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Quantum dynamics of a driven damped harmonic oscillator in Heisenberg picture: exact results and possible generalizations

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In the framework of the Heisenberg picture, an alternative derivation of the reduced density matrix of a driven dissipative quantum harmonic oscillator as the prototype of an open quantum system is investigated. The reduced density matrix for different initial states of the combined system is obtained from a general formula, and different limiting cases are studied. Exact expressions for the corresponding characteristic function in quantum thermodynamics and Wigner quasi-distribution function are found. A possible generalization based on the Magnus expansion of the evolution operator is presented.

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  1. 1.

    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

  2. 2.

    U. Weiss, Quantum Dissipative Systems, 2nd edn. (World Scientific, Singapore, 1999)

  3. 3.

    A.O. Caldeira, An Introduction to Macroscopic Quantum Phenomena and Quantum Dissipation (Cambridge University Press, Cambridge, 2014)

  4. 4.

    J. Schwinger, J. Math. Phys. 2, 407 (1961)

  5. 5.

    R.P. Feynman, F.L.J. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963)

  6. 6.

    A.O. Caldeira, A.J. Leggett, Physica A 121, 587 (1983)

  7. 7.

    H. Grabert, P. Schramm, G.-L. Ingold, Phys. Rep. 168, 115 (1988)

  8. 8.

    M. Carlesso, A. Bassi, Phys. Rev. A 95, 052119 (2017)

  9. 9.

    A. Lampo, Quantum Brownian motion revisited: extensions and applications. Doctorial thesis, (Universitat Politècnica de Catalunya, Catalunya) (2018)

  10. 10.

    A. Lampo, S.H. Lim, M.Á. Garc ía-March, M. Lewenstein, Quantum 1, 30 (2017)

  11. 11.

    W.T. Coffey, YuP Kalmykov, J.T. Waldron, The Langevin Equation, 2nd edn. (World Scientific, Singapore, 2005)

  12. 12.

    C.H.G. Bessa, V.B. Bezerra, E.R. Bezerra de Mello, H.F. Mota, Phys. Rev. D 95, 085020 (2017)

  13. 13.

    E. Cobanera, P. Kristel, C.M. Smith, Phys. Rev. B 93, 245422 (2016)

  14. 14.

    P. Massignan, A. Lampo, J. Wehr, M. Lewenstein, Phys. Rev. A 91, 033627 (2015)

  15. 15.

    D.J. Kraft, R. Wittkowski, B. ten Hagen, K.V. Edmond, D.J. Pine, H. Löwen, Phys. Rev. E 88, 050301(R) (2013)

  16. 16.

    A. Chakrabarty, A. Konya, F. Wang, J.V. Selinger, K. Sun, Q.-H. Wei, Phys. Rev. Lett. 111, 160603 (2013)

  17. 17.

    X.Q. Jiang, B. Zhang, Z.W. Lu, X.D. Sun, Phys. Rev. A 83, 053823 (2011)

  18. 18.

    N. Lambert et al., Nat. Phys. 9, 10 (2013)

  19. 19.

    S. Gröblacher, A. Trubarov, N. Prigge, G.D. Cole, M. Aspelmeyer, J. Eisert, Nat. Commun. 6, 7606 (2015)

  20. 20.

    A. Pomyalov, D.J. Tannor, J. Chem. Phys. 123, 204111 (2005)

  21. 21.

    J. Gemmer, M. Michel, G. Mahler, Quantum Thermodynamics, 2nd edn. (Springer, Berlin, 2009)

  22. 22.

    S. Gasparinetti, P. Solinas, A. Braggio, M. Sassetti, New J. Phys. 16, 115001 (2014)

  23. 23.

    M. Carrega, P. Solinas, A. Braggio, M. Sassetti, U. Weiss, New J. Phys. 17, 045030 (2015)

  24. 24.

    W. Dou, M.A. Ochoa, A. Nitzan, J.E. Subotnik, Phys. Rev. B 98, 134306 (2018)

  25. 25.

    R.S. Whitney, Phys. Rev. B 98, 085415 (2018)

  26. 26.

    K. Funo, H.T. Quan, Phys. Rev. Lett. 121, 040602 (2018)

  27. 27.

    M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, J. Eisert, Phys. Rev. Lett. 120, 120602 (2018)

  28. 28.

    M.A. Ochoa, N. Zimbovskaya, A. Nitzan, Phys. Rev. B 97, 085434 (2018)

  29. 29.

