Damped oscillation in quantum optics
A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity.[1][2][3] Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.
Mathematical treatment
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A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is
![{\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{0}{\frac {{\hat {\sigma }}_{z}}{2}}+\hbar g\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dda04f5f7d9fc969297382f0fc0f4cbebe827516)
where
is the Pauli z spin operator for the two eigenstates
and
of the isolated two level system separated in energy by
;
and
are the raising and lowering operators of the two level system;
and
are the creation and annihilation operators for photons of energy
in the cavity mode; and
![{\displaystyle g={\frac {\mathbf {d} \cdot {\hat {\mathcal {E}}}}{\hbar }}{\sqrt {\frac {\hbar \omega }{2\epsilon _{0}V}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/346fd8986cd78eaf1470e3e397445fc31b27d1df)
is the strength of the coupling between the dipole moment
of the two level system and the cavity mode with volume
and electric field polarized along
.
[4]
The energy eigenvalues and eigenstates for this model are
![{\displaystyle E_{\pm }(n)=\hbar \omega \left(n+{\frac {1}{2}}\right)\pm {\frac {\hbar }{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}=\hbar \omega _{n}^{\pm }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5795a533c51eb24401f4be039f61367659aabdb0)
![{\displaystyle |n,+\rangle =\cos \left(\theta _{n}\right)|g,n+1\rangle +\sin \left(\theta _{n}\right)|e,n\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/09cfef3a1e6b7416038070b38716d9a1b9c7cbf5)
![{\displaystyle |n,-\rangle =\sin \left(\theta _{n}\right)|g,n+1\rangle -\cos \left(\theta _{n}\right)|e,n\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c284e08a77b227a5dcdf15123299a9a304593648)
where
is the detuning, and the angle
is defined as
![{\displaystyle \theta _{n}=\tan ^{-1}\left({\frac {g{\sqrt {n+1}}}{\delta }}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1803d6f32dc54f29bd759d76e039da967aa1755)
Given the eigenstates of the system, the time evolution operator can be written down in the form
![{\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }&=\sum _{|n,\pm \rangle }\sum _{|n',\pm \rangle }|n,\pm \rangle \langle n,\pm |e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|n',\pm \rangle \langle n',\pm |\\&=~e^{i(\omega -{\frac {\omega _{0}}{2}})t}|g,0\rangle \langle g,0|\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{+}t}(\cos {\theta _{n}}|g,n+1\rangle +\sin {\theta _{n}}|e,n\rangle )(\cos {\theta _{n}}\langle g,n+1|+\sin {\theta _{n}}\langle e,n|)}\\&~~~+\sum _{n=0}^{\infty }{e^{-i\omega _{n}^{-}t}(-\sin {\theta _{n}}|g,n+1\rangle +\cos {\theta _{n}}|e,n\rangle )(-\sin {\theta _{n}}\langle g,n+1|+\cos {\theta _{n}}\langle e,n|)}\\\end{aligned}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0b3ab35af4284d12ad27b89c14cdb24b061b8d)
If the system starts in the state
, where the atom is in the ground state of the two level system and there are
photons in the cavity mode, the application of the time evolution operator yields
![{\displaystyle {\begin{aligned}e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle &=(e^{-i\omega _{n}^{+}t}(\cos ^{2}{(\theta _{n})}|g,n+1\rangle +\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )+e^{-i\omega _{n}^{-}t}(-\sin ^{2}{(\theta _{n})}|g,n+1\rangle -\sin {\theta _{n}}\cos {\theta _{n}}|e,n\rangle )\\&=(e^{-i\omega _{n}^{+}t}+e^{-i\omega _{n}^{-}t})\cos {(2\theta _{n})}|g,n+1\rangle +(e^{-i\omega _{n}^{+}t}-e^{-i\omega _{n}^{-}t})\sin {(2\theta _{n})}|e,n\rangle \\&=e^{-i\omega _{c}(n+{\frac {1}{2}})}{\Biggr [}\cos {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {\delta ^{2}-4g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|g,n+1\rangle +\sin {{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}|e,n\rangle {\Biggr ]}\end{aligned}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/386e28ca1964b25e14eee685b5658a768c1d7ef7)
The probability that the two level system is in the excited state
as a function of time
is then
![{\displaystyle {\begin{aligned}P_{e}(t)&=|\langle e,n|e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|g,n+1\rangle |^{2}\\&=\sin ^{2}{{\biggr (}{\frac {t}{2}}{\sqrt {4g^{2}(n+1)+\delta ^{2}}}{\biggr )}}{\biggr [}{\frac {8\delta ^{2}g^{2}(n+1)}{\delta ^{2}+4g^{2}(n+1)}}{\biggr ]}\\&={\frac {4g^{2}(n+1)}{\Omega _{n}^{2}}}\sin ^{2}{{\bigr (}{\frac {\Omega _{n}t}{2}}{\bigr )}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6bbb7fe204ec4ad0b225b1e64d4fa7b006a963e)
where
is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number
is zero, the Rabi frequency becomes
. Then, the probability that the two level system goes from its ground state to its excited state as a function of time
is
![{\displaystyle P_{e}(t)={\frac {4g^{2}}{\Omega _{0}^{2}}}\sin ^{2}{{\bigr (}{\frac {\Omega _{0}t}{2}}{\bigr )}.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/846fa7e35ab60582268dac83a7dc8aea34af6f51)
For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning
vanishes, and
becomes a squared sinusoid with unit amplitude and period
Generalization to N atoms
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The situation in which
two level systems are present in a single-mode cavity is described by the Tavis–Cummings model
[5]
, which has Hamiltonian
![{\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\sum _{j=1}^{N}{\hbar \omega _{0}{\frac {{\hat {\sigma }}_{j}^{z}}{2}}+\hbar g_{j}\left({\hat {a}}{\hat {\sigma }}_{j}^{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{j}^{-}\right)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffbac6f858d0300131ac29e6324a0906b6acd26)
Under the assumption that all two level systems have equal individual coupling strength
to the field, the ensemble as a whole will have enhanced coupling strength
. As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of
.[6]
References and notes
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