc for as long as the photon stays inside the cavity, i.e. for a span of time given byκ−1.The total transfer of momentum is therefore given by pphot = ~kLcκ−1 = ~ωκLcav. The strength of the zero-point momentum fluctuations is given bypZPF =q~mM2ωM = 2x~
ZPF.Taking the ratio of these to quantities leads directly to the quantum parameter:
We see that for a large quantum parameter a single phonon of the cantilever causes a detectable shift of the cavity resonance as well as a single photon causes the cantilever to change its mo-mentum noticeably. Finally we note, that in a recent article Murch et al., introduced a so called granularity parameter to describe the impact of a single photon on the collective motion of ultra-cold atoms [32]. It directly corresponds to the quantum parameter (1.3), as we will see in section (4.3) of this thesis.
In the following section we will discuss the dynamics of the cantilever due to the driving of the cavity both in a quantum mechanical and a classical treatment. The quantum parameter will turn out to be very suitable for this analysis which will focus on the system’s most characteristic quantities, in particular the number of photons in the cavity and the energy of the cantilever’s oscillation. There are a few words to be said about the mechanical oscillation energy.
In the classical picture we can obtain a solution of the oscillation amplitude A as a function of the system parameters. This solution has been given in [21] and will be briefly reviewed at the beginning of the subsequent chapter. The expression for the mechanical oscillation energy follows directly from this solution as EM,cl = 12mωM2 A2. The quantum mechanical treatment on the other hand allows to get EM from the expectation value of the cantilever’s occupation number: EM,qm=~ωMhˆnMi,where we exclude the zero-point energy. We note that there is one peculiarity in our definition of the phonon occupation numberhˆnMi.If we would definenˆM = ˆc†c,ˆ a static displacement of the cantilever, i.e. hxˆMi 6= 0,would already yield a non-zero occupation number, even in the absence of any oscillations. In order to exclude these contributions, we shift the position operator by its expectation value and introducexˆ0M = ˆxM−hˆxMi.Correspondingly we obtain shifted annihilation and creation operatorsˆb0 = 2x1
ZPF(ˆx0M+mωi
MpˆM)andˆb0†= 2x1
ZPF(ˆx0M−
i
mMωMpˆM), where pˆM = i
q~mMωM
2 (ˆb†−ˆb) is the momentum operator of the cantilever. The phonon number operator can now be defined as
nˆM = ˆb0†ˆb0
= 1
4x2ZPF xˆ2M + pˆ2M m2MωM2
+ 1
4x2ZPF hˆxMi2−2ˆxMhˆxMi
= ˆb†ˆb+ 1
4x2ZPF hxˆMi2−2ˆxMhˆxMi (1.5) Its expectation value is given byhˆnMi=hˆb0†ˆb0i=hˆb†ˆbi −4x12
ZPF
hˆxMi2 and directly corresponds to the oscillation energyEM,qm.
In order to obtain a dimensionless quantity for our comparison, we divide the cantilever energy EM by a characteristic classical energy scale of the system. To set this characteristic energy scale, we take the energy E0 = 12mωM2 x2FWHM associated with an oscillation amplitude xFWHM
of the mechanical cantilever which moves the cavity just out of its resonance. It follows that EM/E0 = (A/xFWHM)2 in the classical case, and EM/E0= 4ζ2hˆnMi in the quantum version.
Chapter 2
The optomechanical instability in the quantum regime
In this chapter we focus on the question of how the optomechanical instability changes due to quantum effects. To answer this question at least partially, we will employ a fully quantum me-chanical treatment of the system, based on the numerical solution of a quantum master equation.
We will concentrate on the case of blue-detuned pumping of the cavity, where the cantilever can settle into self-induced oscillations once the input power is increased beyond some threshold value. The results of the quantum mechanical treatment can then readily be compared to the classical solution [21]. Below the threshold of the instability, we can check the results of a simple rate equation approach against the results of the master equation. This rate equation approach captures the amplification behaviour of the coupled system and catches the effects of photon shot noise on the cantilever motion [15]. The full quantum mechanical treatment can describe the crossover from the regime below the threshold of instability to the regime of self-induced oscillation. Moreover, the comparison to the classical solution allows to observe the effects of the quantum fluctuations. In this analysis, the quantum parameterζ =xZPF/xFWHMwill be the most important quantity as it governs the quantum-to-classical transition.
