A. MESSIKH AND T. EL-SHAHAT

Tw o-le v e l atom in a squ e e z e d v ac u u m w ith ® n ite ban d w id th R. TANAS Â ² , Z. FICEK

Department of Physics and Centre for Laser Science, The University of Queensland, Brisbane, Australia 4072

A. MESSIKH AND T. EL-SHAHAT

Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan  ^{, Poland}

( Received 16 September 1997; revision received 17 November 1997 )

Ab strac t. The resonance ¯ uorescence of a two-level atom driven by a coherent laser ® eld and damped by a ® nite bandwidth squeezed vacuum is analysed. We extend the Yeoman and Barnett technique toanon-zerodetuning of the driving ® eld from the atomic resonance and discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of the spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural line- width in order to satisfy the Marko€ approximation. The narrowing of the spectral lines is interpreted in terms of the quadrature-noise spectrum. We ® nd that, depending on the Rabi frequency, detuning and the squeezing phase, di€ erent factors contribute tothe line narrowing. For a strong resonant driving

® eld there is no squeezing in the emitted ® eld and the ¯ uorescence spectrum exactly reveals the noise spectrum. In this case the narrowing of the spectral lines arises from the noise reduction in the input squeezed vacuum. For a weak or detuned driving ® eld the ¯ uorescence exhibits a large squeezing and, as a consequence, the spectral lines have narrowed linewidths. Moreover, the

¯ uorescence spectrum can be asymmetric about the central frequency despite the symmetrical distribution of the noise. The asymmetry arises from the absorption of photons by the squeezed vacuum which reduces the spontaneous emission. For an appropriate choice of the detuning some of the spectral lines can vanish despite that there is no population trapping. Again this process can be interpreted as arising from the absorption of photons by the squeezed vacuum. When the absorption is large it may compensate the spontaneous emission resulting in the vanishing of the ¯ uorescence lines.

1. In trod u c tion

Since the ® rst paper published by Gardiner on spectroscopy with a broadband squeezed vacuum ® eld [1] much work has been done to ® nd new features in the resonance ¯ uorescence and probe absorption spectraof two- and three-level atoms in a squeezed vacuum [2]. Gardiner [1]has shown that in a squeezed vacuum the atomic dipole moment can decay with twodi€ erent rates, one much longer and the other much shorter than that in normal vacuum. In consequence, a subnatural

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² Permanent address: Nonlinear Optics Division, Institute of Physics, Adam Mick-

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linewidth has been predicted in the spontaneous emission spectrum. The addition of a coherent driving ® eld to the problem introduces a strong dependence of the atom dynamics and the ¯ uorescence spectrum on the relative phase between the coherent ® eld and the squeezed ® eld. Carmichael et al. [3] have shown that, depending on the phase, the central peak of the Mollow triplet [4] can either be much narrower or much broader than the natural linewidth of the atom. The narrowing of the central peak relative to its normal vacuum width is possible for a squeezed vacuum with an arbitrary photon number N . However, the sidebands could be narrowed only for a su ciently low photon number ( N < 0 . 125 ) [5, 6]

and for N > 0 . 125 are always broadened compared totheir normal vacuum width.

Thus, the spectrum can be modi® ed quantitatively from the spectrum associated with the normal vacuum. Apart from the quantitative modi® cations, the qualita- tive changes of the ¯ uorescence spectrum have also been predicted. Courty and Reynaud [7] have found that for a certain detuning of the driving ® eld from the atomic resonance the central peak and one of the sidebands can be suppressed due to a population trapping in the dressed state. Smart and Swain [8± 10] have found unusual features in the resonance ¯ uorescence spectra, such as a hole burning and dispersive pro® les. These features, however, appear for Rabi frequencies compar- able to the atomic linewidth and are very sensitive to the various parameters involved.

