The theory of direct laser excitation of nuclear transitions

Abstract

A comprehensive theoretical study of direct laser excitation of a nuclear state based on the density matrix formalism is presented. The nuclear clock isomer $^{229\text {m}}$Th is discussed in detail, as it could allow for direct laser excitation using existing technology and provides the motivation for this work. The optical Bloch equations are derived for the simplest case of a pure nuclear two-level system and for the more complex cases taking into account the presence of magnetic sub-states, hyperfine-structure and Zeeman splitting in external fields. Nuclear level splitting for free atoms and ions as well as for nuclei in a solid-state environment is discussed individually. Based on the obtained equations, nuclear population transfer in the low-saturation limit is reviewed. Further, nuclear Rabi oscillations, power broadening and nuclear two-photon excitation are considered. Finally, the theory is applied to the special cases of $^{229\text {m}}$Th and $^{235\text {m}}$U, being the nuclear excited states of lowest known excitation energies. The paper aims to be a didactic review with many calculations given explicitly.

Appendices

Appendix A: Averaging over fluctuations of the laser field

The influence of phase fluctuations on the excitation of optical transitions has been studied in a number of works, e.g., in Refs. [109, 110]. Here the respective analysis for the particular case of $\delta$-correlated phase fluctuations is presented, described by the equations

$$\begin{aligned} \langle {{\dot{\phi }}}(t) \rangle = 0; \quad \langle {{\dot{\phi }}}(t) {{\dot{\phi }}}(t') \rangle = \varGamma _\ell \delta (t-t'). \end{aligned}$$

(132)

This $\delta$-correlated description takes account for the fact that the typical correlation times of phase fluctuations are significantly shorter than the typical evolution times of the system. The brackets denote the ensemble average and the correlation function defined as [80]

$$\begin{aligned} \langle b(t) \rangle = \text {lim}_{N\rightarrow \infty } \frac{1}{N} \sum _{i=1}^{N} b_i(t) \end{aligned}$$

(133)

and

$$\begin{aligned} \langle b(t)b(t') \rangle = \text {lim}_{N\rightarrow \infty } \frac{1}{N}\sum _{i=1}^{N} b_i(t)b_i(t'), \end{aligned}$$

(134)

respectively. Different indices i denote different systems in the ensemble. This correlation corresponds to a Lorentzian shape of the laser spectrum with $\mathrm{FWHM}=\varGamma _\ell$, as will be shown in the following. First, an averaging of Eq. (21) over these fluctuations is performed. This set of equations is degenerate, but the degeneracy can be waived with the help of the normalization condition $\sum _i \rho _{ii}=1$. It can be written in matrix form as

$$\begin{aligned} {\dot{{\bar{\rho }}}} = \left[ {\mathbb {M}}+ {\mathbb {N}}(t) \right] \cdot {\bar{\rho }}+ {\bar{S}}, \end{aligned}$$

(135)

where the column vector ${\bar{\rho }}$ consists of matrix elements of the density matrix ${\hat{\rho }}$, the matrix ${\mathbb {M}}$ is time-independent, ${\bar{S}}$ is a constant vector appearing due to the normalization condition, and ${\mathbb {N}}(t)$ is a fast fluctuating diagonal matrix proportional to ${\dot{\phi }}$, whose only non-zero elements are $i{\dot{\phi }}$ (corresponding to elements $\rho _{eg}$), and $- i{\dot{\phi }}$ (corresponding to $\rho _{ge}$):

$$\begin{aligned} {\mathbb {N}}(t)= {\dot{\phi }}(t) \, {\mathbb {N}}^0; \quad {\mathbb {N}}^0_{eg,eg}=i; \quad {\mathbb {N}}^0_{ge,ge}=-i\, . \end{aligned}$$

(136)

The solution of Eq. (135) can be represented as a sum of two terms, slowly and rapidly varying in time, as in [24]:

$$\begin{aligned} {\bar{\rho }}(t) = {\bar{\rho }}_0(t)+{\bar{\rho }}^\prime (t). \end{aligned}$$

(137)

