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Atomic emission spectrum including virtual photon transitions in a cavity
15 January
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 (1997) 455-462
Full length article
Atomic emission spectrum including virtual photon transitions in a cavity Peng Zhou a, S. Swain a, Gao-Xiang Li b, Jin-Sheng Peng b a Department ofApplied Marhematics and Theoretical Physics, The Queen’s University of Bevast, Beljast B77 INN. UK b Department of Physics. Huazhong Normal Uniuersity, Wuhan 430070, Chinu Received 29 May 1996; revised version received 29 July 1996; accepted 21 August 1996
Abstract
We investigate the effect of virtual photon processes (due to the antirotating terms in the Hamiltonian) on the time-dependent physical spectrum emitted by a two-level atom interacting with a single-mode quantized field in a perfect cavity. By means of a Heisenberg operator perturbative technique, a generalized emission spectrum is obtained which includes a contribution from the phase-dependent quantum interference between the real and virtual photons. The spectrum is phase sensitive. The effects of different statistics of the initial cavity mode are reported. For the cavity mode in a coherent state, one of the sidebands in the atomic emission spectrum can be reduced whilst the other can be enhanced due to the virtual photon transitions. The effect can be significant, and may be detected using a highly excited Rydberg atom in a microwave cavity with a high quality factor. PACS: 42.50.G~; 42.50.D~; 32.80. - t
1. Introduction
The rotating-wave approximation (RWA) [l] is a frequently employed simplification in quantum optics, laser physics and other branches of resonance phenomena. Under this approximation, the rapidly oscillating terms, which represent the virtual-photon transitions (VPT) that correspond to non-energy-conserving processes, are dropped from the interaction Hamiltonian. The resulting Hamiltonian describes the system quite adequately for many purposes, and may be exactly solvable. For example, the JaynesCummings model (JCM) [2] involving the one-pho0030~4018/97/$17.00 Copyright PII SOO30-4018(96)00553-6
ton interaction of a two-level atom with a single-mode quantized field, can be solved exactly within the RWA, yet retains a great deal of interesting quantum-mechanical features of the atom-field coupling, such as the collapses and revivals of the atomic inversion oscillations, field squeezing [3], and so forth. Some of these predictions have been verified experimentally [4]. Although the RWA is a very good approximation in many respects, the virtual photon contributions it neglects may, in appropriate circumstances, yield significant physical effects. Perhaps the most wellknown of theses effects are the Bloch-Siegert shifts
0 1997 Elsevier Science B.V. All rights reserved.
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P. Zhou et al./ Optics Communications
and multiple quantum resonance [5]. These features are negligible under usual circumstances at optical frequencies, but are significant at microwave frequencies where indeed, they have been experimentally observed [6]. One of us investigated the effect of multiple spontaneous emissions (to all orders) and quantum interference on the resonance fluorescence spectrum, employing the full Hamiltonian of a two-level atom interacting with the radiation field in which the RWA was not made [7]. Stimulated and spontaneous emission and absorption of such a system were also reported by Swain [8] based on nonperturbative continued-fraction methods [9]. Compagno and coauthors [ 101 considered systematically the influences of the virtual photons surrounding a ground-state neutral atom-field system, and showed that the VPT are essential in order to ensure causality, and cannot be neglected in any accurate treatment of photon absorptions [l 11.Atomic level shifts were also discussed from the point of view of virtual-photon processes [ 121. The role of the VPT in a model of photodetection was examined in Ref. [ 131, where Drummond argued that, because the rotating-wave approximation essentially removes all of the detector-radiation coupling from the ground state of the interacting system, it alters the treatment of timevarying fields. The omitted VPT can significantly change the predicted excitation rates on short time scales. Furthermore, he introduced a new technique in which a canonical transformation was used to transform the atom-radiation Hamiltonian to one without VPT, which may remove virtual-photon correlations from the calculation, without any need for the RWA. Vyas and Singh’s results [14] displayed that the VFT may raise the threshold of laser oscillations and broaden the photon-number distribution in a single mode laser. The chaotic behaviour of a strongly driven N-atom system due to the VPT was also analyzed by Milonni, Ackerhalt and Galbraith [15]. Concerning the JCM, the effects of the virtual photons on the atomic inversion, the phase properties and the field squeezing [16,17] were extensively discussed by employing various perturbation techniques. Recently, Crips [ 181has analyzed the validity of the RWA in quantum-electrodynamic (QED) problems, and shown that for a spin-f particle, at rest in a constant magnetic field and for a system
134 (1997) 4X-462
that obeys the electric dipole Am = f 1 selection rules, a choice of a perturbing field that consists of circularly polarized photon can make the RWA unnecessary for the solution of the QED problem, However, a two-level system that obeys the Am = 0 selection rule will require the RWA, regardless of the choice of polarization of the perturbing field. An experimental test of the validity of the RWA has also been proposed by Crips [ 181. The atomic emission spectrum is of fundamental importance in laser spectroscopic techniques. It has been extensively studied both in free space [19], and in a cavity [20]. Recently, Gea-Banacloche, Schlicher and Zubairy [21] have investigated the time-dependent physical spectrum emitted by an atom in an ideal cavity. They found that when the cavity mode is occupied by a squeezed vacuum, the spectrum is independent of the squeezed phase, due to the lack of coherent coupling between the single-photon transition of the atom and the two-photon correlated character of the squeezed vacuum in such a perfectly closed system. This is substantially different behaviour from that of an atom embedded in a squeezed vacuum reservoir (an irreversible system), where the phase-sensitivity plays an important role [22]. All these studies were based on the rotating-wave approximation. The question immediately arises of how different the atomic emission spectrum would be in the presence of the virtual-photon transitions. In this paper we examine the effect of the virtual-photon processes on the time-dependent physical spectrum emitted by an atom interacting with a single-mode quantized field in a perfect cavity. We find that the differences are small, as was to be expected, but that they should be experimentally detectable in certain circumstances. The layout is as follows: In Section 2, we generalize the spectrum for the case in which the virtual-photon transitions are taken into account by means of a perturbative method [17]. The emission spectrum obtained is phase-sensitive because of correlations between real photon and virtual photon transitions. The numerical results of the spectrum are presented in Section 3, for the cavity mode initially in a squeezed vacuum and in a coherent state, respectively. The final part contains a summary and brief discussions of possible schemes to detect the effect of the virtual photon transitions on the spectrum.
