ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2017, Vol. 81, No. 5, pp. 570–574. © Allerton Press, Inc., 2017. Original Russian Text © R.V. Urmancheev, K.I. Gerasimov, M.M. Minnegaliev, S.A. Moiseev, 2017, published in Izvestiya Rossiiskoi Akademii Nauk, Seriya Fizicheskaya, 2017, Vol. 81, No. 5, pp. 616–620.

Photon Echo Area Theorem for Optically Dense Media R. V. Urmancheeva, K. I. Gerasimova, b, M. M. Minnegalieva, and S. A. Moiseeva, b, * a

Kazan Quantum Center, Kazan National Research University, Kazan, 420111 Russia Zavoisky Physical-Technical Institute, Kazan Scientific Center, Russian Academy of Sciences, Kazan, 420029 Russia *e-mail: [email protected]

b

Abstract⎯The McCall–Hahn area theorem for a photon echo in an optically dense medium is generalized in light of an absorption line consisting of several spectrally unresolved optical transitions and characterized by symmetrical form. Certain theoretical results are compared to experimental data obtained for a photon echo in a Tm3+:Y3Al5O12 crystal. DOI: 10.3103/S1062873817050252

INTRODUCTION The McCall–Hahn area theorem [1] was formulated for the interaction between two-level atoms with a short pulse of laser radiation whose duration is shorter than the phase relaxation of atomic transitions. The area theorem in turn allows us to obtain general information about, e.g., the details of interaction between light pulses and an optically dense system of resonant atoms, pointing testifying to the conditions of formation for light solitons in the medium [2]. Based on this approach, the problem of photon echo formation in an optically dense medium was solved in the approximation of small pulse input areas in [3, 4]. This result was later extended to intense short light pulses of arbitrary amplitude [5]. With the development of quantum optics and informatics, interest in the area theorem is again strong [6–11], due partially to optically dense media being promising for use in broadband quantum memory [9, 10, 12]. The area theorem for the signal of a photon echo [5] has been applied to a scheme of optical quantum memory based on photon echoes [9], allowing us to analyze the effect the nonideality of the pulse areas of governing laser pulses has on the accuracy of reconstructing intense light pulses. The emergence of numerous new protocols of optical quantum memory on a photon echo makes substantiating the area theorem [5], its applicability to new photon echo protocols [12], and developing this approach for more complex systems of atoms that are of interest in optical quantum memory highly relevant. The aims of this work were to substantiate experimentally the theoretical approach proposed in [5] for describing the pulse area of a photon echo in an optically dense medium, and to develop it for more general atomic systems. Our experimental results on observing a photon echo in a Tm3+:Y3Al5O12 crystal are com-

pared to an analytical solution using the appropriate experimental parameters of the spectroscopic transition of atoms and the parameters of light pulses during their propagation in the medium. On the basis of this comparison, conclusions are drawn as to the approach’s applicability to specific tasks [5]. As is shown by a comparison to our experimental results, the discrepancies in their description could be due to the complicated structure of the optical transition for which the absorption line consists of several closely spaced overlapping unresolved optical lines. To build a more complete theory that can describe such systems of atoms, we give a generalization of the McCall– Hahn area theorem to the formation of photon echo in an optically dense medium where the resonant line of absorption consists of several closely spaced overlapping optical lines that form a symmetrical contour of the absorption line. This situation is often found in, e.g., crystals containing rare earth ions, which are widely used as carriers of quantum information [12‒15]. For such systems we derive an equation for the pulse area of a photon echo that allows us to consider the difference between dipole moments, and the positions and shapes of overlapping lines, to generalize equations obtained earlier. EXPERIMENTAL AND THEORETICAL RESULTS In [5], we generalized the solution for the area of a photon echo [3, 4] to a case of intense pulses whose duration is short compared to the relaxation time of an atomic transition. In contrast to [1–4], nonlinear features of the behavior of a photon echo that arise at high amplitudes of input pulses were considered. In addition to finding the equation for the pulse area of a photon echo, an analytical solution was found in [5] to describe the evolution of pulse area Ae ( z ) of a photon

