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Control of atomic transitions by the symmetry of excitation pulses
15 November 1995
OPTICS
COMMUNICATIONS Optics Communications 121 (1995) 31-35
ELSEVIER
Control of atomic transitions by the symmetry of excitation pulses N.V. Vitanov, P.L. Knight Optics Section, BlacketrLaboratory, imperialCollege, PrinceConsortRoad,London SW72B.Z UK
Received 5 May 1995; revised version received 19 July 1995
Abstract
It is shown that any N-level atom or molecule coherently driven by laser pulses, whose Rabi frequencies and detunings are arbitrary odd functions of time, is restored completely to its initial state after the pulses have been turned off. Interaction of atoms or molecules with amplitude and phase modulated laser pulses provides a physical example in which this effect can be observed.
The symmetry properties of a coherently driven Nlevel quantum system determine a number of important features of the interaction dynamics. In the simplest case of a two-level system, some general properties following from the symmetry of the Schrodinger Iequation and the driving external field are well known: the independence of the transition probability on the signs of the coupling and the detuning, on phase transformation, time reversal, time translation and time scaling; the conservation of the probability and the Bloch vector length; the presence of Stueckelberg oscillations for double level crossing and form time-symmetric coupling of constant detuning [ 11. Recently, some peculiarities in the case of excitation by asymmetric pulses with constant detuning have been established which generally follow from the lack of symmetry. For e:xample, zero transition probability is impeded (though not forbidden) [ 21 and under certain conditions the populations tend to one-half [ 31. In the case of N levels, symmetry determines various other important general properties. For example, it has been shown that the dynamics of a system possessing the SU( 2) symmetry [4], the Gell-Mann SU( 3) symmetry [5,6] or the Elliott SU( 3) symmetry [ 51 can be reduced to an Ieffective two-level system. Furthermore, the general con0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD/OO30-4018(95)00464-5
ditions under which the dynamic space of an N-level laser-atom interacting system can be factorized into independent subspaces have been found [ 71. Finally, a number of new constants of motion have been discovered, both for three-level systems [ 81 and generally for N-level systems [ 93. As far as the symmetry of the external field is concerned, attention in the literature has mainly been paid to the cases in which the coupling is an even function of time. In the theory of slow atomic collisions this is natural because the coupling is supposed to depend on time only through the internuclear distance which is an even function of time. In optics, the coupling, that is the laser pulse, is most frequently considered to have an envelope which is an even function too, for example, rectangular, hyperbolic-secant, Gaussian, hyper-GausSian, etc. Recently, various authors have treated interaction of atoms and molecules with amplitude and/or phase modulated laser pulses [ lo] ; then both the coupling and the detuning can be made odd functions of time, a case which turns out to contain interesting new physics. In this short communication, we exploit the symmetry of the Schrodinger equation and the interaction to show that whenever the couplings and the detunings
32
N. V. Vitunov, P.L. Knight/Optics
are odd functions of time, an N-level system is restored completely to its initial state after the external fields are turned off although the system may undergo strong excitation during the interaction. Thence, we call this effect a “symmetq-forbidden transition”. We suggest a physical example in which such a situation can be met, namely, when atoms or molecules interact with specifically tailored amplitude and phase modulated laser pulses, and present results of numerical calculations of this situation for two-level atoms. The dynamics of the probability amplitude C,(t>(n=l, 2, . . . . N) of state In) in a coherently driven relaxationlessN-level system is described by the time-dependent Schrodinger equation N d i z C,(t) = L& HJt)CLt)
(n= 1, 2, . . . . N) a
(1) where Hnk= (n 1Hjk) are the matrix elements of the Hamiltonian (assumed Hermitian). For time-dependent couplings which vanish at t--f 31CQ(which is the only possible physical case) and given that the initial values C,,( t -+ - m) are known, of particular importance are the values of the probability amplitudes as t + + ~0.These are related to the initial values via the S matrix C( +a)
=S( +“,
-“)
C( -“)
Communications
121 (1995) 31-35
Let us now suppose that the matrix elements Hnk( t) are odd functions of time %,A -
r) = -H,,(t)
.
