Low group velocity of light without an extra driving laser field

Martin Haas*, Christoph H. Keitel

Theoretische Quantendynamik, Fakult€aat f€uur Mathematik und Physik, Universit€aat Freiburg, Hermann-Herder-Straße 3,

D-79104 Freiburg, Germany

Received 27 September 2002; received in revised form 23 December 2002; accepted 2 January 2003

Abstract

Coherent light pulse propagation is investigated in a medium of three-level atoms with two possibly closely spaced

upper levels without the presence of any auxiliary driving laser fields. We derive an analytic expression for the group

velocity in this system and demonstrate that it may be very low along with a small pulse distortion for a certain range of

intensities. The group velocity is shown to depend sensitively on the upper level splitting and may thus be conveniently

controlled by a magnetic field rather than an extra laser field.

� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Low group velocity; Atomic coherence; Quantum information

1. Introduction

The recent progress in quantum informationtheory [1] has generated numerous seminal results,

especially in quantum communication and quan-

tum computation. Since the carriers of informa-

tion in the employed concepts are generally

quantum systems like electrons or photons, the

controllable manipulation of the involved objects

is of great importance. In the case of light carrying

information, it is rather easy to conceive methodsto transport it, even through the air, whereas the

secure storage and controlled manipulation, like

delay lines for light pulses or direction switching, is

a challenging task. For this reason the velocity oflight has been thoroughly studied and it has re-

cently been shown theoretically and observed ex-

perimentally that it is possible to dramatically slow

down the velocity of the envelope of a propagating

light pulse, the group velocity [2–4,6–8], or even

bring the light to a full stop [5,9,10]. All these

approaches make use of an additional light field to

control the propagation of the probe field, whereasin this communication, we consider a setup where

very low, though not vanishing, group velocities

can be reached without the need of an additional

laser field. Limited control of the group velocity is

shown to be possible through the variation of level

spacings over a wide range, e.g., by the application

of a magnetic field.

Optics Communications 216 (2003) 385–389

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +49-761-2035792; fax: +49-

761-2035883.

E-mail addresses: haas@physik.uni-freiburg.de (M. Haas),

keitel@physik.uni-freiburg.de (C.H. Keitel).

0030-4018/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0030-4018(03)01075-7

2. The system of interest

We investigate a quantum mechanical three-le-

vel-system (see Fig. 1) coupled to one classical la-

ser field, which constitutes the probe field, whosegroup velocity is being studied. At the same time it

also represents the driving field in the sense that it

is the field which causes the dynamics in the sys-

tem.

The interaction Hamiltonian V for this config-

uration in the interaction picture and rotating

wave approximation reads:

V ¼ � 1

2�hðXj1ih3j þ Xj2ih3j þ ðd � DÞj1ih1j

þ ðd þ DÞj2ih2jÞ þH:c: ð1ÞWe assume the dipole matrix elements for both

transitions 1–3 and 2–3 to be equal l13 ¼ l23 ¼: land denote the common Rabi frequency by

X ¼ Ejlj=�h, with E being the electric field ampli-tude of the laser field.

The equations of motion for the matrix ele-

ments of the density operator q including phe-

nomenologically added decay rates can then be

written as follows:

_qq11 ¼ �c13q11 �1

2iXðq13 � q31Þ; ð2Þ

_qq12 ¼ � 1

2ðc13 þ c23Þq12 � 2iDq12

� 1

2iXq13 þ

1

2iXq32; ð3Þ

_qq13 ¼ � 1

2iXðq11 � q33Þ �

1

2iXq12

� 1

2c13q13 þ iðd � DÞq13; ð4Þ

_qq22 ¼ �c23q22 �1

2iXðq23 � q32Þ; ð5Þ

_qq23 ¼ � 1

2iXðq22 � q33Þ �

1

2iXq21

� 1

2c23q23 þ iðd þ DÞq23 ð6Þ

with q33 ¼ 1� q11 � q22, as the system is closed.These equations can be solved analytically in

steady state with qssij (i; j 2 f1; 2; 3g) denoting the

steady state solutions of the corresponding matrix

elements. This assumption generally restricts the

discussion to pulses of duration of order maxðc�113 ;

c�123 Þ or longer. However, for initial conditions closeto steady state, this condition can be considerably

relaxed. The preparation of the system could be

achieved by applying a cw-laser prior to signal

transmission.