    P. Haughian, M. Esposito, T.L. Schmidt, Phys. Rev. B 97, 085435 (2018)

  30. 30.

    J. Lekscha, H. Wilming, J. Eisert, R. Gallego, Phys. Rev. E 97, 022142 (2018)

  31. 31.

    P. Talkner, M. Campisi, P. Hänggi, J. Stat. Mech. 2009, P02025 (2009)

  32. 32.

    M. Esposito, U. Harbola, S. Mukamel, Rev. Mod. Phys. 81, 1665–702 (2009)

  33. 33.

    F. Kheirandish, Eur. Phys. J. Plus 133, 276 (2018)

  34. 34.

    W. Magnus, Commun. Pure Appl. Math. VII(4), 649–673 (1954)

  35. 35.

    W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1975)

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Correspondence to Fardin Kheirandish.


Derivation of Eqs. (2) and (5)

From Hamiltonian Eq. (1) and Heisenberg equations of motion for \(\hat{a}\) and \(\hat{b}_j\), we have

$$\begin{aligned} \dot{\hat{a}}= & {} \frac{1}{i\hbar }[\hat{a},\hat{H}]=-i\omega _0\hat{a}-i\sum _j f_j \hat{b}_j-i K(t). \end{aligned}$$
$$\begin{aligned} \dot{\hat{b}}_j= & {} \frac{1}{i\hbar }[\hat{b}_j,\hat{H}]=-i\omega _j\hat{b}_j-i \bar{f}_j \hat{a}. \end{aligned}$$

The solution of Eq. (69) is

$$\begin{aligned} \hat{b}_j(t)={\hbox {e}}^{-i\omega _jt} \hat{b}_j(0)-i \bar{f}_j \int _0^t {\mathrm{d}}t'\,{\hbox {e}}^{-i \omega _j(t-t')} \hat{a}(t'). \end{aligned}$$

By inserting this solution into Eq. (68), we find

$$\begin{aligned} \dot{\hat{a}}+i\omega _0 \,\hat{a}+\int _0^t {\mathrm{d}}t'\,\chi (t-t')\,\hat{a}(t')= & {} -i\sum _j f_j\,{\hbox {e}}^{-i \omega _jt}\,\hat{b}_j(0)-i\, K(t), \end{aligned}$$

where the response function of the medium is defined by

$$\begin{aligned} \chi (t-t')=\sum _j |f_j|^2\,{\hbox {e}}^{-i\omega _j(t-t')}. \end{aligned}$$

By taking the Laplace transform of both sides of Eq. (71), we obtain

$$\begin{aligned} \tilde{a}(s)= & {} \frac{1}{s+i\omega _0+\tilde{\chi }(s)}\,\hat{a}(0)-i\frac{\tilde{K}(s)}{s+i\omega _0+\tilde{\chi }(s)}\nonumber \\&-\,i \sum _j \frac{f_j\,\hat{b}_j(0)}{(s+i\omega _0+\tilde{\chi }(s))(s+i\,\omega _j)}, \end{aligned}$$

and by taking inverse Laplace transform, we finally find

$$\begin{aligned} \hat{a}(t)=G(t)\,\hat{a}(0)-i\sum _j M_j (t)\,\hat{b}_j(0)-i \zeta (t), \end{aligned}$$

with the Hermitian conjugation

$$\begin{aligned} \hat{a}^{\dag } (t)=\bar{G}(t)\,\hat{a}^{\dag } (0)+i\sum _j \bar{M}_j (t)\,\hat{b}_j^{\dag } (0)+i \bar{\zeta }(t), \end{aligned}$$


$$\begin{aligned}&G(t)=\mathcal {L}^{-1}\bigg [\frac{1}{s+i\omega _0+\tilde{\chi }(s)}\bigg ],\nonumber \\&M_j (t)=f_j\,\int _0^t {\mathrm{d}}t'\,{\hbox {e}}^{-i\omega _j(t-t')}\,G(t'),\nonumber \\&\zeta (t)=\int _0^t {\mathrm{d}}t'\,G(t-t')\,K(t'). \end{aligned}$$

