The Theory of Three-level Photon Echo Using of Rotating Wave Approximation
TThe Theory of Three-level Photon Echo forQuasi-degenerated Levels
Tara Ahmadi ∗ ,Sergei.A Pulkin and Vladimir.A SheftsovDepartment of General Physics,St.Petersburg State University,Petergof,St.Petersburg,RussiaOctober 26, 2018 Abstract
The three-level photon echo has been described in different works[1,2]by using rotating wave approximation but none of them didnt get re-sults which show the effects of detuning frequencies on frequency ofground level of system; also the effect distance between two generatedlevels on signal resonance is being neglected. In this work, we stud-ied a -type system theoretically and numerically. By considering theDopplers effect in environment, we get different equation for polariza-tion of echo signal and its intensity.
Quantum optical data storage is a key element in quantum information pro-cessing such as quantum computing and long-haul quantum communicationsbased on quantum repeaters. However, most modified photon echoes pro-tocols are still limited by lack of homogenous resonance signal to explaintime scales for the different systems. Photon echo spectroscopy is the mostimportant way which can deeply extract microscopic information about thetime scales of molecular and collective dynamics of condensed phase. Pho-ton echo spectroscopy can cover any inhomogeneous broadening in state.The three-level photon echo is made of three different signals in a medium. ∗ [email protected] a r X i v : . [ qu a n t - ph ] M a y mong these levels or dots distributed homogenously resonance frequencies.In the present paper, by using rotating wave approximation, we showed thatproportion of detuning frequencies is equal with proportion of frequenciesbetween degenerated and ground levels. Our method to solve relevant equa-tions is acceptable but simpler than the other methods. This equation canbe used in order to determine the resonance form of echo signal. As hasbeen shown in previously works, signal resonance is depending on frequen-cies between degenerated and ground levels. After finding the relationshipbetween frequencies and their detuning amounts, its obvious to predict anequation for delay time as ratio of frequencies. Polarization of signal usu-ally depends on frequencies between levels and frequencies detuning. Thisrelationship gets an exponentially resonance for signal polarization betweenground level and second degenerated level. By using this results as numer-ical data, we found an exponentially function for intensity as frequencies,signal delay time and distance of ground level from two degenerated levels.As has been shown previously, it is possible to write equation for frequencydetuning between laser fields either as function of frequencies or as functionof frequency distance between ground level and second degenerated and fre-quency detuning between laser field and transition frequency. In the lastsection of paper, the effect of frequency distance between these two levels onthe general intensity of output signal has been proved. It seems that theseresults have strongly consistent with theoretical background and other sameworks. In other words, our results for the effect of frequency distance be-tween ground and low degenerated level can give decay coefficient of signalin different cases of system and position of echo signal in the long timescalefor the case of qusi-degenerated levels can be predicted. As using strength field for transition between degenerated levels and groundlevel, the reaction time of pulses increase which means decay of signalsmemory time. But this system have Λ − type structure in three-level atomswith long-life lower-levels. These lower-levels are fine-structure componentswhich the interaction between them is so weak and is only due to spin-spinstructure. This can help to create a long-living houl for using as informa-tion/signal memory. In this system we have two strong femtosecond pulsesand a detuning pulse. In the real systems, delay time between detuningdriving pulses is 500microsecond which compared to pulse time (100fem-tosecond) increases the signals memory time 10 times more than signals2ime. In order to prove this process we studied a three-level system. Thesystem considered a three-level Λ − type system with a ground state (2). Thetransition2 ↔ ω and Rabbi frequency G c .The tran-sition 2 ↔ ω and Rabbi Frequency g c . The ratesof emission from 1 and 3 are denoted by 2 γ and 2 γ so detuning amountsof the probe and coupling fields are ∆ = ω − ω and ∆ = ω − ω .From the Liouville equation ˙ ρ = i (cid:126) [ H, ρ ] − γρ Where H- Hamiltonian systemof atom and field and ρ - element of density matrix. The matrix densityequations for three-level system interacting with a pulsed laser [4, 5] in asemi classical dipole rotating-wave approximation are ˙ ρ = − γ ρ + 2Λ ρ − V ( ρ + ρ ) + ig c ρ − ig ∗ c ρ ˙ ρ = − γ ρ − V ( ρ + ρ ) + iG c ρ − iG ∗ c ρ ˙ ρ = − γ ρ + 2Λ ρ − V ( ρ + ρ ) + iG c ρ − iG ∗ c ρ ˙ ρ = − ( γ + Λ + i ∆ ) ρ − V ρ + ig p ( ρ − ρ ) − G c ρ ˙ ρ = − [ γ + γ + i (∆ − ∆ − d )] ρ + ig p ρ − iG ∗ c ρ ˙ ρ = − [ γ + Λ − i (∆ − ∆ − d )] ρ − V ρ − ig p ρ + iG c ( ρ − ρ ) (1)Where parameter ρ ii represented the population of i level and ρ ij representedpopulation between levels, V = − D.E laser cos( ω t ) is interaction of mediumwith laser field,D=dipole moment and d= frequency distance between twodegenerated levels. We introduced the Rabi frequencies as g p = g e i Φ p and G c = g e i Φ c where is phase of frequency. There are several situations in thiscase: (cid:40) ϕ p = ϕ c ϕ p (cid:54) = ϕ c In the first situation laser fields are polarized likely andFigure 1: Λ-type system with driving and probing field3n the second they are polarized differently. In general words, can say thatphases have small influence on the photon-echo signal. We calculate the timeof echo for ensemble of atoms in gas; in which occurs non-uniform spread offrequencies with the Maxwell velocity distribution due to the Doppler Effect,which means proportion ∆ ∆ = ω ω is valid where∆ = ω laser − ω , ∆ = ω laser − ω . During the first emission t to t and second t to T (secondpulse and echo), level 3 does not expose to field and phases of atoms in thislevel acquire phase shift ( T − t ) ∗ ( − ) υc . But phase shift at level 2does not occur not during first nor second emission [7]. Therefore for level2 general phase shift during emissions is υ ∗ ω c ( T − t ). Polarization duringemission from 2 −→ P ( t ) ∼ exp[∆( T − t ) − ∆ (cid:48) ( T − t )] or P ( t ) ∼ exp[( − ω ( T − t ) − ( ω − ω )( T − t )) vc ] (2)The echo signal occurs if phase treats to zero, so we can write equation forsignal delay time as: T = t + ω ω ∗ t . By using the elements of densitymatrix and solving Bloch equation for main its elements(x,y,z), equation forpolarizations in x ,y and z directions can be written as P x = P z = 0. Fromthe equation.1, its obvious that P = (cid:88) i − γ ( ρ − ρ ) + iρ ( k i ϑ i − k (cid:48) i ϑ (cid:48) i ) − ig p ( ρ − ρ + ρ ) P = − (cid:88) i γ ( ρ + ρ ) + ik i ϑ i ρ + ig p ( ρ − ρ + iρ ) (3)Where k i υ i and k (cid:48) i υ (cid:48) i are the wave coefficients and frequencies of differentpulses. By using Gaussian probability functions as f (∆ ωt ) for inhomoge-neous distribution of velocity in system, we can write: P y = (cid:88) i P iy = − P (cid:82) + ∞−∞ cos (∆ ωt ) .f (∆ ωt ) d (∆ ω ) P iy = − P i { cos (∆ ω i t ) cos [∆ ω i ( t − ∆ ω i t )]+ sin (∆ ω i t ) sin [∆ ω i ( t − ∆ ω i t )] } (4)In the result of this method P y ∼ P exp [ − ( t − ω ω t ) T ∗ ]. Where T ∗ is half-width Gaussian distribution. 4 Numerical Framework
In this section, we give some numerical simulations by using RWA expressionfor the state population given from Sec .II. The relevant system propertiesare length of the signal pulses of 100 fs, the probabilities of relaxation tran-sitions are denoted by γ and γ ∼ c − , Homogeneous broadeningΓ ∼ c − , γ = 0 , Γ = 0 , ∆ = 50 , ∆ ∆ = 1. When the pulse resonateswith this excited state, dynamics contributions of excited and ground statesinduce signal. The polarization spectrum of nonlinear system is found fromFourier transform of elements of density matrix.Distance between control pulses for weak degenerate levels b =0.05, c=0.10,a=0.02(Signals polarized differently) As it has been shown in figure.3 in-tensity of signal is depends on the delay time exponential. This figure hasbeen gotten by using parameter of distance in equation for detuning probefield [2]. the intensity of echo signal depends on the modulation index ofprocess grating which led to a reduction of the echo signal.Based on [7] dur-ing delay time T, spectral diffusion cases frequency grating to earase andtheir magnitude to decay in time.The results of our simulation are shown infigure.2. They show a maximum in the signal intensity around T.This peakin related to coherant pump-probe signals.It’s obvious that there is no fastdecay for intensity after separation of third signal from beams in time.So wehave good decaying behaviour on long timescales. In this figure we have pe-riodic exponentially decay for signal intensity. The range of quantum beatsdepends on the frequency distance between levels (d). This phenomenonhas been observed experimentally and used in echo spectroscopy to find theweak degeneracy in atoms. Difference between figures of signal intensity intwo cases (case of polarization) is in place of periods. Signal in these twocases occurred likely but in different times. This difference is equal withcoefficient γ . For explain this decrease, we use energy changes of a photonwhich adds to and removes from the beam, due to the decrease of the upperand lower level. For the number of photons N p of the beam˙ N i = dN i dt = − N i ( T ransitionP robability )) = − N i B ij ρ ij (5)Where B is proportionality factor in transition probability between the en-ergy levels (i and j) with degeneracies depend on ω : N = ω ω N − N Forour system B and are equal to 1, so we have˙ N i = − ω ω γ (6)5or spectral energy density:˙ ρ = ∆ E d ˙ N p dV olume = ∆ E ( − ω ω γ ) (7) ρ ( t ) = ρ e ∆ E ( − ω ω γ ) (8)But I ∝ ρ also I = EnergyAreatime = I e ( − ω ω γ ) = I e at The position of photonecho signal is proportional to the detuning time between driving pulses. Wecan get a decay coefficient from this figure as (cid:52) (cid:52) γ = a which has a goodagreement with theory [1].When distance between 1 and 3 is zero means that they have the sameground level, function of I is like δ ( t γ ) - function. Intensity in this casedecrease rapidly and there is only one max point in figure at first part signal.This figure is same as intensity of signal when energy of ground level is zero[18]. You can see that for d = ∞ fairly there are not any quantum beats andwe have a very slow population relaxation compared with theoretical coef-ficient of signal decay. There are some remarkable characteristics of 3-levelphoton echo by using fluorescence excitation measurements in figures.2and 3.A direct relationship of spectral diffusion has been provided by the variationinjections rate measured as a function of time between 1 −→