P. Zhou rt ul. / Ulxic.\ Conltlluni~uriot1.s134 f 1997) 455-462
2. Generalized
physical
spectrum
In order to measure the spectrum of the light emitted by an excited atom, a frequency-sensitive device such as a tunable filter must be placed in front of the detector. Eberly and Wodkiewiez [23] found the time-dependent physical spectrum to be strongly dependent on the parameters of the filter, especially on its width, r. Explicitly the spectrum is given by the form
457
calculations are quite tedious, we only outline the results obtained ‘. In the Heisenberg representation, the perturbation expansion of the atomic lowering operator S_(r) takes the form, S_(r) =S”““(l)
-ilordi[S~\\‘“(f),W(f.--f)]
W(t”-r)],W(f-f)]
+ ... ,
(4)
where x(s+(t,>s_(t,>>,
(1)
where T is the time over which the measurement takes place, and S_(t) is the atomic dipole lowering operator and S+(r) = [S_(t)]’ the raising operator in the Heisenberg representation. In this paper we are concerned with the transient emission spectrum of a two-level atom interacting with a single quantized cavity-mode field in the presence of the virtual-photon processes in a perfect cavity. (We must in fact allow a tiny loss in order to actually observe the spectrum.) The full Hamiltonian of this system (in units of h) may be written as [7,8]
H=HRWA+V.
(2)
with
H RWA
=
v=&s+a+
o,a’a + wAS, + g(S+a +us_).
+ a+S_), (3)
Here H,,, denotes the rotating-wave-approximated Hamiltonian, V represents the virtual-photon processes, a’ and a are the creation and annihilation operators of the cavity-mode field at frequency wC, S: denotes the atomic inversion operator, wA the atomic transition frequency, and g the atom-field coupling constant. For simplicity, we take the atomic transition to be resonant with the cavity mode (i.e., WA
=
(5)
W(f)
(6)
=
U,,,(f)
VU&.4(r)
9
with URWA(t)= exp( - iHRWAf)being the unitary time-development operator with respect to the RWA Hamiltonian [3]. Eq. (4), in fact, is an exact expression although it cannot be given in closed form. The first term, SFwA(t), on the right-hand side is the exact solution for the atomic lowering operator within the RWA, whilst the other terms represent the perturbations due to the virtual-photon processes. Here we assume the effect of the VFT is very small so that it is enough to restrict the expansion to the first two terms - that is, we take into account only the first-order contribution of the VFT. Our approximation for the lowering operator S_(t) is thus of the form s_(t)=SRW*(t)+SVPT(r),
(7)
where S!PT(t>, the contribution of the virtual-photon processes to the atomic transition operator, is given by pyt)
=
-i/ddr’[S~““(t).W(r’-f)].
(8)
Similarly, the time-dependent physical spectrum (1) may be generalized to include the virtual-photon processes to first order:
WC).
full Hamiltonian is not exactly solvable in a closed form. Here we use the approach of Fang and Zhou [ 171, where the RWA Hamiltonian is treated as the “unperturbed” Hamiltonian, and the contribution of the VPT is treated as a small perturbation on the exact solution of the RWA Hamiltonian. As the The
SRWA(I) = UrJw*(t) S- C;Iw*( t),
A(w) =ARWA(w) +AVPT(w),
’ Detailed calculations are available upon request.
(9)
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Communications 134 (1997) 455-462
with
+ e’wc(‘, -
5 -%*9n+ P2
‘2)
2
nRWA( w)
n=O
= 2r~hf’lo=dt,e x
lF(
(SR,WA(
x cos$,( r, - t2) cosp,t,
-(r-iwXT-f,)-(T+ioXT-12)
[
t,)SRWA(
t,)>,
(10)
X
2r
-cosp2(2,
(r-iwXT-f,)-(T+iwXT-12)
Tdr, = /0 / 0 dtz e-
X x
/0
[
+
(11)
is the emission spectrum based on the RWA, while AvpT( w> represents the first-order contribution of the VPT and corresponds to correlations between the real-photon and virtual-photon transitions. Higher-order contributions of the VPT are dropped. We assume the atom is initially in the excited state ( + >, and the cavity-mode fields in an arbitrary quantized state 19 ) = C:= aFnln>, where In) is the photon number state and 9, is associated with the photon number probability distribution 9( n> through 9(n) = 1.Fn12.The resultant atomic dipole-dipole correlation functions are (StwA( r,)SR_WA(r,)> = e’+(‘~-‘~‘~~09( n)cos&( xcos( P,r,)cos( (SR,WA(t$s_““(
t, - t2)
P,r*),
(12)
t2)) + (S,VPT(t,)SRWA( t*))
=e iO,(l,- 12)? sTn*gn+2 PI cosP,t, cos&, n=O fze-i20c7 /
COS&,(T-
t,)
sinp,(r-
t2)dr
‘I
- eiwc(fl-‘z) 2 sT,;$Fn p, cos&t,cos/3,t2 n=O X I
smp,(T-
t2) cos&rdr
.