By inserting Eq. (74) into Eq. (70), we find

$$\begin{aligned} \hat{b}_j(t)= & {} \sum _k \bigg \{\underbrace{{\hbox {e}}^{-i \omega _jt}\,\delta _{jk}-\bar{f}_j \int _0^t {\mathrm{d}}t'\,{\hbox {e}}^{-i\omega _j(t-t')}\,M_{k} (t')}_{\varLambda _{jk} (t)}\bigg \}\,\hat{b}_k (0)\nonumber \\&\underbrace{-i \bar{f}_j \int _0^t {\mathrm{d}}t'\,{\hbox {e}}^{-i\omega _j(t-t')}\,G_{k} (t')}_{\varGamma _j (t)}\,\hat{a} (0)\nonumber \\&\underbrace{- \bar{f}_j \int _0^t {\mathrm{d}}t'\,{\hbox {e}}^{-i\omega _j(t-t')}\,\zeta _{k} (t')}_{\varOmega _j (t)}. \end{aligned}$$


$$\begin{aligned} \hat{b}_j(t)= & {} \sum _k \varLambda _{jk} (t)\,\hat{b}_k (0)+\varGamma _j (t) \,\hat{a} (0)+\varOmega _j (t),\nonumber \\ \hat{b}^\dag _j(t)= & {} \sum _k \bar{\varLambda }_{jk} (t)\,\hat{b}^\dag _k (0)+\bar{\varGamma }_j (t) \,\hat{a}^\dag (0)+\bar{\varOmega }_j (t). \end{aligned}$$

Derivation of Eq. (11)

$$\begin{aligned} \sum _{n=0}^\infty \hat{Q}_{nn} (t)= & {} \sum _{n=0}^\infty \hat{U}^{\dag } (t) \,|n\rangle \langle n|\otimes I_R\,\hat{U} (t),\nonumber \\= & {} \hat{U}^{\dag } (t) \,I_S\otimes I_R\,\hat{U} (t)=\hat{U}^{\dag } (t)\,\hat{U} (t)=I. \end{aligned}$$
$$\begin{aligned} \sum _{n=0}^\infty n^s\,\hat{Q}_{nn} (t)= & {} \sum _{n=0}^\infty n^s\,\hat{U}^{\dag } (t) \,|n\rangle \langle n|\otimes I_R\,\hat{U} (t),\nonumber \\= & {} \hat{U}^{\dag } (t)\sum _{n=0}^\infty n^s\,|n\rangle \langle n|\otimes I_R\,\hat{U} (t),\nonumber \\= & {} \hat{U}^{\dag } (t) (\hat{a}^\dag (0)\,\hat{a}(0))^s\,\hat{U} (t),\nonumber \\= & {} (\hat{a}^\dag (t)\,\hat{a}(t))^s. \end{aligned}$$

Derivation of Eq. (12)

$$\begin{aligned} \hat{Q}_{nn} (t)= & {} \hat{U}^{\dag } (t) |m\rangle \langle n|\otimes I_R\,\hat{U} (t),\nonumber \\= & {} \hat{U}^{\dag } (t)\,\frac{(\hat{a}^\dag (0))^m}{\sqrt{m!}}|0\rangle \langle 0|\frac{\hat{a}(0))^n}{\sqrt{n!}}\otimes I_R\,\hat{U} (t). \end{aligned}$$

On the other hand [35],

$$\begin{aligned} |0\rangle \langle 0|=\sum _{s=0}^\infty \frac{(-1)^s}{s!}\,(\hat{a}^\dag (0))^s(\hat{a}(0))^s; \end{aligned}$$

by inserting Eq. (82) into Eq. (81), we deduce

$$\begin{aligned} \hat{Q}_{nn} (t)= & {} \frac{1}{\sqrt{m! n!}}\,\sum _{s=0}^\infty \frac{(-1)^s}{s!}\,\hat{U}^{\dag } (t)\,(\hat{a}^\dag (0))^{m+s}\,(\hat{a}(0))^{n+s}\otimes I_R\,\hat{U} (t), \nonumber \\= & {} \frac{1}{\sqrt{m! n!}}\,\sum _{s=0}^\infty \frac{(-1)^s}{s!}\,(\hat{a}^\dag (t))^{m+s}(\hat{a}(t))^{n+s}. \end{aligned}$$

Derivation of Eq. (19)

We have

$$\begin{aligned}&\text{ Tr }_R \Big [(\hat{B}^{\dag })^u (\hat{B})^v\,\hat{\rho }_R (0)\Big ]=\bigg (\frac{\partial }{\partial J}\bigg )^u\,\bigg (\frac{\partial }{\partial \bar{J}}\bigg )^v\,\text{ Tr }_R \Big [{\hbox {e}}^{J\hat{B}^{\dag }} \,{\hbox {e}}^{\bar{J}\hat{B}}\,\hat{\rho }_R (0)\Big ]\bigg |_{J=\bar{J}=0}. \end{aligned}$$

