The presence of a cavity is reflected in three cavity-modified parameters: decay rate eff , Rabi frequency eff , and detuning eff . We show that the cavity-modified ...
PHYSICAL REVIEW A
VOLUME 55, NUMBER 1
JANUARY 1997
Cavity-modified Maxwell-Bloch equations for the vacuum Rabi splitting Changxin Wang and Reeta Vyas Physics Department, University of Arkansas, Fayetteville, Arkansas 72701 ~Received 5 March 1996; revised manuscript received 29 August 1996! We derive a set of cavity-modified Maxwell-Bloch equations for a two-level atom in a cavity driven by a coherent field from the single-atom Fokker-Planck equation in the bad-cavity limit. These equations have the same form as the Maxwell-Bloch equations for a two-level atom in free space interacting with a coherent field. The presence of a cavity is reflected in three cavity-modified parameters: decay rate g e f f , Rabi frequency V e f f , and detuning D e f f . We show that the cavity-modified Maxwell-Bloch equations provide an easy way to study cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, and vacuum Rabi splitting. @S1050-2947~96!07512-9# PACS number~s!: 42.50.Ct, 32.80.2t, 42.50.Lc
The radiative properties of atoms in a cavity have been investigated by many authors in the framework of cavity quantum electrodynamics @1#. Fields inside a cavity are subject to boundary conditions and when the dimensions of the cavity are comparable to the wavelength of the radiation, the spectrum and spatial distribution of the electromagnetic field modes are modified. As a consequence, atoms inside such a cavity exhibit many different radiative properties from those in free space, such as cavity-modified spontaneous emission @2#, cavity-induced radiative frequency shifts @3#, and vacuum Rabi splitting @4#. In the laboratory, several experiments have verified the enhancement or inhibition of the spontaneous emission rate of an atom in a cavity @5#. The phenomena of vacuum Rabi splitting in the strong coupling regime have been observed experimentally, for cavities with a large number of atoms, as well as for a small number of atoms @6,7#. In particular, the case of a single atom on the average inside an optical cavity has been studied experimentally @7#. To describe a single atom inside a cavity we derived a generalized second-order Fokker-Planck equation for a single two-level atom interacting with a cavity mode, without using the system size expansion and any truncation @8#. In this paper, we show that this exact single-atom FokkerPlanck equation in the bad-cavity limit leads to a set of cavity-modified Maxwell-Bloch equations. We use these equations to study the cavity-modified spontaneous emission, radiative level shifts, and vacuum Rabi splitting. Our quantum dissipative system consists of a single damped two-level atom with transition frequency v a interacting with a damped cavity mode of resonance frequency v c . The cavity is driven by a coherent external field of frequency v 0 and amplitude «. In a frame rotating at the frequency v 0 of the external field, the behavior of the combined atom-field system is governed by the master equation for the density operator rˆ (t) @8#,
]rˆ 52iD c @ aˆ † aˆ , rˆ # 2iD a @ sˆ z , rˆ # 1g @ aˆ † sˆ 2 2aˆ sˆ 1 , rˆ # ]t 1« @ aˆ † 2aˆ , rˆ # 1
1050-2947/97/55~1!/823~4!/$10.00
S S
sˆ 2 & 52 ^˙ sˆ 1 & 52 ^˙
S
gef f 2
gef f 2
D D
1iD e f f ^ sˆ 2 & 2iV e f f ^ sˆ z & ,
2iD e f f ^ sˆ 1 & 1iV e f f * ^ sˆ z & ,
sˆ z & 52 g e f f ^ sˆ z & 1 ^˙
D
~2!
1 i i 1 V e f f ^ sˆ 1 & 2 V e f f * ^ sˆ 2 & , 2 2 2
where cavity-modified parameters are
S
g e f f 5 g 11
Vef f5
Def f5
S
2C 11 d 2c
D
5g1
igY
A2 ~ 11i d c !
5i
DS
2g 2 k
k 2 1D 2c
,
2g« , k 1iD c
~3!
D
g 2C d c g 2D c d a2 5 D 2 . a 2 11 d 2c k 2 1D 2c
~1!
Here C5g 2 / kg is the single-atom version of the cooperativity parameter, Y 52 A2g e / kg is the dimensionless driving field amplitude, d a 52D a / g is the atomic detuning parameter, and d c 5D c / k is the cavity detuning parameter. Equations ~2! have the same form as the well-known MaxwellBloch equations for a single two-level atom interacting with
55
823
g @ 2 sˆ 2 rˆ sˆ 1 2 sˆ 1 sˆ 2 rˆ 2 rˆ sˆ 1 sˆ 2 # 2
1 k @ 2aˆ rˆ aˆ † 2aˆ † aˆ rˆ 2 rˆ aˆ † aˆ # .