(2)
Then time reversal t + - t in Eqs. ( 1) does not change their form. This implies that Gnk( - t) = Gnk( t) because the amplitudes Gnk( -t) (n = 1, 2, . . ., N) satisfy the same Eqs. ( 1) with the same boundary conditions at t = 0 as Gnk( t) . Therefore G( - t, 0) = G( t, 0) and S( +m, -33) =G(w, =G(a,
0) G+( -m,O)
0) G+(m, 0) =I,
that is, the S matrix is equal to the unit matrix. In other words, the probability amplitudes at t + + UJare equal to their values at t+ - CQand thus, the system is restored into its initial state. This is a general result which does not depend on the particular time dependences of the matrix elements HJ t). This effect can be observed in coherent excitation of atoms or molecules by laser pulses. In optics, it is usually assumed that the levels are numbered in such a way that in the rotating-wave approximation, the nonzero matrix elements can be written as [ l]
&c(f)
=%(t) lfMt)
(
> for n - k = odd number,
= where C(t) 2 , ..., N). time (with convenient
denotes the column-vector of C,,( t) (n = 1, When Hnk( t) are even or odd functions of t = 0 as the reference symmetry point), it is to introduce the evolution matrix G( t, 0)
(n>k) { 0,
for n - k = even number ,
Because of probability conservation, the evolution matrix G( t, 0) should be unitary, G’( t, 0) = G- ’ (t, 0) = G( 0, t), which, together with the obvious requirement G(0, 0) = 1, suggest that det G( t, 0) = 1. It is easytoseethatthekthcolumnofG(t,O),G,,(t) (n= 1, 2 , ..., N), represents the solution of the Schrbdinger equation ( 1) with initial conditions C,(O) = 1, C,(O) =0 (n= 1,2, . . . . N; n # k) at t = 0. The S matrix is given by
where A,(t) is the cumulative detuning of n - 1 SUCcessive lasers from the corresponding sum of n - 1 transition frequencies and a”,(t) = -&.&(t) /fi is the Rabi frequency that couples levels n and k,d,,, being the transition dipole moment and Enk( t) is the electricfield envelope of the pulse driving the n fs k transition. In many cases, the levels are dipole connected only chainwise, Io203o...~N-l~N,andtheHamiltonian matrix is tridiagonal; however, this assumption is unnecessary in the present case. Eq. (2) implies that the Rabi frequencies Q,,Jt) and the detunings A,(t) should be odd functions of time
S( +=,
Q,,( -t)
C(t) =G(t,
-“)
=G(m,
0) C(0)
=G(a,
.
0) G(0,
0) G+( -m,O)
.
-00)
= -&(t),
A,( -t)
= -A,,(t)
.
Physically, this can be realized by using amplitude and phase modulated laser pulses. Consider, for example,
N. V. Vitanov, P.L. Knight/Optics
Communications
121 (1995) 31-35
33
a two-level atom which interacts with two laser pulses with linearly polarized electric fields given by E,(t)
=e&f(t)
cos](~+r])t+@(t)
G(f) =eE,,f(t) cod Cm- $t+
++I , (P(r) - 41 ,
where e is a unit polarization vector, E. is the field amplitude, f( t) is an arbitrary even function vanishing as t+ ia, which describes the pulse envelope, i ?I are the constant detunings of the carrier laser frequencies from the atomic transition frequency w, @(t) is an even function representing the time-dependent part of the phase and the constant 24 is the phase difference between the two fields at t = 0. To simplify the notation, we have omitted all subscripts “12” denoting the: l++ 2 transition. Within the rotating-wave approximation, the effective Rabi frequency that couples the levels is O(t) =2&f(r)
cos(77r++)
,
(3)
