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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen Rand

Print publication date: 2010

Print ISBN-13: 9780199574872

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199574872.001.0001

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Appendix I Derivation of Effective Hamiltonians

Appendix I Derivation of Effective Hamiltonians

Source:
Lectures on Light
Publisher:
Oxford University Press

Appendix I

Derivation of Effective Hamiltonians

To treat the dynamics of fully‐quantized optical interactions consistently in a single reference frame, one can transform the unperturbed atom into the rotating frame of the light. In cavity QED this calls for transformation of the static atomic Hamiltonian into the rotating frame of the cavity mode.

Consider rotating the wavefunction of the atom into the frame of a cavity field at frequency Ωc. This is given by

ψ ( t ) = [ ψ 1 ( t ) exp ( i Ω c t / 2 ) ψ 2 ( t ) exp ( i Ω c t / 2 ) ] .
(I.1)

(p.295) According to Eq. (2.4.12) of Chapter 2, which describes the temporal evolution of ψ for a time‐independent Hamiltonian H 0, the components of ψ′ in (I.1) progress according to

| ψ 1 ( t ) > = C 1 ( t ) exp ( i ω 1 t ) | 1 >
(I.2)
| ψ 2 ( t ) > = C 2 ( t ) exp ( i ω 2 t ) | 2 >
(I.3)

Hence the first component changes to

| ψ 1 ( t ) > = C 1 ( t ) exp ( i Δ t / 2 ) | 1 > ,
(I.4)

where Δ ≡ ω 2ω 1 − Ωc. Here the zero of energy has been chosen such that (ω 1 + ω 2)/2 = 0. Similarly, the other component of the wavefunction transforms to

| ψ 2 ( t ) > = C 2 ( t ) exp ( i Δ t / 2 ) | 2 >
(I.5)

The frequency factor in the argument of the exponential functions in (I.4) and (I.5) can now be interpreted as new effective frequencies of the energy levels in the rotating frame. Hence we can write an effective Hamiltonian in this frame that is given by

H e f f = ħ 2 [ Δ 0 0 Δ ] = ħ 2 Δ σ z ,
(I.6)

where σz is one of the Pauli spin matrices.

It is left to the reader to verify that the use of Eq. (I.6) in the equation of motion for the density matrix gives the same result for the slowly‐varying amplitude ρ̃ 12 of the coherence as that in Eq. (7.7.18). This establishes the validity of calculations based on effective Hamiltonians such as that in (I.6). The derivation above also provides a general approach to deriving simplified Hamiltonians for fully quantized systems in a single reference frame. Another example of this approach was given in Chapter 7 where the effective Hamiltonian method was used to reduce the more serious level of complexity of coherent population transfer in a 2‐field interaction (Eq. (7.3.2)).