Photon echoes in optically dense media

Sergey A. Moiseev, Mahmood Sabooni, and Ravil V. Urmancheev Kazan Quantum Center, Kazan National Research Technical University n.a. A.N.Tupolev-KAI, 10 K. Marx, Kazan 420111, Russia Institute for Quantum Computing, Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada and Department of Physics, University of Tehran, 14399-55961, Tehran, Iran (Dated: November 22, 2019)

Most general theoretical description of the coherent resonant light-atoms interaction in the optical dense media is provided by the well-known pulse area theorem [22][23][24][25][26][27][28][29]. In early works on the primary photon echo in the optically dense medium, it was found it could result in the generation of multiple echo signals [24,30], that became a subject of a long-stay investigation [23,24,[31][32][33][34][35][36][37][38]. An analytic solution for total area of all the echoes was obtained [24,30,33]. That proved that the total pulse area tends asymptotically towards 2π in the depth of the medium if the initial pulse area of two exciting laser pulses exceeds π in accordance with area theorem [22]. The solution concealed the behavior of each individual echo pulses, which remains unknown. The analytic solution for the primary echo pulse area consisted with numerical simulations [35,39] was obtained in [32]. It predicted that the echo pulse area never exceeds π and generally decays in the depth of the medium. This finding again stressed that the formation of the 2π total echo area is an important and yet unsolved problem.
In this letter, we discover the mechanism of selfinduced transparency [22] for two-pulse excitation leading to the formation of many echo pulses. To do that we find the analytic solution for the pulse areas of the arbitrary secondary photon echo signals. The solutions show that the echo signals are excited coherently one after another in a certain area of the medium and then disappear, generating new echo signals creating a sort of a chain excitation, a self-reviving echo sequence. We show that depending on the input pulse areas the echo pulse train forms a multi-pulse analogue to the well-known single pulse 2π optical soliton or a 0π optical breather despite each individual echo pulse area never exceeding π.
Photon echo area theorem. First we reproduce the McCall-Hahn area theorem and derive the general equation for the pulse area of an arbitrary echo pulse starting with the usual reduced set of Maxwell-Bloch equations [24] for the light field and atomic polarization: where r = r(t, z, ∆) = (u, v, w) T is the Bloch vector, each component depending on time t, spatial coordinate z and atomic detuning ∆; P = u − iv -atomic polarization, electric field E(t, z) = ε(t, z) exp[i(kz − ωt)] + c.c. is described by a complex light field envelope ε(t, z) with corresponding Rabi frequency Ω(t, z) = (2d/ )ε(t, z); µ = 4πN d 2 ω/ c, γ = 1/T 2 , T 2 is the coherence lifetime of the atomic transition and ... ≡ well-known solutions [24]: where β = ln{tan[ θ1(0) 2 ]} and κ = tan[ θ2(0) 2 ]/ sin[θ 1 (0)]. Eqs. (2) and (3) can be used to find the total area of all excited photon echoes [24,30,31,33]: This solution predicts that if θ 2 (0) < π, θ 1 (0)+θ 2 (0) > π, the total area of all echo pulses will asymptotically tends to 2π [30]. It leaves however some uncertainty about the mechanism and physics of the photon echo generation, since any information about the particular photon echo signals remains hidden. How exactly different echoes combine into 2π pulse area? What is the contribution of an individual echo? Moreover, if input pulse area θ 1 (0) < π/2, θ 2 (0) > π, Eq. (4) predicts the sum of all echoes to be 0. What will be the different echo signals in this case and does that mean that there will be no echoes? To answer all these questions, we should analyze the generation of each echo signal individually.
Individual echo pulse area. We will now derive the area theorem for an arbitrary individual photon echo signal and to do so we integrate the first of Eqs. (1) over time around the time of echo emission t e , from t 0 = t e − τ /2 to t 1 = t e + τ /2, where τ is the delay between the pulses. By assuming that τ δt, δt being the pulse duration, we arrive to the equation for pulse area where we substitute the formal solution for P from Eq. (1): The key to finding the correct solution is proper handling of the integrals over t in these two terms. One can show that P 0 (∆) and w(t, ∆) can be presented as a sum of several components P 0 = P .. with the total number of the components depending on the echo signal of interest. These components have a from P where n ∈ Z, the phase ϕ n is either 0 or π/2.