where a = - d - eE,/ fa.The parameter 71plays the role of an effective amplitude modulation frequency. For += r/2, the effective Rabi frequency -20&t) sin( qf) is an odd function of time as required. The effective detuning is A (t) = d@( t) /dt and it is supposed to be an odd function of time. This is feasible experimentally by using chirped pulses [ 11,121. If the chirp is produced by self-phase-modulation ’ then, for example, the phase Q(t) can be made proportional to the electric-field amplitude, that is, to f(t) [ 121. Then A(t) =a df(t)/dt
,
s
0.6 -
it a%
0.6 -
u 2 (I)
0.4
$ .Z :: ill
0.2 -
-
(4)
where LYis aconstant. Insofarasf( t) is an even function, A(t) is automatically an odd function. In Fig. 1, the excited state population 1C,( + m) I* of a two-level atom prepared initially in state 11) and interacting with an external laser pulse with a Rabi frequency given by Eq. (3) and a detuning given by Eq. (4) is plotted as a function of the phase 4 for the case of pulses with Gaussian envelopes, f(t) = e- (f/7)2 and with the following parameters: (Y= 20. The characteristic pulse J&=1&, ?)=7-‘, ’ Other ways of producing amplitude and phase modulation exist. For example, an effective Rabi frequency of the form given by Eq. (3) can be achieved by using pulsed amplification of an amplitude modulated cw laser light, while odd A(t) can be realized by using an electrooptic modulator which produces sinusoidal phase modulation.
-
O0
0.2
0.4 @ (units
0.6
0.6
1.0
of x)
Fig. 1. The excited state population ) C,( + m) 1‘in a two-level atom, initially in its ground state 11).plotted as a function of the phase 4 for a Rabi frequency given by Eq. (3) with a Gaussian envelope, f(t)=e-(‘W, and a detuning defined by Fq. (4). The pulse parameters have the following values: f?,,= lOr_‘, n= T-‘, a=20. The characteristic pulse width r, which provides natural units for time and frequency, is set equal to one, r= 1.
width r, which provides natural units for time and frequency, is set equal to one, r= 1. The figure is symmetric because the change ++ r-- 4 is equivalent to time reversal for fi( t) (as mentioned in the beginning, time reversal does not change the final occupation probabilities). At 4 = rr/2, the excited state population possesses a deep minimum because then the effective Rabi frequency is an odd function of time and the symmetry forbids the transition. For values of 4 outside (0.37r, 0.77~)) the excited-state population is almost equal to one because the detuning A (t), being an odd function, passes through the resonance at t = 0 and furthermore, the parameters of the pulses are chosen to ensure adiabatic evolution; this leads to Landau-Zener transitions which explains the large transition probability. The small deviation of ] C,( + m) ]’ from unity is due to the fact that the Gaussian pulses cannot be made adiabatic at all times [ 121. We should note that we have chosen values of the pulse parameters which ensure adiabatic evolution because in this case the symmetry-forbidden transition leads to a pronounced feature in lG(+w2; otherwise, the minimum might be confused with the non-adiabatic Stueckelberg oscillations. Moreover, in the adiabatic regime, the excitation is relatively insensitive to small changes in the interaction parameters: changing Q,, 77or cr would not affect dramatically the results shown in Fig. 1. Finally, Fig. 1
34
N. V. Vitunov. P.L. Knight/Optics
Time
(units of 7)
Fig. 2. The time evolution of the excited state population 1Cz(r) 1’ in a two-level atom, initially in its ground state 1I), interacting with the same external field as in Fig. 1,but for 4= r/2.
