Fast electromagnetic response of a thin film of resonant atoms with permanent dipole

One of the famous models describing field-matter interaction is the Maxwell-Bloch system [1]. It corresponds to an ensemble of two-level atoms whose states alternate due to the electromagnetic field. In a simple approach the electromagnetic field is assumed to be scalar and the operator of dipole transition to only have non-diagonal matrix elements. This model was the base for the description of many coherent nonlinear effects, the coherent pulse propagation (self-induced transparency [2]) and the coherent transient effects (optical nutations, free induction decay, quantum beats, superradiance, photon echo). A detailed review of these phenomena may be found in [3, 4] and in the book [1].

The Maxwell-Bloch system can be extended further than the model of two-level atoms discussed above. In particular one can generalize the resonant atomic model. The model of three-level atom now attracts great interest, because it describes electromagnetic induced transparency, slow light propagation in (three-level) atomic vapor [5] and the coherent population transfer [6]. The coherent interaction of electromagnetic pulses with quantum dots [7, 8] can be described by such a generalized Maxwell-Bloch system. Another generalization of the two-level model is to take into account diagonal matrix elements of the dipole transition operator [12]. In this medium steady state one-half cycle pulses were obtained in the sharp line limit [13]. A new kind of steady state pulse was found, characterized by an algebraic decay of the electric field.

The reduced Maxwell-Bloch equation in the sharp line limit have a zero-curvature representation[14, 15]. A number of results related with the complete integrability of this reduced system have been obtained in [16, 17, 18]. The numerical simulation of the propagation of extremely short pulses [19] shows the existence of an extraordinary breather with non zero pulse square. An analytical expression for this breather has been found in [14, 15] and reproduced in [20] using the Darboux transformation method. Recently Zabolotzki[21] developed the inverse scattering method to find the solution for the isotropic limit of the general model of a two-component electromagnetic field interacting with two-level atoms with a permanent dipole moment without invoking the slowly varying envelope approximation.

The influence of the permanent dipole on parametric processes was studied in [9, 10]. Recently Weifeng et al studied the generation of attosecond pulses in a two-level system with permanent dipole moment[11]. For this, higher harmonics are generated and the spectrum can be extended to the X-ray range. The quantum interference of both even and odd harmonics results in the generation of higher intensity attosecond pulses.

The propagation of ultrashort pulses propagation through a thin film containing resonant atoms located at the interface between two dielectrics is also described by the Maxwell-Bloch equations. However, if the width of the film is less than a wave length [22], the atomic system is compressed into a ”point”. An insightful comment was made in [23, 24], concerning the critical role of the local Lorentz field. This local field induces a nonlinearity so that the thin film of two-level atoms acts as a nonlinear Fabry-Perot resonator. One then expects optical bistability for this device. There are many generalizations of the thin film model, where three (or more) level atoms, two-photon transition between resonant levels and non-resonant nonlinearity were studied. Here we consider a thin film corresponding to two-level atoms with permanent dipole moment [25]. There the author analyzed numerically the pulse propagation through the film taking into account the local field. He showed that a dense film irradiated by a one-circle pulse emits a short response with a delay much longer than the characteristic cooperative time of the atom ensemble. Contrary to [25] we assume that the atoms of the film are rare so that the local field can be neglected.

Recently [28] we introduced a general formalism to describe the interaction of a (linear polarized) electromagnetic pulse with a medium. With this we studied a ferroelectric film which was described by a Duffing oscillator naturally giving a double well potential. Here we generalize this approach to the case of a layer of resonant atoms described by the generalized Bloch equations taking into account the permanent dipole moment. An important feature of the model is that there is a clear separation between the medium and the surrounding vacuum so that no dispersion relation can be written. Instead we obtain a scattering problem where the reflection and transmission coefficients need to be calculated.
After presenting the model in section 2, we compute its equilibria and their stability in section 3. In section 4 we assume an additional periodic modulation around a fixed background field. This enables to do a spectroscopy study of the film so that both the dipole parameter and the coupling parameter can be extracted from the reflection curve. Section 5 concludes the article.

We now study the stationary points of the system (23). Initially before the wave reaches it, the film is at rest. The electromagnetic field $e_{0}$ shifts the film state to a new equilibrium. The system then relaxes back to its original state after the wave has passed. We will examine these new transient states and their stability.

