Coherent writing and reading of information using frequency-chirped short bichromatic laser pulses

Coherent writing and reading of informationusing frequency-chirped short bichromatic laser

pulsesG. P. Djotyan, J. S. Bakos, Zs. Sörlei

Research Institute for Particle & Nuclear Physics XII. Konkoly Thege ut. 29 - 33, H-1525, Budapest, Hungary

[email protected]

Abstract: We propose to use the sensitivity of the population transfer in three-levelΛ-atoms to the relative phases and amplitudes of frequency-chirped shortbichromatic laser pulses for coherent, fast and robust storage and processing ofphase or intensity optical information. The information is being written into theexcited state population which in a second step is transferred in a fast and robustway into a nondecaying storage level. It is shown that an arbitrary superposition ofthe ground states can be generated by controlling the relative phase between thelaser pulses.

© 1999 Optical Society of America OCIS codes: (020.1670) Coherent optical effects; (070.4560) Optical data processing_____________________________________________________________________________________________

References and links

1. E. Arimondo and G. Orriols, “Nonabsorbing atomic coherences by coherent two-photon transitions in a three- level optical pumping,” Nuovo Cimento Lett. 17, 333-338 (1976).2. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one- photon recoil energy by velocity-selective coherent population trapping,” Phys.Rev.Lett. 61, 826-829 (1988).3. J. Lawall, and M. Prentiss, “Demonstration of a novel atomic beam splitter,” Phys.Rev.Lett. 72, 993-996 (1994).4. J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses,” Phys.Rev. A 40, 6741- 6747 (1989).5. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys.Rev.Lett. 67, p.1855-58 (1991).6. M. Weitz, B. C. Young, and S. Chu, “Atomic interferometer based on adiabatic population transfer,” Phys.Rev.Lett. 73, 2563-2566 (1994).7. P. Marte, P. Zoller, and J. L. Hall, “Coherent atomic mirrors and beam splitters by adiabatic passage in multilevel systems,” Phys. Rev.A 44, R4118-R4121.8. R. Unanyan, M. Fleischhauer, B. W. Shore, K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. 155, 144-154 (1998).9. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).10. D. Kosachev, B. Matisov and Yu. Rozhdestvensky, “Coherent population trapping: sensitivity of an atomic system to the relative phase of exciting fields,” Opt.Commun. 85, 209-212 (1991).11. N. V. Vitanov, “Analytic model of a three-state system driven by two laser pulses on two-photon resonance,” J.Phys.B: At. Mol. Opt. Phys. 31, 709-725 (1998).12. C. E. Caroll and F. T. Hioe, “Three-state model driven by two laser beams,” Phys.Rev.A 36, 724-729 (1987).13. G. P. Djotyan, J. S. Bakos, G. Demeter and Zs. Sörlei, “Theory of the adiabatic passage in two-level quantum systems with superpositional initial states,” J. of Modern Opt. 44, 1511-1523 (1997).

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1. Introduction

Producing samples of atoms or molecules whose population resides almost entirely in asingle desired quantum state is the goal of many important applications of the quantumchemistry, quantum optics, mechanical manipulation and cooling of atoms or molecules bylaser radiation. On the other hand, the coherent superpositions of quantum states also playan important role in the above mentioned fields of science with very interesting and useful

(C) 1999 OSA 18 January 1999 / Vol. 4, No. 2 / OPTICS EXPRESS 113#8347 - $15.00 US Received December 07, 1998; Revised January 08, 1999

applications. The population trapping [1] and laser cooling below the recoil limit [2],construction of atomic beam-splitters based on the dark states [3], using the dressed statewith zero quasienergy for complete transfer of the atomic population from one ground stateinto the other one without populating of the intermediate excited state in the scheme of thestimulated Raman adiabatic passage (STIRAP) [4], enhancement of the refractive index of aresonant medium using the appropriate coherent superposition of the ground states [5] is alist of important applications of the coherent superposition quantum states.

