 ### Разбалловка

A1  1.00 An atom emits light with a wavelength $\lambda_0 = 300 \mathrm{nm}$. Using the classical model estimate an emission time $\tau$ (that is, the period of time it takes the atom to emit the energy equal to that of a single photon). This time coincides with the characteristic time, during which the atom emits a photon, by the order of magnitude. All radiation is due to a single electron located at a distance about $a_0 = 0.1 \mathrm{nm}$ from the nucleus. Express your answer in terms of the physical constants, $\lambda_0$, and $a_0$.

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 Correct formula for $\tau$ (up to numerical coefficient) $$\tau = \frac{3\hbar }{k a_0^2 e^2} \left( \frac{\lambda_0}{2\pi} \right)^3$$ 0.5 Correct numerical coefficient 0.25 Numerical answer in the interval $\tau \in (0.5 \cdot 10^{-8},\; 4.0 \cdot 10^{-8})\,s$ (if the formula is correct up to numerical coefficient). 0.25
A2  0.25 Estimate the power $W_s$ of electromagnetic radiation of all $N$ atoms in the spontaneous emission mode, i.e. when the direction of atomic dipole and the phase of its oscillations randomly change from atom to atom. In your answer write down the formula for the power in terms of $N, \omega$, and $\tau$.

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 Answer $W_s = N \frac{\hbar \omega}{\tau}$ 0.25
A3  0.25 Estimate the duration of the spontaneous emission pulse of this system of atoms. Express your answer in terms of the same quantities.

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 Answer $\Delta t_s = \tau$ 0.25
A4  0.50 Estimate the power $W_i$ of electromagnetic radiation of all $N$ atoms in the superradiance mode, i.e. when the direction of atomic dipoles and the phases of their oscillations are the same for all atoms in the excited state. Express your answer in terms of $N, \omega$, and $\tau$.

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 Answer $W_i = N^2 \frac{\hbar \omega}{\tau}$ 0.5
A5  0.25 Estimate the duration of the radiation pulse of the system of atoms in the superradiance mode. Express your answer in terms of the same quantities.

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 Answer $\Delta t_i = \tau/N$ 0.25
B1  0.50 Let the amplitudes of two wave maxima be $E_ {m1}$ and $E_ {m2}$. Find the difference in their propagation speeds $\Delta v$. Express your answer in terms of $n_0, n_2, c, E_{m1}$, and $E_{m2}$.

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 Answer $\Delta v = v_1 -v_2 \approx \frac{c n_2}{n_0^2}\left( E_{m2}^2- E_{m1}^2\right)$ 0.5
B2  2.00 A light pulse with a wavelength in vacuum of $\lambda_0 = 300 \mathrm{nm}$ and a maximum intensity of $I_0 = 3 \cdot 10^9 \mathrm {W / cm^2}$ propagates along the axis of a quartz fiber. Assume the envelope of a time dependence of the electric field squared $E_m^2 (t)$ of the wave to be a parabola. How far (find the distance $s$) does the pulse propagate along the fiber before its spectral width increases by the factor of $K = 200$? Express your answer in terms of $K, \lambda_0, n_2, E_m$ and calculate the numerical value (in meters, rounded to an integer).

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 Correct formula for parabolic envelope of $E_m^2(t)$ 0.1 It is shown that frequency depends on the coordinate in the wavepocket or the number of the maximum linearly 0.8 Analytical answer $s = \frac{K \lambda_0}{8 n_2 E_m^2}$ 0.8 Numerical value $s= 8 m$ 0.3
B3  0.50 What sign should the constant $\beta_2$ have in order for the pulse chirped according to the scheme described above to be compressed in time in this medium? Please, indicate "+" or "-" in your answer. In what follows consider that $\beta_2$ has exactly this sign.

