Markov approximation

The Markov approximation (after Andrei Andrejewitsch Markow ) is an approximation method of quantum optics . In the Markov approximation, it is assumed that a reservoir to which a quantum mechanical system is coupled returns instantaneously to its initial state after coupling . The reservoir clearly has no "memory" and very quickly behaves again as if the interaction with the coupled system had not taken place at all.

The Markov approximation is used, for example, in determining the master equation of damped harmonic oscillators . It is also a useful assumption for the numerical calculation of quantum physical processes in discrete time steps ( coarse graining ).

Derivation

Basics

This section deals with a quantum mechanical system ( index S) and a reservoir (index R) as well as the interaction between the two (index SR). The dynamics of the overall system is described with a Hamilton operator , which is composed of the individual Hamiltonians of the system and the reservoir as well as an interaction Hamiltonian: ${\displaystyle H}$

${\displaystyle H=H_{S}+H_{R}+H_{SR}}$

The density operator of the overall system is . The density operator of the individual system and that of the reservoir are determined by the formation of the partial track via the degrees of freedom of the reservoir or the coupled system: ${\displaystyle \chi }$ ${\displaystyle \rho }$ ${\displaystyle R}$

${\displaystyle \rho =\operatorname {Spur} _{R}(\chi )}$

${\displaystyle R=\operatorname {Spur} _{S}(\chi )}$

The time evolution of is described by the Von Neumann equation : ${\displaystyle \chi }$

${\displaystyle {\dot {\chi }}={\frac {\partial \chi }{\partial t}}={\frac {1}{\mathrm {i} \hbar }}[H,\chi ]}$

Transition to the interaction picture

Now one transforms the Von Neumann equation into the interaction picture (indicated by tilden ) by applying the unitary time evolution operator . ${\displaystyle {\tilde {}}}$ ${\displaystyle U}$

${\displaystyle {\dot {\tilde {\chi }}}=\overbrace {e^{\mathrm {i} /\hbar \left(H_{S}+H_{R}\right)t}} ^{U^{\dagger }}\;{\dot {\chi }}\;\overbrace {e^{-\mathrm {i} /\hbar \left(H_{S}+H_{R}\right)t}} ^{U}}$

The unitary transformation at the transition to the interaction picture absorbs the rapid time development due to the undisturbed terms and , so that only the coupling of system and reservoir is responsible for the time development of . Because is a function of , commutes with ; but not with . It follows: ${\displaystyle H_{S}}$ ${\displaystyle H_{R}}$ ${\displaystyle {\tilde {\chi }}}$ ${\displaystyle U}$ ${\displaystyle H_{S}+H_{R}}$ ${\displaystyle U}$ ${\displaystyle H_{S}+H_{R}}$ ${\displaystyle H_{SR}}$

${\displaystyle {\dot {\tilde {\chi }}}={\frac {1}{\mathrm {i} \hbar }}\left[{\tilde {H}}_{SR},{\tilde {\chi }}\right]}$

This differential equation can be formally passed through

${\displaystyle {\tilde {\chi }}(t)=\chi (0)+{\frac {1}{\mathrm {i} \hbar }}\int _{0}^{t}\left[{\tilde {H}}_{SR}(\tau ),{\tilde {\chi }}(\tau )\right]\mathrm {d} \tau }$

solve and is used again in the Von Neumann equation:

${\displaystyle {\dot {\tilde {\chi }}}={\frac {1}{\mathrm {i} \hbar }}\left[{\tilde {H}}_{SR}(t),{\tilde {\chi }}(0)\right]-{\frac {1}{\hbar ^{2}}}\int _{0}^{t}\left[{\tilde {H}}_{SR}(t),\left[{\tilde {H}}_{SR}(\tau ),{\tilde {\chi }}(\tau )\right]\right]\mathrm {d} \tau }$

With a further iteration step, the following still exact solution for . ${\displaystyle {\tilde {\chi }}(t)}$

${\displaystyle {\tilde {\chi }}(t)={\tilde {\chi }}(0)+{\frac {1}{\mathrm {i} \hbar }}\int _{0}^{t}\left[{\tilde {H}}_{SR}(\tau ),{\tilde {\chi }}(0)\right]\mathrm {d} \tau -{\frac {1}{\hbar ^{2}}}\int _{0}^{t}\int _{0}^{\tau }\left[{\tilde {H}}_{SR}(\tau ),\left[{\tilde {H}}_{SR}(\tau '),{\tilde {\chi }}(\tau ')\right]\right]\mathrm {d} \tau '\mathrm {d} \tau }$

