Controlling the motion of an atom in an optical cavity

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28 3.3. Connection to measurable quantities and ˜σ − j |Ψ〉 =ãi |Ψ〉 =0. (3.16) Inserting this into Eq. (3.13) yields a † iσ − j = 〈ai〉 ∗ σ − j . (3.17) The same proof can be used for any normally ordered operator product of atomic and cavity raising and lowering operators. Thus, in the harmonic-oscillator model, the expectation value of such a product is the product of expectation values of the single raising and lowering operators. For products which are not normally ordered, the operator commutator relations can be used to order the product normally. In the harmonic-oscillator model, the commutators at equal time are Yi,Y † j = δij, [Yi,Yj] = Y † i ,Y † j =0. (3.18) Thus, the steady state of any combination of raising and lowering operators can be calculated, with the restrictions mentioned when introducing the harmonic-oscillator model before Eq. (3.5). 3.3 Connection to measurable quantities In this section, the relation between some quantities which can be measured experimentally and the parameters and operators in the quantum mechanical model are investigated. The equations in this section are also valid outside the model of coupled harmonic oscillators. The power transmitted through the cavity can be deduced from the expectation values of the photon number operators for the modes, a † iai. The flux of energy out of all cavity modes which is caused by the non-perfectly reflecting mirrors is given by Nc Ptot =2¯hωl i=1 κi a † iai . (3.19) For a Fabry-Perot type cavity consisting of two identical mirrors, the energy flux through the back mirror is half of Ptot. Of the photons associated with this flux, only T / (T + L) are transmitted through the mirror, where T is the power transmittance of the mirrors, and L are the losses. The other photons are lost. Thus, the transmitted power is Nc T Ptrans =¯hωl κi a T + L i=1 † iai . (3.20) The power of the incident pump light for mode i, Pin,i, is related to the pump parameter of mode i, ηi, by T + L |ηi| Pin,i =¯hωl T 2 . (3.21) 28 κi
3. Quantum description 29 Here, perfect mode matching between the pump beam and the mode is assumed. For non-perfect mode matching, the left hand side of the equation has to be multiplied by the fraction of the pump light which has perfect overlap with the mode, see also the discussion near Eq. (4.2). Equation (3.21) can be deduced by comparing the steady state transmission of an empty cavity in the classical model, see Eq. (2.30) for Neff =0,andin the quantum model, see Eq. (3.38) for geff = 0 and Eq. (3.20). In some cases, especially if more than one mode is interacting with the atom, it is interesting to know the intensity in the cavity as a function of position. For example, the intensity and the relative phases of the light field in the different modes can be measured by observing the interference pattern of the modes. This is nothing else but the spatial dependence of the intensity of the total light field. Also, the intensity distribution of the transmitted light is closely connected to the intensity distribution in the cavity: If the transmittance of the outcoupling mirror is uniform over the mirror, the intensity pattern just after the mirror is the same as the intensity pattern in the cavity just before the outcoupling mirror. The electric field operator for the cavity light field is given by (MS99, chapter 13) where E (r,t)=E + (r,t)+E − (r,t) , (3.22) E + Nc ¯hωl (r,t)= ai (t) ψi (r) e 2ɛ0V −iωt , (3.23) i=1 E − = E +∗ , (3.24) and V is the mode volume, see Eq. (2.22). The intensity operator of the light field inside the cavity, I (r), is given by (Car93, Eq. (5.37)) : I (r,t)=¯h ωlc V Nc i,j=1 ψ ∗ i (r) ψj (r) a † i (t) aj (t) , (3.25) As was mentioned in the introduction, the atoms are subject to light forces. The position of the atoms determines the coupling constants, which in turn determine the steady state of the system. Thus, the light forces can have a large influence on the time evolution of the system. The operator describing the cavity light force acting on atom j is given by (HGHR98, Eq. (30)) Nc Fcav,j = −¯h i=1 (∇jgij) a † iσ − j + ∇jg ∗ ij σ + j ai . (3.26) Here, ∇j denotes the gradient with respect to the position of the jth atom, rj. Notethat Fcav,j contains both radiation pressure and dipole force. 29
Page 1: Technische Universität München Ma
Page 5 and 6: Contents 1 Introduction 1 2 Classic
Page 7 and 8: Bibliography 127 Publications 133 D
Page 9 and 10: Chapter 1 Introduction In the field
Page 11 and 12: 1. Introduction 3 microwave photons
Page 13 and 14: 1. Introduction 5 cavity is extende
Page 15 and 16: Chapter 2 Classical description An
Page 17 and 18: 2. Classical description 9 a phase
Page 19 and 20: 2. Classical description 11 corresp
Page 21 and 22: 2. Classical description 13 The ψn
Page 23 and 24: 2. Classical description 15 Here,
Page 25 and 26: 2. Classical description 17 This sh
Page 27 and 28: y [w0] 4 2 0 -2 a -4 -4 -2 0 x [w0]
Page 29 and 30: Chapter 3 Quantum description In th
Page 31 and 32: η2 η1 η2 vj γj rj 3. Quantum de
Page 33 and 34: np 3 2 1 0 … √ √ 3η 3κ √
Page 35: and ⎛ ⎜ Inoise = ⎜ ⎝ Fmode,
Page 39 and 40: ∆c,1/(2π)[MHz] and 20 0 -20 -40
Page 41 and 42: ∆c/(2π)[MHz] 30 20 10 0 -10 -20
Page 43 and 44: 3. Quantum description 35 non-linea
Page 45 and 46: = 3. Quantum description 37 Nc |ψ
Page 47 and 48: 3. Quantum description 39 All the a
Page 49 and 50: m 〈Yi〉 ∗ 0 〈Yl〉 ∞ 0 δ
Page 51 and 52: elements of the diffusion matrix: 3
Page 53 and 54: 3. Quantum description 45 between E
Page 55 and 56: ∆c/(2π)[MHz] 30 20 10 0 -10 -20
Page 57 and 58: The potential is given by σ + j
Page 59 and 60: 3. Quantum description 51 vector is
Page 61 and 62: Chapter 4 Measurement of the atomic
Page 63 and 64: 4. Measurement of the atomic positi
Page 67 and 68: g eff [g0] 1 0.8 0.6 0.4 0.2 0 4. M
Page 75 and 76: Chapter 5 Experimental set-up In th
Page 77 and 78: laser 25 cm weak Rb cloud cavity La
Page 79 and 80: 5. Experimental set-up 71 requires
Page 81 and 82: 5. Experimental set-up 73 the AOM a
Page 83 and 84: 5. Experimental set-up 75 Hermite-G
Page 85 and 86: 5. Experimental set-up 77 cooperati
Page 87 and 88:
Reflection Grating Single- Photon-
Page 89 and 90:
modes can be built in the near futu
Page 91 and 92:
Chapter 6 Control of the atomic mot
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6. Control of the atomic motion 85
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cavity transmission [pW] 1 0 4 3 2
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Amplitude [db] 40 35 30 25 20 15 10
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Transmittance Transmittance 0.7 0.6
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Chapter 7 Conclusion and outlook In
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Appendix A Algorithm to determine t
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A. Algorithm to determine the exit
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Bibliography [ACDF94] Kyungwon An,
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Bibliography 129 [GR99] Markus Gang
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Bibliography 131 [PFM + 00] P.W.H.
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Publications P. Münstermann, T. Fi
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Danksagung Mein Dank gilt zuallerer
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Controlling the motion of an atom in an optical cavity

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