Another spectroscopic feature accessible to experimental veri® cation is the probe absorption spectrum. Mollow [11] has predicted that the absorption spectrum of a weak ® eld probing a system of two-level atoms driven by an o€ - resonant laser ® eld consists of one absorption and one emission component at the Rabi sidebands and a small dispersion-like component at the centre of the spectrum. The emission component indicates that, in one sideband, stimulated emission outweighs absorption, so that the probe beamis ampli® ed at the expense of the driving ® eld. The probe ® eld can be ampli® ed due to the population inversion between the dressed states despite the fact that there is no population inversion between the bare atomic states. Apart from the ampli® cation at one of the Rabi sidebands, the absorption spectrum also exhibits ampli® cation on one side of a small dispersion-like structure centred at laser frequency [12], the physical origin of which comes from the interference between absorption and emission processes and is not associated with any population inversion because the transition occurs between equally populated states both in the bare and in the dressed-atom basis [13]. This ampli® cation, however, vanishes when the atom is driven by a resonant laser ® eld. Ampli® cation without population inversion has become a subject of intensive research in recent years [14]. For a resonant driving

® eld, however, the probe absorption spectrum exhibits dispersion-like pro® les at the Rabi sidebands of a relatively small amplitude. The features have been interpreted in terms of the dressed-atom description of the ® eld± atom interaction [15].

The asymmetry of the absorption spectrum is not only crucial in obtaining lasing without inversion but, for example, is also important in laser cooling [16].

Cirac and Zoller [17] have shown that the spectrum of ¯ uctuations of the dipole force, which is proportional to the absorption spectrum, can be asymmetric, even for a resonant cooling laser ® eld, when the two quadratures of the atomic dipole decay at di€ erent rates. This is exactly the situation that occurs when a two-level atom is damped by a squeezed vacuum [18]. Apart from the asymmetry the

s

absorption spectrum can exhibit a strong emission peak (ampli® cation ) at the central frequency which is not attributed to population inversion in either the bare-atom or the dressed-atom picture [19].

Most of the studies dealing with the problem of atwo-level atom in a squeezed vacuum assume that the squeezed vacuum is broadband, i.e. the bandwidth of the squeezed vacuumis much larger than the atomic linewidth and the Rabi frequency of the driving ® eld. Experimental realizations of squeezed states [20± 23], however, indicate that the bandwidth of the squeezed light is typically of the order of the atomic linewidth. The most popular schemes for generating squeezed light are those using a parametric oscillator operating below threshold, the output of which is asqueezed beamwith abandwidth of the order of the cavity bandwidth [24, 25].

There are two types of squeezed ® eld that can be generated by such a parametric oscillator. If the oscillator works in adegenerate regime, the squeezed ® eld has the pro® le with the maximum of squeezing at the central frequency and a small squeezing far from the centre. In the non-degenerate regime, the pro® le has two peaks at frequencies symmetrically displaced from the central frequency. For strong driving ® elds and ® nite bandwidth of squeezing this means that the Rabi sidebands can feel quite di€ erent squeezing than the central line. A realistic description of radiative properties of the two-level atom in such a squeezed ® eld must thus take into account the ® nite bandwidth of the squeezed ® eld.

First studies of the ® nite-bandwidth e€ ects have been performed by Gardiner

et al. [24], Parkins and Gardiner [26]and Ritsch and Zoller [27]. The approaches were based on stochastic methods and numerical calculations, and were applied to analyse the narrowing of the spontaneous emission and absorption lines. The fundamental e€ ect of narrowing has been con® rmed, but the e€ ect of ® nite bandwidth was to degrade the narrowing of the spectral lines rather than enhance it. Later, however, numerical simulations done by Parkins [28, 29] demonstrated that for strong driving ® elds a ® nite bandwidth of squeezing can have a positive e€ ect on the narrowing of the Rabi sidebands. He has found that there is a di€ erence between the two types of squeezed light generated in either the degenerate or non-degenerate regime of the parametric oscillator. In the former case it is possible tonarrow either both of the Rabi sidebands or the central peakof the ¯ uorescent spectrum, while in the latter case simultaneous narrowing of all three spectral peaks is possible.