From can also be determined by tracing the reservoir : ${\displaystyle {\tilde {\chi }}(t)}$ ${\displaystyle {\tilde {\rho }}(t)}$

${\displaystyle {\begin{aligned}{\tilde {\rho }}(t)&=\operatorname {Spur} _{R}\left({\tilde {\chi }}(t)\right)\\&=\operatorname {Spur} _{R}\left({\tilde {\chi }}(0)\right)+{\frac {1}{\mathrm {i} \hbar }}\int _{0}^{t}\underbrace {\operatorname {Spur} _{R}\left(\left[{\tilde {H}}_{SR}(\tau ),{\tilde {\chi }}(0)\right]\right)} _{=0}\mathrm {d} \tau -{\frac {1}{\hbar ^{2}}}\int _{0}^{t}\int _{0}^{\tau }\operatorname {Spur} _{R}\left(\left[{\tilde {H}}_{SR}(\tau ),\left[{\tilde {H}}_{SR}(\tau '),{\tilde {\chi }}(\tau ')\right]\right]\right)\mathrm {d} \tau '\mathrm {d} \tau \end{aligned}}}$

Assumptions on the reservoir / Born approximation

For the sake of simplicity, it is assumed that the effects of the operators of the reservoir who couple to the system have no influence on the reservoir at the time on average. So the expected value of vanishes. ${\displaystyle t=0}$ ${\displaystyle {\tilde {H}}_{SR}}$

${\displaystyle \operatorname {Spur} _{R}\left({\tilde {H}}_{SR}R_{0}\right)=0}$

This can be shown by breaking it down into a product of two operators, one of which can be extracted from the track, as in the following section. It follows directly that ${\displaystyle {\tilde {H}}_{SR}}$

${\displaystyle \operatorname {Spur} _{R}\left(\left[{\tilde {H}}_{SR},{\tilde {\chi }}(0)\right]\right)=0}$

Furthermore, it is assumed that the density matrix of the overall system can be factored at the point in time . Now one applies the Born approximation , i. i.e. one neglects contributions of higher orders in : ${\displaystyle \chi (0)=\rho (0)R(0)=\rho (0)R_{0}}$ ${\displaystyle {\tilde {H}}_{SR}}$

${\displaystyle {\tilde {\chi }}(t)\approx {\tilde {\rho }}(t)R_{0}+{\mathcal {O}}({\tilde {H}}_{SR})}$

This finally results for the density matrix of the system:

${\displaystyle {\tilde {\rho }}(t)={\tilde {\rho }}(0)-{\frac {1}{\hbar ^{2}}}\int _{0}^{t}\int _{0}^{\tau }\operatorname {Spur} _{R}\left(\left[{\tilde {H}}_{SR}(\tau ),\left[{\tilde {H}}_{SR}(\tau '),{\tilde {\rho }}(\tau ')R_{0}\right]\right]\right)\mathrm {d} \tau '\mathrm {d} \tau }$

Reservoir-internal correlation functions

The interaction Hamiltonian is written with the help of operators that only affect the coupled system and operators that only affect the reservoir. Thus: ${\displaystyle H_{SR}}$ ${\displaystyle s_{i}}$ ${\displaystyle r_{i}}$

${\displaystyle {\tilde {H}}_{SR}=\hbar \sum _{i}{\tilde {s}}_{i}{\tilde {r}}_{i}}$

This expression for is now inserted into the last equation of the previous section. It should be noted that operators that only act on the coupled system can be pulled out of the track via the degrees of freedom of the reservoir. Furthermore, one uses the invariance of the track under cyclic permutations and thus obtains ${\displaystyle {\tilde {H}}_{SR}}$

${\displaystyle {\begin{aligned}{\tilde {\rho }}={\tilde {\rho }}(0)-\sum _{i,j}\int _{0}^{t}\int _{0}^{\tau }{\Big (}&\left({\tilde {s}}_{i}\left(\tau \right){\tilde {s}}_{j}\left(\tau '\right){\tilde {\rho }}\left(\tau '\right)-{\tilde {s}}_{j}\left(\tau '\right){\tilde {\rho }}\left(\tau '\right){\tilde {s}}_{i}\left(\tau \right)\right){\color {red}\operatorname {Spur} _{R}\left(R_{0}{\tilde {r}}_{i}\left(\tau \right){\tilde {r}}_{j}\left(\tau '\right)\right)}\\+&\left({\tilde {\rho }}\left(\tau '\right){\tilde {s}}_{j}\left(\tau '\right){\tilde {s}}_{i}\left(\tau \right)-{\tilde {s}}_{i}\left(\tau \right){\tilde {\rho }}\left(\tau '\right){\tilde {s}}_{j}\left(\tau '\right)\right){\color {red}\operatorname {Spur} _{R}\left(R_{0}{\tilde {r}}_{j}\left(\tau '\right){\tilde {r}}_{i}\left(\tau \right)\right)}{\Big )}\mathrm {d} \tau '\mathrm {d} \tau \end{aligned}}}$

