Controlling the motion of an atom in an optical cavity

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10 2.2. Classical calculation Phase shift per round trip Cavity Atom Total Transmission a Frequency c Frequency Phase shift per round trip Cavity Atom Total Transmission b Frequency d Frequency Figure 2.3: a), b) Phase shift per round trip of a light field in an empty cavity, phase shift induced by an atom, and resulting phase shift for the cavity with an atom inside as a function of the light frequency. c), d) The transmission of the cavity with an atom drawn against the light frequency. In a) and c), the resonance frequencies of atom and cavity are the same, whereas in b) and d), the resonance frequency of the atom is larger than that of the cavity. The peaks of the transmission are near to laser detunings where the total phase shift per round trip is zero, as indicated by the dotted black lines. here. 2.2 Classical calculation In this section, a Fabry-Perot type cavity is considered. It consists of two identical mirrors which are separated by a distance l, see Fig. 2.1. The mirrors have a power reflectance R, a transmittance T , and losses L, withR + T + L = 1. It is assumed that the mirror reflectance is close to one, and the cavity geometry is such that it sustains a stable mode (Sie86, chapter 19, Eq. (8)). On one side, the cavity is illuminated with monochromatic light characterized by the electric field amplitude Ein and wavelength λ, 10
2. Classical description 11 corresponding to the angular frequency ωl. There are atoms at positions rj inside the cavity, which together cause a small attenuation αatom and a phase shift δatom per round trip of the light. For a simple notation, the electric field, E, is written as a complex field here. The actual electric field is given by the real part of E times a polarisation vector. As a first step, the attenuation and the phase shift induced by the atoms is assumed to be known, and their effect on the cavity transmission will be determined. In a second step, the phase shift and attenuation caused by the atoms will be calculated. In steady state, the electric field behind the first mirror, E1, is the sum of the transmitted part of the incident light field, √ T Ein, and the intracavity field after one round trip. In this round trip, the light is reflected at two mirrors and interacts with the atoms twice. Hence, the amplitude is modified by the factor √ 2 R (1 − αatom), and the phase by the total phase shift, δtot = δcav + δatom, which is the sum of the phase shifts in one round trip caused by detuning from the cavity resonance and the phaseshift caused by the atoms. Thus, E1 = √ T Ein + R (1 − αatom) e −iδtot E1. (2.1) Now, the limit of small losses and phase shifts is applied, i.e it is assumed that 1 −R, αatom and δtot are small as compared to one. This is the case if the atoms have only a small influence on the light field per round trip, and if the frequency of the incident light field is close to a resonance of the empty cavity. Both conditions are well satisfied in the experiment. In this limit, the exponential function in Eq. (2.1) can be expanded linearly: Inserting this into Eq. (2.1) yields e −iδtot =1− iδtot. (2.2) E1 {1 − [1 − (1 −R)] (1 − αatom)(1− iδtot)} = √ T Ein. (2.3) Omitting products of the small quantities 1 −R, αatom and δtot, E1 is calculated to be √ T E1 = Ein. (2.4) (1 −R)+αatom + iδtot Neglecting the small attenuation and phase shift in half a round trip, the electrical field Eout which is transmitted through the cavity is given by √ T E1, thus Eout = T 1 1 −R 1+ αatom Ein. (2.5) δtot + i 1−R 1−R The first factor is a constant that depends only on mirror properties and gives the maximum transmittance for the electric field through the cavity. For L = 0, this factor equals unity, and all incoupled light can be transmitted. The second factor describes the influence of the atoms on the cavity transmission. The magnitude of this influence is given by 11
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: 2. Classical description 9 a phase
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 36: and ⎛ ⎜ Inoise = ⎜ ⎝ Fmode,
Page 37 and 38: 3. Quantum description 29 Here, per
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 65 and 66: 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
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4. Measurement of the atomic positi
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Chapter 5 Experimental set-up In th
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laser 25 cm weak Rb cloud cavity La
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5. Experimental set-up 71 requires
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5. Experimental set-up 73 the AOM a
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5. Experimental set-up 75 Hermite-G
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5. Experimental set-up 77 cooperati
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Reflection Grating Single- Photon-
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modes can be built in the near futu
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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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