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CHAPTER 2. CAVITY QUANTUM ELECTRODYNAMICS 43 ω r e ωr χ χ κ ω r g g ωr κ ∆ ω0 a 1 ωa 2 ωa 3 ωa 2χ + δ Figure 2.5: Spectrum of coupled cavity and atom system in strong dispersive limit. (left) Cavity spectrum when atom is in ground (blue), excited (red) state or not present (dashed black). (right) Atom spectrum when cavity has different (exact) photon numbers. measuring the state of a single atom using the cavity, one has the luxury of using many photons to compensate for a small dispersive shift χ < κ, even in the weak dispersive limit (pink region in Fig. 2.4). Because there is only a single atom in the cavity, single photon precision requires dispersive strong coupling. Perhaps even more interesting is when the Stark shift per photon exceeds the atom decay (χ > γ). In this case the atom is shifted by more than a linewidth for each photon, only responding when driven at a given frequency when a specific number of photons are present in the cavity. By probing the response of the atom at different frequencies one can measure the exact photon number distribution (see section 8.3). The qubit can not only measure the photon number but be coherently controlled by it. If the qubit is made only to respond when a single photon is present this can be considered a conditional-not (CNOT) gate. If multiple atoms in the cavity (which might be far apart from each other) perform a CNOT on the same photon state they can be entangled without ever interacting directly. In the weak dispersive limit (red region in Fig. 2.4), the cavity, though its frequency became atom state-dependent, remained harmonic at photon numbers below ncrit. If the coupling becomes strong enough, even in the dispersive limit well below ncrit (where perturbation theory is still accurate) higher order terms in the expansion of equations 2.4 and 2.7 can become significant compared with the cavity linewidth. A new critical photon number can be defined for when the 4 th order shift γ ω4 a 5 ωa (nχ 2 /∆), changes the resonator frequency by more than a linewidth per photon. ω a
CHAPTER 2. CAVITY QUANTUM ELECTRODYNAMICS 44 nκ = κ∆ κ∆3 = χ2 g4 (2.13) While the wavefunction overlap between atom and cavity is small in absolute terms ((g/∆) ≪ 1), the coupling can be large enough (g ≫ κ) for the cavity to inherit a significant non-linearity. Note that the qubit remains mostly in the ground state. Depending on how the cavity is being used, this could be a blessing or a curse. If linearity is desired, then it is not sufficient to stay below ncrit, one must also keep the number of photons to be much less than nκ. However, an anharmonic oscillator can also be an extremely useful device. It can be used to squeeze light (in a less extreme way than in the resonant case), or as a hysteretic latching measurement of another atom [Siddiqi2006]. If the coupling is stronger still, it is even possible to have nκ < 1 (the orange line in Fig. 2.4). In this limit, the cavity becomes anharmonic on the single photon level, acting more like a many level atom than a harmonic oscillator. As such it can also be used as a “photonic” qubit which is discussed briefly in section 8.3.2. Cavity QED can be used to explore a wide variety interaction types and decoherence effects. All of these regimes have useful properties for quantum information. The resonant interaction between an atom and a single photon can be used to transport quantum information. The dispersive shifts can be used to measure the photon number or qubit state, while the radiative lifetime enhancement can be used to protect it from decoherence. In the strong dispersive regime, one can use the dispersive shifts to measure photons, creating and characterizing exotic states of light, which could be used for communication of quantum information or to be studied in their own right. It might even be possible to make fundamentally new quantum bits based on the non-linearities present in cavity QED. As the phase diagram in figure 2.4 and as chapter 3 will show, superconducting circuits can be made into an excellent CQED system, exhibiting a wealth of interesting physics.
Page 1 and 2: Abstract Circuit Quantum Electrodyn
Page 3 and 4: c○2007 by David Schuster. All rig
Page 5 and 6: friendship. All of my friends from
Page 7 and 8: CONTENTS 4 3.2.3 Split CPB . . . .
Page 9 and 10: CONTENTS 6 8.2.1 AC Stark Effect .
Page 11 and 12: List of Figures 1.1 Relaxation and
Page 13 and 14: LIST OF FIGURES 10 5.3 Resist Profi
Page 15 and 16: LIST OF FIGURES 12 B.1 Radiative de
Page 17 and 18: List of Symbols and Abbreviations
Page 19 and 20: List of Symbols and Abbreviations 1
Page 21 and 22: Chapter 1 Introduction Progress in
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Page 39 and 40: CHAPTER 2. CAVITY QUANTUM ELECTRODY
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CHAPTER 4. DECOHERENCE IN THE COOPE
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CHAPTER 5. DESIGN AND FABRICATION 1
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CHAPTER 6. MEASUREMENT SETUP 142 ω
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CHAPTER 6. MEASUREMENT SETUP 146 ac
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CHAPTER 6. MEASUREMENT SETUP 150 Q(
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Chapter 7 Characterization of CQED
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CHAPTER 7. CHARACTERIZATION OF CQED
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CHAPTER 8. CAVITY QED EXPERIMENTS W
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Chapter 9 Time Domain Measurements
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CHAPTER 9. TIME DOMAIN MEASUREMENTS
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Chapter 10 Future work The biggest
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CHAPTER 10. FUTURE WORK 222 possibl
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CHAPTER 10. FUTURE WORK 224 frequen
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CHAPTER 11. CONCLUSIONS 226 There i
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APPENDIX A. OPERATORS AND COMMUTATI
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APPENDIX B. DERIVATION OF DRESSED S
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APPENDIX B. DERIVATION OF DRESSED S
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Appendix D Recipes This Appendix wi
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APPENDIX D. RECIPES 236 PMMA develo
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APPENDIX D. RECIPES 238 Electron Be
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BIBLIOGRAPHY 240 [Bennett1984] C. H
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BIBLIOGRAPHY 242 [Clauser1974] John
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BIBLIOGRAPHY 244 frequency noise in
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BIBLIOGRAPHY 246 [Kuhn2002] Axel Ku
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BIBLIOGRAPHY 248 controlled by exte
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BIBLIOGRAPHY 250 [Rivest1978] R. L.
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BIBLIOGRAPHY 252 [Walls2006] D. F.
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