Oscillations of Light during a Photon Echo: Observation and Application

Abstract

An ultra high-resolution optical spectroscopy technique is proposed for measuring Zeeman and pseudo-Stark frequency splitting during optical transitions. This approach uses the change in the time shape of the echo signal upon a a weak pulse perturbation that splits the optical transition frequencies of two or more echo-active ion subgroups.

PHOTON ECHO IN A PULSED MAGNETIC FIELD

In [1], we showed for the first time that if nanosecond pulses of a weak magnetic field parallel to the C axis are applied in the intervals between the first and second laser pulses (or between the second laser pulse and the echo signal) in observing light echoes in YLiF₄:Er³⁺ and LuLiF₄:Er³⁺ crystals, oscillations of the echo signal’s intensity are observed that depend on the pulses’ amplitude. The g-factors of the ground and excited states of the Er³⁺ ions in these crystals in a zero static magnetic field can be determined from the period of oscillations.

To understand the experimental results, we should recognize that the Er³⁺ ions in LiYF₄ and LiLuF₄ matrices are Kramers ions, so the lower crystal levels of the ground ⁴I_15/2(I) and excited ⁴F_9/2(I) multiplets have two-fold spin degeneracy (see Fig. 1). Both groups of Er³⁺ ions, having S_z = 1/2 as well as S_z = −1/2 spin projections in the ground state, are excited upon a resonance light pulse.

Fig. 1.

Energy levels for erbium in LiYF₄ and LiLuF₄. Transitions for different laser pulse polarizations are marked by arrows.

In magnetic field H directed parallel to the С axis, the optical frequencies of the transitions of each ion group change by a different signed value, depending on the spin projection:

$$\begin{gathered} \Delta \nu = {{S}_{z}}(g{'} \pm g){{H\beta } \mathord{\left/ {\vphantom {{H\beta } {{{h}_{{{\text{pl}}}}}}}} \right. \kern-0em} {{{h}_{{{\text{pl}}}}}}} = {{S}_{z}}{{Z}_{{\pi ,\sigma }}} \\ {{Z}_{{\pi ,\sigma }}} = (g{'} \pm g){{H\beta } \mathord{\left/ {\vphantom {{H\beta } \hbar }} \right. \kern-0em} \hbar }, \\ g{'} = g{{(}^{4}}{{F}_{{{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-0em} 2}}}}),\,\,\,\,\,g = g{{(}^{4}}{{I}_{{{{15} \mathord{\left/ {\vphantom {{15} 2}} \right. \kern-0em} 2}}}}), \\ \end{gathered} $$

((1))

where β is a Bohr magneton; h_pl is the Planck constant; S_z is the С component of the total orbital (and thus the total spin moment of the Er³⁺ ion); and g'- and g are the g-factor components of the excited ⁴F_9/2(I) and ground ⁴I_15/2(I) states parallel to the C axis. In (1), the positive sign denotes σ- polarization of laser pulses; the negative sign, to π-polarization. Zeeman splitting Z_π,σ corresponds to transitions under the effect of π- and σ-polarized light.

If a magnetic pulse (MP) of amplitude H and length τ_h acts in the intervals between the first and second laser pulses, or between the second laser pulse and the echo signal, the dipole moment for each ion group acquires an additional phase:

$$\alpha = \pm 2\pi \frac{{\partial \nu }}{{\partial H}}H{{\tau }_{{\text{h}}}},\,\,\,\,\frac{{\partial \nu }}{{\partial H}} = \frac{1}{2}\frac{\beta }{{{{h}_{{{\text{pl}}}}}}}(g{'} \pm g).$$

((2))

The total dipole moment is proportional to cosα, and the relative change in the intensity of the photon echo is equal to the square of the angle’s cosine:

$$\frac{I}{{{{I}_{0}}}} = {{\cos }^{2}}\alpha .$$

((3))

Oscillations in the intensity of the photon echo are therefore caused by the phase shift of the precessing dipole moments under the effect of a pulsed magnetic field. Dependence (3) is oscillatory in nature, and the period of oscillations is determined by the area of the magnetic pulse.