Here aˆ and aˆ † are the field annihilation and creation operators; sˆ 1 , sˆ 2 , and sˆ z are the Pauli spin operators describing the two-level atom; g is the atomic spontaneous emission rate; and 2 k is the rate at which the cavity is losing photons. Our model incorporates both the atomic detuning D a (5 v a 2 v 0 ) and the cavity detuning D c (5 v c 2 v 0 ). The master equation ~1! can be transformed into a secondorder Fokker-Planck equation without using system size expansion or any truncation in the bad-cavity limit @8#. By eliminating the field variables adiabatically, we obtained a second-order Fokker-Planck equation containing only the atomic variables. The drift terms of the Fokker-Planck equation lead to the differential equations for the mean values of the atomic operators @see Eqs. ~28! in Ref. @8#!#, which can be expressed in the following form:
© 1997 The American Physical Society
BRIEF REPORTS
824
55
a coherent field in free space @9#. The effect of the cavity on the atom appears in the three modified parameters: decay rate g → g e f f , Rabi frequency V→V e f f , and detuning D→D e f f . The steady-state solutions of Eqs. ~2! are
^ sˆ z & ss 52
g 2e f f 14D 2e f f 1 , 2 2 g e f f 14D 2e f f 12 u V e f f u 2
^ sˆ 1 & ss 5 ~ ^ sˆ 2 & ss ! * 52
iV * e f f ~ g e f f 12iD e f f !
g 2e f f 14D 2e f f 12 u V e f f u 2
.
~4!
Let us now use Eqs. ~2! to study various processes in the presence of a cavity. We first discuss the cavity-modified spontaneous emission rate and radiative energy level shifts of the atom in the absence of an external driving field. We assume v 0 5 v a and «50. As a result, Eqs. ~3! reduce to V e f f 50 ,
g 0e f f 5 g 1 D 0e f f 52
2g 2 k , k 1~ v c2 v a !2 2
~5!
g 2~ v c2 v a ! . k 1~ v c2 v a !2 2
Here the superscript ‘‘0’’ in g e f f and D e f f refers to the absence of a driving field. Substituting Eqs. ~5! into Eqs. ~2! and expressing them in the Schro¨dinger picture, we obtain
F F
G G D
g 0e f f ˙ ˆ 1i ~ v a 1D 0e f f ! ^ sˆ 2 & , ^ s 2 & 52 2 sˆ 1 & 52 ^˙
g 0e f f 2i ~ v a 1D 0e f f ! ^ sˆ 1 & , 2
S
sˆ z & 52 g 0e f f ^ sˆ z & 1 ^˙
~6!
1 . 2
It is interesting to note that in the absence of an external driving field, the inversion is completely decoupled. The equations for ^ sˆ 2 & and ^ sˆ 1 & describe a damped harmonic oscillator. Our results in this limit are similar to the results obtained by the classical model @3,6#. If we compare Eqs. ~6! to the free space equations, the effects of the cavity on the atom appear as follows. ~i! The spontaneous emission rate changes from g to g 0e f f , which contains two terms. The first term represents the spontaneous emission rate of the atom in free space. The second term represents the enhancement due to the presence of the cavity. Thus if the atom is placed in a cavity tuned to the atomic transition frequency ~i.e., v c 5 v a ), it decays with the cavity-enhanced spontaneous emission rate g 12g 2 / k . For large cavity detuning, the decay rate reduces to g @2#, because with an increase in detuning the effective coupling of the atom to the cavity mode decreases. ~ii! The atomic transition frequency changes from v a to v a 1D 0e f f Thus the cavity induces the radiative frequency shift D 0e f f . We can see from Eqs. ~5! that the cavity-induced
FIG. 1. ~a! The cavity-modified spontaneous emission rate and ~b! the cavity-induced radiative frequency shift as a function of detuning v c 2 v a for g520g and k 510g .