Recently, Yeoman and Barnett [30] have proposed an analytical technique for

investigating the behaviour of a coherently driven atom damped by a squeezed

vacuum with ® nite bandwidth. In their approach, they have derived a master

equation and analytic expressions for the ¯ uorescent spectrum for the simple case

of atwo-level atomexactly resonant with the frequencies of boththe squeezed® eld

andthe driving ® eld. Their analytical results agree with that of Parkins [28, 29]and

show explicitly that the width of the central peak of the ¯ uorescent spectrum

depends solely on the squeezing present at the Rabi sideband frequencies. They

have assumed that the atom is classically driven by aresonant laser ® eld for which

the Rabi frequency is much larger than the bandwidth of the squeezed vacuum

though this is still large compared to the natural linewidth. Unlike the conven-

tional theory, based on uncoupled states, it is possible to obtain a master equation

consistent with the Born± Marko€ approximation by ® rst including the interaction

of the atomwith the driving ® eld exactly, and then considering the coupling of this

combined dressed-atom system with the ® nite-bandwidth squeezed vacuum. The

advantage of this dressed-atom method over the more complex treatments based on adjoint equation or stochastic methods [28, 29, 31] is that simple analytical expressions for the spectra can be obtained, thus displaying explicitly the factors that determine the intensities of the spectral features and their widths. Recently, the ideaof Yeoman andBarnett has been extendedby Ficek et al. [32]tothe case of afully quantized dressed-atom model coupled toa® nite bandwidth squeezed ® eld inside an optical cavity. They have studied the ¯ uorescence spectrum under the secular approximation [15] and have found that in the presence of a single-mode cavity the e€ ect of squeezing on the ¯ uorescence spectrum is more evident in the linewidths of the Rabi sidebands rather than in the linewidth of the central component. In the presence of a two-mode cavity and a two-mode squeezed vacuum the signature of squeezing is evident in the linewidths of all spectral lines.

They have alsoestablished that the narrowing of the spectral lines is very sensitive to the detuning of the driving ® eld from the atomic resonance. The dressed-atom method including a detuning of the driving ® eld from the atomic resonance has also been applied to calculate the probe absorption spectra of a driven two-level atom in a narrow bandwidth squeezed vacuum [33]. This method could also be applied to calculate the ¯ uorescence spectrum.

In this paper we extend the Yeoman and Barnett [30] technique to include a non-zero detuning of the driving ® eld from the atomic resonance and derive the master equation for a two-level atom driven by a classical laser ® eld and damped by a ® nite-bandwidth squeezed vacuum. Despite the complexity of the problem, we obtain a quite simple master equation that is valid for arbitrary values of the Rabi frequency and the detuning but for the squeezing bandwidths much greater than the natural linewidth. The corresponding optical Bloch equations for the atomic operators are derived in a standard way from the master equation. We apply the Bloch equations to calculate the ¯ uorescence spectrum and the quad- rature-noise (squeezing ) spectrumof the scattered ® eld. We ® nd that the detuning changes considerably the shape of the resonance ¯ uorescence spectrum and leads to novel spectral features. The squeezing spectrum allows us to interpret in a simple way the mechanism of the linewidth narrowing, hole burning and disap- pearance of the spectral lines. We ® nd that for a strong resonant driving ® eld the

¯ uorescence ® eld does not exhibit any squeezing but the spectral lines can be signi® cantly narrowed. When the atom is driven by a weak laser ® eld the

¯ uorescence ® eld exhibits a large squeezing which leads to further narrowing of the spectral lines and hole burning. Moreover, we ® nd that for some detunings the number of lines in the ¯ uorescence spectrumdoes not correspond tothe number of lines in the noise spectrum in contrast to what one could expect that the

¯ uorescence spectrum should reveal the noise spectrum.

2. Maste r e qu ation for atom ic syste m

We consider a two-level atom driven by a detuned monochromatic laser ® eld and damped by a squeezed vacuum with ® nite bandwidth. Applying the approach of Yeoman and Barnett [30], which is an extension of the model proposed by Carmichael and Walls [34] and Cresser [35], we derive a master equation of the system which includes squeezing bandwidth e€ ects. In this approach, we ® rst performthe dressing transformation toinclude the interaction of the atomwith the driving ® eld and next we couple the resulting dressed-atom to the narrow

s

bandwidth squeezed vacuum ® eld. We derive the master equation under the Marko€ approximation which requires the squeezing bandwidth to be much greater than the atomic linewidth, but not necessarily greater than the Rabi frequency of the driving ® eld and the detuning. For simplicity, we assume that the squeezing properties are symmetric about the central frequency of the squeezed ® eld which, in turn, is exactly equal to the laser frequency. Our model di€ ers from that of Yeoman and Barnett in adding a non-zero detuning which, as we shall see later, leads to interesting e€ ects not yet discussed in the literature.