In [1], however, we did not consider a case where the magnetic pulse is applied during echo signal generation. If this should happen, the shape and intensity of the photon echo signal change strongly because of the Zeeman splitting of the ground and excited states of the erbium ion [2]. This is discussed below.

The experimental details were thoroughly described in [1], so we now present only the main points and some differences. The photon echo was observed in the reverse mode upon the ⁴I_15/2 → ⁴F_9/2 transition of an Er³⁺ ion at a temperature of 2 K. The polarization of the laser pulses could be set parallel (π‑polarization) or perpendicular (σ-polarization) to the C axis using a half-wave plate. The LuLiF4:Er3⁺ and YLiF₄:Er³⁺ samples were placed in solenoids consisting of two coaxial coils of seven turns each. The focused radiation of the first and second laser pulses was directed to the 1 mm wide gap between the coils. Optical axis C of the samples coincided with the solenoid axes. The magnetic field for each solenoid was calculated using the MatLab software.

When the magnetic pulse was applied during echo signal generation, the main difference with [1] was that the duration of the magnetic pulse did not matter as long, as it was greater than the duration of the photon echo signal.

If the direction of the field is constant, the above holds for an amplitude-varying magnetic field as well. A time-dependent phase in this case develops that describes the temporal evolution of the wave function in the magnetic field, which can be expressed as an integral of the perturbation over time [2, 3]. The relative phase shift between the groups of optically excited dipoles with spin projection ±1/2 in the ground state becomes non-zero for the duration of a pulse,

$$\varphi \left( t \right) = \int\limits_{{{t}_{0}}}^t {Z\left( {t{'}} \right)} dt{'} = \left[ {g{'} \pm g} \right]\frac{\beta }{\hbar }\int\limits_{{{t}_{0}}}^t {H\left( {t{'}} \right)} dt{'}{\text{,}}$$

((4))

and does not depend on the location of the dipole frequency in a nonuniform line. In (4), t₀ is the starting time of a magnetic pulse. The echo signal develops intense oscillations according to phase φ(t):

$$I(t) = {{I}_{0}}(t)(1 + \cos {{\varphi (t))} \mathord{\left/ {\vphantom {{\varphi (t))} 2}} \right. \kern-0em} 2},$$

((5))

where I₀(t) is the intensity of the echo when there is no magnetic pulse.

These oscillations of the photon echo signal were observed in LuLiF₄ samples with erbium concentrations of 0.025 and 0.1%, and in YLiF₄ samples with erbium concentrations of 0.1%, depending on the laser radiation’s polarization. Figure 2 shows an oscillogram illustrating the onset of photon echo oscillations when a magnetic pulse is applied during echo signal generation.

Fig. 2.

Oscillations of the photon echo signal in LiLuF₄:Er³⁺crystal that occured when a magnetic field pulse of 13.3 G was introduced for the π-polarized laser pulses.

It follows from (4) and (5) that the period of the photon echo signal’s oscillation is determined only by the amplitude of the magnetic pulse at any time and not by its duration. Knowing amplitude H and the period of the photon echo signal’s oscillation, we can obtain the (g' + g) values for σ-polarization or the (g' – g) values for π-polarization of a laser pulse from a single oscillogram. To improve the accuracy of the experimental results, oscillograms of the photon echo oscillations were recorded for different laser pulse polarizations and magnetic pulse amplitudes. The inverse of the oscillations period was plotted with respect to H [3], and we obtained