For $e_{0}$ constant, the system of equations (21-23) has the fixed point $(r_{1}^{\ast},0,r_{3}^{\ast})$ where $r_{1,3}^{\ast}$ satisfy

$$(1+\mu e_{0})r_{1}^{\ast}+e_{0}r_{3}^{\ast}=0.$$

(27)

Assuming $r_{2}^{\ast}\neq 0$ leads to a contradiction. From (19) it follows that

$$r_{1}^{\ast 2}+r_{3}^{\ast 2}=1$$

(28)

Combining these two equations we obtain

$$r_{3}^{\ast}=\pm[1+{e_{0}^{2}\over(1+\mu e_{0})^{2}}]^{-1/2},~~r_{1}^{\ast}=-{e_{0}\over 1+\mu e_{0}}r_{3}^{\ast}.$$

(29)

This fixed point corresponds to a stationary polarization and population induced in the two-level atoms by the incident constant field $e_{0}$. When $e_{0}\rightarrow\infty$

$$r_{3}^{\ast}\rightarrow\pm{\mu\over\sqrt{1+\mu^{2}}},~~~r_{1}^{\ast}\rightarrow-{r_{3}^{\ast}\over\mu}.$$

(30)

To understand geometrically the position of the fixed points, we can parametrize the Bloch sphere with $r_{2}^{\ast}=0$ by

$$r_{1}^{\ast}=\sin\theta,~~r_{3}^{\ast}=\cos\theta,$$

yielding the relation

$$\tan\theta=-{e_{0}\over 1+\mu e_{0}}.$$

(31)

The dependence of the values $r_{1,3}^{\ast}$  on the electric field $e_{0}$ is monotonic so there is no bistability. The two fixed points are shown in Fig. 2 for $\mu=\pm 0.1,~\pm 1$. For $\mu$ small (left panel) the fixed points start from $(0,\pm 1)$ and rotate following $(\sin\theta,\cos\theta)$ with $\theta$ given by (31). For $\mu=-1$ and $e_{0}=1$ $\theta=\pi/2+n\pi$ so we obtain $(\pm 1,0)$. Since $r_{3}$ is the difference in the population of the levels, these are equally populated. When $\mu$ is large and positive (top right panel) the fixed points do not change very much when the electric field is increased. On the contrary when $\mu$ is large and negative (bottom right panel) increasing the electric field shifts the fixed points from $(0,\pm 1)$ to $(\pm 1,0)$ for $e_{0}=1$ and about $(\pm 1/\sqrt{2},\pm 1/\sqrt{2})$ for $e_{0}=5$. For this last value of the parameter we can have a strong population inversion.

To study the stability of the fixed point we linearize the modified Bloch equations. We set $r_{1}=r_{1}^{\ast}+\delta r_{1}$, $r_{2}=0+\delta r_{2}$, $r_{3}=r_{3}^{\ast}+\delta r_{3}$. The linearized modified Bloch equations read

	$\displaystyle{\delta r_{1}}_{\tau}$	$\displaystyle=$	$\displaystyle-(1+\mu e_{0})\delta r_{2},$
	$\displaystyle{\delta r_{2}}_{\tau}$	$\displaystyle=$	$\displaystyle(1+\mu e_{0})\delta r_{1}+e_{0}\delta r_{3}+(\alpha\mu/2)r_{1}^{\ast}\delta r_{2}+(\alpha/2)r_{3}^{\ast}\delta r_{2},$		(32)
	$\displaystyle{\delta r_{3}}_{\tau}$	$\displaystyle=$	$\displaystyle-e_{0}\delta r_{2},$

where as usual the $\tau$ subscript indicates the derivative. We introduce the field modified frequency

$$\Omega^{2}=(1+\mu e_{0})^{2}+e_{0}^{2},$$

(33)

and ratio

$$b={\alpha\over 4}(\mu r_{1}^{\ast}+r_{3}^{\ast})={\alpha\over 4}{r_{3}^{\ast}\over 1+\mu e_{0}}=\pm{\alpha\over 4\Omega}.$$

(34)

