Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

More documents

Recommendations

Info

3. Homodyne measurements on a Bose-Einstein condensateFigure 3.1: The homodyne detection scheme that monitors the tunnelling between two spatiallyseparated condensates. One part of the condensate is contained in an optical cavity, the light inwhich is well detuned from any atomic resonance. The output light from the cavity is detected bybalanced homodyne detection.Ω2nd CondensateLASERεχγ1st CondensatePhotodetectorway of describing how quantum correlations develop and evolve is in terms of one system‘measuring’ another is an accident of history, which, while providing an intuitive handlewith which to grasp quantum mechanics, has often lead to confusion about the objectivenature of the quantum world, particularly in popular accounts. The formulation of quantummechanics in terms of developing correlations subsumes the measurement processas one particular type of correlation-inducing coupling, namely the interaction betweena measurement apparatus and a sampled system. Thus while the system studied hereis explicitly a measurement model, it provides insight into symmetry breaking in othersituations as well, not necessarily so easily interpreted as measurement processes.To investigate this symmetry-breaking process, we will use the double-well model introducedin Ch. 2, and once again use the two-mode Hamiltonian (Eq. (2.49)):Ĥ 2 = ΩĴz +2κĴ 2 x. (3.1)61
3. Homodyne measurements on a Bose-Einstein condensate3.2 Homodyne detection schemeFigure 3.1 illustrates the system under investigation here. One of the wells of the doublewellpotential sits in an optical cavity.A coherent field strongly drives the cavity atthe cavity frequency. We will assume that on the timescale of tunnelling oscillations, thecavity is heavily damped. The cavity field thus relaxes to the steady state on a much fastertimescale than the condensate dynamics. This enables us to make an adiabatic eliminationof the cavity dynamics. We can assume that the cavity field is far off resonance from anydipole transitions in the atomic species, so that the effect of the atoms is then entirelydispersive. The presence of the condensate shifts the phase of the cavity field by an amountproportional to the number of atoms in the cavity at any particular time. If the numberof atoms in the cavity oscillates, so will the phase shift. Thus tunnelling of the condensatewill be manifest as a modulated phase shift in the optical field exiting the cavity.To detect this phase shift we consider a homodyne detection scheme.The opticalcavity resides in one arm of an interferometer that combines the exiting light with areference beam of the same frequency. The result falls on a photodetector, which recordsthe photocurrent. If there is a difference in atom number between the two condensates,then coherent tunnelling can occur and the homodyne current will be modulated at thetunnelling frequency.Assuming that the incoming light is detuned from any atomic resonance, the interactionHamiltonian is∫Ĥ I = −d 3 r ˆψ † (r)µg(r)â † â ˆψ(r), (3.2)where â, â † are the cavity field operators, g(r) is the intensity mode function and µ =g 2 d /4∆ is the coupling strength, with dipole coupling constant g d and optical detuning ∆.Averaging over the optical mode function gives the interaction energy in terms of thecondensate field operators c, c † :where χ is the interaction strength.Ĥ I = −χâ † âĉ † 1ĉ1= − N 2 χâ† â − χâ † âĴx, (3.3)If the optical mode has a beam waist w, and aGaussian profile g(r) =cos 2 (kz)exp(−(x 2 + y 2 )/w 2 ) then the interaction strength isχ =µ2 √ 2(r 0 /w) 2 +1 . (3.4)62
Page 1 and 2:
Open Quantum Dynamics of Mesoscopic
Page 3 and 4:
AcknowledgementsMy thanks must firs
Page 5 and 6:
AbstractThe properties of an atomic
Page 7 and 8:
CONTENTS4 Continuously monitored Bo
Page 9 and 10:
List of Figures2.1 Two-mode approxi
Page 11 and 12: LIST OF FIGURES7.5 Angular momentum
Page 14 and 15: LIST OF ABBREVIATIONS AND SYMBOLSI{
Page 17 and 18: 1. Condensation ‘without forces
Page 19 and 20: 1. Condensation ‘without forces
Page 21 and 22: Chapter 2Properties of an atomic Bo
Page 23 and 24: 2. Properties of an atomic Bose con
Page 64 and 65: 3. Homodyne measurements on a Bose-
Page 70: 3. Homodyne measurements on a Bose-
Page 79 and 80: Chapter 4Continuously monitored con
Page 81 and 82: 4. Continuously monitored Bose cond
Page 107 and 108: Chapter 5Weak force detection using
Page 109 and 110: 5. Weak force detection using a dou
Page 111 and 112:
5. Weak force detection using a dou
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Part IIQuantum evolution of a Bose
Page 125 and 126:
6. Quantum effects in optical fibre
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Chapter 7Quantum simulations of eva
Page 143 and 144:
7. Quantum simulations of evaporati
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
A. Stochastic calculus in outlineco
Page 173 and 174:
Appendix BQuasiprobability distribu
Page 175 and 176:
B. Quasiprobability distributions u
Page 177 and 178:
Bibliography[1] Agrawal, G. P. Nonl
Page 179 and 180:
BIBLIOGRAPHY[22] Carter,S.J.Quantum
Page 181 and 182:
BIBLIOGRAPHY[45] Drummond, P. D., C
Page 183 and 184:
BIBLIOGRAPHY[68] Gordon,D.,andSavag
Page 185 and 186:
BIBLIOGRAPHY[92] Jaksch,D.,Gardiner
Page 187 and 188:
BIBLIOGRAPHY[114] Marshall, R. J.,
Page 189 and 190:
BIBLIOGRAPHY[135] Naraschewski, M.,
Page 191 and 192:
BIBLIOGRAPHY[158] Shelby, R. M., Le
Page 193 and 194:
BIBLIOGRAPHY[179] Wiseman, H. M. Qu
Page 195:
. . . but this book is already too
show all

Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?