$$\begin{gathered} \left. \begin{gathered} {{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}_{{{\text{exp}}}}} = 18.069\,\,{\text{MHz}}\,\,{{{\text{G}}}^{{ - 1}}}, \hfill \\ {{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}_{{{\text{exp}}}}} = 9.274\,\,{\text{MHz}}\,\,{{{\text{G}}}^{{ - 1}}}, \hfill \\ \end{gathered} \right\}{\text{YLi}}{{{\text{F}}}_{{\text{4}}}}{\text{:E}}{{{\text{r}}}^{{{\text{3 + }}}}} \hfill \\ \left. \begin{gathered} {{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}_{{{\text{exp}}}}} = 17.716\,\,{\text{MHz}}\,\,{{{\text{G}}}^{{ - 1}}}, \hfill \\ {{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}_{{{\text{exp}}}}} = 9.156\,\,{\text{MHz}}\,\,{{{\text{G}}}^{{ - 1}}}, \hfill \\ \end{gathered} \right\}{\text{LuLi}}{{{\text{F}}}_{{\text{4}}}}{\text{:E}}{{{\text{r}}}^{{{\text{3 + }}}}}{\text{.}} \hfill \\ \end{gathered} $$

((6))

Since the phase shift over a period is 2π, we write an approximation:

$$\begin{gathered} \varphi (t + T) - \phi (t) = 2\pi = \int\limits_t^{t + T} {Z(t')} dt' \approx ZT, \\ Z \approx (g{\text{'}} \pm g){{\beta H} \mathord{\left/ {\vphantom {{\beta H} \hbar }} \right. \kern-0em} \hbar }, \\ \end{gathered} $$

((7))

which actually holds only on flat regions of the magnetic pulse. The Zeeman splitting can now be expressed in terms of the inverse the period of oscillations:

$$Z = (g{\text{'}} \pm g){{\beta H} \mathord{\left/ {\vphantom {{\beta H} \hbar }} \right. \kern-0em} \hbar } \approx {{2\pi } \mathord{\left/ {\vphantom {{2\pi } T}} \right. \kern-0em} T},$$

((8))

and the sum and difference between the g-factors can be expressed in terms of experimentally measured values:

$$\begin{gathered} {{\partial {{Z}_{{\sigma ,\pi }}}} \mathord{\left/ {\vphantom {{\partial {{Z}_{{\sigma ,\pi }}}} {\partial H}}} \right. \kern-0em} {\partial H}} = (g{'} \pm g){\beta \mathord{\left/ {\vphantom {\beta \hbar }} \right. \kern-0em} \hbar } \\ = (g{'} \pm g)\,\,{\text{1}}{\text{.3996(MHz}}\,\,{{{\text{G}}}^{{ - 1}}}{\text{)}} = 2\pi \partial {{\left. {{{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{{\sigma ,\pi }}}}}} \right. \kern-0em} {{{T}_{{\sigma ,\pi }}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{{\sigma ,\pi }}}}}} \right. \kern-0em} {{{T}_{{\sigma ,\pi }}}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}_{{{\text{exp}}}}}. \\ \end{gathered} $$

((9))

From expressions (6) and (9), we easily obtain the difference (for π-polarization), the sum (for the σ-polarization), and the values of the g-factors for the excited and ground states:

$$\begin{gathered} \left. \begin{gathered} g{'} + g = 12.908 \hfill \\ g{'} - g = 6.622 \hfill \\ \end{gathered} \right\}g = {\text{3}}{\text{.143}}{\text{,}}\,\,g{'} = 9.765,\,\,{\text{YLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}}, \hfill \\ \left. \begin{gathered} g{'} + g = 12.658 \hfill \\ g{'} - g = 6.543 \hfill \\ \end{gathered} \right\}g = {\text{3}}{\text{.058}}{\text{,}}\,\,g{'} = 9.60,\,\,{\text{LuLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}}. \hfill \\ \end{gathered} $$

((10))

