Spontaneous emission from nonhermitian perspective: complex scaling of the photon coordinates

ABSTRACT

Spontaneous emission (SE) is usually studied within the framework of quantum dissipation theory. In this paper we pursue a different view of SE. Namely, we take advantage of Nonhermitian Quantum Mechanics (NHQM) and interpret SE as the decay of a Feshbach type resonance. Correspondingly, we aim at calculating the SE decay rate Γ (resonance width) and the associated resonance wavefunction via the complex scaling (CS) method. Standard application of CS in NHQM is based on scaling the dissociative coordinates of the system, $x\mapsto x{\mathrm{e}}^{i\theta }$. In the case of SE, the decay consists in emitting photons. Therefore the CS must be applied on (suitably defined) position coordinates of the photons. Feasibility of such a programme is demonstrated explicitly through choosing the adequate photon coordinates and performing CS of the photon wavefunctions in the position representation. It is anticipated in this way that CS makes the full resonance wavefunction square integrable, and enables thus the calculation of Γ by proceeding in complete analogy with the well established NHQM approaches. As an illustration, the standard Golden Rule formula for Γ is rederived from the CS nonhermitian perturbation theory.

GRAPHICAL ABSTRACT

Keywords:

• Spontaneous emission
• resonance
• nonhermitian quantum mechanics
• complex scaling
• photon wavefunction

Acknowledgments

This work is dedicated by M. Š. to Prof. Nimrod Moiseyev on the occasion of his 70^th birthday. Time is flying, but I will remember forever the unique and stimulating scientific atmosphere of the Nimrod's group at the Technion in Haifa. Nimrod deserves an enormous credit for his enlightened and relaxed approach to science which cultivates enthusiasm and creativity. There are fewer and fewer places of this kind in the world nowadays, since the ‘real’ or ‘slow’ science suffers so much under suffocating commercial and vanity pressures. Fortunately, Nimrod has created for his group a true oasis of freedom, in which scientific work remains exciting and meaningful. A preliminary heuristic idea to study SE via NHQM and CS has been explored in the form of a short computational test project, given by M. Š. to the participants of the Max Planck – Minerva Winterschool on Light Matter Interactions, held at the Technion – Israel Institute of Technology, Haifa, Israel during February 21–27, 2010. The corresponding author (M. Š.) acknowledges the participants listed below for their ability to produce encouraging numerical results, which much later motivated development of the underlying general formalism (namely CS of x-dependent photonic wavefunctions) and its presentation here in this paper. The above-mentioned participants are written out in alphabetical order as follows: Saieswari Amaran, Shachar Klaiman, Axel Lode, Adi Nathan, Bala Pananghat, Vaibhav Prabhudesai, Tahir Shaaran, and Martin Schlessinger.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that Ref. [21] is not dealing at all with CS of the photonic wavefunctions in the position representation.

2 Definition (4(4) $\begin{array}{rl}\hat{B}(t,x)& =-\frac{\mathrm{\partial }\hat{A}(t,x)}{\mathrm{\partial }x}=-i{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}k\mathrm{sgn}\left(k\right)\\ & \sqrt{\frac{\hslash {\omega }_k}{8\pi }}{\mathrm{e}}^{-i({\omega }_{k}t-kx)}\hat{a}\left(k\right)+\mathrm{c}.\mathrm{c}.\end{array}$(4) ) of the magnetic field is consistent with the standard 3D relation $\overrightarrow{B}=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{A}$ (recall that $\overrightarrow{A}$ is pointing here along the z-axis).

3 Subscript ${}_R$ indicates here the radiation field. We have adopted this notation in advance while keeping in mind our forthcoming elaborations which would include also the state vectors of an atom.