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PHOTON ECHO AREA THEOREM FOR OPTICALLY DENSE MEDIA

echo upon propagation in a medium of two-level atoms:

Ae ( z )

( )

Optical density

χz 2 ⎡ 2τ ⎤ γ sinh (1) ⎢ −T2 ⎥ 2 = arctan ⎢e ⎥  , χz 2 2 + γ  ⎥ cosh  β cosh β − ⎢ ⎣ ⎦ 2 where T2 is the time of coherence relaxation upon an atomic transition; parameter χ determines absorption ⎛ ⎛ A ( z ) ⎞⎞ in the medium; parameters β = ln ⎜ tan ⎜ 1 ⎟⎟  and ⎝ ⎝ 2 ⎠⎠ γ = scs ( A1 ( z )) tan ( A2 ( z ))  are determined by the areas of first and second pulses Ak ( z ) ,   k = 1,2. Let us now consider the interaction between light pulses and an ensemble of atoms whose absorption spectrum near the carrier frequency of the light field can be presented as a sum of symmetrically located

(

∑

(

) )

lines G ( Δ ) = G ( Δ − Δ m ) , where Δ m is the m =1 m detuning from the center of the line, and G m ( x ) is the shape of the inhomogeneous broadening of the m-th absorption line. The symmetry of the lines’ arrangement means that for each m-th transition having line shape G m ( Δ − Δ m ) a symmetrical transition exists with line shape G m ' ( Δ − Δ m ' ) = G m ( −Δ + Δ m ) . The described general inhomogeneously broadened line is symmetric relative to central resonance frequency ω0, which corresponds to detuning Δ = 0 and coincides with the carrier frequency of the light pulses. An example of the described line is presented in Fig. 1. The system of Maxwell–Bloch equations can in this case be written as

ω0 – Δ1 ω0 ω0 + Δ1 Frequency Fig. 1. Example of an absorption line consisting of several overlapping symmetrical Lorentz contours of absorption: ω0 is the central frequency coinciding with the frequency of the light beam, and Δi is the detuning of the i-th line of resonance.

M

M

m ⎛ ∂ + n ∂ ⎞ E ( z, t ) = 2i πω0N 0 P21 , ⎜ ⎟ nc ⎝ ∂z c ∂t ⎠ m =1

(2)

∂ P m t, Δ, z = i Δ − γ P m t, Δ, z ) ( ) m ) 21 ( 21 ( ∂t iϕ + m E (t, z ) w m (t, Δ, z ) , ћ

(3)

∂ w t, Δ, z = − 2i ϕ m E t, z ) ( ) m( ∂t ћ × P21m (t, Δ, z ) − P12m (t, Δ, z ) ,

(4)

∑

{

571

}

∞

P21m

= ϕm

∫ d ΔG

m

(Δ − Δ m ) P21m (t, Δ, z ) .

(5)

−∞

As in [2–6], we assume here that the pulse duration is less than the shortest of all of the transverse relaxation times γ −m1 (m = 1, …, N) of atomic transitions. In these formulas, P21m (t, Δ, z ) is the polarization operator on the m-th transition in the Heisenberg representation; w m (t, Δ, z ) = P22m (t, Δ, z ) – P11m (t, Δ, z ) is the inversion of population upon this transition; ϕ m and γ m are the dipole moment and the homogeneous width of the m-th transition, respectively; E k (t, z ) is the slowly varying envelope of the k-th light pulse; N0 is the concentration of atoms interacting with the light; ћ is Planck’s constant; с is the speed of light; and n is the refractive index of the medium. Integrating system of equations (2)–(4) over time in the vicinity of the first and second light pulses, we find the differential equations for the areas of the first two exciting light pulses:

M ⎧∂ π 2ω 0 N 0 2ϕ A z   = − ϕ mG m ( −Δ m ) sin ⎛⎜ m A1 ( z ) ⎞⎟ , ⎪ 1( ) nc m =1 ⎝ ћ ⎠ ⎪∂ z ⎨ M 2 2ϕ 2ϕ π ω0 N 0 ⎪∂ A z =   − ϕ mG m ( −Δ m ) cos ⎛⎜ m A1 ( z ) ⎞⎟ sin ⎛⎜ m A2 ( z ) ⎞⎟ , ( ) 2 ⎪∂ z nc m =1 ⎝ ћ ⎠ ⎝ ћ ⎠ ⎩

∑

∑

where Ak ( z ) =

∫

+∞

−∞

E k (t, z ) dt is the area of the k-th pulse.