By inserting the definitions

$$\begin{aligned} \hat{B}=\sum _k M_k (t)\,\hat{b}_k(0), \nonumber \\ \hat{B}^{\dag }=\sum _k \bar{M}_k (t)\,\hat{b}^\dag _k(0), \end{aligned}$$

into the generating function defined by Eq. (84), we find

$$\begin{aligned}&\text{ Tr }_R \Big [{\hbox {e}}^{J\hat{B}^{\dag }} \,{\hbox {e}}^{\bar{J}\hat{B}}\,\hat{\rho }_R (0)\Big ]=\prod _k \underbrace{\text{ Tr }_k \bigg ({\hbox {e}}^{J \bar{M}_k (t)\hat{b}^\dag _k(0)}{\hbox {e}}^{\bar{J} M_k (t) \hat{b}_k(0)} \frac{{\hbox {e}}^{-\beta \hbar \omega _k\,\hat{b}^\dag _k\hat{b}_k}}{z_k}\bigg )}_{I_k}, \end{aligned}$$

where \(\text{ Tr }_k\) means taking trace over the Hilbert space of the kth oscillator in the environment and \(z_k\) is the corresponding partition function

$$\begin{aligned} z_k=\text{ Tr }_k\big ({\hbox {e}}^{-\beta \hbar \omega _k\,\hat{b}^\dag _k\hat{b}_k}\big ). \end{aligned}$$

Now, we have

$$\begin{aligned}&I_k = \frac{1}{z_k}\sum _{n_k=0}^\infty {\hbox {e}}^{-\beta \hbar \omega _kn_k}\,\langle n_k|{\hbox {e}}^{J \bar{M}_k (t)\hat{b}^\dag _k(0)}{\hbox {e}}^{\bar{J} M_k (t) \hat{b}_k(0)}|n_k\rangle , \nonumber \\&\quad = \frac{1}{z_k}\sum _{n_k=0}^\infty {\hbox {e}}^{-\beta \hbar \omega _kn_k}\sum _{l=0}^{n_k} \frac{|J|^{2l}\,|M_k (t)|^{2l}}{(l!)^2}\langle n_k |(\hat{b}^\dag _k)^l (\hat{b}_k)^l|n_k\rangle , \nonumber \\&\quad = \frac{1}{z_k}\sum _{n_k=0}^\infty {\hbox {e}}^{-\beta \hbar \omega _kn_k}\sum _{l=0}^{n_k} \frac{|J|^{2l}\,|M_k (t)|^{2l}}{(l!)^2}\,\frac{n_k!}{(n_k-l)!}, \nonumber \\&\quad = {\hbox {e}}^{-J\bar{J}\,\frac{|M_k (t)|^2}{1-{\hbox {e}}^{\beta \hbar \omega _k}}}. \end{aligned}$$

Therefore, the generating function is

$$\begin{aligned} \text{ Tr }_R \Big [{\hbox {e}}^{J\hat{B}^{\dag }} \,{\hbox {e}}^{\bar{J}\hat{B}}\,\hat{\rho }_R (0)\Big ]={\hbox {e}}^{-J\bar{J}\,\frac{|M_k (t)|^2}{1-{\hbox {e}}^{\beta \hbar \omega _k}}}={\hbox {e}}^{-J\bar{J}\,\eta (t)}. \end{aligned}$$

By making use of Eq. (84), we finally find

$$\begin{aligned} \text{ Tr }_R \Big [(\hat{B}^{\dag })^u (\hat{B})^v\,\hat{\rho }_R (0)\Big ]= & {} (\partial /\partial J)^u\,(\partial /\partial \bar{J})^v\,{\hbox {e}}^{-J\bar{J}\,\eta (t)}\bigg |_{J=\bar{J}=0},\nonumber \\= & {} \delta _{u v}\,u!\,[\eta (t)]^u. \end{aligned}$$

Derivation of Eq. (26)