2. Differentview of figure.3 shows the importance of distance between two degeneratelevels on signal density.
In this paper we developed a general theory of three-level system by assum-ing rotating wave approximation and gave an analytical solution for problemof frequency distance between ground and low degenerated level. Moreoverin the three-level system we gave some approximate solutions which are im-portant in order to analyze experimental data. We carried out numericalsimulation of an atomic medium to the field of two driving pulse and onesignal pulse in Λ-scheme. It was shown that the time of two long-wavelengththree-pulse echoes is proportional with ratio of detuning of the resonancelines of transitions in a non-uniform distribution. As the result of numericalsimulation for RWA, the dependence of the signal from the delay time inweak degenerate levels. These results are consistent with experiment andtheory. By this time all figures and theories have neglected relationship be-tween signals intensity and detuning amounts of fields, but in this work, weproved linear signature. In general words, photon echo spectra are found to6e strongly dependent on frequency of pump and probe frequencies of sys-tem also their dependence with detuning amounts of frequencies. During thelast part of simulation, are found the importance effect of frequency distancebetween two degenerate levels (d) on the delay time of signal. This depen-dence could help us to make coherent distribution of signal in system. Forexample if have been investigated investigate the characteristics of a quan-tum dot, we find out that if first impulse has time interval of 100fs, we canincrease the echo time by increasing the ratio of detuning time frequenciesand frequency distance between two degenerated levels. Basic on the calcu-lations the ratio of pulse signal to interval between 1 and 2 is T pulse t ∼ − .In other words by using the effect of frequency distance on the time of echosignal occurrence, its possible to increase memory time of this dot to 500s.Increasing memory time of quantum dots is one of the most important tasksof quantum electronics.In order to use this method for quantum dots, signalintensity has to be as strong as to give g p t ∼ π (Impulse area). In otherwords, we have to collect a system from ∼ × atoms. References [1] A.A.Villaeys, Phys.review A ,053418 (2009)[2] S. A. Pulkin, et al,Opt. Spectrosc. ,2 (2002)[3] X.H.Yang, Eur.Phy.D ,253258 (2010)[4] G.X.Li, et al. , arXiv:10020400v1, (2010)[5] H.Freedhoff, J. Opt. Soc. Am. B ,1337 1993[6] A.V. Pisliakov, J. Chem. Phys. , 234505 (2006)[7] A. Piryatinski, Phys.review B , 161404 (2004)[8] N. Christensson, J. Chem. Phys. , 024510 (2009)[9] J. D. Hybl , J. Chem. Phys. , 6606 (2001)[10] R. Agarwal, et al. J. Chem. Phys. , 6243 (2002)[11] A.Brawn, Phys.review A , 013403 (2000)[12] Y.J. Yan and S. Mukamel, J. Phys. Chem , 5160 (1988)[13] Y.J. Yan and S. Mukamel, Phys. Rev. A , 6485 (1990)714] S.Mukamel, principle of nonlinear optical spectrometry, oxford univer-sity press, (1995)[15] D. Abramavicius, et al, Europhys. Lett. , 17005 (2007)[16] P. Hamm and M. T. Zanni, Concepts and Methods of 2D InfraredSpectroscopy,Cambridge University Press, (2011)[17] L.V.Dao, et al. , J.Chem.Phys , 1651057(2004).[18] S.A.Pulkin, et al. , J. Opt. and Spectrosc, ,6 (2007)8 bc AB CFigure 2: intensity of echo signal as a function of delay time9igure 3: Intensity of echo signal as a function of t γ . For frequency dis-tance is d=0 - (dashed), d = ∞∞