The slow term ${\bar{\rho }}_0=\langle {\bar{\rho }}_0 \rangle$ varies with a characteristic time of order of the inverse eigenvalues of the matrix ${\mathbb {M}}$, and the fast term ${\bar{\rho }}'$ varies with a characteristic time of order of the correlation time of the noise terms. It is adopted that the average value of the rapidly oscillating terms vanishes over times much longer than the correlation time of the phase noise terms, but much shorter than the characteristic evolution time of the slow terms; the same is correct for the ensemble average, i.e., $\langle {\bar{\rho }}' \rangle =0$, and ${\bar{\rho }}_0=\langle {\bar{\rho }}_0 \rangle$. Then Eq. (135) gives

$$\begin{aligned} {\dot{{\bar{\rho }}}}_0+\underline{{\dot{{\bar{\rho }}}}'} = {\mathbb {M}}\cdot \left[ {\bar{\rho }}_0+ \underline{{\bar{\rho }}'}\, \right] + \underline{{\mathbb {N}}(t) \left[ {\bar{\rho }}_0+ \underline{{\bar{\rho }}'}\, \right] } + {\bar{S}}. \end{aligned}$$

(138)

The underlined terms fluctuate rapidly, and vanish after having been averaged over the ensemble, as well as over the time longer than the correlation time of the phase noise. The twice underlined term consists of a product of rapidly fluctuating terms and may contain a slowly varying component. This allows to write

$$\begin{aligned} {\dot{{\bar{\rho }}}}_0(t) =&{\mathbb {M}}\cdot {\bar{\rho }}_0(t) + {\bar{S}}+ \langle {\mathbb {N}}(t) {\bar{\rho }}'(t) \rangle , \end{aligned}$$

(139)

$$\begin{aligned} {\dot{{\bar{\rho }}}}'(t) =&{\mathbb {M}}\cdot {\bar{\rho }}'(t) + {\mathbb {N}}(t) {\bar{\rho }}_0(t). \end{aligned}$$

(140)

The solution of Eq. (140) can be written as

$$\begin{aligned} \begin{aligned} {\bar{\rho }}'(t)&= \int \limits _{-\infty }^t e^{{\mathbb {M}}(t-t')} \cdot {\mathbb {N}}(t') \cdot {\bar{\rho }}_0(t') dt' \\&\approx \int \limits _{-\infty }^t e^{{\mathbb {M}}(t-t')} \cdot {\mathbb {N}}(t') dt' \cdot {\bar{\rho }}_0(t), \end{aligned} \end{aligned}$$

(141)

where it was used that ${\bar{\rho }}_0$ is a slow variable. Substituting this into Eq. (139) and using Eqs. (132) and (136), one obtains

$$\begin{aligned} \begin{aligned} {\dot{{\bar{\rho }}}}_0(t)&= \left[ {\mathbb {M}}+ \int \limits _{-\infty }^t \left\langle {\mathbb {N}}(t) \cdot e^{{\mathbb {M}}(t-t')} \cdot {\mathbb {N}}(t') \right\rangle dt' \right] \cdot {\bar{\rho }}_0(t) + {\bar{S}}\\&= \left[ {\mathbb {M}}+ \int \limits _{-\infty }^t \left\langle {\mathbb {N}}(t) \cdot {\mathbb {N}}(t') \right\rangle dt' \right] \cdot {\bar{\rho }}_0(t) + {\bar{S}}\\&= \left[ {\mathbb {M}}+ \frac{\varGamma _\ell }{2}{{\mathbb {N}}^0}^2 \right] \cdot {\bar{\rho }}_0(t) + {\bar{S}}, \\ \end{aligned} \end{aligned}$$

(142)

or

$$\begin{aligned} \langle {\dot{{\bar{\rho }}}}\rangle = \left[ {\mathbb {M}}+ \frac{\varGamma _\ell }{2}{{\mathbb {N}}^0}^2 \right] \cdot \langle {{\bar{\rho }}}\rangle + {\bar{S}}. \end{aligned}$$

(143)

It follows from the form of the matrix ${\mathbb {N}}^0$ [Eq. (136)] that averaging over the fluctuating laser phase leads to the appearance of additional relaxation terms $\varGamma _\ell /2$ for the non-diagonal matrix elements $\rho _{eg}$ and $\rho _{ge}$.