We start from the Hamiltonian of the system which in the rotating-wave and electric-dipole approximations is given by

H = H A + H R + H L + H I , ^{( 1 )}

where

H _A = ¹ ₂ hx _A s z = - ¹ 2 h ¢ s z + ¹ ₂ hx _L s z ( 2 ) is the Hamiltonian of the atom,

H R = ò ¥

0 x b ⁺ ( x ) b ( x ) d x ( 3 ⁾

is the Hamiltonian of the vacuum ® eld,

H L = ¹ ₂ h X [ s + exp ( - ^{i x L t ) + s} - exp ( ix L t ) ] ^{( 4 )}

is the interaction between the atom and the classical laser ® eld, and

H I = i h ò ¥

0 K ( x ) [ s + b ( x ) - ^{b + ( x ) s} - ] ^{d x ( 5 )}

is the interaction of the atom with the vacuum ® eld. In (2 ) ± (5 ) , K ( x ) is the coupling of the atom to the vacuum modes, ^¢ = ^x L - ^{x A} is the detuning of the driving laser ® eld frequency ^x L from the atomic resonance ^x A , and ^s + , ^s - , and ^{s z} are the Pauli pseudo-spin operators describing the two-level atom. The laser driving ® eld strength is given by the Rabi frequency ^X , while the operators b ( x ) and b ⁺ ( x ) are the annihilation and creation operators for the vacuum modes satisfying the commutation relation

[ ^{b ( x ) , b + ( x} _Â ⁾ ] ^{= d ( x - x} _Â ^{) . ( 6 )}

For simplicity, we assume that the laser ® eld phase is equal to zero ( u L = 0 ) . In order to derive the master equation we perform a two-step unitary transformation. In the ® rst step we use the second part of the atomic Hamiltonian (2 ) and the free ® eld Hamiltonian (3 ) to transform to the frame rotating with the laser frequency x L and to the interaction picture with respect to the vacuum modes. After this transformation our system is described by the Hamiltonian

H 0 + H ^r _I ( t ) , ^{( 7 )}

where

H 0 = - ¹ 2 h ¢ s z + ¹ ₂ h ^X ( s + + s - ) , ^{( 8 )}

and

H _I ^r ( t ) = i h ò ¥

0 K ( x ) [ s ₊ b ( x ) exp [ ^{i ( x L - x ) t} ] ^{- b + ( x ) s - exp} [ ^{- i ( x L - x ) t} ] ] ^{d x . ( 9 )}

The second step is the unitary dressing transformation performed with the

Hamiltonian H ₀ , given by (8 ) . The transformation

s 6 ( t ) = exp - ⁱ _h ^{H 0 t} [ ] ^{s 6 exp i h H 0 t} [ ] ^{( 10 )}

leads to the following time-dependent atomic raising and lowering operators

s 6 ( t ) = ¹ ₂ [ ^{s a 6 ( 1 7 ~ ¢ ) s b exp ( i V} _Â ^{t ) 6 ( 1 6 ~ ¢ ) s c exp ( - i X} _Â ^{t )} ] ^{, ( 11 )}

where

s a = ^{~ X ~ X} ( s + + s - ) - ^{~ ¢ s z} [ ] ,

s b = ¹ ₂ ( 1 - ^{~ ¢ ) s +} - ^{( 1 + ~ ¢ ) s} - - ^{~ X s z} [ ] , ^{( 12 )}

s c = ¹ ₂ ( 1 + ~ ¢ ) s + - ^{( 1} - ^{~ ¢ ) s} - + ~ _{X s}

[ z ] ,

are the `dressed’ operators oscillating at frequencies 0, ^X Â ^{and -} X Â , respectively,

and ~ _X = ^X

X Â ^,

~ ¢ = ^¢

X Â ^, X Â ^{= ( X 2 + ¢2 ) 1 / 2 . ( 13 )}

For ^¢ = 0, the transformation (11 ) reduces to that of Yeoman and Barnett [30].