A coherent superposition of quantum states may be created, in principle using a radiofrequency field coupling these states. Alternatively, the STIRAP technique in a three-levelΛ−atom may be modified by maintaining a fixed ratio of Stokes and pump pulse amplitudesat the end of the interaction to create a definite superposition of the ground states assuggested in Refs. [6,7]. This method, however requires an accurate control of the relativestrength of Stokes and pump pulses and, for example inhomogeneous transverse intensitydistribution of the laser beams may cause problems. A method of generation of coherentsuperposition of long lived quantum states robust against small variations of the time andspace parameters of the laser pulses was suggested in Ref.[8]. This method is based on theusual three-level STIRAP scheme with coupling of the intermediate excited state to a forthmetastable state by a third (control) pulse. The final coherent superposition of the groundstates in this tripod-linked system can be governed by adjusting of the relative delay of thecontrol pulse. The relative phase of the components of the superposition is determined by therelative phase of the Stokes and pump pulses.

We propose in this paper an alternative and more simple scheme for generation ofarbitrary coherent superposition of two ground states of a three-level • • atom by usingfrequency-chirped short bichromatic laser pulses (BLP) each being superposition of twopulses of a same shape with different carrier (and, in general different Rabi) frequenciesbeing in Raman resonance with the atom under the conditions of the adiabatic passage (AP)regime of interaction [9]. Each laser pulse of the BLP couples corresponding ground states ofthe • • atom to a common excited state, as it is shown in Fig.1. Due to the identical shapeand chirp of the pulses and the assumed condition of the Raman resonance, the system underconsideration is equivalent to a two-state system consisting of a "bright" superposition of thetwo ground states which is coupled to the excited state by the laser field and of a decoupled"dark" superposition of the ground states which doesn't interact with the laser field. As iswell known, a frequency-chirped laser pulse produces complete transfer of populationsbetween the states of a two-state atom in AP regime of interaction [9]. In our case, thefrequency-chirped BLP produces complete population transfer from the "bright"superposition to the excited state in a time short compared to the relaxation times of the atomleaving unchanged the population of the "dark" superposition state. Since the “bright” and“dark” superposition states depend on the relative phase and relative strength of the pulsesforming the BLP, this dependence will be transferred to the population of the excited state.In the same time an arbitrary coherent superposition of the ground states (an arbitrary Ramanor Zeeman coherence) can be generated in the “dark” superposition of the ground states. Thepopulation of the excited state in a second step has to be transferred in a fast way into a

nondecaying storage level (level 4 in Fig.1) by a subsequent frequency-chirped laser pulseto avoid the decoherence induced by the relaxation processes. It is important to note that the

nondecaying storage level 4 in our scheme is being coupled to the excited state 2 onlyafter the action of the BLP and doesn’t play any role during the interaction of BLP with thethree-level • • atom. So, our scheme differs significantly from the tripod-linked scheme ofRef.[8].


Fast and robust writing, storage and reading of optical information is a task to be solvednowadays, especially when short laser pulses are used as the carriers of the opticalinformation. The sensitivity of the population transfer in a multilevel quantum system to therelative phases of exciting laser fields considered above (see also [6-8,10,11]) may be used tosolve this problem.

We propose in this paper to use the suggested above scheme of generation ofsuperposition states which are sensitive to the relative phase and relative strength of pulsesforming BLP for coherent writing, storage and reading of the phase or intensity opticalinformation. The information which is assumed to be contained in the relative phase orrelative amplitude of the pulses of the BLP is being written in the population of the excitedstate when the population of the “bright” superposition of the ground states is beingcompletely transferred into the excited state by the frequency-chirped BLP. The subsequentfrequency-chirped short laser pulse is used to transfer the population of the excited state (andso, the information written into the population of this state) into the storage nondecaying

level (level 4 in Fig.1). The same laser pulse can be used for reading out the stored

information by exciting the atom from level 4 . It is worth noting that the informationwriting and reading processes are fast and robust in the proposed scheme due to theshortness of the applied laser pulses and the robustness of the population transfer producedby the frequency-chirped pulses in the AP regime of interaction. As was mentioned above, the three-level problem under consideration reduces to theinteraction of a single laser pulse with an equivalent two-level system. There exists a numberof exact analytic solutions to the problem of interaction of short laser pulses (with chirpedfrequencies as well) with two-level systems for certain pulse shapes and modulations in time[11,12]. We use here the simpler approximate solutions corresponding to the AP regime[9,13].