For n = 0, P (n) 0 and w (n) are rapidly oscillating functions of ∆ near the echo pulse emission time t e since τ δt. Averaging over ∆ leads to that only the slowly varying terms P (0) 0 and w (0) contribute to the echo pulse area in Eq. (5). After using P 0 (∆) = P (0) 0 , we simply integrate the first term by taking into account: t1 t0 dte −i∆(t−te) → 2πδ(∆) (this limit is valid assuming no temporal overlapping between the all light pulses). In the second term, we switch the order of temporal integrals and arrive to the integral similar [24,25]: where we have also taken into account that w (0) (t , ∆) and G(∆) are even functions of ∆. Thus Eq. (5) comes to: where To findw(t) and to integrate Eq. (7), we write the Bloch equation set for the case ∆ = 0, and since γδt 1, we also ignore relaxation during the pulses and only consider the relaxation between the pulses:ṽ where θ(t) = t t0 Ω(t)dt,ṽ(t) is a resonant part of the phased coherence,w 0 =w(t 0 ). Equation (7) can now be integrated, and we obtain the general equation for an arbitrary echo pulse area: After transition to η = tan θ 2 we get a linear equation This equation describes the pulse area of a chosen echo signal given the phasing coherenceṽ 0 in a presence of spectral uniform inversionw 0 . Equation (9) comes down to findingṽ 0 andw 0 for each echo signal. But whatever they may be, we note that |θ| never exceeds π. Below we analyze the analytic solutions for the pulse areas of all the echo signals.
Phasing components of polarization and inversion Under a multipulse excitation a two level system engages in two processes: it is either interacting with the electric field of the applied pulse, or it is left to its own devices and experiences free oscillations decaying as e −γt . In the assumed timescales of these processes the influence of the pulse with area θ can be written as a rotation of the Bloch vector around u-axis: And free nutation is described with another rotation matrix this time around w-axis: In case of multi-pulse excitation, each pulse and each free oscillation period contribute to the resulting coherence, so after n pulses we get r((n − 1)τ ) = T (θ n )U (τ )...T (θ 2 )U (τ )T (θ 1 ) r(0), assuming the pulses are equidistant. To investigate the dynamic properties of polarization and inversion we leave the time dependence in the last multiplier, so r(t) = U [t − (n − 1)τ ] r((n − 1)τ ). Only the phasing part of the Bloch vector will contribute to the echo formation. The contributing part has then the following form: T . Let us now apply the described method to calculate the areas of primary and secondary echoes excited in the optically dense medium.
Both polarization and inversion consist of two components characterized by different physical meanings. The first termṽ 1 = 1 2 Γ 2 τ sin θ 1 sin θ 2 sin θ e1 , andw 1 = −Γ 2 τ sin θ 1 sin 2 θ2 2 sin θ e1 are proportional to sin θ 1 and vanish when the first pulse is absorbed. They are responsible for stimulated photon echo generated by incoming pulses and primary echo pulse. The other two componentsṽ 2 = Γ 2 τ cos θ 1 sin θ 2 sin 2 θe1 2 andw 2 = − cos θ 1 cos θ 2 cos θ e1 proportional to cos θ 1 correspond to the secondary two-pulse photon echo created by the second pulse and the primary echo pulse.
The next excited echoes analysis follows the same procedure but requires more calculations sinceṽ 0 andw 0 have more terms with each step. In the Supplementary materials, we introduce the phasing polarization and inversion components for the third and the fourth echoes and discuss the physical meaning of different components. Using those expressions calculate their pulse areas by solving Eq. (9).
We will now use the solution for pulse areas to clarify the mechanism of the total 2π pulse area formation when θ 1 (0) + θ 2 (0) > π. Figure 1 shows the spatial behavior of the area of incoming pulses, echo pulses and total area depending on the optical density of the medium for θ 1 (0) = 0.1π, θ 2 (0) = 0.999π. We see that incoming pulses excite primary and secondary echoes that in turn excite subsequent echos. Each echo pulse is born, propagates and eventually dies out within a finite spatial interval. However the total area of all existing pulses behaves strictly in accordance with McCall-Hahn area theorem Eq. (4). This is realized due to the precise spatial consistency of all the echo pulses. In the case θ 1 (0), θ 2 (0) < π presented in Figure 1, the total area remains equal to 2π.
