Counterintuitive Versus Regular Inversionless Gain in a Coherently Prepared Ladder Scheme 1

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1 ISSN 54-66X, Laser Physics,, Vol., No. 7, pp. 5. Pleiades Publishing, Ltd.,. Original Text Astro, Ltd.,. RUBRIC Counterintuitive Versus Regular Inversionless Gain in a Coherently Prepared Ladder Scheme D. Braunstein a, *, E. Smolik a, **, G. A. Koganov b, c, ***, Y. Biton b, ****, and R. Shuker c, ***** a Physics Department, Shammon College of Engineering, P. O. Box 45, Beer sheva, 84 Israel b Physics Department, Ben Gurion University of the Negev, P. O. Box 653, Beer sheva, 845, Israel; c Eilat Campus, Ben Gurion University of the Negev, Hatmarim Blv, Eilat 88, Israel * doronb@sce.ac.il ** elag@sce.ac.il *** quant@bgu.ac.il **** yaacov@bgu.ac.il ***** shuker@bgu.ac.il Received November, ; in final form, November 9, Abstract In this research we study lasing without population inversion from a three-level atom interacting with two laser fields, in the ladder or cascade scheme. We investigate counterintuitive sequencing as well as regular sequencing of the time of laser fields application. In a counterintuitive sequence scheme a short probe pulse is introduced prior to the application of the coupling field, in contrast, to a standard sequence scheme, where both fields are introduced at the same time. The influence of varying the probe pulse width and time delay between the initiation of probe and coupling fields on transient probe gain is investigated. The calculations indicate the potential of producing gain without inversion via counterintuitive sequence scheme. DOI:.34/S5466X33. INTRODUCTION Recently counterintuitive effect in lasing and amplification without inversion (LWI) was studied []. Quantum interferences, LWI and absorption with inversion were studied in a variety of schemes [ 8]. Experimental realization of such schemes have been demonstrated [9 ]. The key mechanism, which is common to most of the proposed schemes, is the utilization of external coherent fields that induce quantum coherence and interference in multi level systems. In particular, it was shown that if coherence is established between certain atomic states, different absorption channels may interfere destructively leading to the reduction or even to the cancelation of absorption [3]. In recent publications we have investigated gain without inversion in a three level V and ladder schemes in the dressed and bare state pictures using two CW external coherent fields [4, 5]. The inversionless gain obtained at the transient regime for the ladder scheme is significantly higher than in the steady state one, especially at the first Rabi cycle. Dressed states analysis of a ladder scheme involving an incoherent pump in a CW regime was discussed in [6, 7]. Timedependent analytical solution for the atom-field density matrix has been obtained, and spectra and fine shape of the probe transition were calculated. In particular, enhanced refractive index accompanied by vanishing absorption was found. In a recent paper [8] The article is published in the original. we have studied a counterintuitive gain without inversion in Λ-scheme utilizing incoherent pump along with CW coherent pump and a Gaussian shaped probe field. It was found that counterintuitive sequence of fields exhibits stronger gain without inversion compared to that achieved in a regular sequence. In the present paper we introduce time dependent study of pulsed probe and drive fields. This is done in a ladder scheme of three level system in the bare representation. Introducing a short pulse probe field prior to the drive field (counterintuitive sequencing) exhibits enhancement gain without inversion in a CW scheme. This is counterintuitive to what would be expected. Counterintuitive coherent control studies have recently shown the importance of the scheme in a number of subjects including population transfer [9 3], quantum computation [5 7], and others. Here we show the advantage of such process in the ladder scheme for LWI. This is expected in other schemes of LWI and in electromagnetically induced transparency (EIT) as well. Advanced preparation of the quantum state of the probe transition by advanced turn on of the probe field in three level system, results in an enhancement of the probe related effects. This is the so called counterintuitive sequence of applying the probe and drive fields. Undoubtedly, the drive field is an essential one in generating the gain on the probe transition. Turning on the drive field alone without the probe field actually results in Rabi oscillation of the populations and