Note that this way of determining the g-factors allows for some error in measuring the impulse amplitude of the current and the oscilloscope’s time base. However, the relation between the sum of the g-factors of the excited and ground states and their difference does not depend on the magnitude of the magnetic field or the period of oscillation. This allows us to exclude such errors in, e.g., determining the g-factor of an excited state with a known value of the ground state’s g-factor. Using the measurement results in (6), we obtain

$$\begin{gathered} \frac{{{{{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\sigma }}}}} \right. \kern-0em} {{{T}_{\sigma }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}}_{{{\text{exp}}}}}}}{{{{{\left. {\partial {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 {{{T}_{\pi }}}}} \right. \kern-0em} {{{T}_{\pi }}}}} \right)} {\partial H}}} \right. \kern-0em} {\partial H}}} \right|}}_{{{\text{exp}}}}}}} = \frac{{g{'} + g}}{{g{'} - g}} \\ = \left\{ \begin{gathered} 1.95,\,\,{\kern 1pt} \;{\text{YLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}},\,\,g = 3.14 \Rightarrow g{'} = 9.755, \hfill \\ 1.93,\,\,\;{\text{LuLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}},\,\,g = 3.06 \Rightarrow g{'} = 9.61. \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

((11))

The g-factors for the excited state therefore virtually coincide with the results presented above (10). This confirms the accuracy of our calculations for a magnetic field in the coils of a solenoid.

The obtained g-factors for the ground state are in good agreement with the literature data. Values $g{'}$ for YLiF₄:Er³⁺ differ considerably among authors. In the literature, there is no data on the g-factor of excited state ⁴F_9/2 in LuLiF₄:Er³⁺. Its value is based on the intensity of photon echo modulation in [1], where several factors affecting the results were ignored:

$$\begin{gathered} {\text{YLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}} \Rightarrow g = {\text{3}}{\text{.14}}\,\,{\text{[4]}}{\text{,}} \\ g{'} = 9.84 \pm 0.25\,\,\,[5],\,\,\,\,g{'} = 9.61\,\,[6], \\ {\text{LuLi}}{{{\text{F}}}_{4}}{\text{:E}}{{{\text{r}}}^{{{\text{3}} + }}} \Rightarrow g = {\text{3}}{\text{.09}}\,\,{\text{[7]}}{\text{,}}\,\,\,\,g{'} = 10.3\,\,[1]. \\ \end{gathered} $$

((12))

PHOTON ECHO IN A PULSED ELECTRIC FIELD

Modulation of the intensity and oscillations in the shape of the signal of a photon echo in a pulsed magnetic field were first observed by the authors. The modulation of the photon echo intensity in a pulsed electric field was observed in [8, 9], where the parameters of the pseudo-Stark splitting of Er³⁺ and Eu³⁺ ions were obtained for YalO₃ crystals. However, oscillations in the shape of a photon echo’s signal in a pulsed electric field had never been detected earlier [10].

Let us consider the possibility of oscillations in the intensity of a photon echo occurring when a pulsed electric field is switched on, and the conditions needed for this.

Optical electric dipole transitions in ions that occur between states of the same parity $\left| 1 \right\rangle$ and $\left| 2 \right\rangle$ become possible if the local symmetry of the ion is not centrally symmetrical [11]. The odd component of crystal field V_u then adds to the wave function the high energy states of the opposite parity:

$$\begin{gathered} \left| {\tilde {1}} \right\rangle = \left| 1 \right\rangle + \sum\limits_u {\left| u \right\rangle } {{\left\langle {u\left| {{{V}_{u}}} \right|1} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {u\left| {{{V}_{u}}} \right|1} \right\rangle } {({{E}_{u}}}}} \right. \kern-0em} {({{E}_{u}}}} - {{E}_{1}}), \\ \left| {\tilde {2}} \right\rangle = \left| 2 \right\rangle + \sum\limits_u {\left| u \right\rangle } {{\left\langle {u\left| {{{V}_{u}}} \right|2} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {u\left| {{{V}_{u}}} \right|2} \right\rangle } {({{E}_{u}}}}} \right. \kern-0em} {({{E}_{u}}}} - {{E}_{2}}), \\ {{d}_{{21}}} = \left\langle {\tilde {2}\left| {\vec {d}} \right|\tilde {1}} \right\rangle = \sum\limits_u {\left\langle {2\left| {\vec {d}} \right|u} \right\rangle } {{\left\langle {u\left| {{{V}_{u}}} \right|1} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {u\left| {{{V}_{u}}} \right|1} \right\rangle } {({{E}_{u}}}}} \right. \kern-0em} {({{E}_{u}}}} - {{E}_{1}}) \\ + \,\,\sum\limits_u {\left\langle {2\left| {{{V}_{u}}} \right|u} \right\rangle } {{\left\langle {u\left| {\vec {d}} \right|1} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {u\left| {\vec {d}} \right|1} \right\rangle } {({{E}_{u}}}}} \right. \kern-0em} {({{E}_{u}}}} - {{E}_{2}}). \\ \end{gathered} $$

