Puntos de Venta

atoms coupled to the cavity mode [201]. When detecting the phase fluctuations in the transmitted light with a homodyne detection scheme, the signal at Ωm splits (cf.

Figure6.3b), but the (suppressed) scattered light at the carrier frequency exhibits no splitting. It is important to note that the splitting in the displacement spectrum is not observed unless gm > κ/

√

2 , owed to the finite width of the peaks.

Due to the requirements on the cavity bandwidth and the detuning, the parameter regime in which NMS may appear implies cooling. In turn, for a positive detuning (which entails amplification) the observation of NMS is prevented by the onset of the parametric instability [179]. Therefore, a discussion of NMS cannot be decoupled from an analysis of the associated cooling. We also show below that the CRT in the interaction lead to the quantum limit of backaction cooling [199,198].

6.2.1. Experimental observation of parametric normal mode

splitting

Shortly after the publication of the results presented in the previous section (cf. reference [2]), parametric NMS was observed by Gr¨oblacher and co-workers [210]. The authors grew a reflective mirror pad on a doubly clamp beam with a vibrational frequency of Ωm = 2π×947 kHz. The oscillating mirror formed one end of a 25 mm

Fabry-P´erot cavity with a finesse of 14,000 corresponding to an energy decay rate of 215 kHz and thus being well within the RSB parameter regime. Using a strong driving laser beam of∼10 mW, the authors could achieve optomechanical coupling rates up togm ≈2π×460 kHz and observe optomechanical normal mode splitting.

Likewise Teufel and co-workers achieved strong coupling with an electromechanical device [211, 212]. The authors used a drum-like, deformable capacitor, coupled to a microwave resonator with resonance frequencies of Ωm = 2π ×10.69 MHz and

ω0 = 2π×7.5 GHz respectively. The microwave resonator feature a total decay rate

106 6. Cavity opto-mechanics

As a result of the low loss rate, compared to the mechanical resonance frequency, normal mode splitting can be very well resolved in the system.

On the other hand we will show in the following section that the very same feature limits the cooling performance of the device.

6.3. Dynamical backaction cooling

In this section we use the approximate eigenfrequencies to perform contour inte- gration on the normal ordered mechanical spectrum in order to obtain the final occupancy of the mechanical oscillator

nf = hˆa†m(τ)ˆam(0)i

τ=0 . (6.6)

In this treatment we take both the thermal and the vacuum noise of the driving resonator into account. A finite value for ¯np may be relevant for electromechanical

systems, considering that 1 GHz ˆ=50 mK [187,191].

Within our approximation scheme we can introduce a formal parameter that tags the CRT and expand nf in its powers. To zeroth order the poles are determined

by the approximate eigenfrequencies ω(0)± ,−ω (0)∗

± given in Equation 6.4, and it is

straightforward to evaluate n(0)_f (including Γm). To second order we use instead

the poles ω_±(0) +ω_±(2),−ω_±(0)∗ −ω(2)∗_± . Subsequently, n(2)_f is expanded in the small parameters gm/Ωm, κ/Ωm, and |δ|/Ωm up to second order with Γm →0. Both n

(0)

f

andn(2)_f do not contain terms linear inδ, allowing one to directly minimize the result with respect toδ by setting δ →0. This yields

n(0)_f = ¯nm Γm κ g2 m+κ2 g2 m+ Γmκ + g 2 m g2 m+ Γmκ ¯ np, (6.7) n(2)_f = ¯nm Γm κ g2 m 4Ω2 m + ¯ np+ 1 2 κ2 _{+ 2g}2 m 8Ω2 m . (6.8)

The final occupancy nf = n

(0)

f +n

(2)

f consists of three contributions. One is pro-

portional to the occupancy of the thermal bath ¯nm and displays linear cooling for

Γm gm κ, i.e., nf ≈ _gΓ2m

m/κn¯m. Whengm approaches κ, deviations from the lin-

ear cooling regime become apparent. Indeed, the final occupancy is always limited by nf & n¯mΓ_κm, which implies that the largest temperature reduction is bound by

the cavity decay rateκ 1. This is equivalent to the condition

Qm >n¯m

Ωm

κ

for ground state cooling. It is noted that operation in the deeply RSB regime is advantageous to avoid photon-induced heating [194], entailing that the condition on the mechanicalQm is therefore more stringent.

1_{Note thatn}(0)

f follows from the classical rate equations for two resonant oscillators (frequency

6.3 Dynamical backaction cooling 107 0.01 0.1 1 10 100 1.0 0.5 5.0 0.1 10.0 0.01 0.1 1 10 100 1.0 0.5 5.0 0.1 10.0 _n p=5 0.2 np= 1 np= 0 np= np=5 0.2 np= 1 np= 0 np= limiting occupancy nf Doppler limit RSB RSB Doppler limit nm= 1000 nm= 100

Figure 6.4.: Dynamical backaction cooling of a mechanical mode. The final occupancy of a dynamically cooled mechanical mode is plotted for different parameter settings. On the left hand side the initial thermal occupancy ofnm = 100 is in principle

sufficient to cool the mechanics below one phonon to its quantum ground state. Here a finite “temperature” of the drive limitsnf. For a quantum limited drive, i.e.,np= 0,

the minimum is defined by the so called quantum limitκ2/16Ω2_m. On the right hand side a higher initial temperature of the mechanics shows a different behavior. Here it is the cavity decay rate κ that sets the cooling limit via nf ≥ nmΓm/κ. This

is exemplified by the rise of nf towards smaller values of κ/Ωm, i.e., deeper in the

RSB regime. It is interesting to note, that it any case there exists an optimal cavity

bandwidthκ that allows for maximum cooling.

A second contribution is proportional to the finite occupancy of the driving circuit (¯np) and corresponds to heating from thermal noise in its input. It implies that it is

impossible to cool below the equilibrium occupation of the resonator. If we assume that the mechanical and electromagnetic baths are at the same temperatureTm, it

entails nf ≥ n¯mΩ_ωm_p. Last, there is a term in n

(2)

f that is temperature-independent

and corresponds to heating from quantum backaction noise. This term determines the quantum limit to the final occupancy and agrees with references [199, 198]. Interestingly, in the present analysis the quantum limit arises from the CRT. We note that the trade-off between the quantum limit and the cavity bandwidth limitation leads to an optimal value for κ. Consistent results are obtained with a covariance matrix approach [1].

Finally, we consider appreciable cooling (nf ¯nm so that we can take Γm → 0 in

the denominator of Equation6.8) and optimizen(0)_f +n(2)_f with respect togm, which

yields nopt ≈n¯m Γm κ + ¯np + κ2 16Ω2 m + s ¯ nmΓmκ(¯np+1/2) Ω2 m (6.9) at the coupling rate

gopt = 4

q

4¯nmΓmκΩ2m/[¯np+1/2+ ¯nmΓm/κ]).

rates Γmandκ(Γmκ), and coupled via heat diffusion with a rateg2m/κ. In this picture, the

deviation from linear cooling corresponds to heat diffusion from the cavity to the mechanical oscillator.

108 6. Cavity opto-mechanics

In the ground state cooling regime, the first three terms of Equation 6.9 always give the correct order of magnitude. Thus, a comparison ofgopt with the condition

gm > κ/2 implies that optimal ground state cooling leads to NMS only when the

thermal noise (cf. first term in Equation6.9) is at least comparable to the quantum backaction noise (cf. third term in Equation6.9).