From (32) we can get the equation for $\delta r_{2}$

$${\delta r_{2}}_{\tau\tau}-2b{\delta r_{2}}_{,\tau}+\Omega^{2}\delta r_{2}=0.$$

(35)

The characteristic equation is

$$\lambda^{2}-2b\lambda+\Omega^{2}=0,$$

whose roots are

$$\lambda_{1,2}=b\pm i\sqrt{\Omega^{2}-b^{2}}=\pm{\alpha\over 4\Omega}\pm i\Omega\sqrt{1-{\alpha^{2}\over 16\Omega^{4}}}$$

Then if $b<0$ and $\Omega^{2}>b^{2}$ the fixed point is stable. Depending on $\alpha$ two cases occur. Consider first a small $\alpha$ so that $\Omega^{2}>b^{2}$. Then if $\mu>0$, $b>0$ if $r_{3}^{\ast}>0$. Then the fixed point such that $r_{3}^{\ast}>0$ is unstable. Conversely the fixed point such that $r_{3}^{\ast}<0$ is stable. If $1+\mu e_{0}<0$ which occurs for $\mu<0$ and large fields $e_{0}$ the situation is reversed. The fixed point such that $r_{3}^{\ast}>0$ is stable while the one such that $r_{3}^{\ast}<0$ is unstable. The oscillation frequency of the orbit as it approaches the fixed point is given by the imaginary part of $\lambda$

$$\omega^{\ast}=\Omega\sqrt{1-{\alpha^{2}\over 16\Omega^{4}}}$$

(36)

.

When $\Omega^{2}>b^{2}$ corresponding to a large $\alpha$, there is no imaginary part of $\lambda$. Even for very large $\alpha$ the stability remains unchanged because for the first term $b={\alpha\over 4\Omega}$ will always dominate the second one $\Omega\sqrt{{\alpha^{2}\over 16\Omega^{4}}-1}$. Therefore the fixed point such that $r_{3}^{\ast}<0$ is stable (resp. unstable) for $\mu>0$ (resp. $1+\mu e_{0}<0$). In this case there are no oscillations around the fixed point.

To conclude, for all values of $\alpha$, the stability is shown in Fig. 3 in the parameter space $(\mu,e_{0})$.

We now assume a constant field $e_{0}$ applied to the film together with a small harmonic wave. It is then possible to examine how these waves of different frequencies get scattered by the film. Using this spectroscopic analysis of the film, we will see that one can recover the atomic dipolar parameter $\mu$ and the coupling parameter $\alpha$.

For a constant $e_{0}$, the system (14,17) becomes

	$\displaystyle\delta e_{\tau\tau}-\delta e_{\zeta\zeta}$	$\displaystyle=$	$\displaystyle\alpha\delta_{D}\left(x\right)\delta r_{2,\tau},$		(38)
	$\displaystyle\delta r_{1,\tau}$	$\displaystyle=-$	$\displaystyle(1+\mu e_{0})\delta r_{2},$		(39)
	$\displaystyle\delta r_{2,\tau}$	$\displaystyle=$	$\displaystyle(1+\mu e_{0})\delta r_{1}+e_{0}\delta r_{3}+(\mu r_{1}^{\ast}+r_{3}^{\ast})\delta e,$		(40)
	$\displaystyle\delta r_{3,\tau}$	$\displaystyle=$	$\displaystyle-e_{0}\delta r_{2}.$		(41)

where we temporarily use $\delta_{D}$ to define the Dirac delta function.

We separate time and space

$$\delta e=E(\zeta)e^{-i\omega\tau},~~~~\delta r_{1,2,3}=R_{1,2,3}e^{-i\omega\tau},$$

(42)

and obtain the system

	$\displaystyle E_{\zeta\zeta}+\omega^{2}E$	$\displaystyle=$	$\displaystyle i\omega\alpha\delta\left(\zeta\right)R_{2},$		(43)
	$\displaystyle i\omega R_{1}$	$\displaystyle=$	$\displaystyle(1+\mu e_{0})R_{2},$		(44)
	$\displaystyle-i\omega R_{2}$	$\displaystyle=$	$\displaystyle(1+\mu e_{0})R_{1}+e_{0}R_{3}+(\mu r_{1}^{\ast}+r_{3}^{\ast})E,$		(45)
	$\displaystyle i\omega R_{3}$	$\displaystyle=$	$\displaystyle e_{0}R_{2}.$		(46)