Under the transformation (11 ) the interaction Hamiltonian takes the form

H I ( t ) = i h ò ¥

0 K ( x ) [ s + ( t ) b ( x ) exp [ ^{i ( x L - x ) t} ] ^{- b + ( x ) s - ( t ) exp} [ ^{- i ( x L - x ) t} ] ] ^{d x .}

( 14 ⁾ The master equation for the reduced density operator q of the system can be derived using standard methods [36]. In the Born approximation the equation of motion for the reduced density operator is given by [36]

¶ q ^D ¶ t = - ¹ _h 2 ò ^{t 0 Tr R [ H I ( t ) , [ H I ( t - ¿ ) ,} q _R ( 0 ^{) q D (} t - ^{¿ )} ]] ^{{ } d ¿ , ( 15 )}

where the superscript D stands for the dressed picture, q R ( 0 ⁾ is the density operator for the ® eld reservoir, Tr R is the trace over the reservoir states and the Hamiltonian H I ( t ) is given by (14 ) . We next make the Marko€ approximation [36]

by replacing q ^D ( t - ^{¿ ) in (15} ) by q ^D ( t ) , substitute the Hamiltonian (14 ) and take the trace over the reservoir variables. We assume that the reservoir is in asqueezed vacuum state in which the operators b ( x ) and b ⁺ ( x ) satisfy the relations [1]

Tr R [ ^{b ( x ) b + ( x} _Â ⁾ ] ⁼ [ ^{N ( x ) + 1} ] ^{d ( x - x} _Â ^{) ,}

Tr R [ ^{b + ( x ) b ( x} _Â ⁾ ] ^{= N ( x ) d ( x - x} _Â ^{) ,}

Tr R [ ^{b ( x ) b ( x} _Â ⁾ ] ^{= M ( x ) d ( 2 x L - x - x} _Â ^{) , ( 16 )}

where N ( x ) and M ( x ) are the parameters describing the squeezing and that the carrier frequency of the squeezed ® eld is equal to the laser frequency x L . In the Marko€ approximation we can extend the upper limit of the integration over ^¿ to in® nity and next perform necessary integrations using the formula

ò ¥

0 exp ⁽ 6 i ^{² ¿ )} d ^¿ = p ^{d ( ² )} 6 i P 1

² , ^{( 17 )}

s

where _P means the Cauchy principal value. After lengthy calculations we obtain the master equation which in the frame rotating with the laser frequency x _L can be written as

Ç q = ¹ ₂ ig ^d [ ^{s z , q} ]

+ ¹ ₂ g ^{N ~} ( 2 s + q s - - ^s - ^{s + q} - ^{q s} - ^{s +} ) + ¹ ₂ g ^{( N ~ +} 1 ⁾ ( 2 ^s _- ^{q s} + - ^{s + s} - ^q - ^{q s + s} - )

- g ^{M ~} s + q s + - g ^{M ~ *} s - ^{q s} -

- ¹ 2 i ^X [ ^{s + + s - , q} ] ^{+ 1} 4 i ^{( b} [ ^{s + ,} [ ^{s z , q} ]] ^{- b *} [ ^{s - ,} [ ^{s z , q} ]] ^{) , ( 18 )}

where g is the natural atomic linewidth,

N ~ = N ( x _L + ^X Â ^{) + 1 2 ( 1 - ~ ¢2 )} g ⁿ , ^{( 19 )}

M ~ = ( | ^{M ( x} L + ^X Â ^{) | + i ~ ¢} d M ) ^{exp ( i u )} - ¹ 2 ( 1 - ^{~ ¢2 ) ( g n} - ^{i d n )} , ^{( 20 )}

d = ^¢ g ^{+ ~ ¢} d N + ¹ ₂ ( 1 - ^{~ ¢2 )d n} , ^{( 21 )}

b = g ^~ X d N + d M exp ⁽ i ^{u )} - ^{i ~ ¢} g ⁿ - ^{i d n ( )} [ ] , ^{( 22 )}

g ⁿ = N ( x _L ) - ^{N ( x} L + ^X Â ^{) - | M (} x _L ) | - | ^{M ( x} L + ^X Â ^{) | ( ) cos} u , ^{( 23 )}

d n = | ^{M ( x} L ) | - | ^{M ( x} L + X Â ^{) | ( ) sin} u , ^{( 24 )}

d N = 1

p ^P ò ¥ - ¥

N ( x )