((13))

Electric dipole transitions are allowed. For the same reason, the diagonal components of the electric dipole moment are distinct from zero; i.e., the energy levels shift by a value that depends linearly on the amplitude of the local electric field, $\alpha \vec {E}\,:$

$$\Delta {{E}_{i}} = - \alpha \vec {E}{{\vec {d}}_{{ii}}},\,\,\,\,i = 1,2,$$

((14))

where Е is the external electric field and α accounts for polarization effects caused by the ion shift and the electron cloud deformations in the crystal.

If the crystal lattice is centrally symmetrical, we require substitution centers in the lattice such that their local symmetry lacks inversion and the the odd component of the crystal field has the opposite sign. This is true for, e.g., ruby, where two groups of paramagnetic chrome ions are in nonequivalent positions. As follows from (13), the energy levels of groups A and B of Сr³⁺ ions with positive and negative odd components of the crystal field shift in an electric field by $\Delta {{E}_{i}}(A) = - \Delta {{E}_{i}}(B)$ values with the same magnitude but the opposite sign (see Fig. 3), and the optical transition frequency is split:

$$\begin{gathered} \delta \nu = \pm {{Z{\text{*}}} \mathord{\left/ {\vphantom {{Z{\text{*}}} 2}} \right. \kern-0em} 2},\,\,\,\,Z^{*} = 2\frac{{\partial \nu }}{{\partial E}}E, \\ \frac{{\partial \nu }}{{\partial E}} = \alpha {{\left[ {{{d}_{{22}}}(A) - {{d}_{{11}}}(A)} \right]} \mathord{\left/ {\vphantom {{\left[ {{{d}_{{22}}}(A) - {{d}_{{11}}}(A)} \right]} {(2\pi \hbar )}}} \right. \kern-0em} {(2\pi \hbar )}}. \\ \end{gathered} $$

((15))

Fig. 3.

Ruby crystal lattice structure where the chrome ions are located at two nonequvalent positions A and В. The energy levels of chrome ions in a constant electric field are shown on the right. Solid lines represent the upper and lower Kramers doublets for one of the nonequvalent Cr³⁺positions; the dashed lines, the doublets for the other nonequvalent position.

Here, the plus and minus signs correspond to A and B Сr³⁺ ions, and $Z{\text{*}}$ is the pseudo-Stark frequency. It is equal to the frequency difference between the optical line components produced due to line splitting in a pulsed electric field.

If the electric field amplitude is time-dependent, the relative phase shift between the A and B groups of optically excited dipoles is distinct from zero during a pulse, and the intensity of the light radiated during the echo flash oscillates:

$$I(t) = {{I}_{0}}(t){{\left\langle {\cos \frac{{\varphi (t)}}{2}} \right\rangle }^{2}},$$

((16))

where I₀(t) is the intensity of the light radiated when there is no electric pulse, and $\left\langle {} \right\rangle$ denotes averaging over the optically excited volume.