shows that the phase 4 enables an effective control of the interaction dynamics: the transition probability varies from almost one at 4 = 0 and d, = rr, where the pulse envelope is symmetric, to zero at += 7r/2, where the pulse envelope is antisymmetric. To a certain extent, the deep minimum seen in Fig. 1 and interpreted as due to a symmetry-forbidden transition, is a surprising property given that the two-level system does interact with the external field and the probability amplitudes do change during the interaction. The symmetry of Eqs. (1) and of the external field, however, imposes a complete recovery of the initial conditions upon completion of the interaction. This can be seen in Fig. 2 where we have shown the time evolution of the excited state population 1C,(t) 1’ in a two-level system, initially in its ground state 1l), interacting with the same external field and with the same parameters &,, 17and (Yas in Fig. 1 but for $= n-/2. Note that 1C,(t) l* is an even function of time; that is why the system returns into its initial state when the external field is over. In terms of the Bloch terminology, the Bloch vector evolves from an initially downward orientation, accomplishes some rotation on the Bloch sphere until reaching a certain position at t = 0 and then it returns to its initial downward orientation, moving backward on exactly the same trajectory. It should be pointed out that the pulses considered by us are zero-r pulses, that is, their envelope area is equal to zero. In the case of exact resonance, such pulses
Communications
I21 (1995) 31-35
produce no excitation, of course; this is in agreement with our results inasmuch as exact resonance can be viewed as a particular case of odd detuning discussed here. For constant non-zero detuning, however, zerorr pulses do produce non-zero excitation in general, in which they substantially differ from the chirped pulses studied here. Finally, we note that this effect is strictly valid within the rotating-wave approximation only. The presence of counter-rotating terms in Eqs. ( 1) violates the symmetry and, in general, leads to non-zero transition probabilities. However, in the case of optical transitions and for not extremely intense laser pulses, the rotatingwave approximation is well justified and the deviations of the transition probabilities from zero are very small: in the two-level example considered above, we have calculated that including counter-rotating terms for an atomic transition frequency of w= 103r- ’ leads to lCZ(+~)(2=7.3X10-6 at Cp=rr/2 and does not change markedly the results shown in Figs. 1 and 2. In conclusion, we have shown that the symmetry of the Schrijdinger equation and the interaction leads to the existence of symmetry-forbidden transitions: whenever the couplings and the detunings are odd functions of time, an N-level system is restored completely to its initial state after the external fields are turned off. This property is independent on the particular time dependences of the couplings and the detunings. We have suggested a physical case in which this effect can be observed, namely, the interaction of atoms or molecules with amplitude and phase modulated laser pulses, and we have presented an example of numerical calculations for two-level atoms. This work was supported in part by the UK Engineering and Physical Sciences Research Council and by the European Union. N.V.V. acknowledges the Royal Society for its support through the award of a Postdoctoral Fellowship.
References [ I ] B.W. Shore, The Theory of Coherent
Atomic Excitation (Wiley, New York, 1990). [ 21 A. Bambini and P.R. Berman, Phys. Rev. A 23 ( 198 I ) 2496; E.J. Robinson, Phys. Rev. A 24 ( 1981) 2239; N.V. Vitanov, J. Phys. B 27 (1994) 1351; 28 (1995) L19. [3] N.V. Vitanov and P.L. Knight, J. Phys. B 28 (1995) 1905.
N. V. Vitanov, P.L. Knight/Optics
