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Adiabatic inversion with light pulses
PHYSICS
Volume 27A. number 7
ADIABATIC
INVERSION
LETTERS
WITH
26 August 1968
LIGHT
PULSES
$
E. B. TREACY United Aircraft
Research
Laboratories,
East Hartford,
Connecticut 06108, USA
Received 5 July 1968
Adiabatic inversion of the populations between a pair of levels connected by electric dipole transitions is predicted when the system is subjected to a strong light pulse in which the carrier frequency is swept through the transition resonance.-
To demonstrate the possibility and consequences of adiabatic inversion [l] for electric dipole transitions, we use a geometrical representation of the Schrbclinger equation due to
Feynman et al. [2] which provides a lucid description of the resonant response of a two-level system to perturbations. In this representation, the time-dependent equation for q(t) = a(t)qa + + b(t&, is rewritten dr/dt = 0 X r where the three components of r are respectively ab* + a*b,i(ab* -a*b), and u*a - b*b, while those of w are - ti-lyEX, - A-lyEy, and ti-l(W, - wb). Here, y stands for the (real) matrix element & (= - p&) of the dipole moment operator between the states with energy levels Wa (upper state) and wb (lower). Consider the perturbation due to a light pulse with electric field components E, = A(t) cos $(t) and Ey = A(t) sin $(t) where 4(t) = o,t + ipt2, A(t) is the pulse envelope, and w = ~-l(~c - wb). The pulse carrier frequency is swept at a constant rate )J through the resonant frequency of the two-level system. The motion of r is best visualized by referring r to a new frame of axes (I’, 2’, 3’) that is turned through an angle e(t) about the J-axis of the old (1,2,3) frame. (Axes 3 and 3’ coincide.) The new components of the state vector obey the transformed equation dr’/dt = = O’ X r' where Wi = - W-‘VA(~), CL$= 0, and w3 = - pt. According to this equation, r' precesses at a rate ( OJ’1 about the instantaneous o’ vector as shown in fig. 1. If the system is initially in its upper state (b = 0) and the perturbation starts from below $ This research is part of Project DEFENDER under the joint sponsorship of the Advanced Research Projects Agency, the Office of Naval Research, and the Department of Defense.
Fig. 1. Vector diagram illustrating adiabatic inversion in the rotating frame. As the vector (07 sweeps along the curve, r' follows by precessing around it. The case illustrated is for positive 1-1and the atom or molecule initially in the upper state. With positive ,U and the system initially in the lower state, r-fis directed oppositely to that in the figure.
resonance, both r' and o’ are initially pointing up (along the 3’-axis). Owing to the positive p, ok’ tips over as A(t) builds up from zero and r' precesses about o’ as ok’ sweeps along the curve with shape A(t). Finally both r ’ and cl)’ are pointing downwards, so that a is approximately zero, and thus a transition has been made to the lower state. The condition for adiabatic inversion is that the angle between w’ and r' remains small, and this requires a precession rate 10’ 1 large compared to the rate at which 0’ tips owing to << ti(vlJ2/4r
for a pulse length T. A further con421
Volume 27A, number 7
PHYSICS
LETTERS
26 August 1968
level systems. For positive y the induced x- and y-polarizations are proportional to ~1 and ~2 respectively. Referring to fig. 1 we see that the in-phase polarization is negative with increasing frequency sweep (p > 0) for the systems all initially in the upper state. Reversing the population difference or the frequency sweep makes the inphase polarization positive. This asymmetry will affect the phase velocity and hence the stability of the propagating pulses. Adiabatic inversion and its associated propagation effects seem not to have been observed yet with optical pulses. It might prove to be a useful technique for studying relaxation effects in molecular systems.
dition is the absence of dephasing collisions during the pulse. The vector f’ stays aligned close to w’ for n > 0 with the system initially in the upper state or for n < 0 and the system initially in the lower state. Reversing either the sign of p or the initial state makes r’ stay directed almost opposite to O’ during the sweep. Again inversion of populations occur between the two levels. The theory is identical for right or left circular polarizations with electric dipole transitions since the (1,2,3) space is abstract and the orientation of axes can always be so chosen that the motion of a in the (1,2) plane is in the positive sense. The selection rule is obviously Am = 1 for absorption with right circular polarization, etc. Electric dipole transitions with Am = 0 similarly show no polarization preferences although rotating axes are convenient for the description. There is an asymmetry in the sign of the inphase induced polarization in an ensemble of two-
References 1. A. Abragam, The principles of nuclear magnetism (Oxford University Press, 1961). 2. R. P. Feynman, F. L. Vernon and R. W.Hellwarth, J. Appl. Phys. 28 (1957) 49.