We note, that the main results of this chapter have already been discussed in:
• Max Ludwig, Björn Kubala, Florian Marquardt: “The optomechanical instability in the quantum regime”, New Journal of Physics, volume 10, 095013.
2.1 Classical solution
In the following we will briefly review the classical treatment of the system as given in ( [21]). It allows to find an analytic solutions for the coupled cavity and cantilever dynamics. In particular one can find the amplitude of the self-induced oscillations as a function of the system parameters.
The Hamiltonian (2.16) introduced in the previous chapter allows to readily derive the Heisen-berg equations of motion for the cavity operator ˆa and the cantilever position operator x.. Toˆ investigate the purely classical dynamics of the coupled cavity-cantilever system, we replace the operator ˆa(t) by the complex light amplitude α(t) and the position operator of the cantilever xˆ by its classical counterpart. We thus arrive at:
α˙ = [i(∆ +g xM
xZPF)−κ
2]α−iαL (2.1)
11
x¨=−ω2Mx+ ~g
mxZPF|α|2−ΓMx˙M. (2.2)
Here fluctuations (both the photon shot noise as well as intrinsic mechanical thermal fluctuations) have been neglected, to obtain the purely deterministic classical solution. The variables t,x and α can be rescaled [21] as˜t=ωMt; ˜α=iαωM/(2αL); ˜x=gx/(ωMxZPF), so that the coupled
Crucially, the quantum parameterζ cannot and does not feature in these equations.
Apart from a static solution x(t) ≡ const, this system of coupled differential equations can show self-induced oscillations. In such solutions, the cantilever conducts an approximately sinu-soidal oscillation at its eigenfrequency,x(t)≈x+A¯ cos(ωMt). The light amplitude then shows the dynamics of a damped, driven oscillator, which is swept through its resonance, see equation (2.1);
an exact solution for the light amplitudeα(t)can be given as a Fourier series containing harmonics of the cantilever frequencyωM [21]:
The dependence of oscillation amplitude,A, and average cantilever position,x, on the dimen-¯ sionless system parameters can be found by two balance conditions: Firstly, the total force on the cantilever has to vanish on average, and, secondly, the power input into the mechanical oscillator by the radiation pressure on average has to equal the friction loss.
The force balance condition determines the average position of the oscillator, yielding an implicit equation forx,¯
h¨xi ≡0 ⇔ mω2Mx¯=hFradi= ~g
mxZPFh|α(t)|2i, (2.6) where the average radiation force,hFradi is a function of the parameters x¯and A.
The balance between the mechanical power gain due to the light-induced force,Prad=hFradxi,˙ and the frictional lossPfric= ΓMx˙2 follows from
hx¨˙xi ≡0 ⇔ hFradxi˙ = ΓMhx˙2i. (2.7) For each value of the oscillation amplitudeAwe can now plot the ratio between radiation power input and friction loss, Prad/Pfric = hFradxi/(Γ˙ Mhx˙2i), after eliminating x¯ using equation 2.6.