The resulting steady state matrix elements can

then be used to calculate the macroscopic complex

susceptibility v of a sample of atoms

v ¼ a1

Xqss13

�þ qss

23

�; ð7Þ

a ¼ N jlj2

�0�h; ð8Þ

where N represents the atom density and jlj theabsolute value of the common transition dipolematrix element of the transitions 1–3 and 2–3. It is

related to the frequently used oscillator strength fijof the transition i–j for differing i; j 2 f1; 2; 3gaccording to

fij ¼8p2

3

m0mhe2

jlijj2 ð9Þ

Fig. 1. The system under consideration. A laser field with fre-

quency x couples to the transitions 1–3 and 2–3. The coupling

strength is related to the Rabi frequency X, the upper levels are

split by 2D. The rates c13 and c23 specify the spontaneous decay

rates to level j3i from levels j1i and j2i, respectively. d ¼ x � x0

is the laser detuning with respect to x0, which is the frequency

separation between level j3i and the middle of the upper levels.

386 M. Haas, C.H. Keitel / Optics Communications 216 (2003) 385–389

with electron rest mass m0, electron charge e and

transition frequency m [13]. In all following plots

we measure frequencies in MHz and use light with

k ¼ 500 nm. We work with an oscillator strength

of f13 ¼ f23 ¼ 0:2 and atom density of N ¼ 3:9 �1013 cm�3 such that the pre-factor a equals 1 MHz

in Eq. (8) in these units.The complex refractive index n of the medium

then follows from the relation [12]:

n2 ¼ ðn0 þ in00Þ2 ¼ 1þ v ð10Þwith real n0 and n00.

Substituting the result for the real part of the

refractive index and its derivative into the well-

known formula for the group velocity [7]

vg ¼c

n0ðx0Þ þ x0on0ox jx0

; ð11Þ

we obtain the following expression for vg of a light

pulse, here restricted to the case c13 ¼ c23 ¼: c and

pulse center frequency chosen to be x0:

n0ðx0Þ ¼ffiffiffiffiffiffiffiffiffiffiffi1þ b2

r; ð12Þ

on0

ox

����x0

¼ � 1

ab

ffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffi1þ b

pc4�

� 16D4

þ 2ðc2 þ 4D2ÞX2 þ X4�

ð13Þ

with the abbreviations:

a ¼ 1

aðc2 þ 4D2 þ X2Þ ðc2

�þ 4D2Þ2

þ ð5c2 þ 4D2ÞX2 þ 4X4�; ð14Þ

b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2cðc2 þ 4D2 þ X2Þ2

a

!2vuut : ð15Þ

This calculation holds also in cases where D is

small and absorption becomes appreciable, be-

cause using Eq. (10) the real part of the refractive

index n0 also contains contributions from the

imaginary part of the susceptibility v00, which is

related to absorption. For most practical cases,however, one would want to keep absorption small

and then the following approximations can be

made: v00ðx0Þ � 0, n0ðx0Þ � 1. In this case, the

expressions simplify and only the derivative of

the real part v0 of the susceptibility enters into the

expression for the group velocity [11]

vg ¼c

1þ x0

2

ov0

ox jx0

ð16Þ

and we obtain for ov0

ox jx0, here again for brevity

limited to the case c13 ¼ c23 ¼: c and center fre-

quency x0:

ov0

ox

����x0

¼ � 4

ac4�

� 16D4 þ 2X2ðc2 þ 4D2Þ þ X4�:

ð17ÞAlso in this case we have n00 ¼ v00=2 with v00 given by:

v00 ¼ 2

acðc2 þ 4D2 þ X2Þ2: ð18Þ

3. Discussion

In what follows we investigate in detail the

group velocity of a light pulse and the corre-sponding pulse delay as a function of the various

control parameters. When one evaluates the group

velocity for small values of the detuning D, it

reaches very low values as depicted in Fig. 2.

However, since in an undriven system also ab-

sorption is very high at resonance, absorption ef-

fects have to be taken into account.

From the definition of the complex refractiveindex n the characteristic length

L ¼ c2x0n00ðx0Þ

ð19Þ

Fig. 2. Group velocity vg according to Eq. (11) as a function of

small level splittings D; the other parameters are

c13 ¼ c23 ¼ 0:1 MHz, X ¼ 0:1 MHz and a ¼ 1 MHz.

M. Haas, C.H. Keitel / Optics Communications 216 (2003) 385–389 387

is the distance, over which the intensity of the

travelling light of frequency x0 decreases by a

factor of e in an absorbing medium. As a conse-

quence, if one considers n00 as a function of the

level splitting (see Fig. 3) and requires the attenu-

ation lengthL to be of the order of centimeters, n00

has to be at about 4� 10�6. Therefore, D has to beof the order of 100 MHz considering a system with

parameters c13 ¼ c23 ¼ 0:1 MHz (all frequencies in

this paper being angular frequencies 2pm).Not only is absorption as such an undesirable

effect here, it also plays a role in the question of

whether the group velocity can be identified with

the signal velocity. In the limit D � c the as-

sumptions n0ðxÞ ¼ n0ðx0Þ þ ðx � x0Þ on0

ox jx0and n00

being small are very well matched. Thus the group

velocity can also be considered as the actual ve-

locity of information carried by the light pulse

under consideration. In other cases the evaluation

of the signal velocity can be much more complex

and the two velocities have to be clearly distin-

guished [14]. The diverging and negative group

velocities from Fig. 2 therefore, do not imply thatthe signal velocity exceeds c or becomes negative.