The case of θ 2 (0) > π really helps to highlight the benefits of looking at the individual echo signal rather than at the sum of all echo signals. The second incoming pulse is big enough to form a 2π-soliton on its own, and McCall-Hahn area theorem predicts that the sum of all echoes will equal 0π. The impression could be that after some point in the medium there are no echoes at all. The real picture however is much more vivid, there are many hidden echoes with nontrivial areas working together to comply with the McCall-Hahn area theorem. Figure 2 showcases this echo pulses' behavior for θ 1 (0) = 0.1π, θ 2 (0) = 1.001π. Each two of the subsequent echoes have opposite phases, so they are canceling each other in a dynamical equilibrium, resulting in 0π total pulse area at any point of the medium. Figure 2 also shows that the primary echo assists the formation of the 2π total area, which would otherwise be much further in the medium.
The echo areas in Figs. 1,2 behave very similar, differing only in their spatial delays. This is the case, when we can neglect the stimulated echo terms in Eqs. (14) and find a highly accurate approximate analytic solution for each pulse area. For example we write for the secondary echo area (z > z 1 ): where θ e1 is given in Eq. (13) with the initial pulse areas taken at the transition point z 1 : (θ 1 (0), θ 2 (0)) → (θ 2 (z 1 ), θ e1 (z 1 )). By doing so we assume that at z = z 1 the first pulse was successfully absorbed by the media and neglect polarization and inversion components acquired at z < z 1 . This solution for θ e2 is shown as a dashed lines in Figs. 1,2.
Experimental issues. It is interesting to discuss the experimental detection of photon echo train generation and what it can lead to. As it is seen in Figs. 1,2, one can experimentally observe only 2 or 3 light pulses at the output of the optical density medium, while other pulses will be highly suppressed. Herein in media with higher optical densities, we will see only higher order echo pulses, char-  acterized experimentally by later emission times. The photon echo experiments in such media are quite typical for many quantum memory protocols. In particular, interesting opportunity is to try detecting the spatial evolution of photon echo inside such media, for example in the rare-earth ions doped crystals [9,40,41]. One possible candidate for high optical density and large Rabi frequency is 4 I 9/2 − 4 F 3/2 transition of Nd 3+ :YVO 4 at 897.705 nm with dipole moment d = 9.16 × 10 −32 C.m. Considering P = 100 mW and beam radius of r = 1µm one could reach to Ω ∼ 250 MHz. The π-pulses can be as brief as several nanoseconds which is much shorter than T 2 . These pulses are spatially squeezed in the medium up to 4 orders of magnitude by the group velocity reduction in the presence of a spectral hole in the optical transition [42], this would allow to observe spatial evolution of the solitons and echo pulses inside the medium.
It is worth noting that only soliton-like pulses can propagate through the medium without changing their temporal form and transferring atoms to their initial state. Accordingly, the photon echo pulses in the generated train will be stretch in time and ultimately overlap with each other deep in the medium forming a single 2π-soliton in case of Fig. 1. Similar the stretching echo pulses will asymptotically form a 0π-breather, for the case of Fig. 2. In the core of these transformations lies conservation laws of Maxwell-Bloch Eqs. (1) [43].
Conclusion. To summarize, we complete the longlasting derivation of the photon echo area theorem, providing an analytic solution for the pulse area of any desired photon echo signal. The approach works for an arbitrary incoming pulse areas, provided that they are short in comparison with the relaxation time T 2 . We showcase the power of this approach by specifying in detail the picture of the two-pulse echo excitation of an optically dense medium in two previously undervalued cases: θ 1 (0) < 1, θ 2 (0) ( ) π. We explore the mechanism of 2π total area formation, provided by the self-reviving echo train excited deep in the medium. Thus the hidden beauty of the McCall-Hahn area theorem prediction reveals itself in precise spatial consistency of multiple echo pulses in the inhomogeneously broadened optically dense media.
The developed approach of photon echo pulse area theorem can be applied for general analysis of various schemes photon echo including protocols of quantum memory in free space, waveguide and in cavity assistant schemes (of optical and microwave regions) for two-and three-level atomic systems etc., that opens new opportunities for further research. Next important step could be to generalize and extend the results acquired here for multi-pulse excitation using powerful inverse scattering transform, as was done in [44] for McCall-Hahn area theorem.
The reported study was funded by Russian Foundation for Basic Research according to the research project no.17-52-560009.