To clarify the physical meaning of $\varGamma _\ell$, the power spectral density $S_E(f)$ of the field is calculated:

$$\begin{aligned} \begin{aligned} E(t)&=E_{0}\left( e^{-i(\omega _\ell t+\phi (t))}+ e^{i(\omega _\ell t+\phi (t))}\right) \\&=2E_{0} \cos (\omega _\ell t+\phi (t)). \end{aligned} \end{aligned}$$

(144)

According to the Wiener–Khinchin theorem [80], the power spectral density $S_E(f)$ of the signal E(t) may be expressed through its autocorrelation function $R_E(\tau )$ as

$$\begin{aligned} S_E(f)=&\int \limits _{-\infty }^{\infty } e^{-2 \pi i f \tau } R_{E}(\tau ) d \tau , \quad \mathrm{where}\end{aligned}$$

(145)

$$\begin{aligned} R_E(\tau )=&\langle E(t) E(t+\tau ) \rangle . \end{aligned}$$

(146)

Here $f=2\pi \omega$ is the ordinary frequency. Suppose that the phase fluctuations are Gaussian, then, using the relation for Gaussian random phases

$$\begin{aligned} \langle \exp \left[ i(\phi (t)-\phi (t^\prime ))\right] \rangle = \exp \left[ -\frac{\langle (\phi (t)-\phi (t^\prime ))^2\rangle }{2}\right] ,\nonumber \\ \end{aligned}$$

(147)

after some algebra, the autocorrelation function $R_E(\tau )$ can be expressed as

$$\begin{aligned} R_E(\tau )= 2 E_{0}^2 \exp \left[ -\frac{\varGamma _\ell |\tau |}{2}\right] \cos (\omega _\ell \tau ). \end{aligned}$$

(148)

Substituting Eq. (148) into Eq. (146), the power spectral density $S_E(f)$ is obtained in the form:

$$\begin{aligned} S_E(f)= & {} 2 E_0^2 \left[ \frac{\varGamma _\ell /2}{ \left( \frac{\varGamma _\ell }{2}\right) ^2+\left( 2 \pi f + \omega _\ell \right) ^2} \right. \nonumber \\&\quad + \left. \frac{\varGamma _\ell /2}{ \left( \frac{\varGamma _\ell }{2}\right) ^2+\left( 2 \pi f - \omega _\ell \right) ^2} \right] . \end{aligned}$$

(149)

Therefore, the phase fluctuations given in Eq. (132) correspond to a Lorentzian spectrum, and $\varGamma _\ell$ is the full linewidth at half maximum.

Appendix B: Derivation of the optical Bloch equations for a solid-state environment

As usual, for the derivation of the optical Bloch equations, the quantum optical master equation in Lindblad form, Eq. (1), is used as the starting point. For the considered case of the solid-state environment, the Lindblad superoperator ${\mathcal {L}}[{\hat{\rho }}]$ has a more complicated form, since it describes, besides the spontaneous decay of the excited level sub-states to the ground level sub-states, also various other relaxation processes due to interaction with the environment. It can be represented as

$$\begin{aligned} {\mathcal {L}}[{\hat{\rho }}]={\mathcal {L}}_\mathrm{sp}[{\hat{\rho }}]+{\mathcal {L}}_\mathrm{od}[{\hat{\rho }}]+{\mathcal {L}}_\text {mix,e}[{\hat{\rho }}] +{\mathcal {L}}_\text {mix,g}[{\hat{\rho }}]+{\mathcal {L}}_\mathrm{md}[{\hat{\rho }}], \nonumber \\ \end{aligned}$$