2 BRAUNSTEIN et al. ω ba ω ba γ cb γ ba Ω G coherences on the drive transition. Intuitively one expects that the effect of the probe field requires the existence and turn on of the drive field initially. However, quantum mechanically, initial turn on of the probe field prepares the quantum coherence on the probe transition, in our case, ρ ab. This counterintuitive sequencing results in enhancement of the probe effect, which is efficient transfer of population in the stimulated Raman adiabatic passage (STIRAP) case and, as seen in this paper, in enhanced probe gain in the LWI case. As will be seen below, we find that the counterintuitive enhancement of the gain on the probe field depends quite strongly on the time elapse between turning on the drive field and the advanced turn on of the probe field. There are many realistic atomic systems which can be considered as ladder-type transition schemes. As an example of possible experimental realization of such a scheme, one can consider cold 87 Rb atoms with 5S / 5P 3/ (78.7 nm) as a probe transition, and 5P 3/ 5D 5/ ( nm) as a drive transition.. LEVEL SCHEME AND DENSITY MATRIX EQUATION OF MOTION Consider the closed three level Ladder-type system illustrated in Fig.. The transition c b of frequency ω bc is driven by a strong single mode, coupling laser of frequency ω L. A weak probe laser field of frequency ω p is tuned to the transition a b of frequency ω ba. γ ba and γ cb are the spontaneous emission rates of state b and c, respectively. The states a and c are not directly coupled, i.e., dipole transition forbidden. No incoherent pumping mechanism, is Δ L ω ca ω ba Fig.. Level diagram for the ladder scheme. γ ba and γ cb are the spontaneous emission rates of state b and c, respectively. Ω and G are the drive and probe fields Rabi frequencies of the relevant transitions. Δ p c b c applied to the system. We use a master equation for the density operator similar to that presented in [4], adjusted to account for the current model. It reads: i γ ρˆ - [ ˆ, ρˆ ] ba ( ---- s ) ( + s ) ( ρˆ ρˆ s ) ( ) = [ + + s ] ( () + γ ba s ) ( ) γ ρˆ s cb ( s ) ( + s ) ( ρˆ ρˆ s ) ( ) [ + + s ] ( + γ cb s ) ( ρˆ s ) +. Here, ρˆ is the density operator, ˆ Ge iδ p t ( ) s + Ωe iδ L t ( ) = ( + s + ) + H.C. () is the Hamiltonian in the rotating wave approximation ( ) and in the interaction representation, and s + = b a, ( ) () i () i s = c b, = ( ) + s s +, i =,, 3 are the atomic rasing and lowering operators for the transitions a b, b c, respectively. Ω, G are the classical complex Rabi frequencies associated with the coupling and probe laser field. Δ L = ω cb ω L and Δ p = ω ca ω p, are the driving (coupling) laser and probe detunings. Projection of the master equation over the atomic bare representation basis yields the equations of motion for the density matrix elements: ρ aa = ig( ρ ba ρ ab ) + γ ba ρ bb, (3a) ρ ab = -( γ ba iδ p )ρ ab iωρ ac + ig( ρ bb ρ aa ), (3b) ρ ac = -[ γ cb i( Δ p + Δ L )]ρ ac iωρ ab + igρ bc, (3c) ρ bb = γ ba ρ bb + γ cb ρ cc (3d) + ig( ρ ab ρ ba ) + i( Ωρ cb Ωρ bc ), ρ bc = -( γ ba + γ cb iδ L )ρ bc (3e) + igρ ac + iωρ ( cc ρ bb ), ρ cc = γ cb ρ cc + iωρ ( bc ρ cb ). (3f) In obtaining equation (3a) (3f) we have removed the explicit time dependence via introducing the new variables, ρ ab ρ ab e iδ p t, ρ ac ρ ac e i ( Δ L + Δ p )t = =, ρ bc e Δ L = t, ρ ii = ρ ii. ρ bc (4) 3. RESULTS AND DISCUSSION Figure illustrates the order and timing of field application. In a counterintuitive application of fields the probe field is turned on prior to the coupling field. This is counterintuitive since one can expect that the LASER PHYSICS Vol. No. 7