2. The mathematical formalism

We consider interaction of BLP consisting of two linearly polarised laser pulses having sameshape f t( ) with a three-level Λ−atom having two ground states 1 and 3 and excited state

2 , as it is shown in Fig.1, where also is depicted a nondecaying (metastable) state 4 whichis assumed to be not coupled to the Λ -system by BLP. This state will be used only fortransfer of the population of the excited state (for information storage, see below) at a secondstep, by applying a short frequency-chirped laser pulse coupling the states 2 and 4 afterthe interaction with the BLP.

Fig.1. The scheme of the atomic system.

The laser pulses forming the BLP have complex amplitudes

A t f t A A A i1 2 1 20

1 20

1 20

1 2, ,( )

,( )

,( )

,( ) ( ) ; exp[ ]= = Φ and time-dependent frequencies ω L t1( ) and

I1>

I2>

I3>

I4>


ω L t2 ( ) . The pulse A t1( ) couples the states 1 and 2 , and the pulse A t2( ) couples thestates 3 and 2 . The carrier frequencies of both pulses of the BLP are assumed to bechirped in time in the same way and the BLP duration is assumed to be much shorter than allrelaxation times of the atom. It allows as to deal with the Schrödinger equation for theamplitudes a jj , , , = 1 2 3 of the states of the • • atom in Fig.1.

It is useful (for reduction of the three-state problem to a two-state one) to introduce avector C , whose components ci i, , ,= 12 3 are proportional to the amplitudes ai i, , , = 12 3 of the

atomic states:

c a c a c a i t t1 1 1

2

2

2

1 3 2 1

2

2

2

3 2 2 21= + = + =Ω Ω Ω Ω Ω Ω* */ ( ) ; / ( ) ; exp[ ( ) ] ε ,

where the Rabi frequencies 212 32Ω Ω( ) and 2t t( ) with complex amplitudes Ω1 and Ω2 are

introduced: Ω Ω Ω Ω12 112

1 32 232

20

2 2( ) ( ) ( ) ( ) ( ) ( ) ( )t f t f t

dA t f t f t

dA= = = = and (0)

! !

with

dij being the dipole moment matrix element for the laser induced transition from state j to

state i , (i j, , ,= 12 3). ε ω ω21 1 21( ) ( ) ,t tL= − ε ω ω23 2 23( ) ( )t tL= − are the detunings from

the one-photon resonances with ω ω21 23 and being the resonant transition frequenciesbetween the corresponding states. We assume, that the condition of the Raman resonance hasbeen fulfilled: ε ε ε21 23( ) ( ) ( )t t t≡ = . In what follows, we assume the same linear temporal

chirp of the carrier frequencies of the pulses forming BLP: ω ω βLj Ljt t( ) ( )= +0 2 , j = 12, ,

where ωLj( )0 are the central frequencies and 2β is the speed of the chirp, see Fig.1.

We obtain the following equation for the vector C from the Schrödinger equation for theprobability amplitudes a jj , , , = 1 2 3 , [12]:

d

dti HC C=

^

(1)

The Hamiltonian H^

in the rotating wave approximation is:

^

( )

( ) / ( )

/

/

H f tt f t

=

+

+ +

+

0 0

0 0

1

2

1

2

2

2

1

2

2

2

1

2

2

2

2

2

1

2

2

2

Ω Ω Ω

Ω Ω Ω Ω

Ω Ω Ω

ε

;

One can introduce the following g ( )+ and g ( )− amplitudes of the coupled to the excitedstate by the BLP “bright” and uncoupled “dark” superpositions of the ground states havingamplitudes a1 (c1) and a3 (c3):


g a a c c

a a c c

( ) * *( ) / ;

( ) / ( / / ),

+ = + + = +

= − + = −

Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω Ω Ω

1 1 2 3 12

22

1 3

2 1 1 3 12

22

1 2 1 12

3 22

g(-)

and the excited state amplitude e c= 2 .We obtain the following set of equations for the new state amplitudes using Eq.(1):

d

dtg iF t e

d

dte i t e iF t g

d

dtg

( ) ( )

( )

( ) ; ( ) ( ) ;+ +

−

= − =

=

ε

0,

(2)

where F t f t( ) ( )= +Ω Ω12

22

.

The first two equations in Eq.(2) describe an equivalent two-level system with the “bright”

ground state amplitude g ( )+ being coupled to the excited state with amplitude e by the BLP.

