In the absence of an external trapping potential

In the absence of an external trapping potential, the dipole potential (4.16) is the only source for a trapping of the atoms. The atomic cloud will then have an equilibrium position that is given by the minimum of the dipole potential. We assume the atoms to be “high-field seekers”

(ωL−ωa,res < 0), i.e. they are attracted to regions of high intensity. As a consequence, the equilibrium position hxˆ_ai= ¯x_a lies at a local maximum of the electromagnetic field intensity, i.e.

k(¯x_a−x¯_M−L) = (2n+ 1)·π/2, n Z.Here we already introduced the equilibrium positions of the cantilever (¯xM) and of the atomic CM position (¯xa). The equilibrium position of the cavity field (ˆc) is denoted by¯c.We can now shift the operatorsxˆ_a,xˆ_M andˆcby their respective steady state solutions:

ˆc= ¯c+δˆc xˆM = ¯xM +δxˆM =

q

~/2mωM(¯b+δˆb) +h.c.= ¯xM +x_ZPF(δˆb^†+δˆb)

xˆa = ¯xa+δxˆa (4.17)

Accordingly, the dipole interaction term the dipole interaction term reduces to Vˆdip=U0Nsin²(π

2 +kδxˆa−kδxˆM)(¯c+δˆc)(¯c^∗+δˆc^†). (4.18)

ωa CM oscillation frequency of atomic cloud

10⁵Hz

|¯c|² average photon number

in the cavity

100

g²₀

∆ca = ^U0

~ atom-cavity coupling −10³Hz

ωdip=^p2U0k²|¯c|²/ma strength of the dipole potential

3·10⁴Hz

N number of atoms 10⁵

m_a mass of a single atom 10⁻²⁵kg

x^(N)_a,0 =^p~/2N m_aω_a zero-point width of the CM oscillation

10⁻¹⁰m

ωM frequency of the

cantilever

10⁵Hz

mM mass of the cantilever 10⁻¹⁰kg

x_ZPF=^p~/2m_Mω_M zero-point width of the cantilever

2·10⁻¹⁵m

Γ_M damping rate of the

cantilever motion

1 Hz = 10⁻⁵ω_M

ω_cav cavity resonance

frequency

10¹⁵Hz

L cavity length 10³λ_L≈10⁻³m

g_M =−^ω_L^cavx_ZPF cantilever-cavity coupling 10³Hz

κ cavity decay rate 10⁶Hz

Table 4.1: Assumed parameters for the model of figure (4.3): The estimates on the mechanical properties of the atomic cloud are based on the experiments in Berkeley [32] and Zürich [33]. The assumed properties of the cantilever are in the range of those reached in basic optomechanical setups [10, 11, 12]. The discussion of these parameters can be found in section 4.4, only the definition of ω_dip, the strength of the dipole potential, is given in section 4.5. Note that the estimate of the center of mass oscillation frequency ωa refers to the case, when an additional trapping potentialVˆtrap is present.

4.5. COUPLING CONSTANTS OF THE LINEARIZED MODEL 45 In the linearized model, the motional excursions of both the cantilever and the atomic cloud are assumed to be small, i.e. k^qhδxˆ²_Mi 1 and k^phδˆx²_ai 1. Under these two assumptions, we can approximate the dipole potential (4.18) around a minimum and arrive at

Vˆ_dip'U0N(1−(kδxˆa−kδxˆM)²)(¯c+δˆc)(¯c^∗+δˆc^†). (4.19) Let us now discuss the contributions of this expression step by step:

The term quadratic in δxˆ²_a takes account of the harmonic confinement of the atoms in the dipole potential of the cavity field:

Vˆ_dip = N m_a

2 ω²_dipδxˆ²_a, (4.20)

where the trapping frequency of the dipole potential is given by

ω_dip² = 2|U₀|k²|¯c|²/m_a. (4.21) The validity of the harmonic approximation of the dipole potential is based on the tight confinement of the atomic cloud around its equilibrium position, i.e. k^phδxˆ²_ai 1. This regime is referred to as the Lamb-Dicke regime in atomic physics. As the cloud of atoms is only subject to the harmonic potential of equation (4.20), its CM oscillation frequencyωa is given by the frequency of the dipole potential, ω_dip. We can attribute a zero-point amplitude to the CM motion of the atomic cloud. x^(N_a,0⁾=^p~/2N maωa.The position operator of the “super-atom” can be expressed in terms of the annihilation and creation operators,ˆaandˆa^†,as xˆa=x^(N)_a,0 (ˆa^†+ ˆa).

For the parameters of the estimate given above (see table 4.1), the frequency of the dipole potential has a value of ω_dip ≈ 3·10⁴Hz. Note that the dipole frequency is a useful quantity, even in the presence of an additional external potential. In that case it does no longer determine the CM oscillation frequencyωa of the atomic cloud. Still the coupling constants will depend on the ratioωa/ω_dip,as we will see below.

Next we examine the the term quadratic inδxˆ²_M,

−U₀N k²|¯c|²δxˆ²_M =: ~δω_M,c

x²_ZPF δxˆ²_M, (4.22)

It yields a shift of the cantilever frequency due to the presence of the atoms and additionally produces terms of the formδˆb²+h.c., i.e. terms that involve the annihilation (creation) of two cantilever phonons at the same time.

The coupling between the atomic cloud and the cantilever is taken account of by the term U₀N k²|¯c|²δxˆ_aδxˆ_M =:~g_a,M(δˆa^†+δˆa)(δˆb^†+δˆb), (4.23) where we introduced g_a,M = N k²|¯c|²x^(N_a,0⁾x_ZPFU₀/~. We will refer to this term as the direct coupling between the cantilever and the atomic motion. Note that the coupling constant is proportional to the square root of the number of atoms, as x^(N)_a,0 ∝ 1/√

N . This is exactly the direct coupling between the cantilever and the atomic motion that we already discussed above.

Finally, the termU₀N δˆcδˆc^†corresponds to a shift of the cavity resonance due to the presence of the atoms at the antinodes. Note that it would be absent for if the atoms were “low-field seekers” .

All terms of higher than quadratic order in δxˆa, δxˆM, δˆc and δˆc^†, are neglected in this approach. Therefore no coupling between the atomic cloud and the cavity field appears in this linearized system.

We can complete the linearization procedure by inserting the operators in the form of equations (4.17) into all parts of the Hamiltonian (4.14) and arrive at

Hˆ = −~∆δˆc^†δcˆ

+ ~ω_Mδˆb^†δˆb+~ωaδˆa^†δˆa + ~δωM,c(2δˆb^†δˆb+δˆb²+δˆb^†2) + ~g_a,M(δaˆ+δˆa^†)δˆb+h.c.

+ ~g_M¯c^∗(δˆb+δˆb^†)δˆc+h.c.. (4.24) The first two lines of this expression comprise the basic contributions of the harmonic oscil-lators, i.e. the driven cavity, the cantilever and the atomic CM motion. The frequency shift of the cantilever and the two-phonon processes are contained in the third line. The last two lines finally present the direct coupling between the atomic motion and the cantilever, and the basic optomechanical coupling of the cantilever and the coupling via radiation pressure.

Note that we rescaled the detuning parameter ∆ → ∆− _∆^g20

caN −g_Mx¯_M/x_ZPF in order to include the static shift of the cavity resonance due to the presence of the atoms and the equilibrium position of the cantilever.

The definitions of the parameters are given in table (4.2) and the scheme of figure (4.4) illustrates the coupling mechanisms of this Hamiltonian.