141 F.T. Hioe, J. Opt. Sot. Am. B 4 (1987) 1327. 151 F.T. Hioe, Phys. Rev. A 28 (1983) 879; F.T. HioeandJ.H.Eberly, Phys. Rev.A29 (1984) 1164. [6] F.T. Hioe, Phys. Rev. A 32 (1985) 2824; J. Opt. Sot. Am. B 5 (1988) 859. [7] F.T. Hioe. Phys. Rev. A 30 ( 1984) 3097. [8] H.R. Gray, R.M. Whitley, CR. Stroud Jr., Optics Lett. 3 (1978) 218; M.J. Konopnicki and J.H. Eberly, Phys. Rev. A 24 (1980) 2567; F.T. Hioe and J.H. Eberly, Phys. Rev. A 25 ( 1982) 2168; F.T. Hioe. Phys. Rev. A 29 ( 1984) 3434; H.P.W. Gottlieb, Phys. Rev. A 26 ( 1982) 3713; 32 (1985) 6.53; J.N. Elgin. Phys. Lett. A 80 (1980) 140. [9] F.T. Hioe and J.H. Eberly, Phys. Rev. Lett. 47 (1981) 838; F.T. Hioe and C.E. Carroll, Phys. Rev. A 37 (1988) 3000; J. Oreg and S. Goshen. Phys. Rev. A 29 (1984) 3205. lo] P. Thomann, J. Phys. B 9 (1976) 2411; 13 (1980) 1111; R.E. Silverans, G. Borghs, P. De Bisschop and M. van Hove, Phys. Rev. Lett. 55 (1985) 1070;
Communications
]ll
]J2
121 (1995) 31-35
35
W.M. Ruyten, Phys. Rev. A 40 ( 1989) 1447; 42 ( 1990) 4226; 42(1990)4246;46(1992)4077; H. Freedhoff and Z. Chen, Phys. Rev. A 41 ( 1990) 6013; Z. Ficek and H.S. Freedhoff, Phys. Rev. A 48 ( 1993) 3092; C.K. Law and J.H. Eberly, Phys. Rev. A 43 (1991) 6337; N. Nayak and G.S. Agxwal, Phys. Rev. A 31 (1985) 3175; G.S. Agarwal. Y. Zhu. D.J. Gauthier and T.W. Mossberg, J. Opt. Sot. Am. B 8 (1991) 1163; G.S. Agarwal and W. Harshawardhan. Phys. Rev. A 50 ( 1994) R4465; Y. Zhu, Q. Wu, A. Lezama, D.J. Gauthier and T.W. Mossberg, Phys. Rev. A 41 ( 1990) 6574; Q. Wu, D.J. Gauthier and T.W. Mossberg, Phys. Rev. A 49 (1994) Rl519; 50 (1994) 1474. B. Broers, H.B. van Linden van den Heuvel andL.D. Noordam, Phys. Rev. Lett. 69 ( 1992) 2062; D. Goswami and W.S. Warren, J. Chem. Phys. 101 (1994) 6439; C.W. Hillegas, J.X. Tull, D. Goswami, D. Strickland and W.S. Warren, Optics Lett. 19 ( 1994) 737. D. Goswami and W.S. Warren, Phys. Rev. A 50 (1994) 5190.
OPTICS
COMMUNICATIONS Optics Communications 121 (1995) 31-35
ELSEVIER
Control of atomic transitions by the symmetry of excitation pulses N.V. Vitanov, P.L. Knight Optics Section, BlacketrLaboratory, imperialCollege, PrinceConsortRoad,London SW72B.Z UK
Received 5 May 1995; revised version received 19 July 1995
Abstract
It is shown that any N-level atom or molecule coherently driven by laser pulses, whose Rabi frequencies and detunings are arbitrary odd functions of time, is restored completely to its initial state after the pulses have been turned off. Interaction of atoms or molecules with amplitude and phase modulated laser pulses provides a physical example in which this effect can be observed.
The symmetry properties of a coherently driven Nlevel quantum system determine a number of important features of the interaction dynamics. In the simplest case of a two-level system, some general properties following from the symmetry of the Schrodinger Iequation and the driving external field are well known: the independence of the transition probability on the signs of the coupling and the detuning, on phase transformation, time reversal, time translation and time scaling; the conservation of the probability and the Bloch vector length; the presence of Stueckelberg oscillations for double level crossing and form time-symmetric coupling of constant detuning [ 11. Recently, some peculiarities in the case of excitation by asymmetric pulses with constant detuning have been established which generally follow from the lack of symmetry. For e:xample, zero transition probability is impeded (though not forbidden) [ 21 and under certain conditions the populations tend to one-half [ 31. In the case of N levels, symmetry determines various other important general properties. For example, it has been shown that the dynamics of a system possessing the SU( 2) symmetry [4], the Gell-Mann SU( 3) symmetry [5,6] or the Elliott SU( 3) symmetry [ 51 can be reduced to an Ieffective two-level system. Furthermore, the general con0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD/OO30-4018(95)00464-5
ditions under which the dynamic space of an N-level laser-atom interacting system can be factorized into independent subspaces have been found [ 71. Finally, a number of new constants of motion have been discovered, both for three-level systems [ 81 and generally for N-level systems [ 93. As far as the symmetry of the external field is concerned, attention in the literature has mainly been paid to the cases in which the coupling is an even function of time. In the theory of slow atomic collisions this is natural because the coupling is supposed to depend on time only through the internuclear distance which is an even function of time. In optics, the coupling, that is the laser pulse, is most frequently considered to have an envelope which is an even function too, for example, rectangular, hyperbolic-secant, Gaussian, hyper-GausSian, etc. Recently, various authors have treated interaction of atoms and molecules with amplitude and/or phase modulated laser pulses [ lo] ; then both the coupling and the detuning can be made odd functions of time, a case which turns out to contain interesting new physics. In this short communication, we exploit the symmetry of the Schrodinger equation and the interaction to show that whenever the couplings and the detunings