x + ^X Â ^{d x , ( 25 )}

d M = 1

p ^P ò ¥ - ¥

| ^{M ( x )} |

x + ^X Â ^{d x , ( 26 )}

and ^u is the phase of squeezing ( M ( x ) = | ^{M ( x )} | exp ⁽ i ^{u )} ) . In the derivation of equation (18 ) we have assumed that the phase ^u does not depend on frequency [37], and we have included the divergent frequency shifts (the Lamb shift ) to the rede® nition of the atomic transition frequency [36]. Moreover, we have assumed that the squeezed vacuum is symmetric about the central frequency x L , so that

N ( x _L - ^X Â ^{) = N (} x _L + ^X Â ⁾ , and a similar relation holds for M ( x ) .

The master equation (18 ) has the standard form known from the broadband squeezing approaches with the new e€ ective squeezing parameters N ^~ and M ^~ given by (19 ) and (20 ) . There are alsonew terms, proportional to b which are essentially narrow bandwidth modi® cations tothe master equation. All the narrowbandwidth modi® cations are determined by the parameters g ⁿ , d n and the shifts d N and d M

de® ned in (23 ) ± (26 ) . These parameters become zero when the squeezing band- width goes to in® nity.

The squeezing induced shifts d N and d M depend on the explicit form of N ( x )

and | ^{M ( x )} | . There are two types of squeezed ® eld that can be generated by the

parametric oscillator. For a degenerate parametric oscillator (DPO ) the squeezing

properties are described by [24]

N ( x ) = ^{¸ 2} - ^¹2

4 1

x ² + ^¹2 - _{x 2 +} ¹ _{¸ 2}

[ ] ^{, ( 27 )}

| ^{M ( x )} | = ^{¸ 2} - ^¹2

4 1

x ² + ^¹2 + 1

x ² + ^{¸ 2}

[ ] ^{, ( 28 )}

while for a non-degenerate parametric oscillator (NDPO ) the frequency depen- dence is given by [25]

N ( x ) = ^{¸ 2} - ^¹2

8 1

( x - ^{a ) 2 + ¹2 + ( x + a 1 ) 2 + ¹2}

[

- ₍ ¹

x - ^{a ) 2 + ¸ 2} - ₍ ¹

x + a ) ² + ¸ ² ] ^{, ( 29 )}

| ^{M ( x )} | = ^{¸ 2} - ^¹2

8 1

( x - ^{a ) 2 + ¹2 + ( x + a 1 ) 2 + ¹2}

[

+ 1

( x - ^{a ) 2 + ¸ 2 + ( x + a 1 ) 2 + ¸ 2} ] ^{, ( 30 )}

where x = x - ^{x L , and ¸ and ¹} are related to the cavity damping rate, g c , and the real ampli® cation constant, ^² , of the parametric oscillator according to

¸ = g c + ^² ,

¹ = g c - ^{² . ( 31 )}

The parameter a is characteristic of a two-mode squeezed ® eld generated by the non-degenerate parametric oscillator and represents the displacement from the central frequency of the squeezing at which the two-mode squeezed vacuum is maximally squeezed.

The Cauchy principal values of the integrals (25 ) and (26 ) can be evaluated using the contour integration which gives

d N = d ^¹ - ^{d ¸} ,

d M = d ^¹ + d ¸ , ^{( 32 )}

where the formof d ^¹ and d ¸ depends on the type of squeezing being considered and is explicitly given by:

(i ) for the degenerate case

d ^¹ = g X Â ^¸

2 - ^¹2

4 1

¹ ( X Â ^{2 + ¹2 ) , ( 33 )}

d ¸ = g X Â ^¸

2 - ^¹2

4 1

¸ ( X Â ^{2 + ¸ 2 ) , ( 34 )}

and (ii ) for the non-degenerate case with a = ^X Â

s

d ^¹ = g X Â ^¸

2 - ^¹2

4 1

¹ ( 4 ^X Â ^{2 + ¹2 ) , ( 35 )}

d ¸ = g X Â ^¸

2 - ^¹2

4 1

¸ ( 4 ^X Â ^{2 + ¸ 2 ) . ( 36 )}

From the master equation (18 ) we easily derive the optical Bloch equations for the mean values of the atomic operators

k Ç s - l ⁼ - g ^{( 1} ₂ + ~ N - ^{i d )} _k ^s - l - g ^{~ M} k s + l ^{+ i 2 X} k s z l ,

k Ç s z l ^{= i (} X + b ^* ) k s - l - ^{i ( X + b )} _k ^{s +} _l - ^{g ( 1 + 2 ~ N )} _k ^{s z} _l - ^{g . ( 37 )}

The equation for k s ₊ l is obtained as Hermitian conjugate of equation for k s - l ^.