We have

$$\begin{aligned} \langle n|\hat{\rho }_S (t)|m\rangle= & {} \frac{1}{\sqrt{n!m!}}\,\sum _{s=0}^\infty \frac{(-1)^s}{s!}\sum _{p=0}^{s+m}\sum _{r=0}^{s+n}\sum _{u=0}^{\min (p,r)}\,\left( {\begin{array}{c}s+m\\ p\end{array}}\right) \left( {\begin{array}{c}s+n\\ r\end{array}}\right) \left( {\begin{array}{c}p\\ u\end{array}}\right) \left( {\begin{array}{c}r\\ u\end{array}}\right) u!\,[\eta (t)]^u\nonumber \\&\times \,[\bar{G}(t)]^{s+m-p}[G(t)]^{s+n-r}[\bar{\zeta } (t)]^{p-u}[\zeta (t)]^{r-u}\,(i)^{p-r}\nonumber \\&\times \,\sqrt{\frac{k!}{(k-s-n+r)!}}\sqrt{\frac{k!}{(k-s-m+p)!}}\,\delta _{m-p,n-r},\nonumber \\= & {} \frac{i^{m-n}}{\sqrt{n!m!}}\,\sum _{s=0}^\infty \frac{(-1)^s}{s!}\sum _{r=r_{\mathrm{min}}}^{s+n}\sum _{u=0}^r\,\left( {\begin{array}{c}s+m\\ m-n+r\end{array}}\right) \left( {\begin{array}{c}s+n\\ r\end{array}}\right) \left( {\begin{array}{c}m-n+r\\ u\end{array}}\right) \left( {\begin{array}{c}r\\ u\end{array}}\right) \nonumber \\&\times \,u!\,[\eta (t)]^u\, \frac{k!}{(k-s-n+r)!}\,|G(t)|^{2(s+n-r)}(\bar{\zeta })^{m-n+r}\,(\zeta )^r\,|\zeta |^{-2u},\nonumber \\= & {} \frac{(i\bar{\zeta })^{m-n}}{\sqrt{n!m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!}\sum _{r=r_{\mathrm{min}}}^{s+n}\frac{k!}{(k-s-n+r)!}\,\left( {\begin{array}{c}s+m\\ m-n+r\end{array}}\right) \left( {\begin{array}{c}s+n\\ r\end{array}}\right) \nonumber \\&\times \,|G(t)|^{2(s+n-r)}|\zeta |^{2r}\,\sum _{u=0}^r\left( {\begin{array}{c}m-n+r\\ u\end{array}}\right) \left( {\begin{array}{c}r\\ u\end{array}}\right) \,u!\,\bigg [\frac{\eta (t)}{|\zeta (t)|^2}\bigg ]^u, \end{aligned}$$

where \(r_{\mathrm{min}}=\max (s+n-k,0)\).

Derivation of Eq. (30)

$$\begin{aligned} \langle F(\hat{a}^\dag (0),\hat{a}(0))\rangle _t= & {} \text{ Tr }_S [F(\hat{a}^\dag (0),\hat{a}(0))\,\hat{\rho }_S (t)], \nonumber \\= & {} \text{ Tr }_S [F(\hat{a}^\dag (0),\hat{a}(0))\,\text{ Tr }_R\hat{\rho }(t)],\nonumber \\= & {} \text{ Tr } [F(\hat{a}^\dag (0),\hat{a}(0))\,\hat{\rho }(t)],\nonumber \\= & {} \text{ Tr } [F(\hat{a}^\dag (0),\hat{a}(0))\,\hat{U} (t)\,\hat{\rho }(0)\,\hat{U}^{\dag } (t)],\nonumber \\= & {} \text{ Tr } [\hat{U}^{\dag } (t)\,F(\hat{a}^\dag (0),\hat{a}(0))\,\hat{U} (t)\hat{\rho }(0)],\nonumber \\= & {} \text{ Tr } [F(\hat{a}^\dag (t),\hat{a}(t))\,\hat{\rho }(0)], \end{aligned}$$

Derivation of Eq. (34)

In the absence of dissipation (\(f_k=0,\,\,\,k=0,1,2,\ldots \)), we have from Eq. (21)

$$\begin{aligned}&\tilde{\chi } (s)=0 \rightarrow G(t)={\hbox {e}}^{-i\omega _0 t},\nonumber \\&\eta (t)=0. \end{aligned}$$