To assess the strength of the relevant constants in this model, we turn towards the as-sumed values of table (4.1) again. The direct coupling between atoms and cantilever, ga,M, can be rewritten as g_a,M = (ω_a/2)^pN m_aω_a/m_Mω_M. Its strength is therefore limited by the ratio between the mass of the atomic cloud, N ma, and the mass of the cantilever mM.We see that the ratio of the direct coupling term and the atomic CM oscillation frequency is given by g_a,M/ω_a ≈^pN m_aω_a/m_Mω_M ≈10⁻⁵ for realistic parameters. This shows that the huge differ-ence in mass impedes a strong direct coupling between the between the atomic cloud (∼10⁻²⁰kg)

ωa=^p2|U₀|k²|¯c|²/ma 3·10⁴Hz

g_M =−^ωcav_L x_ZPF 10³Hz

ga,M = ^ω₂^{aqN m}_m ^aωa

MωM 10⁻¹Hz

∆_N =N_∆^g20

ca 10⁷Hz

δωM,c=−N k²x²_ZPF|¯c|²U0/~ 10⁻⁷Hz

Table 4.2: Definition of the coupling constants and frequency shifts that appear in the Hamiltonian (4.24): The CM oscillation frequencyω_a,the coupling between the cantilever and the cavity via radiation pressuregM, the direct coupling between the cantilever and the atomic motionga,M,and finally the frequency shifts of the both the cavity resonance (∆N) and of the cantilever (δωM,c) due to the presence of the atoms. The estimates on the right-hand side are based on the values of table (4.1).

4.5. COUPLING CONSTANTS OF THE LINEARIZED MODEL 47

a b

atoms atoms

cavity cavity

cantilever cantilever

Figure 4.4: Scheme of the coupling mechanisms in a linearized version of the proposed setup (4.3).

(a) If the atoms are located around the local minimum of the dipole potential Vˆ_dip, no coupling between the atoms’ collective mode (ˆa) and the cavity field (ˆc) appears in the Hamiltonian (4.24).

Nevertheless, the atomic CM position is linearly coupled to the position of the cantilever (ˆb) with a coupling strength given byga,M.The cantilever itself is subject to both the harmonic force due to its suspension (V_h.o.) and the radiation pressure∝ˆc^†c.ˆ The latter results in a cavity-cantilever coupling constantg_M =−^ω_L^cavx_ZPF.(b) In the presence of an additional trapping potential (V_trap), the atoms’ equilibrium position is shifted. Subsequently, a coupling term between the atoms and the cavity field appears in the Hamiltonian (4.28). The cavity-cantilever coupling gets an additional contributiong_M,c due to the presence of the atoms.

and the cantilever (∼ 10⁻¹⁰kg). The coupling constant g_a,M has a value that is about a tenth of the cantilever dampingΓ_M rate that we assumed in table 4.1.

The shift of the cavity frequency due to the atoms, ∆_N ≈ N_∆^g2 To get a rough estimate we assume that the atoms form a homogeneous medium and spread over a volume of V ≈ w²₀xa,0, where w0 is the waist of the Gaussian beam and xa,0 denotes influence of the atoms on the shape of the cavity field is small. This proves the assumption of section (4.2) right where we used the intensity profile of the empty cavity to determine the spatial structure of the cavity-atom coupling ^g_∆^20(x)

ca .

The shift of the cantilever frequency due to its coupling to the atoms is vanishingly small: δω_M,c≈ 10⁻⁷Hz for the chosen parameters. However, there appears another, stronger frequency shift, δωM = g_M² |¯c|²(_ω ¹

M+∆ −_ω ¹

M−∆), that stems from the coupling between the cantilever and the cavity. This contribution is commonly referred to as the optical spring and we will derive the expression forδω_M via second order perturbation theory in section (4.6).

We have seen that in the linearized system without an additional trapping, the atomic motion couples only to the cantilever. This coupling, denoted by g_a,M, is rather weak, though. The coupling between the atomic motion and the cavity is of higher order and does not appear in this approach. However, this type of coupling might be exploited to measure the Fock states of the

atomic CM mode, which we will discuss in section (4.9).

Im Dokument Optomechanics in the Quantum Regime (Seite 49-54)