The “dark” ground state with amplitude g ( )− does not interact with the BLP, as it followsfrom the third equation in Eq.(2). It is well known that complete transfer of the populations of two-level atom takes place asa result of interaction with a frequency-chirped laser pulse in the AP regime [9]. It meansthat the population of the “bright” state will be completely transferred to the excited state and

we have for the final amplitude g fin( )+ at the end of the laser pulse:

g c cfin fin fin( )+ = + =1 3 0, (3)

if the atom initially was in the ground state: e cin in= =2 0. The subscripts in fin and standfor the initial (t− > −∞) and the final (t− > ∞ ) values. We obtain for the probability amplitudes a fin1 and a fin3 of the ground states of our original

three-level Λ−atom at the end of the laser pulse using Eq.(3):

Ω Ω1 1 2 3* *a a

fin fin= − (4)

In the simplest case of the same Rabi frequencies (Ω Ω1 2= ) of the laser pulses, we obtain

for the final populations nifin and the phases φ ifin (i=1,3) of the ground states:

n nfin fin1 3= ; and φ φ φ π1 3 13 12fin fin fin− = = +∆ ∆Φ , (5)

where the populations and phases of the states are introduced as follows:

a n i jj j j= exp[ ],φ = , ,12 3 and ∆Φ Φ Φ12 1 2= − is the relative (difference) phase of the laser

pulses.

We obtain for the final population n fin2 of the excited state after interaction with the BLP

(in the case of Ω Ω1 2= ):

n n nfin in in in2 1 1 13 121

21 2 1= + − +[ cos( ) ]∆ ∆Φφ (6)


As it follows from Eq.(6), the probability of excitation of the Λ− atom by frequency-chirpedBLP in AP regime strongly depends on the relative phases ∆φ13in and ∆Φ12, when the initial

state of the Λ−atom is a superposition of the ground states 1 and 3 .

The dependence of the final population n fin2 on the initial phase Φ ∆ ∆Φin in= +φ13 12 is

depicted in Fig.2.

Fig.2. Dependence of the excited state population nfin2

on the phase Φin

for different values of population

n in1 of the ground state 1 : n in1 = 1 (1); .8 (2); .6 (3); .5 (4).

There is complete transfer of the populations of the Λ−atom from the ground states tothe excited state: n fin2 1= , when the initial phase Φin = 0 and• there is no excitation at all,

n nfin in2 2 0= = (dark state) at Φin = π in the case of equal initial population of the ground

states, when n nin in1 3 1 2= = / . The population of the excited state does not depend on phases when the atom is in a singleground state initially (n in1 =1): n fin2 1 2= / and n nfin fin1 3 1 4= = / .

As it follows from Eq.(4), we have in the case of the different Rabi frequencies (Ω Ω1 2≠ )

and the atom in the ground state 1 initially, n in1 1= :

n fin2 22

12

1 1= +/ ( / )Ω Ω , (7)

which means that the final population of the excited state is a function of the relativestrengths of the pulses forming the BLP.

3. Results of the numerical simulations

We have solved numerically the set of Eqs.(1) for BLP having durations much shorter thanthe relaxation times of the system to verify the above obtained conclusions.

The envelope of the BLP has been taken as a Gaussian function: f t t L( ) exp[ / ]= − 2 22τ(with τ L being the duration of the BLP), and the same linear chirp has been assumed for thepulses forming the BLP.


The dependence of the population of the excited state on the phase Φ ∆ ∆Φin in= +φ13 12 is

clearly seen in Figs.3 for the Λ−atom being initially in the superposition of the ground states:There is effective excitation of the atom when Φin = 0 (Fig. 3a) and the excitation of the atomis suppressed when Φin = π (Fig. 3b). Note that in the latter case we have a “dark” state in

case of equal initial populations of the two ground states (for Ω Ω1 2= ).

(a) (b)Fig.3. Time dependence of the populations for n in n in1 7 3 3= =. , . at: (a) Φin = 0 , (b) Φin = π

and Ω Ω Ω1 2= = . The parameters applied are: Ωτ βτL L= =52

5, ; green-n t1( ) , blue-n t3( ), red-n t2( ).