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URMANCHEEV et al. Cryostat D1 M

M

L2(500 mm) λ/2

P

R = 4% L3(200 mm)

L4(150 mm)

AOM 1 L5(300 mm) L1(500 mm) AOM 2 L6(300 mm) D2 RF pulse generator AOM 1

Oscilloscope

Ti:Sp laser AOM 2

Fig. 2. Scheme of the experimental setup: detectors D1 and D2; lenses L1–L6 (focal lengths are indicated in parentheses); mirrors M; polarizer P; acousto-optic modulators AOM 1, 2. The sample temperature in the cryostat was 4.5 K.

As in [2], it is considered in (6) that the initial inversion of the population at the carrier frequency of light pulses prior to the action of the second pulse is 2ϕ equal to − cos ⎛⎜ m A1 ( z ) ⎞⎟ . ⎝ ћ ⎠ To obtain the equation describing the behavior of the pulse area of an echo signal in the medium, we integrate Eqs. (2)–(4) over a time interval containing the time of the emission of a two-pulse echo signal (i.e., near moment t = 2τ), considering the new initial conditions describing the properties of the phasing polarization and inversion of the population in the system of atoms before the propagation of the echo signal. Our calculations yield the differential equation for the pulse area of a two-pulse photon echo signal, using the symmetry of the absorption line: M

{

∂ A z =  − μ ϕ G −Δ −e −2γ mτ sin ⎡θ m z ⎤ m m( m) e( ) ⎣ 1 ( )⎦ ∂z m =1

∑

× sin 2 ⎡1 θ 2m ( z )⎤ cos 2 ⎡1 θ em ( z )⎤ ⎢⎣2 ⎥⎦ ⎢⎣2 ⎥⎦ m m m + cos ⎡⎣θ1 ( z )⎤⎦ cos ⎡⎣θ 2 ( z )⎤⎦ sin ⎡⎣θ e ( z )⎤⎦ ,

(7)

}

here μ is a constant depending on the medium’s prop2ϕ erties, θ im ( z ) = m Ai ( z ) , Ai ( z ) are the new denotaћ tion for the pulse areas of the first, second, and echo pulses (i = 1, 2, e). Note that an analytical solution to Eq. (7) generally cannot be obtained, except in the special case of (1). However, the use of Eqs. (5)–(7) for the pulse areas of exciting light pulses and photon echoes allows us to study their behavior in an optically dense medium by numerically solving only equations of the first order on z using the dipole moments and

the spectral line parameters upon each of the atomic transitions. Our experimental investigations of the dependences of the area of a standard two-pulse echo on the areas of the first or second input pulses were conducted on the installation whose scheme is shown in Fig. 2. As the source of continuous coherent radiation with a wavelength of 793.309 nm, we used a tuneable single-frequency Ti:Sp laser with a radiation spectral bandwidth of <10 kHz/s. The laser radiation is focused on acousto-optic modulator AOM 1 using the system of lenses L1 and L2. The RF signal also entering the AOM 1 determines the formation of pulses from the continuous laser radiation. During the pulse action, the beam is deviated by the angle corresponding to the first order of diffraction. The light in the first diffraction order is thus repeated in the form of the RF signal. In the described experiment, the pulse duration was 400 ns, and the delay between each was 1300 ns. The light pulses received after passing through the half-wave plate and polarizer were focused using lenses L3 and L4 on the crystal held in a Montana Instruments cryostat at a temperature of 4.5 K. A small part of the light was directed onto detector D1 by a beam splitter was to measure the areas of the input pulses. After passing through the crystal, the light was focused onto acousto-optic modulator AOM 2 by the system of lenses L5 and L6, used as the shutter opened after the input pulses passed through for the duration of the echo’s pulse. This protected sensitive detector D2 from the powerful input pulses. We chose a Y3Al5O12:Tm3+ crystal with the concentration of thulium ions С = 10% as our object of research. The width of the absorption line corresponding to the 3H6(1)–3H4(1) transition was 0.3 nm.