(150)

where ${\mathcal {L}}_\mathrm{sp}$ contains spontaneous decay of the excited states to the ground states and was already defined in Eq. (70). ${\mathcal {L}}_\mathrm{od}$ models the loss of coherence between excited and ground states due to some random process shifting the levels with respect to each other. In the considered example this is introduced by the laser light with bandwidth $\varGamma _\ell$ and ${\mathcal {L}}_\mathrm{od}$ takes the form

$$\begin{aligned} {\mathcal {L}}_\mathrm{od}[{\hat{\rho }}]= & {} - \frac{\varGamma _\ell }{4} \left[ {\hat{\rho }}-\left( \sum _e {\hat{\sigma }}_{ee} - \sum _g {\hat{\sigma }}_{gg} \right) {\hat{\rho }}\left( \sum _e {\hat{\sigma }}_{ee} - \sum _g {\hat{\sigma }}_{gg} \right) \right] . \nonumber \\ \end{aligned}$$

(151)

${\mathcal {L}}_\text {mix,e}[{\hat{\rho }}]$ and ${\mathcal {L}}_\text {mix,g}[{\hat{\rho }}]$, respectively, describe transitions between different magnetic sub-states due to non-controllable fluctuations of the local magnetic field created, e.g., by spin changes in the environment as well as variations of the local electric field gradient caused by thermal motion of the atoms in the solid-state environment. The population mixing between different sublevels of the ground ($i,i'=g,g'$) or excited ($i,i'=e,e'$) states is defined by the Lindblad operator

$$\begin{aligned} {\mathcal {L}}_\text {mix, i}[{\hat{\rho }}]=\sum _{i, i'}\gamma _{ii'}\left( {\hat{\sigma }}_{ii'}{\hat{\rho }}{\hat{\sigma }}^\dagger _{ii'}-\frac{1}{2}{\hat{\rho }}{\hat{\sigma }}^\dagger _{ii'}{\hat{\sigma }}_{ii'}-\frac{1}{2}{\hat{\sigma }}^\dagger _{ii'}{\hat{\sigma }}_{ii'}{\hat{\rho }}\right) , \nonumber \\ \end{aligned}$$

(152)

where $\gamma _{gg'}$ and $\gamma _{ee'}$ denote the decay rates between different sub-states of the same level. ${\mathcal {L}}_\mathrm{md}$ takes account for loss of coherence between different magnetic sub-states caused by the same reasons. This can be described by

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_\mathrm{md}[{\hat{\rho }}]&=\sum _{g\ne g^\prime }\frac{{\tilde{\gamma }}_{gg'}}{2} \left[ ({\hat{\sigma }}_{gg}-{\hat{\sigma }}_{g'g'}){\hat{\rho }}({\hat{\sigma }}_{gg}-{\hat{\sigma }}_{g'g'}) \right. \\&\quad \left. - \frac{1}{2} \left\{ ({\hat{\sigma }}_{gg}+{\hat{\sigma }}_{g'g'}),{\hat{\rho }}\right\} \right] \\&\quad + \sum _{e\ne e^\prime }\frac{{\tilde{\gamma }}_{ee'}}{2} \left[ ({\hat{\sigma }}_{ee}-{\hat{\sigma }}_{e'e'}){\hat{\rho }}({\hat{\sigma }}_{ee}-{\hat{\sigma }}_{e'e'}) \right. \\&\quad \left. - \frac{1}{2} \left\{ ({\hat{\sigma }}_{ee}+{\hat{\sigma }}_{e'e'}),{\hat{\rho }}\right\} \right] , \end{aligned} \end{aligned}$$

(153)

where ${\tilde{\gamma }}_{gg'}$ and ${\tilde{\gamma }}_{ee'}$ denote the decoherence rates between different sub-states. The complete set of sub-state optical