32
N. V. Vitunov, P.L. Knight/Optics
are odd functions of time, an N-level system is restored completely to its initial state after the external fields are turned off although the system may undergo strong excitation during the interaction. Thence, we call this effect a “symmetq-forbidden transition”. We suggest a physical example in which such a situation can be met, namely, when atoms or molecules interact with specifically tailored amplitude and phase modulated laser pulses, and present results of numerical calculations of this situation for two-level atoms. The dynamics of the probability amplitude C,(t>(n=l, 2, . . . . N) of state In) in a coherently driven relaxationlessN-level system is described by the time-dependent Schrodinger equation N d i z C,(t) = L& HJt)CLt)
(n= 1, 2, . . . . N) a
(1) where Hnk= (n 1Hjk) are the matrix elements of the Hamiltonian (assumed Hermitian). For time-dependent couplings which vanish at t--f 31CQ(which is the only possible physical case) and given that the initial values C,,( t -+ - m) are known, of particular importance are the values of the probability amplitudes as t + + ~0.These are related to the initial values via the S matrix C( +a)
=S( +“,
-“)
C( -“)
Communications
121 (1995) 31-35
Let us now suppose that the matrix elements Hnk( t) are odd functions of time %,A -
r) = -H,,(t)
.
(2)
Then time reversal t + - t in Eqs. ( 1) does not change their form. This implies that Gnk( - t) = Gnk( t) because the amplitudes Gnk( -t) (n = 1, 2, . . ., N) satisfy the same Eqs. ( 1) with the same boundary conditions at t = 0 as Gnk( t) . Therefore G( - t, 0) = G( t, 0) and S( +m, -33) =G(w, =G(a,
0) G+( -m,O)
0) G+(m, 0) =I,
that is, the S matrix is equal to the unit matrix. In other words, the probability amplitudes at t + + UJare equal to their values at t+ - CQand thus, the system is restored into its initial state. This is a general result which does not depend on the particular time dependences of the matrix elements HJ t). This effect can be observed in coherent excitation of atoms or molecules by laser pulses. In optics, it is usually assumed that the levels are numbered in such a way that in the rotating-wave approximation, the nonzero matrix elements can be written as [ l]
&c(f)
=%(t) lfMt)
(
> for n - k = odd number,
= where C(t) 2 , ..., N). time (with convenient
denotes the column-vector of C,,( t) (n = 1, When Hnk( t) are even or odd functions of t = 0 as the reference symmetry point), it is to introduce the evolution matrix G( t, 0)
(n>k) { 0,
for n - k = even number ,
Because of probability conservation, the evolution matrix G( t, 0) should be unitary, G’( t, 0) = G- ’ (t, 0) = G( 0, t), which, together with the obvious requirement G(0, 0) = 1, suggest that det G( t, 0) = 1. It is easytoseethatthekthcolumnofG(t,O),G,,(t) (n= 1, 2 , ..., N), represents the solution of the Schrbdinger equation ( 1) with initial conditions C,(O) = 1, C,(O) =0 (n= 1,2, . . . . N; n # k) at t = 0. The S matrix is given by
where A,(t) is the cumulative detuning of n - 1 SUCcessive lasers from the corresponding sum of n - 1 transition frequencies and a”,(t) = -&.&(t) /fi is the Rabi frequency that couples levels n and k,d,,, being the transition dipole moment and Enk( t) is the electricfield envelope of the pulse driving the n fs k transition. In many cases, the levels are dipole connected only chainwise, Io203o...~N-l~N,andtheHamiltonian matrix is tridiagonal; however, this assumption is unnecessary in the present case. Eq. (2) implies that the Rabi frequencies Q,,Jt) and the detunings A,(t) should be odd functions of time
S( +=,
Q,,( -t)
C(t) =G(t,
-“)
=G(m,
0) C(0)
=G(a,
.