3 COUNTERINTUITIVE VERSUS REGULAR INVERSIONLESS GAIN 3 Intensity Intuitive zone Δτ p Probe τ L Coupling Counterintuitive zone Δτ L Time Fig.. Counterintuitive and regular (intuitive) field sequencing. In a counterintuitive situation the probe field is applied prior to the coupling field. The order is reversed for the regular or intuitive sequencing. Both fields are taken as rectangular pulses. Probe and coupling initiation times are zero and τ L. Δτ L and Δτ p are the corresponding widths. system ought to be prepared first by the strong coupling field and only than the probe field should be initiated. The probe field pulse (rectangular in shape) is turned on at t = while the coupling field pulse (rectangular in shape) is delayed τ L with respect to it (see Fig. ). The width of the probe and coupling pulses are Δτ p, and Δτ L, respectively. In order to account for the situation described above we take the coupling and probe Rabi frequencies in the form: Ω = Ω H( t τ L )H( τ L + Δτ L t), (5) G = G H()H t ( τ p t), where Ω, G are the Rabi frequencies and H(t) is the Heaviside unit step function. We solve numerically the equations of motion (3a) (3f), subjected to the initial conditions ρ aa () =, ρ ij = for i, j a (i.e., all population is taken to initially reside in the ground state and no pre-preparation of the system is assumed) along with the conservation of population condition, ρ ii =. System parameters are taken to account for a non-inversion situation on the probe transition, i.e., ρ bb ρ aa < (for specific values see figure captions). In addition we set Δ L = Δ p = and γ ba = γ cb = γ. In our notation gain in the probe transition b a, corresponds to (ρ ab ) >. Figure 3 displays the imaginary part of coherence ρ ab as a function of normalized time (in units of γ ) for the case of CW probe field (Δτ p ). The coupling field width, Δτ L, is γ. Several curves are plotted corresponding to various values of the time delay, τ L, with the latter being displayed above each curve. The black thick line corresponds to a situation where the coupling field (driving field) ends before the probe field is applied (τ L < and Δτ L < τ L ). As can be expected only absorption is present. Common to all curves is the absorption present until the initiation time of the Im[ρ ab ], arb. units Time, in units of /γ Fig. 3. Imaginary part of ρ ab as a function of time (shown in units of /γ) at various values of time delay τ L between the CW probe field and the drive pulse. Parameters used in this calculation are: Ω = 6.3γ, G =.63γ, Δτ L = /γ. The delay time τ L (in units of /γ) is shown above each curve. The thick black curve corresponds to the situation when the drive pulse (negative τ L ) ends before the beginning of the probe field. probe field (all curves coincide with the black curve for times t < τ L ). At t = τ L, the gain/absorption line profile of the probe transition changes from absorption to a peak of gain. The maximum gain starts to decrease before the probe pulse dies out and because the probe pulse is narrow (Δτ p = γ ), no additional peaks of gain are formed. After the probe pulse ends, the line profile changes abruptly, absorption gets deeper and coincides at steady state with the black line (no probe is present). As can be observed the maximum gain peak increases and becomes saturated as τ p increases. This is to be expected since that at the time of incipience of the probe pulse the system has acquired a noticeable coherence between states a and b. The saturation of the gain peak is of course a consequence of the fact that for long time delays there is no significant change in the coherence ρ ab at the time the probe pulse is initiated. Figure 4 shows the results of calculation of (ρ ab ) for the situation where both fields are taken as rectangular pulses. Probe and coupling width are Δτ p = γ, and Δτ L =.75γ. Several curves are plotted corresponding to probe time delay, τ L, starting from γ to 7γ in increments of γ. Other system parameters are denoted in figure caption. In Fig. 4a τ L = γ and thus corresponds to an intuitive sequence of field application. However, since the coupling field width is smaller than γ (Δτ L =.75γ ) it dies before the probe field is initiated. Hence, absorption on the probe transition is present. In Fig. 4b both fields start LASER PHYSICS Vol. No. 7

4 4 BRAUNSTEIN et al... (a). (b). (c) (d). (e). (f) (g). (h). (i) Fig. 4. Imaginary part of ρ ab as a function of time (shown in units of γ ) at various values of time delay τ L between the probe and the drive pulses. Parameters used in this calculation: Ω = 6.3γ, G =.63γ, Δτ p = /γ, Δτ L =.75/γ. Delay time τ L (in units of /γ): (a), (b), (c), (d), 3 (e), 4 (f), 5 (g), 6 (h), 7 (i) (a). (b). (c) (d) (e). (f) (g) (h) (i) Fig. 5. Imaginary part of ρ ab as a function of time (shown in units of γ ) at various values of time delay τ L between the probe and the drive pulses. Parameters used in this calculation: Ω = 6.3γ, G =.63γ, Δτ p = /γ, Δτ L = 3/γ. Delay time τ L (in units of /γ): 3 (a), (b), (c), (d), (e), (f), 3 (g), 4 (h), 5 (i). LASER PHYSICS Vol. No. 7