fZei2w~rsin/3z(r‘I
rl)cos&,( T- t2) dr
-
f2)
COSp~t2
f1e-i2”‘~Tsinp,r cos&(r-
t,) dr I
+ eidft-fz)
ARWA(w)
X
7
c
0
w)
=
I f2e -i20
l? n-0
9n*,
2%
P2
1
x cos&( t, - t2) cosp,t,
X
/0
f’ei2wc’sin/3,( r-
I,) cos&r dr
- cos&( c, - t*) COSQ, X
I f’ei20crsinP,rcos&(r0
t2) dr
1, (13)
where &=gfi (k=0,1,2,3). It is easily seen from Eq. (12) that the atomic dipole correlation function contains no phase information under the RWA, when the atom is initially in the excited state (or the ground state). However, if the effect of the virtual-photon processes is taken into account, the resulting correlation is composed of the phase-independent term, Eq. (12), arising from real-photon processes, and the phase-sensitive term, Eq. (13), which represents the quantum interference between the real photon and the virtual photon transitions. In fact, the phase-dependent contribution is always associated with products representing twophoton processes, Fjn.Fn: 2 or Fn*Fn+2, and a factor oscillating rapidly with the double cavity frequency, e * i2wcf. These two-photon influences will, as expected, modify the results of Gea-Banacloche, Schlicher and Zubairy [21] when the cavity mode is initially in a squeezed vacuum. However, when the cavity initially contains a pure photon number state, the effect of the virtual photons disappears.
P. Zhou et ul. / Optics Communications 134 (1997) 455-462
Substituting Eq. (12) into Eq. (101, we have ARwA(,)
459
tons. However, the probability amplitude for the coherent field state is given by
= $~+Q)[lG(&,PJ
3/2 gcoh n
=
e-ii/2
(18)
+G( -PO, P,)t2 +IG( PO? -Pi> +G(-PO?
-P,)l’j~
(141
where G( f x, + y) is defined as ei(Ji.r+y)T_e-rT G( fx,
-+Y)
=
r+i(A+x+y)
’
(15)
with A = w, - w, a scanning frequency referenced to the cavity frequency. This spectrum is the same as Eq. (4) of Ref. [2 I], the transient emission spectrum under the RWA on resonance. The analytical expression of AvpT(w) may be also obtained by inserting Eq. (13) into Eq. (11). However, we are here interested in numerical results. For convenience, we renormalise all parameters by the coupling constant in our numerical calculations; that is, we take
whilst the emission spectrum is scaled as gA(a).
3. Numerical results and discussions In what follows, we assume that the cavity mode is initially either in a squeezed vacuum state, or in a coherent state. The corresponding probability amplitude of the photon distribution for the squeezed vacuum is of the form [21],
=(-I) YSV 2n+1=
@&-)“4(_z-)“‘2e-in~, 0,
(17)
where ii = sinh*r is the squeezed photon number, r is the squeezing parameter (r = 0 for ordinary vacuum) and 4 is the squeezed phase. Clearly, this state is associated with correlations between pairs of pho-
where 71and 4 are the mean photon number and the phase of the coherent field. The atomic transient spectrum is strongly dependent on the photon statistics of the initial cavity mode. Before presenting numerical results, we wish to point out some restrictions on the values of the parameters which may be taken. Our approach is based on the perturbative calculation of the virtualphoton transition contribution to the RWA emission spectrum, which corresponds to the first-order expansion in terms of the parameter g6 &=20,=-L
6 2w
( 19)
Thus it requires E < 1 (i.e., 6 -+zW,). However, we cannot assume E to be too small, or the effects due to virtual transition will be negligible. Hence we consider values of E in the range 0.01 ,< E ,< 0.1. For a given value of the perturbative parameter, for example EN 0.01, one has n - 4 X 10m4 Z,“. In most cavity experiments, however, WCmay be very large even with strong coupling [24] (is, m IO4 106). This means that the photon number inside the cavity must be very large in order to observe the effect of the virtual photon emissions. However, it is probably much more difficult to store large numbers of photons inside the cavity than to reduce Tjc, the ratio of the cavity frequency to the coupling constant. Therefore, we restrict ourselves to the case of small photon numbers and W,. We will discuss possible schemes for realizing this scenario later. The transient spectrum is also very sensitive to the passband width of the filter detector r [21,23]. However, our interest here is on the effect of the virtual photon transitions on the spectrum, so we fix the values of T and r. Explicitly, we set gT = 20 and g-ir= 0.5 throughout our numerical calculations. Fig. 1 presents the atomic emission spectrum for the cavity mode initially in a squeezed vacuum state with E = 0.01 in the frames (a) and (b), and E = 0.1 in the frames (c) and cd), where the solid curves are
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134 (1997) 455-462
Fig. 2. Same as Fig. 1, but with E = 0.1 (ii = 1 and Tj, = 51, for 4 = n/2 in the frame (a) and r$ = T in the frame (b).
