B+(z) = exp [-- 2'/~o] A+.FA+,. B_(z) ==: exp [2i~o] A_FA_,. _~(z) = 4Zz(% 2 + z2) -~ . The operators As shift the argument z by -5 iw, according to the formula.
LETT]~3R:E AL NUOVO CIMENTO
VOL. 15, N. 17
24 Aprile 1976
Nonperturbative Calculation of the Bloch-Siegert Shift. Z. BIALYNICKA-BIRULA
Institute el Physics Polish Academy el Sciences - Warsaw, .Poland I. B IALYNICKI-]3IF~ULA
I~tstitute el Theoretical Physics, Warsaw U~iversity - Warsaw, .Poland Department el Physics, University el Pittsburgh - Pittsburg, Penn. 15260 (rieevuto il 26 Gennaio 1976)
In this letter we shall t)rcsent a new f o r n m l a for the (~relative population ~ comp o n e n t Pa of the density m a t r i x for a two-level system i n t e r a c t i n g w i t h a single mode of radiation. The calculatiou is based on an i t e r a t i v e procedure which is suggested by the phase average representation for the density m a t r i x in q u a n t u m electrodynamics. Our i t e r a t i v e procedure near the resonance sums up all leading t e r m s in the first stop, all n e x t - t o - l e a d i n g t e r m s in the second step, etc. I t seems to be similar to a m e t h o d proposed recently by G()XTIr:R, RAtL~A,~ and TRAtIIN (t) for calculating m u l t i p h o t o n transition amplitudes. The second step of our i t e r a t i v e procedure, when developed i n t o the pertm'bation series, reproduces all t e r m s calculated so far for the Bloch-Siegert shift. IIowever, already in this second step the accuracy of our result for t h e m a i n resonance, as tested ",t the wtlue co0 = 0 against the e x a c t result, is four orders of m a g n i t u d e better t h a n the accuracy of the 10-th--order p e r t u r b a t i v e expansion. A t t h e three-photon resonance our formula agrees with the e x a c t result for w o = 0 within half a perc(;nt, whereas the p e r t u r b a t i v e formula given b y AlibthI) and I~I~LL()U(m (2) does not even reproduce correctly the shape of the curve. Our s t a r t i n g point is the following rclation bctwecn the QE D reduced density m a t r i x for the a t o m i c system amt the corresponding semi-classical density m a t r i x 2~
(1)
o~ED(t)
if, 0
where oQED(t)-= r]'l'fiehl{~)(t)} and -a ( ) describes the evolution of the a t o m i c system in the presence of an e x t e r n a l electroma,gnetic tield. The p o t e n t i a l of this field has
(~) Y. GONTIER, N. K.. R?tH'MAN aItd M. TRAIIIN: Phys. Rev. L e t t . , 3 4 , 779 (1975); 341 (1975). (t) b'. AIL~rAD a n d I{. K . B v r , r , o u o u : .I. Phy,~. B, 7, L'275 (1974).
Phys. L e t t . , 5 4 A ,
627
62~
Z. B I A L Y N I C K A - B I R U L A
and
I. B I A L Y N I C K I - B I n U L A
the form
~4~.'(z,t) = V~A(.+'(x,t) cxp [-i~0]+ V~A(,,-'(x,t)exp[i~],
(2)
where A~+)(x, t) are positive- and negative-frequency parts (analytic signals) of the photon mode function. Formula (1) follows from the phase average representation for QED transition amplitudes derived by us earlier (3) and has been obtained under the assumption that initially there are no correlations between the atomic system and the radiation field and that the field is in the n-photon state (n >> 1). F u r t h e r calculations will be carried out for a two-level system interaetingswith a monochromatic field of the frequency co. The density matrix 0soL(t) is decomposed into a sum of P a u l i matrices
(3)
oscr(t) = ~ + p ( t ) a .
The components of p(t) satisfy the following standard set of equations:
l Pl(t) -P2(z) =
(4)
-- ~o0P2(t),
4). cos (wt + q~)ps(t),
%pdt)--
pa(t) = 4)t cos (ogt + ~0)p2(t) . We will define the Laplace transform of p(t)
Nz) =-;dt exp [--z t ] p ( t ) .
(5)
Assuming the following initial conditions: pl(t = O) = O, p~(t -- 0) = 0 and ps(t = O) = p, we obtain a formal solution for /33(z) in the form (6)
~3(z) =
l~(z)p,
where
R(z) = (A(z) + B+(z) + B_(z))-~,
A(z) --- z + A+.FA_ + A_FA+, (7)
B+(z) = exp [-- 2'/~o]A+.FA+, B_(z) ==: exp [2i~o]A _ F A _ ,
_~(z) = 4Zz(%2 +
z2) -~ .