0) G(0,
0) G+( -m,O)
.
-00)
= -&(t),
A,( -t)
= -A,,(t)
.
Physically, this can be realized by using amplitude and phase modulated laser pulses. Consider, for example,
N. V. Vitanov, P.L. Knight/Optics
Communications
121 (1995) 31-35
33
a two-level atom which interacts with two laser pulses with linearly polarized electric fields given by E,(t)
=e&f(t)
cos](~+r])t+@(t)
G(f) =eE,,f(t) cod Cm- $t+
++I , (P(r) - 41 ,
where e is a unit polarization vector, E. is the field amplitude, f( t) is an arbitrary even function vanishing as t+ ia, which describes the pulse envelope, i ?I are the constant detunings of the carrier laser frequencies from the atomic transition frequency w, @(t) is an even function representing the time-dependent part of the phase and the constant 24 is the phase difference between the two fields at t = 0. To simplify the notation, we have omitted all subscripts “12” denoting the: l++ 2 transition. Within the rotating-wave approximation, the effective Rabi frequency that couples the levels is O(t) =2&f(r)
cos(77r++)
,
(3)
where a = - d - eE,/ fa.The parameter 71plays the role of an effective amplitude modulation frequency. For += r/2, the effective Rabi frequency -20&t) sin( qf) is an odd function of time as required. The effective detuning is A (t) = d@( t) /dt and it is supposed to be an odd function of time. This is feasible experimentally by using chirped pulses [ 11,121. If the chirp is produced by self-phase-modulation ’ then, for example, the phase Q(t) can be made proportional to the electric-field amplitude, that is, to f(t) [ 121. Then A(t) =a df(t)/dt
,
s
0.6 -
it a%
0.6 -
u 2 (I)
0.4
$ .Z :: ill
0.2 -
-
(4)
where LYis aconstant. Insofarasf( t) is an even function, A(t) is automatically an odd function. In Fig. 1, the excited state population 1C,( + m) I* of a two-level atom prepared initially in state 11) and interacting with an external laser pulse with a Rabi frequency given by Eq. (3) and a detuning given by Eq. (4) is plotted as a function of the phase 4 for the case of pulses with Gaussian envelopes, f(t) = e- (f/7)2 and with the following parameters: (Y= 20. The characteristic pulse J&=1&, ?)=7-‘, ’ Other ways of producing amplitude and phase modulation exist. For example, an effective Rabi frequency of the form given by Eq. (3) can be achieved by using pulsed amplification of an amplitude modulated cw laser light, while odd A(t) can be realized by using an electrooptic modulator which produces sinusoidal phase modulation.
-
O0
0.2
0.4 @ (units
0.6
0.6
1.0
of x)
Fig. 1. The excited state population ) C,( + m) 1‘in a two-level atom, initially in its ground state 11).plotted as a function of the phase 4 for a Rabi frequency given by Eq. (3) with a Gaussian envelope, f(t)=e-(‘W, and a detuning defined by Fq. (4). The pulse parameters have the following values: f?,,= lOr_‘, n= T-‘, a=20. The characteristic pulse width r, which provides natural units for time and frequency, is set equal to one, r= 1.