10
-70 ::
Fig. 1. The atomic emission spectrum, gn(a), as a function of a= ( wC- o)/g, for the cavity mode initially in a squeezed vacuum state with the phase Q,= 0. We set the perturbative parameter E = 0.01 with (a) 3i= 0.1, 7jc= 15.8 and (b) Ii= 0.5, I&=35; and E=O.I with (c) Ii=], Z&=5 and (d) H=5, ZJ = 11. The values T = 20 and 7 = 0.5 are taken throughout. In all graphs the solid curves represent the emission spectrum under the RWA, whilst the dash curves is the total spectrum including the virtual photon transitions. The total spectrum has been raised by 0.5 units for clarity.
spectrum under the RWA, whilst the dashed ones indicate the total emission spectrum including the contribution of the virtual photon transitions. For clarity, we have given the full spectrum a vertical displacement. One finds from the first two frames that when E = 0.01, the virtual photon transitions make no noticeable contribution, and the total and RWA spectra are identical and symmetric. However, the virtual photon processes have substantial effects on the spectrum when E increases. For example, with E = 0.1 in frames (c) and (d), the spectra are different. One peak is enhanced whilst the other is reduced, the total spectrum is asymmetric, and it is slightly shifted due to the Bloch-Siegert shift [25] which arises from the virtual photon transitions. One can also see that the overall magnitude of the virtual photon effects on the spectrum depends only on the perturbative parameter E, regardless of the individual values of 3 and Z,, whilst Z determines the spectral shape. When E =B 1 the spectrum shows a single peak at the cavity frequency. (Graphs are not presented here.) We examine the phase sensitivity of the spectrum in Fig. 2 for E = 0.1 with Z = 1 and W, = 5. Comparing with Fig. l(c) where $I = 0, one finds that the
the
spectrum is shifted toward frequencies greater than the cavity frequency when 4 = rr, whilst the spectrum is almost the same as the RWA result when 4 = 7r/2. These departures from the RWA spectrum are due to quantum interference between the real and virtual photon transitions. Noting that the photon emitted by the atom is continually reabsorbed and reemitted in the current system, which is ideal and closed, the reaction of the atom back on the initially squeezed vacuum destroys the squeezed character very rapidly [21]. Therefore, the initial squeezed vacuum state does not persist. As a consequence, the time-dependent emission spectrum is substantially different from the steady-state resonance fluorescence spectrum of the open (irreversible) system of a driven atom interacting with a multimode squeezed vacuum [22].
-10
0
10
A Fig. 3. Same as Fig. 1, but with the cavity mode initially in a coherent state with the phase C#J = 0, and different E: (a) E = 0.01 with ii=lOand &=158,(b) c=O.Ol with ii=20and GC= 223.6, k> E = 0.1 with R = 10 and WC= 15.8, and (d) E = 0.1 with ii = 20 and W, = 22.3. The total spectrum has been raised by 0.2 units for clarity.
P. Zhou et al./
Optics Communications I34 (1997) 455-462
;o~~ . 0
A
10
0
10
A
Fig. 4. Same as Fig. 3, but with E = 0.158 (in = 10 and Ti&= IO), for 4 = 0 in tbe frame (a) and 4 = w/2 in the frame (b).
Next, we show in Fig. 3 and Fig. 4 the emission spectrum when the cavity mode is initially in a coherent state, with the initial mean photon number inside the cavity rather larger than those in Fig. 1 and Fig. 2. Again, when the perturbative parameter E is very small, for instance, E = 0.01 in the frames (a> and (b), the effect of the virtual photon transitions on the emission spectrum is not noticeable. However, on increasing E, for example to 6 = 0.1 in the frames (c) and cd), the virtual photon processes make a substantial contribution to the emission spectrum. (We note that if E is too large the higher-order effects of the virtual photons must be included.) One of the sidebands is reduced in height, while the other is increased. The extent of the reducing and increasing is sensitive to the phase, as shown in Fig. 4 where the perturbative parameter is E = 0.158 with ~=lOand~,=lOand~=Oin4(a)and~=~/2 in 4(b). The spectra are shifted in different directions for 4 = 0 and rr/2. This shows that atomic emissions can be controlled by the phase of the cavity mode through quantum interference.
4. Summary We have derived the generalized emission spectrum emitted by a two-level atom interacting with a single-mode quantized field in a perfect cavity, including the first-order contribution of the virtualphoton processes, which is phase sensitive. The effect of the virtual photon transitions on the spectrum is dependent on the perturbative parameter E. The virtual photons make negligible contribution to the spectrum when E < 0.01. In general, the spectrum is shifted slightly, due to the Bloch-Siegert shift resulting from the virtual photon emissions. For the cavity mode initially in a coherent state, the height of one
461
of the sidebands of the emission spectrum is reduced while the other is increased, if virtual photon effects are included. The reduction and enhancement of the sidebands can be controlled by the phase of the cavity mode. Our results are obtained based on the assumption of a perfect cavity, and the effects will be reduced if small losses are included. (A very small amount of loss is of course necessary for the observation of the spectrum.) Recent developments with the one-atom maser [26] have demonstrated that it is possible to realize experimentally a single atom interacting with a single-mode quantized field in a nearly perfect cavity. However, the problem with detecting the effect of virtual photon transitions is that the ratio WC is very large experimentally in the optical frequency domain. In these circumstances, the effect disappears for the small cavity photon numbers which can be achieved experimentally. However, the effect is most likely observed from a highly excited Rydberg atom in a superconducting microwave cavity with high quality factor Q. where the cavity losses are negligible [4]. The highly excited Rydberg atom has a very large principal quantum number m [27]. Correspondingly, the electric dipole atom-field coupling constant g between neighbouring atomic levels is very large and scales as m2, whilst the orbital frequency of the Rydberg electron decreases like ms3, so thewavelength of a transition between neighbouring levels is in the microwave domain. The ratio Oc = w,/g can be substantially reduced. In addition, the lifetime for spontaneous transitions is also very large and is proportional to m3 or m5 for low- and high-angularmomentum Rydberg states respectively. Thus the spontaneous decay can be neglected. By these means, it should be possible to observe the effects of the antirotating terms.
Acknowledgements
This work is supported by the United Kingdom EPSRC, by the EC, by a NATO Collaborative Research Award, and by the National Natural Scicence Foundation of China. P.Z. wishes to thank the Queen’s University of Belfast for financial support.