De® ning the Hermitian operators ^s x and ^s y as

s x = ¹ ₂ ( s - + ^{s +} ) ,

s y = 1

2i ^{( s -} - ^{s + )} , ^{( 38 )}

we get from (37 ) the following equations of motion for the atomic polarization quadratures

k Ç s x l ⁼ - ^{g ( 1} 2 + ~ N + Re ^~ M ) k s x l - ^{g Im ~ M + d} ( ) k s y l ,

k Ç s y l ⁼ - g Im ^{~ M} - ^d ( ) k s x l - g ^{( 1} 2 + ~ N - ^{Re ~ M )} _k ^{s y} _l ^{+ 1} 2 X k s z l ,

k Ç s z l ^{= 2Im} b k s x l - ^{2 ( X + Re b )} _k ^{s y} _l - g ⁽ 1 + 2 ^~ N ) k s z l - g ^.

( 39 ⁾

3. Station ary lin e s h ape

The Bloch equations (39 ) can be easily solved for the steady-state values of the atomic variables, and the result is given by

k s x l ^{ss = 1 2} g X g ⁽ Im ^~ M+ d )

d ,

k s y l ^{ss =} - ¹ 2 g X g ¹ ₂ + ~ N+ Re ^{~ M} ( )

d ,

k s z l ^{ss =} - g g ^{2 ( 1 4 + ~ N ( ~ N +} 1 ⁾ - | ^{~ M} | ² + d ² )

d ,

( 40 )

where

d = g ^{3 (} 1 + 2 N ^~ ) ( ¹ _{4 + ~} N ( N ^~ + 1 ⁾ - | ^{M ~} | ² + d ² )

+ g X [ ^{( 1 2 + ~ N + Re M ~ ) ( X + Re b ) + Im b ( Im M ~ + d )} ] ^{. ( 41 )}

In ® gure 1, we plot I ( ¢ ) = k s z l ^{ss +} 1 which expresses the steady-state ¯ uorescence light intensity in terms of the squeezing parameters, the Rabi frequency ^X , and the detuning ¢ . The expression is also known as the absorption spectrum of the driving ® eld or stationary lineshape [38]. In both cases of the broadband and narrow-band squeezed vacuum, the absorption spectrum is a Lorentzian whose bandwidth depends on the bandwidth of the squeezed vacuum. Moreover, for

u = p / 2 and a narrow-bandwidth squeezed vacuum the maximum of the Lor-

entzian is shifted towards negative detunings. The shift is proportional to Im M ^~

and is de® nitely a narrow-bandwidth feature in the absorption spectrum.

The steady-state solutions (40 ) exhibit more interesting features. For a resonant driving ® eld ( ^¢ = 0 ) , we ® nd from (20 ) that

Im M ^~ + d = | ^{M ( x} A ) | sin ^u , ^{( 42 )}

indicating that even for ^¢ = 0 the k s x l ^ss component of the Bloch vector can have a non-zero steady-state solution provided the phase ^u is di€ erent from 0 or p and there is a non-zero squeezing at the atomic resonance. This e€ ect can lead to unequal populations of the dressed states of the system[39]. In order toshow this, we diagonalize the Hamiltonian (8 ) , and ® nd the following dressed states of the system

| 1 l ⁼ ( ) ^{1 + ~} ₂ ^{¢ 1 / 2 | g l +} ( ) ^{1 - 2 ~ ¢ 1 / 2 | e l ,}

| 2 l ⁼ - ¹ - ^{~ ¢}

( ) 2 ^{1 / 2 | g l +} ( ) ^{1 + ~ 2 ¢ 1 / 2 | e l ( 43 )}

with the energies E 1 = h X Â ^{/ 2 and E 2 = - h} X Â ^{/ 2, and | g} l ^and | ^e _l are the ground and the excited state of the atom, respectively. The populations of the dressed states can be expressed in terms of the expectation values k s x l ^{ss and} k s z l ^{ss as}