The operators As shift the argument z by -5 iw, according to the formula
a+l(z) = l(z + io~) . The operator R(z) obeys the following operator equation:
(s) (')
/~(z)=/-X(z)-
A-*(z)(B+(z)
+ B_(z))1~(z) .
I. B I ~ I ~ 3 K I - B I ] a U L * and Z. BIar.Y'NICKA-Bn~wL*: Phys. R4v. A, 8, 3146 (1973).
NONt'J'II~TI'I~BATIVJ': CAI,C[H,ATION OF 'fill': I)I,()C[[-SII.:GJs
~[{II"T
~
Now we recall that pa(z) (or R(z)) is to bc averaged over the phase according to eq. (1) and tha~ only terms independent of ~0will contribute. This observation leads in a natural way to our approximation procedure which consists in successively iterating eq. (8) and separating terms independent of ~ after each step. In this way we obtain consecut i v e approximate values Ro(z ), l~l(z ), etc. of R(z) (averaged over the phase): I~ o =
A -l ,
I~ I : . ( R o x - - B + I ~ o B _ - - B _ / r
(9)
1~.., : : (R~ -I - - B + R o B ~/r
II 3
s
-I ,
R o B - - - .B_ R o B _ _ R u B + _ R o B + ) - x ,
(I~7,.I-- B + R o B ~ _ I g I B + IC,o B + I : g 2 B _ I ; ~ o I ~ _ R I B _ . R o B _ - - 11 R o I 3 _ R I B _ _ R . , B _ R 2 B + R o B _ R I B + R o B + ) - I
, etc.
This procedure is rapidly convergent and R~(z) is already very good for calculating the Bloch-Siegert shift for lowest-order resonances. To calculate this shift we take the time-independent component of ps(t) which is equal to zR(z)p at z -- 0, and then we find the condition for its vanishing. We omit here simple but tedious calculations which lead to the final result and write down only the condition for the vanishing of .~R2(z) at z - 0 (10)
64xe(lly ~
38Oy2-{ 2625) t- 8x4(59y s - - 3 7 4 5 y 4 ~- 64625y 2 - - 1 9 6 875) -t f 24 x2(3 y~-- 262 yS ยง 7280y4 _ 69 850y2 .~_ 1 ] 8 125) ~_L 3(y10_ 109yS ,_ 4074y~_. 62266y 4 =- 333 925y 2--275625) -- 0 .
We have used here dimensionless ratios x -- 2/w and y = Wo/W. From eq. (10) one can easily reproduce all perturbative results for the Bloch-Siegert shifts at the one-photon and three-photon resonances in agreement with AHMAD and BULT.OUGH (2) and IIlo~ and EB~RLr (4). The perturbative result is very good at the main resonance but it fails for the three-photen resonance in the region of large values of x 2, which is tile effective expansion parameter.
0j! 0
......i 0.5
~/~
1.0
1.5
F i g . ]. - T h e B [ o e h - S i e g c r t s h i f t for t h e t h r e c - p h o t o R r e s o n a n c e . T h e solid l i n e r e p r e s e n t s o u r r e s u l t . T h e d a s h e d line r e p r e s e n t s t h e p e r t u r b a t i v e r e s u l t of AI~IAD a n d BULLOUGH.
(')
F . T. llIOE a n d J . H. EBERLY: P h y s . R e v . ,4, 11, 1358 (1975).
630
Z. B I A . L ' Y N I C K A - B I R U L A
a n d I. B I A . L Y N I C K I - B I R U L A
The exact solution of eq. (10) for the main resonance does not differ appreciably from the 10-th-order pcrturbative result but it reproduces for y = 0 (exactly known point on the curve) the position of the first zero of the Bessel function up to the 7-th significant figure: our result for
y : O:
exact value for y : O:
x-
0.6012064,
x : 0.601 206389.
The exact solution of eq. (10) at the three-photon resonance is plotted in fig. 1 and compared with the 6-th-order perturbative result given by AHMAD and BULLOUGH. Our solution for the thrco-photon resonance at the point y : 0 reproduces the position of the second zero of the Bessel function within 0.5%. Results obtained from R~(z) for lowest resonances are in perfect agreement with numerical results obtained by ST~NHOLM(5) with the continuous-fraction method. However we expect t h a t for higher resonances we would have to go to the next step in our approximation procedure. The approximation scheme described in this letter is quite general and we intend to apply it also to the transitrion amplitudes in the resonance region.
(s) S. STENIIOLM: J. Phys. B, 5, 878, 890 (1972).