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1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 (1997) 455-462
Full length article
Atomic emission spectrum including virtual photon transitions in a cavity Peng Zhou a, S. Swain a, Gao-Xiang Li b, Jin-Sheng Peng b a Department ofApplied Marhematics and Theoretical Physics, The Queen’s University of Bevast, Beljast B77 INN. UK b Department of Physics. Huazhong Normal Uniuersity, Wuhan 430070, Chinu Received 29 May 1996; revised version received 29 July 1996; accepted 21 August 1996
Abstract
We investigate the effect of virtual photon processes (due to the antirotating terms in the Hamiltonian) on the time-dependent physical spectrum emitted by a two-level atom interacting with a single-mode quantized field in a perfect cavity. By means of a Heisenberg operator perturbative technique, a generalized emission spectrum is obtained which includes a contribution from the phase-dependent quantum interference between the real and virtual photons. The spectrum is phase sensitive. The effects of different statistics of the initial cavity mode are reported. For the cavity mode in a coherent state, one of the sidebands in the atomic emission spectrum can be reduced whilst the other can be enhanced due to the virtual photon transitions. The effect can be significant, and may be detected using a highly excited Rydberg atom in a microwave cavity with a high quality factor. PACS: 42.50.G~; 42.50.D~; 32.80. - t
1. Introduction
The rotating-wave approximation (RWA) [l] is a frequently employed simplification in quantum optics, laser physics and other branches of resonance phenomena. Under this approximation, the rapidly oscillating terms, which represent the virtual-photon transitions (VPT) that correspond to non-energy-conserving processes, are dropped from the interaction Hamiltonian. The resulting Hamiltonian describes the system quite adequately for many purposes, and may be exactly solvable. For example, the JaynesCummings model (JCM) [2] involving the one-pho0030~4018/97/$17.00 Copyright PII SOO30-4018(96)00553-6
ton interaction of a two-level atom with a single-mode quantized field, can be solved exactly within the RWA, yet retains a great deal of interesting quantum-mechanical features of the atom-field coupling, such as the collapses and revivals of the atomic inversion oscillations, field squeezing [3], and so forth. Some of these predictions have been verified experimentally [4]. Although the RWA is a very good approximation in many respects, the virtual photon contributions it neglects may, in appropriate circumstances, yield significant physical effects. Perhaps the most wellknown of theses effects are the Bloch-Siegert shifts
0 1997 Elsevier Science B.V. All rights reserved.
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P. Zhou et al./ Optics Communications
and multiple quantum resonance [5]. These features are negligible under usual circumstances at optical frequencies, but are significant at microwave frequencies where indeed, they have been experimentally observed [6]. One of us investigated the effect of multiple spontaneous emissions (to all orders) and quantum interference on the resonance fluorescence spectrum, employing the full Hamiltonian of a two-level atom interacting with the radiation field in which the RWA was not made [7]. Stimulated and spontaneous emission and absorption of such a system were also reported by Swain [8] based on nonperturbative continued-fraction methods [9]. Compagno and coauthors [ 101 considered systematically the influences of the virtual photons surrounding a ground-state neutral atom-field system, and showed that the VPT are essential in order to ensure causality, and cannot be neglected in any accurate treatment of photon absorptions [l 11.Atomic level shifts were also discussed from the point of view of virtual-photon processes [ 121. The role of the VPT in a model of photodetection was examined in Ref. [ 131, where Drummond argued that, because the rotating-wave approximation essentially removes all of the detector-radiation coupling from the ground state of the interacting system, it alters the treatment of timevarying fields. The omitted VPT can significantly change the predicted excitation rates on short time scales. Furthermore, he introduced a new technique in which a canonical transformation was used to transform the atom-radiation Hamiltonian to one without VPT, which may remove virtual-photon correlations from the calculation, without any need for the RWA. Vyas and Singh’s results [14] displayed that the VFT may raise the threshold of laser oscillations and broaden the photon-number distribution in a single mode laser. The chaotic behaviour of a strongly driven N-atom system due to the VPT was also analyzed by Milonni, Ackerhalt and Galbraith [15]. Concerning the JCM, the effects of the virtual photons on the atomic inversion, the phase properties and the field squeezing [16,17] were extensively discussed by employing various perturbation techniques. Recently, Crips [ 181has analyzed the validity of the RWA in quantum-electrodynamic (QED) problems, and shown that for a spin-f particle, at rest in a constant magnetic field and for a system
134 (1997) 4X-462
that obeys the electric dipole Am = f 1 selection rules, a choice of a perturbing field that consists of circularly polarized photon can make the RWA unnecessary for the solution of the QED problem, However, a two-level system that obeys the Am = 0 selection rule will require the RWA, regardless of the choice of polarization of the perturbing field. An experimental test of the validity of the RWA has also been proposed by Crips [ 181. The atomic emission spectrum is of fundamental importance in laser spectroscopic techniques. It has been extensively studied both in free space [19], and in a cavity [20]. Recently, Gea-Banacloche, Schlicher and Zubairy [21] have investigated the time-dependent physical spectrum emitted by an atom in an ideal cavity. They found that when the cavity mode is occupied by a squeezed vacuum, the spectrum is independent of the squeezed phase, due to the lack of coherent coupling between the single-photon transition of the atom and the two-photon correlated character of the squeezed vacuum in such a perfectly closed system. This is substantially different behaviour from that of an atom embedded in a squeezed vacuum reservoir (an irreversible system), where the phase-sensitivity plays an important role [22]. All these studies were based on the rotating-wave approximation. The question immediately arises of how different the atomic emission spectrum would be in the presence of the virtual-photon transitions. In this paper we examine the effect of the virtual-photon processes on the time-dependent physical spectrum emitted by an atom interacting with a single-mode quantized field in a perfect cavity. We find that the differences are small, as was to be expected, but that they should be experimentally detectable in certain circumstances. The layout is as follows: In Section 2, we generalize the spectrum for the case in which the virtual-photon transitions are taken into account by means of a perturbative method [17]. The emission spectrum obtained is phase-sensitive because of correlations between real photon and virtual photon transitions. The numerical results of the spectrum are presented in Section 3, for the cavity mode initially in a squeezed vacuum and in a coherent state, respectively. The final part contains a summary and brief discussions of possible schemes to detect the effect of the virtual photon transitions on the spectrum.