frequency shift D 0e f f depends nonlinearly on the detuning between the atom and the cavity. When the cavity is resonant with the atom, D 0e f f 50. The cavity effects discussed above are shown in Fig. 1, in which the spontaneous emission rate g 0e f f and the cavityinduced frequency shift D 0e f f are plotted as a function of the detuning v c 2 v a . Recently, spontaneous emission rate and the cavity-induced frequency shift have been measured experimentally @3# and our results obtained from the cavity QED calculations agree with these experimental measurements. In the presence of an external driving field, Eqs. ~2! describe the dynamics of an atom in a driven cavity. We use the three effective parameters to study the double peak structure in the steady-state atomic inversion and intensity transmission function. The interaction of the single atom with cavity field splits the first degenerate excited state of the atom-cavity system into two nondegenerate states @1,4#. This is referred to as vacuum Rabi splitting. The presence of vacuum Rabi splitting is reflected in the parameter D e f f . As the pump field frequency v 0 is scanned, the effective detuning D e f f vanishes whenever v 0 is resonant with one of the vacuum Rabi split levels. Hence we can study the vacuum Rabi splitting effect by imposing the resonance condition D e f f 50 in Eqs. ~2!. We show that the solutions of the equation D e f f 50 provide the positions of the peaks in the steady-state atomic inversion. For simplicity, we first discuss the case of zero detuning between the atom and the cavity v a 5 v c . The condition D e f f 50, with the help of Eqs. ~3!, leads to D e f f 5~ v a2 v 0 !2
g 2~ v a2 v 0 ! 50 . k 21~ v a2 v 0 !2
~7!
BRIEF REPORTS
55
FIG. 2. The atomic inversion ^ sˆ z & ss as a function of detuning ( v s 2 v a )/ g for parameters k 510g , C550, Y 55, and ~a! v c 5 v a , ~b! v c 5 v a 115g .
This equation has three solutions given by
v 00 5 v a
2 2 and v 6 0 5 v a 6 Ag 2 k .
~8!
The solutions v 6 0 result from the atom-field interaction. This interaction splits the common resonance frequency ( v a 5 v c ) of the atom-cavity system into two symmetric resonance frequencies v 6 with energy level shifts 0 6 Ag 2 2 k 2 for g. k . Note that for g, k we have only one real solution. In this case we only see a single peak in atomic inversion. Since the cavity decay rate is much larger than the atomic decay rate ( k @ g ) in the bad-cavity limit, the energy level shifts do not depend on the atomic decay rate g . In the strong coupling limit g@ k , the energy shifts reduce to the well known value of 6g @4#. When the external driving field is resonant with one of these two frequencies, we see a maximum in atomic inversion. In Fig. 2~a!, we plot the steadystate atomic inversion 2 ^ s z & ss as a function of the detuning ( v 0 2 v a )/ g for v a 5 v c , and g. k . In Fig. 2, ^ s z & ss exhibits two symmetric peaks, one approximately at v 0 5 v a 1 Ag 2 2 k 2 and the other at v 0 5 v a 2 Ag 2 2 k 2 . These positions are marked in the graph by small bars. We note that two symmetric peaks are not exactly located at v6 0 . This is because the heights and positions of peaks also depend on g e f f and V e f f , which are functions of the detunings as well. Between these two peaks, the atomic inversion has a minimum at v 0 5 v a . This corresponds to the solution v 00 5 v a in Eq. ~7!. This is not a resonance frequency of the atom-cavity system for g. k . For g, k , v 00 5 v a is the only resonance frequency as noted above. Thus using the cavitymodified Maxwell-Bloch equations we find that the interest-
825
FIG. 3. The cavity transmission function T( v s ) as a function of detuning ( v s 2 v a )/ g for parameters k 510g , C530, and ~a! v c 5 v a , ~b! v c 5 v a 115g . The solid curve is for Y 50.1 and the dashed curve is for Y 510.
ing doublet feature in the atomic inversion, first reported in Ref. @8#, is due to the vacuum Rabi splitting. The case of nonzero atom cavity detuning ( v a Þ v c ) is more complicated. To calculate the energy shifts induced by the vacuum Rabi splitting effect, we solve the following cubic equation D e f f 50. This equation leads to two resonance frequencies v 6 0 , which are not symmetric about v c or about v a . As a result, as shown in Fig. 2~b!, atomic inversion exhibits two asymmetric peaks. The peak on the left with a larger cavity detuning v c 2 v a is higher than the other peak on the right. The reason for this asymmetry can be traced to the dependence of the effective decay rate g e f f and the effective Rabi frequency V e f f on the cavity detuning. This can be seen as follows. We can solve Eqs. ~4! for D e f f 50, to obtain
F S DG
^ sˆ z & peaks 52 112
uVe f fu gef f
2 21
.
~9!
This equation shows that the peak in atomic inversion depends on the ratio u V e f f u / g e f f . Now from Eqs. ~3! we see that g e f f is proportional to ( k 2 1D 2c ) 21 for (g, k )@ g while u V e f f u is proportional to ( k 2 1D 2c ) 21/2. As a consequence, g e f f decreases faster than u V e f f u as the cavity detuning D c 5 v c 2 v 0 increases, and the ratio u V e f f u / g e f f , therefore, increases with an increase in the cavity detuning. This explains why the peak on the left with a larger cavity detuning is higher and has a narrower linewidth than the peak on the right corresponding to larger detuning @see Fig. 3~b!#.