The expressions marked in red are correlation functions of the reservoir. They only depend on the time difference and can be developed into exponential functions that oscillate at different frequencies . ${\displaystyle \Delta \tau =\tau -\tau '}$ ${\displaystyle \omega _{\mu \nu }}$

${\displaystyle {\begin{aligned}\operatorname {Spur} _{R}\left(R_{0}{\tilde {r}}_{i}\left(\tau \right){\tilde {r}}_{j}\left(\tau '\right)\right)&=\operatorname {Spur} _{R}\left(R_{0}{\tilde {r}}_{i}\left(\Delta \tau \right){\tilde {r}}_{j}\left(0\right)\right)\\&=\operatorname {Spur} _{R}\sum _{\mu }\left(p_{\mu }|\mu \rangle \langle \mu |{\tilde {r}}_{i}\left(\Delta \tau \right){\tilde {r}}_{j}\left(0\right)\right)\\&=\sum _{\mu }p_{\mu }\langle \mu |{\tilde {r}}_{i}\left(\Delta \tau \right){\tilde {r}}_{j}\left(0\right)|\mu \rangle \\&=\sum _{\mu }\sum _{\nu }p_{\mu }\left|R_{\mu \nu }\right|^{2}e^{i\omega _{\mu \nu }t}\end{aligned}}}$

In the second line it was assumed that the reservoir is in an eigenstate of and therefore only has diagonal entries . Here and is the energy difference between two eigenstates of the reservoir. ${\displaystyle H_{R}}$ ${\displaystyle R_{0}}$ ${\displaystyle p_{\mu }}$ ${\displaystyle R_{\mu \nu }=\langle \mu |r|\nu \rangle }$ ${\displaystyle \hbar \omega _{\mu \nu }=\hbar \left(\omega _{\mu }-\omega _{\nu }\right)=E_{\mu }-E_{\nu }}$

Perform the Markov approximation

The Markov approximation now consists in assuming that a large number of energy levels are present in the reservoir, the associated correlation functions of which interfere destructively to zero as soon as is greater than the internal time scales of the reservoir. It is assumed that the correlation functions can be approximated as delta functions . ${\displaystyle \Delta \tau }$

${\displaystyle \operatorname {Spur} _{R}\left(R_{0}{\tilde {r}}_{i}(\tau ){\tilde {r}}_{j}(\tau ')\right)\propto \delta \left(\tau -\tau '\right)}$

This turns the integral into a convolution with a delta function. As a result, in the last equation of the previous section can be replaced by. The approximation with a delta function clearly means that the reservoir instantly returns to its initial state after the system has been coupled . This takes place on a time scale that is significantly shorter than the one on which the relevant processes of the connected system run. In the frequency domain, the correlation function of the reservoir is therefore much broader than that of the coupled system. This clear separation of two time scales is the essential condition for the validity of the Markov approximation. ${\displaystyle \rho (\tau ')}$ ${\displaystyle \rho (\tau )}$

If the behavior of is to be investigated, one can assume in the Born-Markow approximation that the reservoir returns to its initial state after the coupling of the system (Born approximation) and that this happens almost immediately (Markov approximation). ${\displaystyle \rho }$

Individual evidence

1. ↑ HJ Carmichael: Statistical Methods in Quantum Optics I . Springer, Berlin / Heidelberg / New York 1999, ISBN 3-540-54882-3 , pp. 5-8 .
2. ^ F. Haake: Statistical Treatment of Open Systems by Generalized Master Equations . In: G. Höhler (Ed.): Quantum Statistics in Optics and Solid-State Physics, Springer Tracts in Modern Physics . tape 66 . Springer, Berlin / Heidelberg 1973, ISBN 978-3-662-39407-6 , pp. 120 ff . ( limited preview in Google Book search).
3. ^ Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Grynberg: Atom Photon Interactions - Basic Processes and Interactions . Wiley-VCH, Berlin / Heidelberg / New York 2004, ISBN 978-0-471-29336-1 , pp. 263 ff .