P. Zhou rt ul. / Ulxic.\ Conltlluni~uriot1.s134 f 1997) 455-462
2. Generalized
physical
spectrum
In order to measure the spectrum of the light emitted by an excited atom, a frequency-sensitive device such as a tunable filter must be placed in front of the detector. Eberly and Wodkiewiez [23] found the time-dependent physical spectrum to be strongly dependent on the parameters of the filter, especially on its width, r. Explicitly the spectrum is given by the form
457
calculations are quite tedious, we only outline the results obtained ‘. In the Heisenberg representation, the perturbation expansion of the atomic lowering operator S_(r) takes the form, S_(r) =S”““(l)
-ilordi[S~\\‘“(f),W(f.--f)]
W(t”-r)],W(f-f)]
+ ... ,
(4)
where x(s+(t,>s_(t,>>,
(1)
where T is the time over which the measurement takes place, and S_(t) is the atomic dipole lowering operator and S+(r) = [S_(t)]’ the raising operator in the Heisenberg representation. In this paper we are concerned with the transient emission spectrum of a two-level atom interacting with a single quantized cavity-mode field in the presence of the virtual-photon processes in a perfect cavity. (We must in fact allow a tiny loss in order to actually observe the spectrum.) The full Hamiltonian of this system (in units of h) may be written as [7,8]
H=HRWA+V.
(2)
with
H RWA
=
v=&s+a+
o,a’a + wAS, + g(S+a +us_).
+ a+S_), (3)
Here H,,, denotes the rotating-wave-approximated Hamiltonian, V represents the virtual-photon processes, a’ and a are the creation and annihilation operators of the cavity-mode field at frequency wC, S: denotes the atomic inversion operator, wA the atomic transition frequency, and g the atom-field coupling constant. For simplicity, we take the atomic transition to be resonant with the cavity mode (i.e., WA
=
(5)
W(f)
(6)
=
U,,,(f)
VU&.4(r)
9
with URWA(t)= exp( - iHRWAf)being the unitary time-development operator with respect to the RWA Hamiltonian [3]. Eq. (4), in fact, is an exact expression although it cannot be given in closed form. The first term, SFwA(t), on the right-hand side is the exact solution for the atomic lowering operator within the RWA, whilst the other terms represent the perturbations due to the virtual-photon processes. Here we assume the effect of the VFT is very small so that it is enough to restrict the expansion to the first two terms - that is, we take into account only the first-order contribution of the VFT. Our approximation for the lowering operator S_(t) is thus of the form s_(t)=SRW*(t)+SVPT(r),
(7)
where S!PT(t>, the contribution of the virtual-photon processes to the atomic transition operator, is given by pyt)
=
-i/ddr’[S~““(t).W(r’-f)].
(8)
Similarly, the time-dependent physical spectrum (1) may be generalized to include the virtual-photon processes to first order:
WC).
full Hamiltonian is not exactly solvable in a closed form. Here we use the approach of Fang and Zhou [ 171, where the RWA Hamiltonian is treated as the “unperturbed” Hamiltonian, and the contribution of the VPT is treated as a small perturbation on the exact solution of the RWA Hamiltonian. As the The
SRWA(I) = UrJw*(t) S- C;Iw*( t),
A(w) =ARWA(w) +AVPT(w),
’ Detailed calculations are available upon request.
(9)
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with
+ e’wc(‘, -
5 -%*9n+ P2
‘2)
2
nRWA( w)
n=O
= 2r~hf’lo=dt,e x
lF(
(SR,WA(
x cos$,( r, - t2) cosp,t,
-(r-iwXT-f,)-(T+ioXT-12)
[
t,)SRWA(
t,)>,
(10)
X
2r
-cosp2(2,
(r-iwXT-f,)-(T+iwXT-12)
Tdr, = /0 / 0 dtz e-
X x
/0
[
+
(11)
is the emission spectrum based on the RWA, while AvpT( w> represents the first-order contribution of the VPT and corresponds to correlations between the real-photon and virtual-photon transitions. Higher-order contributions of the VPT are dropped. We assume the atom is initially in the excited state ( + >, and the cavity-mode fields in an arbitrary quantized state 19 ) = C:= aFnln>, where In) is the photon number state and 9, is associated with the photon number probability distribution 9( n> through 9(n) = 1.Fn12.The resultant atomic dipole-dipole correlation functions are (StwA( r,)SR_WA(r,)> = e’+(‘~-‘~‘~~09( n)cos&( xcos( P,r,)cos( (SR,WA(t$s_““(
t, - t2)
P,r*),
(12)
t2)) + (S,VPT(t,)SRWA( t*))
=e iO,(l,- 12)? sTn*gn+2 PI cosP,t, cos&, n=O fze-i20c7 /
COS&,(T-
t,)
sinp,(r-
t2)dr
‘I
- eiwc(fl-‘z) 2 sT,;$Fn p, cos&t,cos/3,t2 n=O X I
smp,(T-
t2) cos&rdr
.