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PHOTON ECHO AREA THEOREM FOR OPTICALLY DENSE MEDIA

Area of output pulse

0.5

1.5

2.5 3.5 Area of input pulse

4.5

Fig. 3. Experimental (symbols) and theoretical (lines) dependences: area of an individual pulse after passing through the crystal on its input area (u, dotted line); area of the photon echo on the input area of the second pulse at a constant area of the first pulse (d, dashed-and-dotted line); area of the photon echo on the input area of the first pulse at a constant second pulse (r, solid line).

The optical density was chosen experimentally by setting the frequency of the laser on a specific area of the absorption line profile. Three dependences were measured in our experiment: (a) area A1 ( z = L )  of an individual pulse after it passed through the crystal on its input area A1 ( z = 0) (the white squares in Fig. 3); (b) the area of the photon echo on the input area of the second pulse at a constant area of the first pulse (the black circles in Fig. 3); (c) the area of the photon echo on the input area of the first pulse at a constant second pulse (the black diamonds in Fig. 3). Our experimental curves were then approximated by the theoretical dependences obtained in [5]. Formula (1) was used to calculate the pulse area of a photon echo; for the passing of an individual pulse, we used [2]

(

)

χL − ⎡ ⎤ A1 ( z = L ) =  2 a rctan ⎢exp 2 tan 1 A1 ( z = 0) ⎥ . (8) 2 ⎥⎦ ⎣⎢

In our calculations, the data on the relaxation time and optical density of a sample were determined experimentally: T2 = 500 ns and χ L = 1.83 , respectively. Thus, only two parameters were varied: the gain coefficients of detectors D1 and D2, the values of which were determined by approximating the experimental data according to the least squares method. The theoretical curves and the corresponding experimental dependences are presented in Fig. 3. Note that experimental data show the nonlinear behavior of photon echo pulse area (1) when the pulse areas of the

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second pulse are close to π. If the first two sets of experimental data are described well by theoretical curves (the corresponding theoretical curves are indicated by dashed and dashed-and-dotted lines), then for the third set (black diamonds represent the experimental data; the solid line, theoretical values) the deviation from the corresponding theoretical curve is considerable. Theoretically, the graph in this case describes only the general behavior and position of the maximum of the experimental dependence of the behavior of the echo signal’s pulse area. We attribute this discrepancy to the need to consider the more complex structure of absorption lines, along with the finite duration of the first and second pulses, plus the corresponding influence of relaxation processes during the action of pulses. Further experimental and theoretical studies are required to account for these factors.

CONCLUSIONS A theoretical and experimental study of the pulse area of a two-pulse photon echo in an optically dense system of atoms was performed. The experiments were conducted using a Y3Al5O12:Tm3+ crystal upon the 3H (1)–3H (1) transition, for different values of the 6 4 intensity of the first and second exciting light pulses. The obtained experimental data were compared to the analytical solution for the pulse area of photon echo (1). It was found that many of the experimental results were satisfactorily described by the currently accepted theory. The theoretical curve describing the behavior of the pulse area of a photon echo as function of the input area of the first pulse at a constant second pulse describes the experimental data only approximately. The explanation of the latter data requires further theoretical studies that consider the characteristic parameters of the light pulses and the spectroscopic parameters of a substance. We attribute the observed discrepancies to the need to consider the finite duration of the first and second pulses, at which the influence of the relaxation processes during the pulse action becomes important, and to the importance of considering more complex structures of the absorption line. To solve the second problem, we propose a generalization of the McCall–Hahn area theorem to the formation of photon echo in an optically dense medium in which the resonant absorption line consists of several closelyspaced overlapping optical lines that form a symmetrical contour of the absorption line. Our equations allow us to describe more accurately the interaction between light pulses and the formation of photon echoes in media with a continuous absorption line structure (i.e., in crystals with rare-earth ions). ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation, grant no 14-12-01333.

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