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The second approach we consider is to simulate the physical feedback cooling dis- cussed in the previous section by post-processing the measurement data obtained in the absence of any feedback. This approach is based on the idea presented by Harris et al.[92], that, for a linear system with known dynamics and satisfying the

−0.50 −0.25 0.00 0.25 0.50 0.75 Displacement

(a.u.) _{Feedback cooling}

Without signal _Displacement

√_Energy 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Time (ms) −0.50 −0.25 0.00 0.25 0.50 0.75 Displacement

(a.u.) _{Feedback cooling}

With signal

Figure 4.10: Sample traces without and with signal in the presence of feedback cooling, showing both the displacement and estimated energy. The units are arbitrary, and the square root energy is scaled to displacement units. The shaded orange and purple areas indicate when feedback cooling and signal are applied, respectively.

4.3. SENSITIVITY ENHANCEMENT 87 property that all unknown forces are independent of position, any linear feedback protocol can be simulated via post-processing.

We refer the reader to Harris et al.[92] for the full derivation of the possibility of this simulation, but as mentioned in Section 4.1 it can be understood intuitively. Consider the simplified case where we apply a single impulsive feedback signal at a particular time, based on some prior measurements of the system. Rather than actually applying this feedback, we could instead simply calculate the feedback we would have applied, and predict the response of the system to that feedback in the absence of all other forces. By linearity and the assumption of position- independence of unknown forces, we can then simply add this response to the observed trajectory to obtain the exact trajectory we would have observed had feedback been applied. For more realistic feedback protocols involving more than a single pulse we apply the same argument, but now when calculating the feedback we would have applied at each timestep we also take into account the response of the system to feedback applied at previous timesteps.

More precisely, letx(t) be the position of the oscillator at timetin the absence of any feedback. Lety(t) be the trajectory we would have observed had feedback been applied. We will show how y can be determined from x.

Since we are interested in simulating the cold damping described in Sec- tion 4.2.4 we consider the particular case of a linear feedback protocol depending only on the oscillator velocity at the current time, and assume without loss of generality that the mean oscillator position is 0. The feedback force at timetcan thus be written as Ffb(t) =G(t) ˙y(t), whereG(t) is the gain at time t. To apply the theory of Harris et al.[92] we rewrite this in the kernel form

Ffb(t) =

Z t

0

−G(t) ˙δ(τ ₋t)x(τ)dτ, (4.15)

where we have introduced the derivative of the delta function ˙δ.

Denoting byχm(t) the (Fourier transform of) the mechanical susceptibility as in Section 4.2.5 we may now define the transfer function

h(t, τ) :=

Z ∞

−∞−

G(t0) ˙δ(τ ₋t0)χm(t₋t0)dt0, (4.16)

which describes the response of the oscillator at time t to the feedback force resulting from its position at time τ. The theory of Harris et al.[92] now asserts

the following relationship between x and y:

y(t)−

Z t

0

h(t, τ)y(τ)dτ =x(t). (4.17)

Again, this result is quite intuitive: it simply states that the signal in the absence of feedback can be obtained by considering the signal with feedback and sub- tracting the combined response of the system to all prior feedback. In practice we must solve this Fredholm equation for y (given x), and the simplest way to achieve this is by temporal discretisation. Specifically, we sample x, y and h at some large number of timesteps (in our case we use 1000), and collect the values into vectors xand y and a matrix H, which yields the matrix equation

y₋Hy=x (4.18)

y= (I ₋H)−1_{x, (4.19)}

where I is the appropriate identity matrix. This equation allows us to determine y, the measurement record we would have obtained had we used feedback, from x, the measurement record we actually observed in the absence of feedback.

To apply this method in our scenario we simply need to specify the feedback gain G(t) for our particular feedback protocol. As in Section 4.2.4 we consider an effective gain of the form −gγm, where g is a fixed gain factor, but now we switch off the feedback after a particular time t0. That is,

G(t) =    −gγm t _≤t0 0 t > t0. (4.20)

With this choice, the transfer function h becomes

h(t, τ) = Z t0 −∞ gγmδ˙(τ₋t0)χm(t₋t0)dt0 =    −gγmχ˙m(t₋τ) τ _≤t0 0 τ > t0. (4.21)

From this we may determine the matrixH and thus the transformation mapping the uncooled measurement record x to the virtually cooled record y. From y we determine the SNR via energy averaging, exactly as in the physical feedback

4.3. SENSITIVITY ENHANCEMENT 89 cooling case.

The only remaining step is to specify values for the relevant parameters, namely ωm, γ and g. We determined these as follows. First, the parameters ωm and γ were roughly estimated from the raw data, and then the gaing was varied in order to maximise the associated SNR. The estimates of ωm and γ were then repeatedly adjusted, interleaved with optimisation ofg, in order to maximise the peak SNR. The final values we used were ωm/2π = 339.61 kHz, γ/2π= 0.54 kHz and g = 55.

Sample traces filtered with this method are shown in Fig. 4.11. We see that the virtual cooling is effective at damping the motion while turned on. After the cooling is turned off we observe similar behaviour to the physical feedback cooling case: with no signal the oscillations gradually increase in amplitude back to what we would observe with no cooling, while with signal the oscillator quickly responds to the force and the oscillations closely match what we would have observed without any feedback. Therefore, as with physical feedback cooling, we expect an enhancement in SNR in the period immediately after the cooling is switched off, since this corresponds to reduced noise but largely unaffected signal.