Bloch equations in case of laser excitation in a solid-state environment is then obtained in the form [52, 76]

$$\begin{aligned} \begin{aligned} {\dot{\rho }}_{ee}&= -i \sum _{g} \frac{\varOmega _{eg}}{2} \Big [ \rho _{ge}-\rho _{eg}\Big ] - \sum _{g} \varGamma _{ge} \rho _{ee}\\&+ \sum _{e'} \gamma _{ee'} (\rho _{e'e'}-\rho _{ee}); \\ {\dot{\rho }}_{ge}&=-i\sum _{{e'}} \frac{\varOmega _{e'g}}{2} \rho _{e'e} +i\sum _{{g'}} \frac{\varOmega _{eg'}}{2} \rho _{gg'}\\&-\rho _{ge}(i\varDelta \omega _{eg}+{\tilde{\varGamma }}_{ge}); \\ {\dot{\rho }}_{gg}&= -i \sum _{e} \frac{\varOmega _{eg}}{2}\Big [\rho _{eg}-\rho _{ge}\Big ] + \sum _{e} \varGamma _{ge} \rho _{ee}\\&+ \sum _{g'} \gamma _{gg'} (\rho _{g'g'}-\rho _{gg}); \\ {\dot{\rho }}_{ee'}&= -i \sum _{g} \Big [ \frac{\varOmega _{eg}}{2} \rho _{ge'}- \frac{\varOmega _{e'g}}{2} \rho _{eg} \Big ]\\&- \rho _{ee'} (i \omega _{ee'} + {\tilde{\varGamma }}_{ee'}), \quad e \ne e'; \\ {\dot{\rho }}_{gg'}&= -i \sum _{e} \Big [ \frac{\varOmega _{eg}}{2} \rho _{eg'}- \frac{\varOmega _{eg'}}{2} \rho _{ge} \Big ]\\&-\rho _{gg'}(i \omega _{gg'} + {\tilde{\varGamma }}_{gg'}), \quad g \ne g'. \end{aligned} \end{aligned}$$

(154)

As previously, $\omega _{ij}$ and $\omega _{eg}$ were defined as $\omega _{ij}=\omega _i-\omega _j$ and $\varDelta \omega _{eg}=\omega _\ell -\omega _{eg}$. This time, the decay rates of the coherences are obtained in the form:

$$\begin{aligned} {\tilde{\varGamma }}_{ge}&=\sum _{g'}\frac{\varGamma _{g'e}+\gamma _{g'g}}{2} +\sum _{e'}\frac{\gamma _{e'e}}{2}+\frac{\varGamma _\ell }{2} \nonumber \\&\quad +\sum _{g''\ne g}\frac{{{\tilde{\gamma }}}_{g g''}+{{\tilde{\gamma }}}_{g'' g}}{4} +\sum _{e''\ne e}\frac{{{\tilde{\gamma }}}_{e e''}+{{\tilde{\gamma }}}_{e'' e}}{4},\nonumber \\ {\tilde{\varGamma }}_{ee'}&=\sum _{g} \frac{\varGamma _{ge}+\varGamma _{ge'}}{2} +\sum _{e''}\frac{\gamma _{e''e}+\gamma _{e''e'}}{2} \\&\quad +\frac{{{\tilde{\gamma }}}_{ee'}+{{\tilde{\gamma }}}_{e'e}}{2}+ \sum _{e''\ne e} {{\tilde{\gamma }}}_{ee''} +\sum _{e''\ne e'} {{\tilde{\gamma }}}_{e''e'},\nonumber \\ {\tilde{\varGamma }}_{gg'}&=\sum _{g''}\frac{\gamma _{g''g}+\gamma _{g''g'}}{2} +\frac{{{\tilde{\gamma }}}_{gg'}+{{\tilde{\gamma }}}_{g'g}}{2}\nonumber \\&\quad +\sum _{g''\ne g} {{\tilde{\gamma }}}_{gg''} +\sum _{g''\ne g'} {{\tilde{\gamma }}}_{g''g'}.\nonumber \end{aligned}$$

(155)

In the case that $\varOmega _{eg}^2/{\tilde{\varGamma }}_{ge}$ is small compared to the energy differences $\omega _{gg'}$ and $\omega _{ee'}$, or in comparison with the decoherence rates ${\tilde{\varGamma }}_{ee'}$ and ${\tilde{\varGamma }}_{gg'}$, again it is possible to neglect the two-photon coherences $\rho _{ee'}$ and $\rho _{gg'}$ and Eq. (154) transforms to Eq. (85).