From the second and fourth equations we get

$$R_{1}=-i\frac{(1+\mu e_{0})}{\omega}R_{2},~~~~\ R_{3}=-i\frac{e_{0}}{\omega}R_{2}.$$

(47)

Plugging these expression into (43) results in

$$R_{2}=i\frac{(\mu r_{1}^{\ast}+r_{3}^{\ast})\omega E}{\omega^{2}-\Omega^{2}},$$

(48)

where $\Omega$ is given by (33). The substitution of the previous equation into (38) leads to the non standard eigenvalue problem

$\displaystyle E_{xx}+\omega^{2}\left[1+\frac{\alpha(\mu r_{1}^{\ast}+r_{3}^{\ast})\delta(x)}{\omega^{2}-\Omega^{2}}\right]E=0.$

(49)

As usual we assume a scattering experiment so that the field is given by

$\displaystyle E$

$\displaystyle=$

$\displaystyle\left\{\begin{array}[]{l l}Ae^{i\omega x}+Be^{-i\omega x},&x<0,\\ Ce^{i\omega x},&x>0.\end{array}\right.$

(52)

At the film $x=0$ the field is continuous and its gradient satisfies the jump condition

$\displaystyle\left\{\begin{array}[]{l l}E(0+)=E(0-)=E(0)\\ E_{x}(0+)-E_{x}(0-)+\beta E(0)=0.\end{array}\right.$

(55)

where

$$\beta=\frac{\alpha(\mu r_{1}^{\ast}+r_{3}^{\ast})\omega^{2}}{\omega^{2}-\Omega^{2}}$$

Writing the two conditions (55) using the left and right fields results in

$$\left\{\begin{array}[]{l l}A+B=C\\ i\omega(C-A+B)+\beta C=0.\end{array}\right.$$

(56)

The solution is

$$B=\frac{i\beta}{2\omega}\frac{1}{1+\frac{i\beta}{2\omega}}A,~~~~C=\frac{1}{1+\frac{i\beta}{2\omega}}A$$

The fixed point is such that $(1+\mu e_{0})r_{1}^{\ast}+e_{0}r_{3}^{\ast}=0$, so that $\mu r_{1}^{\ast}+r_{3}^{\ast}=-\frac{r_{1}^{\ast}}{e_{0}}$, so the reflection and transmission coefficients are

$$R={B\over A}=-\frac{i\alpha\omega(r_{1}^{\ast}/e_{0})}{2(\omega^{2}-\Omega^{2})+i\alpha\omega(r_{1}^{\ast}/e_{0})}$$

(57)

$$T={C\over A}=\frac{(\omega^{2}-\Omega^{2})}{(\omega^{2}-\Omega^{2})+i\alpha\omega(r_{1}^{\ast}/2e_{0})}$$

(58)

where the reflection and transmission coefficients satisfy $|R|^{2}+|T|^{2}=1$ From (29) we have

$$r_{1}^{\ast}/e_{0}=\pm 1/\Omega,$$

(59)

so that the transmission coefficient is

$$T=\frac{2\Omega(\omega^{2}-\Omega^{2})}{2\Omega(\omega^{2}-\Omega^{2})\pm i\alpha\omega}.$$

(60)

The modulus squared of the transmission and reflection coefficients are then

$$|T|^{2}=\frac{4\Omega^{2}(\omega^{2}-\Omega^{2})^{2}}{4\Omega^{2}(\omega^{2}-\Omega^{2})^{2}+\alpha^{2}\omega^{2}},$$

(61)

$$|R|^{2}=\frac{\alpha^{2}\omega^{2}}{4\Omega^{2}(\omega^{2}-\Omega^{2})^{2}+\alpha^{2}\omega^{2}},$$

(62)

Let us examine how $|R|^{2}$ depends on the different parameters. It attains its maximum 1 for $\omega=\Omega$ and decays at infinity as ${1\over\omega^{2}}$. The half-width of the resonance, such that $|R|^{2}(\omega_{h})=1/2$ can be easily obtained. Solving the quadratic equation for $\omega_{h}^{2}$ we get

$$\omega_{h}^{2}=\Omega^{2}\pm{\alpha\over 2}\sqrt{1+{\alpha^{2}\over 16\Omega^{4}}}+{\alpha^{2}\over 8\Omega^{2}}\approx\Omega^{2}\pm{\alpha\over 2},$$