*****
MijSSBAUER
EFFECT
FOLLOWING
K. A. HARDY, Department
of Physics,
COULOMB
EXCITATION
OF
183W
*
D. C. RUSSELL and R. M. WILENZICK Tulane
University,
Received
New Orleans,
Louisiana,
USA
28 June 1968
The Mdssbauer effect of the gamma radiation from the 46.5 keV state of 183W has been observed following Coulomb excitation with 3.3 MeV protons. The spectrum is a single line with maximum absorption 8.1 f f 0.57~ at 77’K and half width 4.72 f 0.24 cm/set.
We have observed recoilless emission and absorption of gamma radiation from the 46.5 keV (t-) first excited state of 183W following Coulomb excitation. The Mossbauer effect has been observed previously [l-5] in this state using a 183Ta source which has a five day half life. This short lifetime, which severely limits the usefulness of 183W for Mossbauer studies is avoided using the Coulomb excitation technique. The low energy of the transition and the large mass of this nucleus results in a large value for the recoil-free fraction (0.63 at 77OK). In this experiment, the Mbssbauer effect of * Research tion.
422
supported
by the National Science
Founda-
the 46.5 keV gamma radiation of 183W was measured after population of the level by Coulomb excitation with a 0.6 IJA beam of 3.3 MeV protons from the Tulane Van de Graaff accelerator. The target and absorber were natural tungsten metal foils with a thickness of 8.8 mg/cm2 of 183W. In order to enhance the recoil-free fraction, the target and absorber were cooled to 77’K. The cryogenic system, designed for use with either liquid nitrogen or liquid helium, consisted of a 10 liter helium dewar mounted on an aluminium chamber which had entry ports and flanges on all six faces. The arrangement is similar to that used by Stevens et al. [6], and will be described in detail in a future article. The target was mounted to an aluminium cold
Volume 27A. number 7
ADIABATIC
INVERSION
LETTERS
WITH
26 August 1968
LIGHT
PULSES
$
E. B. TREACY United Aircraft
Research
Laboratories,
East Hartford,
Connecticut 06108, USA
Received 5 July 1968
Adiabatic inversion of the populations between a pair of levels connected by electric dipole transitions is predicted when the system is subjected to a strong light pulse in which the carrier frequency is swept through the transition resonance.-
To demonstrate the possibility and consequences of adiabatic inversion [l] for electric dipole transitions, we use a geometrical representation of the Schrbclinger equation due to
Feynman et al. [2] which provides a lucid description of the resonant response of a two-level system to perturbations. In this representation, the time-dependent equation for q(t) = a(t)qa + + b(t&, is rewritten dr/dt = 0 X r where the three components of r are respectively ab* + a*b,i(ab* -a*b), and u*a - b*b, while those of w are - ti-lyEX, - A-lyEy, and ti-l(W, - wb). Here, y stands for the (real) matrix element & (= - p&) of the dipole moment operator between the states with energy levels Wa (upper state) and wb (lower). Consider the perturbation due to a light pulse with electric field components E, = A(t) cos $(t) and Ey = A(t) sin $(t) where 4(t) = o,t + ipt2, A(t) is the pulse envelope, and w = ~-l(~c - wb). The pulse carrier frequency is swept at a constant rate )J through the resonant frequency of the two-level system. The motion of r is best visualized by referring r to a new frame of axes (I’, 2’, 3’) that is turned through an angle e(t) about the J-axis of the old (1,2,3) frame. (Axes 3 and 3’ coincide.) The new components of the state vector obey the transformed equation dr’/dt = = O’ X r' where Wi = - W-‘VA(~), CL$= 0, and w3 = - pt. According to this equation, r' precesses at a rate ( OJ’1 about the instantaneous o’ vector as shown in fig. 1. If the system is initially in its upper state (b = 0) and the perturbation starts from below $ This research is part of Project DEFENDER under the joint sponsorship of the Advanced Research Projects Agency, the Office of Naval Research, and the Department of Defense.