4. Writing/reading of phase and intensity information

The atomic system under consideration may be used for construction of arbitrary coherentsuperpositions of the ground states and for writing/reading and storage of phase information.As it follows from Eq.(5), the relative phase of the probability amplitudes of the ground statesis equal to ∆ ∆Φφ π13 12fin

= + at the end of the BLP. So, an arbitrary phase difference ∆φ13in

(arbitrary value of the Raman, or Zeeman coherence) may be generated by controlling thephase difference ∆Φ12 of the pulses forming the BLP.

As it follows from Eq.(6), the population n fin2 of the excited state is a function of the phase

∆Φ12. So, the phase information contained in the BLP may be written in the population n fin2

of the excited state if the atoms are prepared initially in the same coherent superposition ofthe ground states, for e.g., in a “dark” state (with the same initial parameters n in1 , n in3 and∆φ13in

). This information however will be distorted due to the spontaneous decay from the

excited state during the decay time of this state. One of the way to preserve this informationis to transfer the population of the excited state 2 (with the information written therein)into the additional nondecaying metastable state 4 , see Fig.1, by acting with a subsequentfrequency-chirped short laser pulse in the AP regime of interaction. Reading of the phaseinformation stored in this nondecaying state may be produced by acting by the same chirpedlaser pulse which transforms the phase information into the population of the excited state.The latter may be detected for example, by analysing the spontaneous or stimulated emissionfrom this state. According to Eq.(7), where the atoms are assumed to be optically pumped into the groundstate 1 initially, the population of the excited state resulting from the interaction with the


frequency-chirped BLP is a function of the relative strength Ω Ω22

12

/ . So, optical

intensity information (image) contained in the relative intensity of the pulses forming theBLP may be written into the population n fin2 of the excited state. One have to store this

information (image) to preserve it from distortion due to the spontaneous decay by transferthe population of the excited state into the nondecaying state (for e.g., into the state 4 inFig.1). This may be done just like to the case of the considered above phase informationstorage using a subsequent frequency-chirped short laser pulse. The reading of the storedinformation may be produced also in the same way as that one of the phase informationreading.

5. Conclusions

In conclusion, the results of analysis of the interaction of short frequency-chirped BLP withthree-level Λ-atom has been presented. The dependence of the excitation probability on therelative phase and relative strength of the pulses forming the BLP has been proposed in thispaper to be used for the generation of coherent superpositions of the ground states witharbitrary controllable values of Raman or Zeeman coherences, as well as for writing andreading of phase or intensity optical information. The physics of the proposed technique is asfollows. The frequency-chirped short BLP in the AP regime of interaction produces completetransfer of population of the “bright” superposition of the ground states into the excited stateleaving unchanged the population of the “dark” superposition state. Since the “bright” and“dark” states depend on the relative phase and relative strength of the BLP’s pulses, thephase and intensity information contained in the BLP are transferred to the excited state withthe population of the “bright” state or are written in the population of the “dark”superposition in the ground state. This information writing is produced by BPL with durationmuch shorter compared to the atomic relaxation times. It has been shown, that the atoms have to be prepared in the coherent superposition of theground states initially before using them for writing the phase information. They have to beprepared in one of the ground state to be used for the intensity information writing. Thestorage of information in both cases is produced in a second step by a subsequent frequency-chirped short laser pulse which in a fast way transfers the population of the excited state (andhence, the information written therein) into one of the nondecaying state of the atom. Thereading of the stored information is produced by excitation of atoms by the same frequency-chirped short laser pulse from this state. It is worth noting that the information writing and reading are fast and robust in theproposed scheme. These processes have time scales equal to those of the laser pulses, whichdurations may be chosen to be very short. The restriction on the duration of the laser pulses ismainly connected with the conditions for the AP regime of interaction [9,13]. The robustnessof these processes is coming from the robustness of the population transfer in quantumsystems produced by the frequency-chirped laser pulses in the AP regime of interaction: It iswell known that the effectiveness of this transfer may be near to 100% and is insensitive tothe shapes and the transverse intensity distributions of the laser pulses, as well as to exactresonance conditions.

Acknowledgements

The authors wish to thank Prof. K. Bergmann for stimulating discussions.This work was supported by the Research Fund (OTKA) of the Hungarian Academy ofSciences.


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Coherent writing and reading of information using frequency-chirped short bichromatic laser pulses