BRIEF REPORTS
826
The cavity-modified Maxwell-Bloch equations ~2!, along with the so-called adiabatic formula and its conjugate @8#, also enable us to obtain the steady-state intensity transmission function for the cavity @1,4,7#. Recently the vacuum Rabi splitting effects have been investigated for the case of ¯ 51 by direct spectroscopic meaaverage atomic number N surements @1,7#. To calculate the transmitted steady-state response function T( v 0 ) for our single-atom system in the bad-cavity limit, we first express the mean cavity field as @8#
^ aˆ & 5
An s @ 2 A2C ^ sˆ 2 & 1Y # , 11i d c
~10!
where n s 5 g 2 /8g 2 is the saturation photon number. Using Eq. ~4! and Eq. ~10!, we obtain the steady-state intensity transmission function T( v 0 ) as T~ v0!5 5 5
U U ^ aˆ & ss
2
An s Y
UU X Y
2
1 11 d 2c
U
12
U
2C @~ 2C112 d a d c ! 2i ~ d a 1 d c !# 2 , Y 2 1 ~ 2C112 d a d c ! 2 1 ~ d a 1 d c ! 2 ~11!
where X is the intracavity field amplitude. For an empty cavity (C50), the cavity intensity transmission function T( v 0 ) reduces to 1/(11 d 2c ) and shows a single peak at v 0 5 v c as expected. On resonance ( v a 5 v c 5 v 0 ), Eq. ~11! leads to the following equation for the transmission function:
F
G
Y 2 1 ~ 112C ! 2 . Y 2 1 ~ 112C ! 2
55
This equation shows that for a weak external driving field (Y !1), the effect of a single atom on the cavity transmission function is very large and a resonant single intracavity atom can reduce the transmission by a factor of (112C) 2 @1#. For a large driving field (Y @1) the cavity transmission function approaches 1. In Fig. 3, we plot the intensity transmission function T( v 0 ) for ~a! zero atom-cavity detuning ( v a 5 v c ) and ~b! nonzero atom-cavity detuning ( v a Þ v c ). The transmission spectra are split into two symmetric peaks when the cavity and the atom are in resonance @see Fig. 3~a!#. We note that the positions of the two peaks can be determined by the solutions of D e f f 50, and that they are almost independent of the external driving field Y . When the cavity is detuned from the atomic transition ~i.e., v c Þ v a ), Fig. 3~b! shows that the two peaks of the transmission spectra are asymmetric. The peak on the right with a smaller cavity detuning v c 2 v 0 is higher than the other on the left. This is in contrast with the behavior of the atomic inversion shown in Fig. 2~b!. As the atom-cavity detuning ( v c 2 v a ) increases, the peak on the right comes closer to v c and becomes higher while the peak on the left becomes farther away from v c and lower. For large atom-cavity detunings, the cavity effectively decouples from the atom, and hence the cavity transmission reduces to one peak at v 0 5 v c of the empty cavity. In conclusion, we have shown that the new cavitymodified Maxwell-Bloch equations can be used to study cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, vacuum Rabi splitting, and other related effects.
~12!
The authors wish to thank Dr. G. S. Agarwal, Dr. H. J. Carmichael, and Dr. S. Singh for many helpful discussions.
@1# Cavity Quantum Electrodynamics, edited by P.R. Berman ~Academic, Boston, 1994!. @2# E.M. Purcell, Phys. Rev. 69, 681 ~1946!. @3# D.J. Heinzen and M.S. Feld, Phys. Rev. Lett. 59, 2623 ~1987!. @4# J.J. Sanchez-Mondragon, N.B. Narozhny, and J.H. Eberly, Phys. Rev. Lett. 51, 550 ~1983!; G.S. Agarwal, ibid. 53, 1732 ~1984!. @5# P. Goy, J.M. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903 ~1983!; R.G. Hulet, E.S. Hilfer, and D. Klepp-
ner, ibid. 55, 2137 ~1985!. @6# M. G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble, and H.J. Carmichael, Phys. Rev. Lett. 63, 240 ~1989!; Y. Zhu, D. J. Gautheir, S.E. Morin, Q. Wu, H.J. Carmichael, and T.W. Mossberg, ibid. 64, 2499 ~1990!. @7# R.J. Thompson, G. Rempe, and H.J. Kimble, Phys. Rev. Lett. 68, 1132 ~1992!. @8# C. Wang and R. Vyas, Phys. Rev. A 51, 2516 ~1995!. @9# W.H. Louisell, Quantum Statistic Properties of Radiation ~Wiley, New York, 1973!.
T~ v0!5