$$\begin{aligned} \langle n|\hat{\rho }_S (t)|m\rangle= & {} \frac{1}{\sqrt{n! m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!}\sum _{p=0}^{s+m}\sum _{r=0}^{s+n}\,\left( {\begin{array}{c}s+m\\ p\end{array}}\right) \left( {\begin{array}{c}s+n\\ r\end{array}}\right) \,{\hbox {e}}^{-i\omega _0 t (s+n-r)}{\hbox {e}}^{i\omega _0 t (s+m-p)}\nonumber \\&\times \, \bar{\zeta }^p \,\zeta ^r \,\alpha ^{s+n-r} \,\bar{\alpha }^{s+m-p}\,(i)^{p-r},\nonumber \\= & {} \frac{1}{\sqrt{n! m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!}|\alpha |^{2s}\,\alpha ^n\,\bar{\alpha }^m\,{\hbox {e}}^{-i n \omega _0 t} {\hbox {e}}^{i m \omega _0 t}\nonumber \\&\times \, \sum _{p=0}^{s+m}\left( {\begin{array}{c}s+m\\ p\end{array}}\right) {\hbox {e}}^{-i p \omega _0 t}\,(i\bar{\zeta })^p\,(\bar{\alpha })^{-p}\,\sum _{r=0}^{s+n}\left( {\begin{array}{c}s+n\\ r\end{array}}\right) {\hbox {e}}^{i r \omega _0 t}\,(-i\zeta )^r\,\alpha ^{-r},\nonumber \\= & {} \frac{1}{\sqrt{n! m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!} |\alpha |^{2s}\,\alpha ^n\,\bar{\alpha }^m\,{\hbox {e}}^{-i n \omega _0 t} {\hbox {e}}^{i m \omega _0 t}\nonumber \\&\times \, \bigg (1+\frac{i\bar{\zeta }{\hbox {e}}^{-i\omega _0 t}}{\bar{\alpha }}\bigg )^{s+m}\bigg (1-\frac{i\zeta }{\alpha }{\hbox {e}}^{i\omega _0 t}\bigg )^{s+n},\nonumber \\= & {} \frac{1}{\sqrt{n! m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!} |\alpha -i\zeta \,{\hbox {e}}^{i\omega _0 t}|^{2s}\,(\bar{\alpha } \,{\hbox {e}}^{i\omega _0 t}+i\bar{\zeta })^m\,(\alpha \,{\hbox {e}}^{-i\omega _0 t}-i\zeta )^n,\nonumber \\= & {} \frac{(\alpha \,{\hbox {e}}^{-i\omega _0 t}-i\zeta )^n(\bar{\alpha } \,{\hbox {e}}^{i\omega _0 t}+i\bar{\zeta })^m}{\sqrt{n! m!}}\,{\hbox {e}}^{-|\alpha \,{\hbox {e}}^{-i\omega _0 t}-i\zeta |^2}. \end{aligned}$$

Vanishing of coherency in Eq. (41)

In large-time limit, we have \(G(t)\rightarrow 0\) and for the non-diagonal matrix elements of the reduced density matrix (\(n\ne m\)) we easily see that the maximum degree of G(t) in denominator is smaller than its degree in numerator, since

$$\begin{aligned} 2r=2(s+\min (m,n))<2s+m+n,\,\,\,(m\ne n); \end{aligned}$$

therefore, the non-diagonal elements tend to zero and accordingly there is no coherency in the preferred number basis \(|n\rangle \).

Derivation of Eq. (55)

In large-time limit \(G(t)\rightarrow 0\), so by setting \(s+n-r=0\) and \(s+m-r=0\) we find \(n=m\), leading to

$$\begin{aligned} \langle n|\hat{\rho }_S (t)|m\rangle= & {} \delta _{n,m}\,\frac{1}{\sqrt{n! m!}}\sum _{s=0}^\infty \frac{(-1)^s}{s!}\,(s+n)!\,[\eta (t)]^{s+n},\nonumber \\= & {} \delta _{n,m}\,\frac{\eta (t)^n}{\sqrt{n! m!}}\,\sum _{s=0}^\infty (-1)^s\,\frac{(s+n)!}{s!}\,[\eta (t)]^{s},\nonumber \\= & {} \delta _{n,m}\,\frac{\eta (t)^n}{n!}\,\frac{n!}{[1+\eta (t)]^{n+1}}\nonumber \\= & {} \delta _{n,m}\,\frac{[\eta (t)]^n}{[1+\eta (t)]^{n+1}}. \end{aligned}$$

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Kheirandish, F. Quantum dynamics of a driven damped harmonic oscillator in Heisenberg picture: exact results and possible generalizations. Eur. Phys. J. Plus 135, 243 (2020). https://doi.org/10.1140/epjp/s13360-020-00264-4

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