5 COUNTERINTUITIVE VERSUS REGULAR INVERSIONLESS GAIN 5 at the same time (τ L = ) and a small peak of gain develops. Since the coupling field terminates relatively fast due to its small width, absorption commences. For further coupling field delay the gain peak drifts to longer time and increases in its intensity (see Fig. 4c). As the coupling field delay is increased more the gain peak drifts to longer times and decreases in intensity. This is attributed to the fact that at longer times the coupling field meets the system with a relatively small value of coherence on the probe transition. Figure 5 displays results similar to those presented in the former figure, however the coupling pulse width is greater now and equals to Δτ L = 3γ. Figure 5a is identical to that presented in part (a) of the previous figure. In Fig. 5b the coupling pulse starts prior to probe pulse, τ L = γ and since its width is Δτ L = 3γ there is a time of duration γ for which both fields overlap. A narrow gain peak of small intensity develops and after both fields have died the system returns to absorption. Further increase of the time delay (Figs. 5c and 5d) results in additional peaks of gain and the probe transition exhibits the usual Rabi oscillations. In Fig. 5d the coupling field appears at the time as the probe field terminates. At this time the system has acquired a substantial coherence and hence the system exhibits approximately 5 fold larger gain then in the previous case. Furthermore, due to the relatively broad coupling pulse, three gain peaks appear now. Further increase of the time delay τ L results in the same number of gain peaks, however with smaller intensity. This is due to the fact that at long times any coherence present in the system has washed out. 5. CONCLUSIONS In this study we showed that gain without the requirement of population inversion can be achieved in the ladder system utilizing pulsed coupling and prob laser fields, in a counterintuitive order of application. This is a consequence of quantum coherence and interference among multilevel schemes utilizing external coherent fields. The role played by the time delay between the fields as well as their width was explored. REFERENCES. S. Ya. Kilin, K. T. Kapale, and M. O. Scully, Phys. Rev. Lett., 736 (8).. M. O. Scully, Shi-Yao Zhu, and A. Gavrielides, Phys. Rev. Lett. 6, 83 (989). 3. A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev. Lett. 66, 54 (99). 4. J. Mompart, C. Peters, and R. Corbalan, Phys. Rev. A 57, 63 (998). 5. G. Vemuri, K. V. Vasavada, and G. S. Agarwal, Phys. Rev. A 5, 38 (995). 6. V. Ahufinger, R. Shuker, and R. Corbalan, Appl. Phys. B: Laser Opt. 8, 67 (5). 7. G. A. Koganov and R. Shuker, Laser Phys. 9, 73 (9). 8. M. A. Erukhimova, Laser Phys. 7, 656 (7). 9. A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, Phys. Rev. Lett. 75, 499 (995).. G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, Phys. Rev. Lett. 76, 53 (996).. J. Kitching and L. Hollberg, Phys. Rev. A 59, 4685 (999).. Y. Zhu and J. Lin, Phys. Rev. A 53, 767 (996). 3. G. Grynberg, M. Pinard, and P. Mandel, Phys. Rev. A 54, 776 (996). 4. D. Braunstein and R. Shuker, Phys. Rev. A 64, 538 (). 5. D. Braunstein and R. Shuker, Phys. Rev. A 68, 38 (3). 6. D. Braunstein and R. Shuker, Laser Phys. 9, 9 (9). 7. D. Braunstein and R. Shuker, Laser Phys. 8, 37 (8). 8. R. Shuker, A. Har-Tal, and G. A. Koganov, Laser Phys., 954 (). 9. J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, Phys. Rev. A 4, 674 (989).. C. E. Carroll and F. T. Hioe, Phys. Rev. Lett. 68, 353 (99).. B. Broers, H. B. van Linden, van den Heuvell and L. D. Noordam, Phys. Rev. Lett. 69, 6 (99).. S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 7, 3637 (993). 3. A. Raczynski, A. Rezinerska, and J. Zaremba, Phys. Rev. A 63, 54 (). 4. J. Martin, B. W. Shore, and K. Bergmann, Phys. Rev. A 54, 556 (996). 5. M. B. Plenio and P. L. Knight, Phys. Rev. A 53, 986 (996). 6. V. S. Malinovsky and I. R. Sola, Phys. Rev. A 7, 434 (4). 7. Z. Kis and F. Renzoni, Phys. Rev. A 65, 338 (). 8. O. Kocharovskaya and P. Mandel, Phys. Rev. A 4, 53 (99). LASER PHYSICS Vol. No. 7