4 Relations (24(24) $\begin{array}{rl}& =-\frac{4i}{\hslash c}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}x\mathcal{E}\left(x\right){\hat{\mathcal{A}}}^{†}\left(x\right)|\mathrm{vac}{⟩}_R.\end{array}$(24) )–(23(23) $\begin{array}{rl}|{\mathrm{\Xi }}_1{⟩}_R& =+\frac{4i}{\hslash c}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}x\mathcal{A}\left(x\right){\hat{\mathcal{E}}}^{†}\left(x\right)|\mathrm{vac}{⟩}_R\end{array}$(23) ) actually have a neat manifestly covariant analogue. Recall that $|{\mathrm{\Xi }}_1{⟩}_R$ is a covariant entity much like $\sqrt{{\omega }_k}\hat{a}\left(k\right)$. Within a manifestly covariant formulation one would replace (24(24) $\begin{array}{rl}& =-\frac{4i}{\hslash c}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}x\mathcal{E}\left(x\right){\hat{\mathcal{A}}}^{†}\left(x\right)|\mathrm{vac}{⟩}_R.\end{array}$(24) )–(23(23) $\begin{array}{rl}|{\mathrm{\Xi }}_1{⟩}_R& =+\frac{4i}{\hslash c}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}x\mathcal{A}\left(x\right){\hat{\mathcal{E}}}^{†}\left(x\right)|\mathrm{vac}{⟩}_R\end{array}$(23) ) by prescription $\begin{array}{rl}& {\mathrm{\Xi }}_1{⟩}_R=-{\int }_{\mathrm{\Sigma }}\mathrm{d}{\mathrm{\Sigma }}^{\mu }\left(x\right){\mathrm{\partial }}_{\mu }{\mathcal{A}}^{\nu }\left(x\right){\hat{\mathcal{A}}}_{\nu }^{†}\left(x\right)|\mathrm{vac}{⟩}_R\\ & =+{\int }_{\mathrm{\Sigma }}\mathrm{d}{\mathrm{\Sigma }}^{\mu }\left(x\right){\mathcal{A}}^{\nu }\left(x\right){\mathrm{\partial }}_{\mu }{\hat{\mathcal{A}}}_{\nu }^{†}\left(x\right)|\mathrm{vac}{⟩}_R.\end{array}$ Here Σ is a 1D spacelike surface (curve) foliating our (1,1) spacetime, see e.g. Ref. [40] for more information.

5 By ‘reasonable’ we mean here Schwartzian test functions, i.e. smooth functions of x which fall off to zero at large $|x|$ faster than any power of x. When applying $\mathfrak{d}$ on operators, one assumes equally ‘reasonable’ behaviour of their matrix elements.

6 Discarding the boundary term is legitimate here, because physical state vectors do not populate highly ultraviolet modes of $k\to \pm \mathrm{\infty }$.

7 We have tacitly assumed here that both ${\mathcal{A}}_1\left(x\right)$ and ${\mathcal{E}}_1\left(x\right)$ are analytic functions of the complex spatial coordinate.

8 Gaussian UV cutoff of a length gauge coupling Hamiltonian is implemented also in Ref. [21] for the same purpose.

9 Vector $|\mathrm{\Psi }⟩$ does not contain any attached subscripts, since it includes all the degrees of freedom, both the atomic ones and the field ones.

10 Perturbation theory supports this claim, see the next Subsubsection 4.2.2 and especially equations (99(99) $\begin{array}{rl}& |{\mathrm{{\rm Y}}}_{\left(\mathrm{PT}1\right)}^{\left(2\right)}{⟩}_R\\ & =i\mathfrak{D}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}k\sqrt{\frac{\hslash {\omega }_k}{8\pi }}\frac{{\mathrm{e}}^{-k^2/4\xi }}{\hslash ({\omega }_A+{\omega }_k)}{\hat{a}}^{†}\left(k\right)|\mathrm{vac}{⟩}_R.\end{array}$(99) ), (101(101) $\begin{array}{rl}{\mathcal{E}}_{1,\mathrm{PT}1}^{\left(2\right)}\left(x_1\right)& {=}_R⟨\mathrm{vac}|\hat{\mathcal{E}}\left(x_1\right)|{\mathrm{{\rm Y}}}_{\left(\mathrm{PT}1\right)}^{\left(2\right)}{⟩}_R\\ & =-\frac{\mathfrak{D}}{4\pi }{\int }_0^{+\mathrm{\infty }}\mathrm{d}k\frac{{\omega }_k}{{\omega }_A+{\omega }_k}\\ & \times {\mathrm{e}}^{-k^2/4\xi }\cos \left(k{x}_1\right).\end{array}$(101) ) together withFigure 2.