The time-dependent phase can be expressed as an integral of the perturbation over time (15), as was done in [2, 3] for a magnetic pulse. If t₀ is the starting time of the electric pulse, the phase change over a period of 2π can be written as

$$\begin{gathered} \left\langle {\varphi (t + T)} \right\rangle - \left\langle {\varphi (t)} \right\rangle = 2\pi \\ = 2\pi \int\limits_t^{t + T} {\left\langle {Z{\text{*}}(t{'})} \right\rangle } dt' \approx 2\pi \left\langle {Z{\text{*}}} \right\rangle T, \\ \end{gathered} $$

((17))

and the pseudo-Stark frequency can be expressed in terms of the period of inverse oscillation:

$$\left\langle {Z{\text{*}}} \right\rangle = 2\frac{{\partial \nu }}{{\partial E}}\left\langle E \right\rangle \approx {1 \mathord{\left/ {\vphantom {1 T}} \right. \kern-0em} T}.$$

((18))

The photon echo in ruby occurred during the ²E → ⁴A₂ transition of the R₁ line of the chrome ions in the crystal at a temperature of 2 K. Ruby samples with different contents of Cr nuclei isotopes d = 4.2 mm (Cr⁵³) and d = 4.5 mm (Cr⁵²) thick were placed in a capacitor [10]. Nanosecond voltage pulses V with amplitudes of up to 260 V were applied to the capacitor plates. The electric field in the sample was calculated using the Comsol software. For this experimental geometry, it was found to be $\left\langle E \right\rangle = 0.987V{{d}^{{ - 1}}}.$

When an electric pulse is applied simultaneously with a photon echo flash, the shape of the echo response changed considerably and oscillations occurred in the echo signal’s intensity. Using value 1/Т inverse to the period of photon echo oscillations, the pseudo-Stark splitting parameters were determined for different electric field values and ruby samples (see Fig. 4):

$$\begin{gathered} {{2\partial \nu } \mathord{\left/ {\vphantom {{2\partial \nu } {\partial E}}} \right. \kern-0em} {\partial E}} = \partial {{\left( {{1 \mathord{\left/ {\vphantom {1 T}} \right. \kern-0em} T}} \right)} \mathord{\left/ {\vphantom {{\left( {{1 \mathord{\left/ {\vphantom {1 T}} \right. \kern-0em} T}} \right)} \partial }} \right. \kern-0em} \partial }\left\langle E \right\rangle \\ = \left. {0.214\,\,{\text{MHz}}\,\,{\text{V}}\,\,{\text{c}}{{{\text{m}}}^{{{\text{--1}}}}}} \right\}{{{\text{ }}}^{{{\text{52}}}}}{\text{Cr}},\,{{\,}^{{{\text{53}}}}}{\text{Cr}}. \\ \end{gathered} $$

((19))

Fig. 4.

Dependence of value 1/Т inverse to the period of photon echo oscillation on the magnitude of a pulsed electric field 40 ns long for different samples of ruby.

Generally speaking, the ${{2\partial \nu } \mathord{\left/ {\vphantom {{2\partial \nu } {\partial E}}} \right. \kern-0em} {\partial E}}$ parameter for ruby was obtained earlier by other authors; however, despite the use of strong electric fields of ~10⁵ V cm⁻¹, the measurements were not precise, since the obtained values were scattered: ${{2\partial \nu } \mathord{\left/ {\vphantom {{2\partial \nu } {\partial E}}} \right. \kern-0em} {\partial E}} = {\text{0}}.176\,\,{\text{[11]}}{\text{,}}$${{2\partial \nu } \mathord{\left/ {\vphantom {{2\partial \nu } {\partial E}}} \right. \kern-0em} {\partial E}} = {\text{0}}{\text{.220}} \pm {\text{0}}{\text{.016}}\,\,{\text{[12]}}{\text{.}}$

CONCLUSIONS

It is possible to substantially alter the resulting dipole moment of a system by controlling the relative phase of excited dipoles with small perturbations. With a photon echo, this can be apparent both as oscillations in the intensity of the photon echo and as oscillations in the shape of the echo response. In this work, the g-factors of the ground and excited states, and the pseudo-Stark splitting of optical lines, were measured for different samples from the period of oscillations and known amplitudes, and the electric parameters of ions were determined with high degrees of precision. Note that because of the high accuracy of the obtained results, the magnetic and electric fields we used were several orders of magnitude weaker than those employed in traditional techniques.