(63)

for large $\Omega$. Therefore the half-width is

$$\omega_{h}\approx\Omega\pm{\alpha\over 4\Omega}.$$

(64)

Fig. 7 shows the square of the modulus $|R|^{2}$ for a fixed $\Omega$ ($e_{0}=1,~\mu=1$). As expected the half-width is proportional to $\alpha$. Notice how the resonance becomes asymmetric for large $\alpha$ indicating that all higher order terms in (63) should be considered.

When $\mu$ is varied $\Omega$ varies so $|R|^{2}$ will be shifted. Fig. 8 shows $|R|^{2}$ for $\mu=-1,0$ and 1. As expected for $\mu<0$ (resp. $\mu>0$) the resonance is shifted towards low (resp. high) frequencies. For $\mu=-1$, the higher order terms in (63) should be taken into account and the resonance is not symmetric. When $\mu=1$, the higher order terms can dropped and the resonance curve becomes symmetric.

The amplitude of the incoming pulse $e_{0}$ will also change the resonance by shifting $\Omega$. Fig. 9 shows $|R|^{2}$ for $e_{0}=0.1,1$ and 2. As expected, for $e_{0}=0.1$ the resonance is asymmetric. It becomes symmetric for $e_{0}\geq 1$.

The reflection curve $R(\omega)$ allows to estimate the atomic parameter $\mu$ and coupling parameter $\alpha$ through the formulas (33) and (63). The position of the resonance gives $\Omega$ and from there one can compute $\mu$. We have

$$\mu=-{1\over e_{0}}\pm{\sqrt{\Omega^{2}-e_{0}^{2}}\over e_{0}}.$$

(65)

The coupling parameter $\alpha$ can then be obtained from (63).

We consider the electromagnetic pulse propagation thought the thin film containing the two-level atoms with taking account both non-diagonal and diagonal matrix elements of the operator of the dipole transition between resonant energy levels. The exact solution of the wave equation allows to derive the modified Bloch equations. Thus the problem of propagating extremely short (one- or few-cycle) pulses through a thin film is reduced to the analysis of a system of nonlinear ordinary differential equations on the Bloch sphere.

In the presence of a constant background field the equilibrium state of the system film/field is changed. These new equilibrium states are the fixed points of the modified Bloch equations. They depend on the field amplitude, the difference of diagonal elements of the operator of the dipole transition (dipole parameter) and the coupling constant. The stability analysis of these fixed points indicates which states will be attained by the system. For the stable states the population difference is negative or positive depending on the sign of the dipole parameter. When the film is illuminated by an electromagnetic field it reaches the new stable state with a typical relaxation time which depends on the field amplitude, the dipole parameter and the coupling constant. Using our estimate, experimentalists could measure the dipole parameter.

In the last part of the article we considered that in addition to a constant background, the thin film is irradiated by a small harmonic field of frequency $\omega$. This spectroscopy analysis yields the reflection and transmission coefficients of the film as a function of $\omega$. As expected the film is completely opaque i.e. its reflection coefficient is equal to 1 for the field modified transition frequency $\Omega$ given by (33). We found that the width of the resonant curve is proportional to the coupling constant. The position of the resonance depends on the dipole parameter and the ground field.

The modern progress in nano-technology allows to produce thin films of different features and the investigation of such features is attractive. Our study shows that the fast electromagnetic response of a thin film could be used in experiments to measure intrinsic parameters of generalized atoms (quantum dots, meta-atoms, molecules …) and their coupling parameter to the field.

Acknowledgments

One of the authors (A.I.M.) is grateful to the Laboratoire de Mathématiques, INSA de Rouen for hospitality and support. Elena Kazantseva thanks the Region Haute-Normandie for a Post-doctoral grant. Jean-Guy Caputo thanks the Centre de Ressources Informatiques de Haute-Normandie for access to computing ressources. The authors express their gratitude to S.O. Elyutin for enlightening discussions.