Question

5 The dispersive Jaynes-Cummings Hamiltonian: 20 points In class, we introduced the Jaynes-Cummings Hamiltonian ?_{JC} = -frac{hbaromega_{qb}}{2}hat{?}_z + hbaromega_0 left(hat{a}^daggerhat{a} + frac{1}{2} ight) + hbar g (hat{?}_+hat{a} + hat{?}_-hat{a}^dagger) (9) which describes the quantum interaction of a two-level system and a harmonic oscillator and is an important model in cavity and circuit QED. In the dispersive limit, (g ? |Delta|, Delta = omega_{qb} - omega_0), the interaction leads to a shift of the oscillator frequency that depends on the state of the qubit, allowing the measurement of the oscillator frequency to serve as a readout of the qubit state. This effect can be made explicit by applying the unitary transformation ? = exp left[ frac{g}{Delta} (hat{?}_+hat{a} - hat{?}_-hat{a}^dagger) ight] (10) to the Hamiltonian. a) Compute the transformed Hamiltonian ??_{JC}?^dagger using the Baker-Hausdorf lemma to 2nd order in g/Delta. Argue that the transformed Hamiltonian you calculate has the properties described above. b) Compute the energy spectrum of your transformed Hamiltonian and confirm that it agrees with the dispersive spectrum presented in class. Note: The minus sign in the first term of eqn. 7 determines which eigenvector of the matrix rep. of hat{?}_z is consider to be the ground state. The matrix reps. of hat{?}_- and hat{?}_+ must be consistent with this.

          5 The dispersive Jaynes-Cummings Hamiltonian: 20 points
In class, we introduced the Jaynes-Cummings Hamiltonian
?_{JC} = -frac{hbaromega_{qb}}{2}hat{?}_z + hbaromega_0 left(hat{a}^daggerhat{a} + frac{1}{2}
ight) + hbar g (hat{?}_+hat{a} + hat{?}_-hat{a}^dagger) (9)
which describes the quantum interaction of a two-level system and a harmonic oscillator and is an important model in cavity and circuit QED. In the dispersive limit, (g ? |Delta|, Delta = omega_{qb} - omega_0), the interaction leads to a shift of the oscillator frequency that depends on the state of the qubit, allowing the measurement of the oscillator frequency to serve as a readout of the qubit state. This effect can be made explicit by applying the unitary transformation
? = exp left[ frac{g}{Delta} (hat{?}_+hat{a} - hat{?}_-hat{a}^dagger) 
ight] (10)
to the Hamiltonian.
a) Compute the transformed Hamiltonian ??_{JC}?^dagger using the Baker-Hausdorf lemma to 2nd order in g/Delta. Argue that the transformed Hamiltonian you calculate has the properties described above.
b) Compute the energy spectrum of your transformed Hamiltonian and confirm that it agrees with the dispersive spectrum presented in class.
Note: The minus sign in the first term of eqn. 7 determines which eigenvector of the matrix rep. of hat{?}_z is consider to be the ground state. The matrix reps. of hat{?}_- and hat{?}_+ must be consistent with this.

$5 The dispersive Jaynes-Cummings Hamiltonian: 20 points In class, we introduced the Jaynes-Cummings Hamiltonian ?JC = -frachbaromegaqb2hat?z + hbaromega0 left(hata^daggerhata + frac12 ight) + hbar g (hat?+hata + hat?-hata^dagger) (9) which describes the quantum interaction of a two-level system and a harmonic oscillator and is an important model in cavity and circuit QED. In the dispersive limit, (g ? |Delta|, Delta = omegaqb - omega0), the interaction leads to a shift of the oscillator frequency that depends on the state of the qubit, allowing the measurement of the oscillator frequency to serve as a readout of the qubit state. This effect can be made explicit by applying the unitary transformation ? = exp left[ fracgDelta (hat?+hata - hat?-hata^dagger) ight] (10) to the Hamiltonian. a) Compute the transformed Hamiltonian ??JC?^dagger using the Baker-Hausdorf lemma to 2nd order in g/Delta. Argue that the transformed Hamiltonian you calculate has the properties described above. b) Compute the energy spectrum of your transformed Hamiltonian and confirm that it agrees with the dispersive spectrum presented in class. Note: The minus sign in the first term of eqn. 7 determines which eigenvector of the matrix rep. of hat?z is consider to be the ground state. The matrix reps. of hat?- and hat?+ must be consistent with this.$

Added by Sheila L.

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