width r, which provides natural units for time and frequency, is set equal to one, r= 1. The figure is symmetric because the change ++ r-- 4 is equivalent to time reversal for fi( t) (as mentioned in the beginning, time reversal does not change the final occupation probabilities). At 4 = rr/2, the excited state population possesses a deep minimum because then the effective Rabi frequency is an odd function of time and the symmetry forbids the transition. For values of 4 outside (0.37r, 0.77~)) the excited-state population is almost equal to one because the detuning A (t), being an odd function, passes through the resonance at t = 0 and furthermore, the parameters of the pulses are chosen to ensure adiabatic evolution; this leads to Landau-Zener transitions which explains the large transition probability. The small deviation of ] C,( + m) ]’ from unity is due to the fact that the Gaussian pulses cannot be made adiabatic at all times [ 121. We should note that we have chosen values of the pulse parameters which ensure adiabatic evolution because in this case the symmetry-forbidden transition leads to a pronounced feature in lG(+w2; otherwise, the minimum might be confused with the non-adiabatic Stueckelberg oscillations. Moreover, in the adiabatic regime, the excitation is relatively insensitive to small changes in the interaction parameters: changing Q,, 77or cr would not affect dramatically the results shown in Fig. 1. Finally, Fig. 1
34
N. V. Vitunov. P.L. Knight/Optics
Time
(units of 7)
Fig. 2. The time evolution of the excited state population 1Cz(r) 1’ in a two-level atom, initially in its ground state 1I), interacting with the same external field as in Fig. 1,but for 4= r/2.
shows that the phase 4 enables an effective control of the interaction dynamics: the transition probability varies from almost one at 4 = 0 and d, = rr, where the pulse envelope is symmetric, to zero at += 7r/2, where the pulse envelope is antisymmetric. To a certain extent, the deep minimum seen in Fig. 1 and interpreted as due to a symmetry-forbidden transition, is a surprising property given that the two-level system does interact with the external field and the probability amplitudes do change during the interaction. The symmetry of Eqs. (1) and of the external field, however, imposes a complete recovery of the initial conditions upon completion of the interaction. This can be seen in Fig. 2 where we have shown the time evolution of the excited state population 1C,(t) 1’ in a two-level system, initially in its ground state 1l), interacting with the same external field and with the same parameters &,, 17and (Yas in Fig. 1 but for $= n-/2. Note that 1C,(t) l* is an even function of time; that is why the system returns into its initial state when the external field is over. In terms of the Bloch terminology, the Bloch vector evolves from an initially downward orientation, accomplishes some rotation on the Bloch sphere until reaching a certain position at t = 0 and then it returns to its initial downward orientation, moving backward on exactly the same trajectory. It should be pointed out that the pulses considered by us are zero-r pulses, that is, their envelope area is equal to zero. In the case of exact resonance, such pulses
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produce no excitation, of course; this is in agreement with our results inasmuch as exact resonance can be viewed as a particular case of odd detuning discussed here. For constant non-zero detuning, however, zerorr pulses do produce non-zero excitation in general, in which they substantially differ from the chirped pulses studied here. Finally, we note that this effect is strictly valid within the rotating-wave approximation only. The presence of counter-rotating terms in Eqs. ( 1) violates the symmetry and, in general, leads to non-zero transition probabilities. However, in the case of optical transitions and for not extremely intense laser pulses, the rotatingwave approximation is well justified and the deviations of the transition probabilities from zero are very small: in the two-level example considered above, we have calculated that including counter-rotating terms for an atomic transition frequency of w= 103r- ’ leads to lCZ(+~)(2=7.3X10-6 at Cp=rr/2 and does not change markedly the results shown in Figs. 1 and 2. In conclusion, we have shown that the symmetry of the Schrijdinger equation and the interaction leads to the existence of symmetry-forbidden transitions: whenever the couplings and the detunings are odd functions of time, an N-level system is restored completely to its initial state after the external fields are turned off. This property is independent on the particular time dependences of the couplings and the detunings. We have suggested a physical case in which this effect can be observed, namely, the interaction of atoms or molecules with amplitude and phase modulated laser pulses, and we have presented an example of numerical calculations for two-level atoms. This work was supported in part by the UK Engineering and Physical Sciences Research Council and by the European Union. N.V.V. acknowledges the Royal Society for its support through the award of a Postdoctoral Fellowship.
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