Fig. 1. Vector diagram illustrating adiabatic inversion in the rotating frame. As the vector (07 sweeps along the curve, r' follows by precessing around it. The case illustrated is for positive 1-1and the atom or molecule initially in the upper state. With positive ,U and the system initially in the lower state, r-fis directed oppositely to that in the figure.
resonance, both r' and o’ are initially pointing up (along the 3’-axis). Owing to the positive p, ok’ tips over as A(t) builds up from zero and r' precesses about o’ as ok’ sweeps along the curve with shape A(t). Finally both r ’ and cl)’ are pointing downwards, so that a is approximately zero, and thus a transition has been made to the lower state. The condition for adiabatic inversion is that the angle between w’ and r' remains small, and this requires a precession rate 10’ 1 large compared to the rate at which 0’ tips owing to << ti(vlJ2/4r
for a pulse length T. A further con421
Volume 27A, number 7
PHYSICS
LETTERS
26 August 1968
level systems. For positive y the induced x- and y-polarizations are proportional to ~1 and ~2 respectively. Referring to fig. 1 we see that the in-phase polarization is negative with increasing frequency sweep (p > 0) for the systems all initially in the upper state. Reversing the population difference or the frequency sweep makes the inphase polarization positive. This asymmetry will affect the phase velocity and hence the stability of the propagating pulses. Adiabatic inversion and its associated propagation effects seem not to have been observed yet with optical pulses. It might prove to be a useful technique for studying relaxation effects in molecular systems.
dition is the absence of dephasing collisions during the pulse. The vector f’ stays aligned close to w’ for n > 0 with the system initially in the upper state or for n < 0 and the system initially in the lower state. Reversing either the sign of p or the initial state makes r’ stay directed almost opposite to O’ during the sweep. Again inversion of populations occur between the two levels. The theory is identical for right or left circular polarizations with electric dipole transitions since the (1,2,3) space is abstract and the orientation of axes can always be so chosen that the motion of a in the (1,2) plane is in the positive sense. The selection rule is obviously Am = 1 for absorption with right circular polarization, etc. Electric dipole transitions with Am = 0 similarly show no polarization preferences although rotating axes are convenient for the description. There is an asymmetry in the sign of the inphase induced polarization in an ensemble of two-
References 1. A. Abragam, The principles of nuclear magnetism (Oxford University Press, 1961). 2. R. P. Feynman, F. L. Vernon and R. W.Hellwarth, J. Appl. Phys. 28 (1957) 49.
*****
MijSSBAUER
EFFECT
FOLLOWING
K. A. HARDY, Department
of Physics,
COULOMB
EXCITATION
OF
183W
*
D. C. RUSSELL and R. M. WILENZICK Tulane
University,
Received
New Orleans,
Louisiana,
USA
28 June 1968
The Mdssbauer effect of the gamma radiation from the 46.5 keV state of 183W has been observed following Coulomb excitation with 3.3 MeV protons. The spectrum is a single line with maximum absorption 8.1 f f 0.57~ at 77’K and half width 4.72 f 0.24 cm/set.
We have observed recoilless emission and absorption of gamma radiation from the 46.5 keV (t-) first excited state of 183W following Coulomb excitation. The Mossbauer effect has been observed previously [l-5] in this state using a 183Ta source which has a five day half life. This short lifetime, which severely limits the usefulness of 183W for Mossbauer studies is avoided using the Coulomb excitation technique. The low energy of the transition and the large mass of this nucleus results in a large value for the recoil-free fraction (0.63 at 77OK). In this experiment, the Mbssbauer effect of * Research tion.
422
supported
by the National Science
Founda-
the 46.5 keV gamma radiation of 183W was measured after population of the level by Coulomb excitation with a 0.6 IJA beam of 3.3 MeV protons from the Tulane Van de Graaff accelerator. The target and absorber were natural tungsten metal foils with a thickness of 8.8 mg/cm2 of 183W. In order to enhance the recoil-free fraction, the target and absorber were cooled to 77’K. The cryogenic system, designed for use with either liquid nitrogen or liquid helium, consisted of a 10 liter helium dewar mounted on an aluminium chamber which had entry ports and flanges on all six faces. The arrangement is similar to that used by Stevens et al. [6], and will be described in detail in a future article. The target was mounted to an aluminium cold