This is shown in figure 2.1. Power balance is fulfilled if this ratio is one, corresponding to the contour line Prad/Pfric = 1. If the power input into the cantilever by radiation pressure is larger
2.1. CLASSICAL SOLUTION 13
detuning
1 0 3 2
0
cantilever energy
-1 100
power fed into the cantilever
1 2
−1 0 1 2 3
Figure 2.1: Classical self-induced oscillations of the coupled cavity-cantilever system. The radiation pressure acting on the cantilever provides an average mechanical power input ofPrad. The ratio Prad/Pfric of this powerPrad vs. the loss due to mechanical friction,Pfric, is shown as a function of the detuning∆ and the cantilever’s oscillation energyEM, at fixed laser input power P. The oscillation energy EM = mωM2 A2/2 is shown in units of E0, where EM/E0 = (A/xFWHM)2. Self-induced oscillations requirePrad =Pfric. This condition is fulfilled along the horizontal cut at Prad/Pfric = 1(see black line and the inset depicting the same plot, viewed from above). These solutions are stable if the ratioPrad/Pfric decreases with increasing oscillation amplitude A. The blue regions at the floor of the plot indicate thatPrad is negative, resulting in cooling. The cavity decay rate isκ = 0.5ωM, the mechanical damping is chosen as ΓM/ωM = 1.47·10−3, and the input power asP = 6.05·10−3; these parameters are also used in figures 2.2, 2.3, 2.4, and 2.6, and will be referred to asΓ∗M andP∗.
than frictional losses (i.e., for a ratio larger than one), the amplitude of oscillations will increase, otherwise it will decrease. Stable solutions (dynamical attractors) are therefore given by that part of the contour line where the ratio decreases with increasing oscillation amplitude (energy), as shown in figure 2.1.
Changing the (dimensionless) mechanical damping rateΓM/ωM will scale the plot in figure 2.1 along the vertical axis, so that the horizontal cut at one yields a different contour line of stable solutions [a changed input power P gives a similar scaling, but leads to further changes in the solution, as P also enters the force balance condition, equation (2.6)]. Decreasing mechanical damping or increasing the power input will increase the plot height in figure 2.1, so that the amplitude/energy of oscillation of the stable solution increases.
While the surface or contour plots in figure 2.1 allow a discussion of general features of the self-induced oscillations, such as the multistabilities discussed in Ref. [21], a slightly different representation of the classical solution is more amenable to an easier understanding of the particular dynamics of the system for a certain set of fixed system parameters. Figure 2.2 shows the cantilever energy EM,cl= 12mωM2 A2 in terms of the classical energy scaleE0 = 12mωM2 x2FWHM as function of driving P and detuning ∆/ωM. These are the parameters that can typically be varied in a given experimental setup.
For sufficiently strong driving, self-induced oscillations appear around integer multiples of the cantilever frequency, ∆ ≈ nωM. For a cavity decay rate κ = 0.5ωM assumed in figure 2.2, the different bands are distinguishable at lower driving; for larger κ (or for stronger driving), the various ‘sidebands’ merge. For the lower-order sidebands, the nonzero amplitude solution connects continuously to the zero amplitude solution, which becomes unstable. This is an example of a (super-critical) Hopf bifurcation into a limit cycle.
The vertical faces, shown gray in figure 2.2, for ∆ ≈ 2ωM and ∆ ≈ 3ωM are connected to the sudden appearance of attractors with a finite amplitude. For example, while approaching the detuning of∆ = 2ωM at fixed P (the solid line in figure 2.2 refers to P = 1.47·10−3), a finite amplitude solution appears, althoughA = 0 remains stable. In Ref. [21] the existence of higher-amplitude stable attractors and, correspondingly, dynamic multistability were discussed.
2.1. CLASSICAL SOLUTION 15
energy
0
detuning driving
cantilever
1 2
3 0
0.015
100
strength
Figure 2.2: Cantilever oscillation energy EM ∝ A2 versus detuning ∆and laser input power P.
This plot (in contrast to figure 2.1) shows only the stable oscillation amplitude, but as a function of variable input power. The particular value P∗ corresponding to figure 2.1, and the resulting profile of oscillation amplitudes are indicated by a black line. The green floor of the plot indicates regions without self-induced oscillations. The other system parameters are as in figure 2.1. The continuous onset of the self-oscillations in the sidebands at ∆/ωM = 0,1 (which merge for the present parameter values) represents a super-critical Hopf bifurcation, fromA= 0 toA6= 0. At higher sidebands, an attractor with a finite A6= 0 appears discontinuously, whileA = 0 remains a stable solution.