11 Perturbation theory supports this anticipation, see the Subsubsections 4.3.2 and 4.3.3 below.

12 Footnote¹¹ applies also in the context of the present paragraph.

13 We wish to stress in this context that our nonhermitian formulation given here does not contain any approximations, in contrast to the conventional master equation approaches to SE.

14 An integrand of (117(117) $\begin{array}{rl}{E}_{\mathrm{PT}2,\theta }^{\left(1\right)}& =-\frac{{\mathfrak{D}}^2}{4\pi c}{\int }_0^{+\mathrm{\infty }}\mathrm{d}\omega {\mathrm{e}}^{-i\theta }\frac{\omega {\mathrm{e}}^{-i\theta }}{{\omega }_A+\omega {\mathrm{e}}^{-i\theta }}{\mathrm{e}}^{-{\omega }^2{\mathrm{e}}^{-2i\theta }/2\xi c^2}\\ & =-\frac{{\mathfrak{D}}^2}{4\pi c}{\int }_0^{(+\mathrm{\infty }{\mathrm{e}}^{-i\theta })}\mathrm{d}\omega \frac{\omega }{{\omega }_A+\omega }{\mathrm{e}}^{-{\omega }^2/2\xi c^2}.\end{array}$(117) ) contains just a single pole $\omega =-{\omega }_A$ in the complex ω-plane.

Hence the used integration contour $\{|\omega |{\mathrm{e}}^{-i\theta },|\omega |\in (0,+\mathrm{\infty }\left)\right\}$ can be replaced by the positive real ω-axis.

15 In the above Subsubsection 4.3.1 we have mentioned a critical value of ${\theta }_{\mathrm{crit}}=\arctan \left(\mathrm{\Gamma }/\right(2E_R\left)\right)$. Within the present framework of nonhermitian PT there is no such ${\theta }_{\mathrm{crit}}$, since the relevant unperturbed state $|{\mathrm{\Psi }}_{\left(\mathrm{PT}0\right)}^{\left(2\right)}⟩=|2{⟩}_A|\mathrm{vac}{⟩}_R$ is bound, and gains a nonzero width (imaginary part of the resonance energy) only due to the perturbation (which is proportional to $\mathfrak{D}$).

16 Exactly the same phenomenon is observed in case of the well established NHQM perturbation theory for the Feshbach type resonances [18,19,28].

17 The same conclusion would emerge also in the standard NHQM perturbation theory for Feshbach type resonances [18,19,28].

18 Quantity $E_{\mathrm{PT}2}^{R,\left(2\right)}=\left(137\right)$ is then interpreted as the associated shift of the resonance position.

19 We recall in this context that resonances are ill defined within the hermitian quantum mechanics [19]. The hermitian GR treatment has correspondingly its methodogical drawbacks. Hence, an agreement of the nonhermitian PT2 decay rate (138(138) ${\mathrm{\Gamma }}_{\mathrm{PT}2}^{\left(2\right)}=\frac{{\mathfrak{D}}^2}{2c}{\omega }_A{\mathrm{e}}^{-{\omega }_A^2/2\xi c^2}.$(138) ) with the GR should be interpreted as a NHQM based justification of the GR, rather than as a GR based check of NHQM. This by the way applies not only in the case of SE but also for any other kind of resonances (e.g. of Feshbach type).

Additional information

Funding

D. Š. was supported by the Program COST (European Cooperation in Science and Technology), titled “Towards understanding and modelling intense electronic excitation”, action number CA17126.