Rotating wave approximation

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The English term rotating wave approximation (RWA, dt. Rotating wave approximation ) describes an approximation method of quantum optics . In this approximation, the influences of rapidly rotating terms in the Hamilton operator of a system are neglected. In this context, fast means fast compared to the lifetimes of atomic states. The rotating shaft approximation is used in numerous models, such as B. in the Jaynes-Cummings model , in equations of motion of the density matrix in optical pumping , to solve the Rabi problem or in magnetic resonance phenomena.

It is justified as long as the system is subject to only a comparatively weak disruption . In addition, the frequency of the light field must be close to the atomic resonance frequency or the detuning must be small compared to the atomic resonance frequency: ${\displaystyle \omega _{L}}$ ${\displaystyle \omega _{a}}$

${\displaystyle \Delta \omega :=\left|\omega _{a}-\omega _{L}\right|\ll \left|\omega _{a}+\omega _{L}\right|\approx 2\omega _{\mathrm {a} }}$

The name of the approximation comes from the transition into a reference system that rotates with the light frequency , in which the Bloch vector of the atom interacting with the light no longer precesses in the case of exact resonance . Then the influences of the rapidly rotating terms can be neglected. ${\displaystyle \omega _{L}}$

Derivation

This section deals with the interactions between an atom, which is considered to be a two-level system , and an electromagnetic field. Both the atom and the photon are described in the second quantization .

Hamiltonian without interaction

The Hamilton operator of the overall system contains a part that describes the atom and the photons individually and without interaction: ${\displaystyle H_{0}}$

${\displaystyle H_{0}=\hbar \omega _{a}\sigma ^{+}\sigma ^{-}+\hbar \omega _{L}a^{\dagger }a}$

Here is the energy difference between the ground state and the excited state of the atom. is the energy of the photon. and are the ascending and descending operators of the atom and and the bosonic creation and annihilation operators for photons. ${\displaystyle \hbar \omega _{a}}$ ${\displaystyle |g\rangle }$ ${\displaystyle |e\rangle }$ ${\displaystyle \hbar \omega _{L}}$ ${\displaystyle \sigma ^{+}=|e\rangle \langle g|}$ ${\displaystyle \sigma ^{-}=|g\rangle \langle e|}$ ${\displaystyle a^{\dagger }}$ ${\displaystyle a}$

Description of the interaction

In addition to , a Hamiltonian provides information about the interactions between photons and atoms. This is made up of the dipole operator and the E-field vector with the polarization . ${\displaystyle H_{0}}$ ${\displaystyle H_{int}}$ ${\displaystyle {\vec {d}}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {\epsilon }}}$

${\displaystyle {\vec {E}}={\vec {\epsilon }}\underbrace {\sqrt {\frac {\hbar \omega _{L}}{\epsilon _{0}V}}} _{E_{0}}\left(a^{\dagger }+a\right)}$

Assuming that the polarization is parallel to the dipole moment, the interaction Hamiltonian can be written as:

${\displaystyle {\begin{aligned}H_{int}&=-{\vec {d}}\cdot {\vec {E}}\\&=-dE_{0}\left(\sigma ^{+}+\sigma ^{-}\right)\cdot \left(a^{\dagger }+a\right)\\&=-dE_{0}\left(\sigma ^{+}a^{\dagger }+\sigma ^{+}a+\sigma ^{-}a^{\dagger }+\sigma ^{-}a\right)\end{aligned}}}$

Due to parity considerations , it was assumed here that and that is real-valued. ${\displaystyle \langle e|{\vec {d}}|e\rangle =0=\langle g|{\vec {d}}|g\rangle }$ ${\displaystyle d=\langle e|{\vec {d}}|g\rangle }$

The time evolution of a quantum mechanical operator is determined in the interaction picture by the unitary time evolution operator : ${\displaystyle A}$ ${\displaystyle U}$

${\displaystyle A(t)=U^{\dagger }A(t=0)U}$

With the Baker-Campbell-Hausdorff formula the following time development of the ascending, relegating and creation and annihilation operators can be shown:

${\displaystyle {\begin{aligned}a(t)&=a(0)e^{-i\omega _{L}t}\\a^{\dagger }(t)&=a^{\dagger }(0)e^{+i\omega _{L}t}\\\sigma ^{-}(t)&=\sigma ^{-}(0)e^{-i\omega _{a}t}\\\sigma ^{+}(t)&=\sigma ^{+}(0)e^{+i\omega _{a}t}\end{aligned}}}$

These time-dependent operators are inserted into the above equation for the interaction Hamiltonian (the zeros in brackets are no longer explicitly written for the sake of clarity).

${\displaystyle H_{int}\approx -dE\left(\sigma ^{+}a^{\dagger }e^{i\left(\omega _{a}+\omega _{L}\right)t}+\sigma ^{+}ae^{i\left(\omega _{a}-\omega _{L}\right)t}+\sigma ^{-}a^{\dagger }e^{i\left(-\omega _{a}+\omega _{L}\right)t}+\sigma ^{-}ae^{-i\left(\omega _{a}+\omega _{L}\right)t}\right)}$

An approximate sign ( ) is used in the last formula because the exponents from the time evolution of the individual operators cannot generally simply be added without interaction. The exact solution can be derived from the Baker-Campbell-Hausdorff formula mentioned above. This first approximation is only valid for a weak coupling ( disturbance ) between atom and electromagnetic field. Effects such as possible degeneracy of the levels, which could be canceled out by a strong field, are neglected. ${\displaystyle \approx }$

The strength of the coupling can be expressed with a coupling constant that must be significantly smaller than the frequency of the electromagnetic field so that the approximation remains meaningful. ${\displaystyle g}$ ${\displaystyle \omega _{L}}$

${\displaystyle g=-{\frac {\left|{\vec {d}}{\vec {E}}\right|}{\hbar }}\ll \omega _{L}}$

Performing the Rotating Wave Approximation

The rotating wave approximation consists in neglecting the rapidly oscillating terms with in the exponent of the function. It is argued that the rapidly oscillating terms average themselves away comparatively quickly , so that they are of no importance on the time scales of the relevant processes such as atomic transitions or the decay of states. In the last equation of the previous section, the neglected terms are the first and last summand, which correspond to an excitation of the atom with simultaneous generation of a photon or the relaxation of the atom into the ground state with simultaneous absorption of a photon. These processes only play a role on very short time scales. ${\displaystyle \left(\omega _{a}+\omega _{L}\right)}$ ${\displaystyle e}$ ${\displaystyle 0}$

If the rapidly rotating terms are included in a more precise calculation, corrections are obtained that explain the Bloch-Siegert effect , for example .

Individual evidence

1. ^ Mark Fox: Quantum Optics - An Introduction . 1st edition. Oxford University Press, New York 2006, ISBN 978-0-19-856672-4 , pp. 189 .
2. ^ Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Grynberg: Atom Photon Interaction - Basic Processes and Applications . 1st edition. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-29336-1 , pp. 361 .
3. ↑ Christopher C. Gerry: Introductory Quantum Optics . 3. Edition. Cambridge University Press, Cambridge / New York 2008, ISBN 978-0-521-52735-4 , pp. 90-93 .
4. ↑ Christopher C. Gerry: Introductory Quantum Optics . 3. Edition. Cambridge University Press, Cambridge / New York 2008, ISBN 978-0-521-52735-4 , pp. 13.92 .
5. ^ Leslie Allen, JH Eberly: Optical Resonance and Two-Level-Atoms . 1st edition. Wiley-Interscience, New York 1975, ISBN 0-471-02327-2 , pp. 47 ff .