fZei2w~rsin/3z(r‘I
rl)cos&,( T- t2) dr
-
f2)
COSp~t2
f1e-i2”‘~Tsinp,r cos&(r-
t,) dr I
+ eidft-fz)
ARWA(w)
X
7
c
0
w)
=
I f2e -i20
l? n-0
9n*,
2%
P2
1
x cos&( t, - t2) cosp,t,
X
/0
f’ei2wc’sin/3,( r-
I,) cos&r dr
- cos&( c, - t*) COSQ, X
I f’ei20crsinP,rcos&(r0
t2) dr
1, (13)
where &=gfi (k=0,1,2,3). It is easily seen from Eq. (12) that the atomic dipole correlation function contains no phase information under the RWA, when the atom is initially in the excited state (or the ground state). However, if the effect of the virtual-photon processes is taken into account, the resulting correlation is composed of the phase-independent term, Eq. (12), arising from real-photon processes, and the phase-sensitive term, Eq. (13), which represents the quantum interference between the real photon and the virtual photon transitions. In fact, the phase-dependent contribution is always associated with products representing twophoton processes, Fjn.Fn: 2 or Fn*Fn+2, and a factor oscillating rapidly with the double cavity frequency, e * i2wcf. These two-photon influences will, as expected, modify the results of Gea-Banacloche, Schlicher and Zubairy [21] when the cavity mode is initially in a squeezed vacuum. However, when the cavity initially contains a pure photon number state, the effect of the virtual photons disappears.
P. Zhou et ul. / Optics Communications 134 (1997) 455-462
Substituting Eq. (12) into Eq. (101, we have ARwA(,)
459
tons. However, the probability amplitude for the coherent field state is given by
= $~+Q)[lG(&,PJ
3/2 gcoh n
=
e-ii/2
(18)
+G( -PO, P,)t2 +IG( PO? -Pi> +G(-PO?
-P,)l’j~
(141
where G( f x, + y) is defined as ei(Ji.r+y)T_e-rT G( fx,
-+Y)
=
r+i(A+x+y)
’
(15)
with A = w, - w, a scanning frequency referenced to the cavity frequency. This spectrum is the same as Eq. (4) of Ref. [2 I], the transient emission spectrum under the RWA on resonance. The analytical expression of AvpT(w) may be also obtained by inserting Eq. (13) into Eq. (11). However, we are here interested in numerical results. For convenience, we renormalise all parameters by the coupling constant in our numerical calculations; that is, we take
whilst the emission spectrum is scaled as gA(a).
3. Numerical results and discussions In what follows, we assume that the cavity mode is initially either in a squeezed vacuum state, or in a coherent state. The corresponding probability amplitude of the photon distribution for the squeezed vacuum is of the form [21],
=(-I) YSV 2n+1=
@&-)“4(_z-)“‘2e-in~, 0,
(17)
where ii = sinh*r is the squeezed photon number, r is the squeezing parameter (r = 0 for ordinary vacuum) and 4 is the squeezed phase. Clearly, this state is associated with correlations between pairs of pho-
where 71and 4 are the mean photon number and the phase of the coherent field. The atomic transient spectrum is strongly dependent on the photon statistics of the initial cavity mode. Before presenting numerical results, we wish to point out some restrictions on the values of the parameters which may be taken. Our approach is based on the perturbative calculation of the virtualphoton transition contribution to the RWA emission spectrum, which corresponds to the first-order expansion in terms of the parameter g6 &=20,=-L
6 2w
( 19)
Thus it requires E < 1 (i.e., 6 -+zW,). However, we cannot assume E to be too small, or the effects due to virtual transition will be negligible. Hence we consider values of E in the range 0.01 ,< E ,< 0.1. For a given value of the perturbative parameter, for example EN 0.01, one has n - 4 X 10m4 Z,“. In most cavity experiments, however, WCmay be very large even with strong coupling [24] (is, m IO4 106). This means that the photon number inside the cavity must be very large in order to observe the effect of the virtual photon emissions. However, it is probably much more difficult to store large numbers of photons inside the cavity than to reduce Tjc, the ratio of the cavity frequency to the coupling constant. Therefore, we restrict ourselves to the case of small photon numbers and W,. We will discuss possible schemes for realizing this scenario later. The transient spectrum is also very sensitive to the passband width of the filter detector r [21,23]. However, our interest here is on the effect of the virtual photon transitions on the spectrum, so we fix the values of T and r. Explicitly, we set gT = 20 and g-ir= 0.5 throughout our numerical calculations. Fig. 1 presents the atomic emission spectrum for the cavity mode initially in a squeezed vacuum state with E = 0.01 in the frames (a) and (b), and E = 0.1 in the frames (c) and cd), where the solid curves are
P. Zhou et d/Optics
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Communications
134 (1997) 455-462
Fig. 2. Same as Fig. 1, but with E = 0.1 (ii = 1 and Tj, = 51, for 4 = n/2 in the frame (a) and r$ = T in the frame (b).