It is rather another sign, that in the region of small

D absorption is not negligible.

For non-vanishing temperatures of the atomic

gas, the Doppler shift has to be taken into account.

For a rubidium gas at room temperature, this shift

amounts to 592 MHz for the most probable

velocity of the atoms and light of wavelength

k ¼ 500 nm. Considering a level splitting of 100

MHz, clearly all features are washed out, and in

particular many atoms are near resonance due to

their motion relative to the laser. As a conse-

quence, absorption increases, but as can be seenfrom Figs. 3 and 4, the absorption properties and

the group velocity are very similar to the non-

Doppler case at a level splitting of D ¼ 100 MHz

for temperatures up to 0.3 K. In the following, we

consider the case of zero Doppler broadening.

In Fig. 4 we obtain low group velocities in the

regime of low absorption. They reach down to

hundreds of meters per second depending on D.This parameter is easily controllable by applica-

tion of an external magnetic field using the Zee-

man effect, as it is also considered in [15]. Apart

from the regime of small D with high absorption,

our results with regard to the group velocity can be

transferred to the actual signal velocity.

Tuning D to very small values increases ab-

sorption drastically and might thus be used toswitch off signal transmission.

The dynamics of the system depends further-

more on the probe field, as opposed to previous

works, where the probe field is small as compared

to the driving field and thus has little influence.

Therefore, here vg as a function of the probe field

intensity has to be investigated. It is obvious from

Fig. 5 that the pulse may start from zero intensity.

Fig. 3. A logarithmic plot of the imaginary part of the refrac-

tive index n00 characterizing absorption as a function of D. Theremaining parameters are c13 ¼ c23 ¼ 0:1 MHz, X ¼ 0:1 MHz

and a ¼ 1 MHz. The solid line indicates no Doppler broaden-

ing. The dashed line represents 10 MHz of Doppler broadening,

corresponding to a temperature of 0.09 K; dotted line, 20 MHz

(0.3 K); dash-dotted line, 30 MHz (0.8 K).

Fig. 4. Group velocity vg following Eq. (16) as a function of

larger level splittings D up to 100 MHz with parameters

c13 ¼ c23 ¼ 0:1 MHz, X ¼ 0:1 MHz and a ¼ 1 MHz. The solid

line indicates no Doppler broadening. The dashed line repre-

sents 10 MHz of Doppler broadening, corresponding to a

temperature of 0.09 K; dotted line, 20 MHz (0.3 K); dash-

dotted line, 30 MHz (0.8 K).

388 M. Haas, C.H. Keitel / Optics Communications 216 (2003) 385–389

However, the Rabi frequency and thus the inten-

sity should not exceed a certain upper limit, in

order to keep vg approximately constant over this

intensity range. This is necessary to ensure that the

low-intensity parts of the pulse do not travel at a

substantially different velocity than the high-in-tensity parts and thus to keep distortion of the

propagating pulse at a minimum.

Another interesting quantity is the pulse delay

achievable in one attenuation length Dt ¼ L=vg�L=c, with both L and vg depending on D. It rep-resents the time the light takes to pass the dispersive

medium reduced by the time the pulse would take to

travel the same distance in vacuum. As can be seenin Fig. 6,Dt is constant over a wide range ofD and in

fact approaches c�1 for large D.Since the delay time Dt does not vary much for

D > 10 as opposed to the group velocity vg, the

length of the medium causing the signal delay is

not constant. Thus for a given delay and medium

length a suitable D can be chosen.

Concluding, we have proposed an alternative

efficient means of slowing down group and signal

velocities of light which may be convenientlycontrolled by magnetic fields rather than coherent

and often intense laser pulses.

Acknowledgements

Funding by Deutsche Forschungsgemeinschaft

(Nachwuchsgruppe within SFB 276) is gratefullyacknowledged. We thank J. Evers for helpful dis-

cussions.

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Fig. 6. Logarithmic representation (abscissa) of the pulse delay

Dt in ls in one attenuation length LðDÞ versus D, with re-

maining system parameters chosen as in Fig. 4.

Fig. 5. Group velocity vg in dependence of both the level

splitting D and the laser Rabi frequency X at decay rates as in

Fig. 4. Shown as solid lines are sections for constant

X ¼ 25 MHz and constant D ¼ 75 MHz, respectively.

M. Haas, C.H. Keitel / Optics Communications 216 (2003) 385–389 389