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 Correct sign $\beta_2<0$ 0.5
B4  1.00 A pulse described in B2 has a duration $\Delta t_0 = 10 \mathrm{ps}$ and an initial spectral width $\Delta\omega_0 \approx 2 \pi / \Delta t_0$ (before chirping) and propagates in the medium described above. Find the distance the pulse should travel in order to achieve the minimum possible duration after chirping with spectrum broadening by the factor of $K = 200$. Express your answer in terms of physical constants, $K, \Delta t_0, \beta_1$, and $\beta_2$ and calculate the numerical value in meters, rounded to an integer.

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 Formula for the group velocity $V_g = \frac{1}{\beta_1 + \beta _2(\omega - \omega_0)}$ 0.25 Formula for the distance $l = \frac{(\Delta t_0)^2 }{2 \pi K |\beta_2|}$ 0.5 Numerical answer $l= 4 m$ 0.25
B5  1.50 Nonlinearity of a medium leads to disappearance of diffraction of a light beam of sufficiently high intensity. Estimate the minimum power of a light pulse $W_c$ at which it does not experience diffraction, i.e. propagates inside a narrow cylindrical channel of constant radius. Express your answer for $W_c$ in terms of physical constants, frequency $\omega_0$, $n_0$, and $n_2$. Assume the intensity distribution over the channel cross section to be approximately uniform. Find the numerical value of the power for a pulse with a wavelength in vacuum $\lambda_0 = 300 \mathrm{nm}$ propagating in quartz. Coefficient $n_0 = 1.47$.

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 Formula for diffraction divergence $\theta_d \approx \lambda/ a$ 0.2 Formula for total internal reflection angle $\sin \alpha_c = n_0/(n_0 + n_2 E_m^2)$ 0.4 Formula for the power $W_c = \frac{\pi \varepsilon_0 c^3}{n_2 \omega_0^2}$ 0.5 Numerical answer $W_c \approx 26 MW$ 0.4
C1  1.00 Propose a method that would allow one to detect an exoplanet with a noticeable inclination of its orbital plane with respect to the line of sight by means of studying the spectrum of its star in the optical range. As an answer name the physical phenomenon underlying your method.

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 Method suggested and correct physical phenomenon is specified 1
C2  1.00 Suppose an exoplanet of mass $m$ orbits a star of mass $M$ in a circular orbit of a radius $R$ and the period of revolution is $T$. The orbital plane is at an angle $\theta$ to the direction to Earth. Estimate the accuracy of the relative frequency measurement, $\Delta \omega / \omega$, required to detect such an exoplanet by your method. In your answer express $\Delta \omega / \omega$ in terms of the fundamental constants, $R, T, \theta, m$, and $M$.

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 Formula for the velocity of the star $v_1 = \frac{2\pi Rm}{TM}$ 0.2 Projection of the velocity on the line of sight $v_1 \cos \theta$ 0.1 Connection between $\Delta \omega/\omega$ and $v/c$ 0.1 Analytical answer $\Delta \omega/\omega = \frac{2\pi R }{T c} \frac{m}{M}\cos \theta$ 0.6
C3  0.25 Assume the mass of the exoplanet and its star to be equal to the mass of Earth and the Sun, respectively. Assume the radius of the circular orbit to be equal to the distance from Earth to the Sun ($R \approx 1.5 \cdot 10^{11} \text{m}$), the angle $\theta = 60^\circ$. The Solar mass is 330,000 times of the Earth's mass, the period of the Earth's revolution around the Sun is 1 year. Find an integer $n$ such that $10^{-n}$ is the accuracy of relative frequency measurement required by your method. Usage of ultrashort (femtosecond) laser pulses makes it possible to measure frequencies in the optical range ($10^{15} \text{Hz}$) with an accuracy of about 10 Hz. Is this accuracy enough to register the exoplanet?

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 Numerical value of $n$: $n \approx 10$. 0.15 Correct statement that precision is enough to detect the exoplanet which does not contradict to the previous results 0.1