10
-70 ::
Fig. 1. The atomic emission spectrum, gn(a), as a function of a= ( wC- o)/g, for the cavity mode initially in a squeezed vacuum state with the phase Q,= 0. We set the perturbative parameter E = 0.01 with (a) 3i= 0.1, 7jc= 15.8 and (b) Ii= 0.5, I&=35; and E=O.I with (c) Ii=], Z&=5 and (d) H=5, ZJ = 11. The values T = 20 and 7 = 0.5 are taken throughout. In all graphs the solid curves represent the emission spectrum under the RWA, whilst the dash curves is the total spectrum including the virtual photon transitions. The total spectrum has been raised by 0.5 units for clarity.
spectrum under the RWA, whilst the dashed ones indicate the total emission spectrum including the contribution of the virtual photon transitions. For clarity, we have given the full spectrum a vertical displacement. One finds from the first two frames that when E = 0.01, the virtual photon transitions make no noticeable contribution, and the total and RWA spectra are identical and symmetric. However, the virtual photon processes have substantial effects on the spectrum when E increases. For example, with E = 0.1 in frames (c) and (d), the spectra are different. One peak is enhanced whilst the other is reduced, the total spectrum is asymmetric, and it is slightly shifted due to the Bloch-Siegert shift [25] which arises from the virtual photon transitions. One can also see that the overall magnitude of the virtual photon effects on the spectrum depends only on the perturbative parameter E, regardless of the individual values of 3 and Z,, whilst Z determines the spectral shape. When E =B 1 the spectrum shows a single peak at the cavity frequency. (Graphs are not presented here.) We examine the phase sensitivity of the spectrum in Fig. 2 for E = 0.1 with Z = 1 and W, = 5. Comparing with Fig. l(c) where $I = 0, one finds that the
the
spectrum is shifted toward frequencies greater than the cavity frequency when 4 = rr, whilst the spectrum is almost the same as the RWA result when 4 = 7r/2. These departures from the RWA spectrum are due to quantum interference between the real and virtual photon transitions. Noting that the photon emitted by the atom is continually reabsorbed and reemitted in the current system, which is ideal and closed, the reaction of the atom back on the initially squeezed vacuum destroys the squeezed character very rapidly [21]. Therefore, the initial squeezed vacuum state does not persist. As a consequence, the time-dependent emission spectrum is substantially different from the steady-state resonance fluorescence spectrum of the open (irreversible) system of a driven atom interacting with a multimode squeezed vacuum [22].
-10
0
10
A Fig. 3. Same as Fig. 1, but with the cavity mode initially in a coherent state with the phase C#J = 0, and different E: (a) E = 0.01 with ii=lOand &=158,(b) c=O.Ol with ii=20and GC= 223.6, k> E = 0.1 with R = 10 and WC= 15.8, and (d) E = 0.1 with ii = 20 and W, = 22.3. The total spectrum has been raised by 0.2 units for clarity.
P. Zhou et al./
Optics Communications I34 (1997) 455-462
;o~~ . 0
A
10
0
10
A
Fig. 4. Same as Fig. 3, but with E = 0.158 (in = 10 and Ti&= IO), for 4 = 0 in tbe frame (a) and 4 = w/2 in the frame (b).
Next, we show in Fig. 3 and Fig. 4 the emission spectrum when the cavity mode is initially in a coherent state, with the initial mean photon number inside the cavity rather larger than those in Fig. 1 and Fig. 2. Again, when the perturbative parameter E is very small, for instance, E = 0.01 in the frames (a> and (b), the effect of the virtual photon transitions on the emission spectrum is not noticeable. However, on increasing E, for example to 6 = 0.1 in the frames (c) and cd), the virtual photon processes make a substantial contribution to the emission spectrum. (We note that if E is too large the higher-order effects of the virtual photons must be included.) One of the sidebands is reduced in height, while the other is increased. The extent of the reducing and increasing is sensitive to the phase, as shown in Fig. 4 where the perturbative parameter is E = 0.158 with ~=lOand~,=lOand~=Oin4(a)and~=~/2 in 4(b). The spectra are shifted in different directions for 4 = 0 and rr/2. This shows that atomic emissions can be controlled by the phase of the cavity mode through quantum interference.
4. Summary We have derived the generalized emission spectrum emitted by a two-level atom interacting with a single-mode quantized field in a perfect cavity, including the first-order contribution of the virtualphoton processes, which is phase sensitive. The effect of the virtual photon transitions on the spectrum is dependent on the perturbative parameter E. The virtual photons make negligible contribution to the spectrum when E < 0.01. In general, the spectrum is shifted slightly, due to the Bloch-Siegert shift resulting from the virtual photon emissions. For the cavity mode initially in a coherent state, the height of one
461
of the sidebands of the emission spectrum is reduced while the other is increased, if virtual photon effects are included. The reduction and enhancement of the sidebands can be controlled by the phase of the cavity mode. Our results are obtained based on the assumption of a perfect cavity, and the effects will be reduced if small losses are included. (A very small amount of loss is of course necessary for the observation of the spectrum.) Recent developments with the one-atom maser [26] have demonstrated that it is possible to realize experimentally a single atom interacting with a single-mode quantized field in a nearly perfect cavity. However, the problem with detecting the effect of virtual photon transitions is that the ratio WC is very large experimentally in the optical frequency domain. In these circumstances, the effect disappears for the small cavity photon numbers which can be achieved experimentally. However, the effect is most likely observed from a highly excited Rydberg atom in a superconducting microwave cavity with high quality factor Q. where the cavity losses are negligible [4]. The highly excited Rydberg atom has a very large principal quantum number m [27]. Correspondingly, the electric dipole atom-field coupling constant g between neighbouring atomic levels is very large and scales as m2, whilst the orbital frequency of the Rydberg electron decreases like ms3, so thewavelength of a transition between neighbouring levels is in the microwave domain. The ratio Oc = w,/g can be substantially reduced. In addition, the lifetime for spontaneous transitions is also very large and is proportional to m3 or m5 for low- and high-angularmomentum Rydberg states respectively. Thus the spontaneous decay can be neglected. By these means, it should be possible to observe the effects of the antirotating terms.
Acknowledgements
This work is supported by the United Kingdom EPSRC, by the EC, by a NATO Collaborative Research Award, and by the National Natural Scicence Foundation of China. P.Z. wishes to thank the Queen’s University of Belfast for financial support.
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