q 11 = ¹ ₂ 1 - ^{~ ¢} _k ^{s z} _l ss ( ) ^{+ ( 1} - ^{~ ¢2 ) 1 / 2} _k ^{s x} _l ss ,

q ₂₂ = ¹ ₂ 1 + ~ ^¢ k s z l ^ss ( ) - ^{( 1} - ^{~ ¢2 ) 1 / 2} _k ^{s x} _l ss . ( 44 )

For a resonant driving ® eld, ¢ ^~ = 0, and the stationary populations of the dressed states depend solely on k s x l ^ss , which, on the other hand, can be non-zero only when the phase u is di€ erent from0 and p and, simultaneously, there is anon-zero

s

10 8 6 4 2 0 2 4 6 8 10

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

D /g

I( D )

Figure 1. Steady-state ¯ uorescence intensity I( ^¢ ) = k s _z l ^{ss +} 1 as a function of ^¢ for

X / g = 5, u = p / 2, ^² / g c = 0 . 5 ( N( x A ) = 1 . 78, | ^{M( x A )} | ^{= 2 . 22} ) , g c / g = 10 (solid line ) ,

and g c / g = 100000 (dashed line ) .

squeezing at the atomic resonance. This suggests that the best candidate toobserve the unequal populations of the dressed states would be a squeezed vacuum produced by a degenerate rather than non-degenerate parametric ampli® er. In

® gure 2, we show the population q 11 as a function of g c / g and ^X / g for ^¢ = 0,

u = p / 2 and a ® xed ^² / g c = 0 . 25, such that N ( x _A ) = 0 . 28 and M ( x _A ) = 0 . 60. The population reaches its maximumvalue q ₁₁ . 0 . 8 for small Rabi frequencies and for g c > 10 g is almost independent of the squeezing bandwidth. The population inversion ⁽ q 11 > 0 . 5 ) seen in ® gure 2 is an example of a nonsecular e€ ect, which appears only for small Rabi frequencies and/or detunings. This is shown in

® gure 3, where we plot the population q ₁₁ as a function of ^X _/ g and ^¢ / g for

u = p / 2 and a ® xed g c / g = 10 and ^² / g c = 0 . 5 ( N ( x A ) = 1 . 78, M ( x A ) = 2 . 22 ) . It is seen that for small Rabi frequencies and ^¢ _. 0 there is a peak in the population which is due to the squeezing. This peak vanishes for large Rabi frequencies and/

or detunings. We note from ® gure 2, that for g c < 10g the population ^q 11 strongly depends on the squeezing bandwidth. However, the study of this behaviour in the regime g c < 10g is forbiddenby the Marko€ approximationmade inthe derivation of master equation (18 ) , which requires the squeezing bandwidth to be much greater than the atomic linewidth. The parameter’s values g c = 10g and ^² / g c = 0 . 5, used in the ® gures, are consistent with the actual experiments on spectroscopy with squeezed light [23] and the Marko€ approximation.

Having available the steady-state populations of the dressed states, we can check whether the population trapping e€ ect, predicted by Courty and Reynaud [7](see also[40] ) for abroadband squeezed vacuum, can be observed for anarrow- bandwidth squeezed vacuum. In ® gure 4, we plot the population q ₁₁ as a function of ^X / g and ^¢ / g for ^u = p , and the DPO output with g c / g = 100000 and

² / g c = 0 . 5. In this case the squeezed vacuum is very broad and, as is seen from

® gure 4, there is a population trapping ( q 11 = 1 ) . Interestingly, the population trapping appears for very large detunings and Rabi frequencies indicating that this

0 20 40 60 80 100

2 0 6 4

10 8 0.5

0.55 0.6 0.65 0.7 0.75 0.8 0.85

g

/g W /g r

Figure 2. Population of the dressed state | ¹ _l as a function of g c / g and ^X / g for ^¢ = 0,

u = p / 2, and ^² / g c = 0 . 25 ( N( x A ) = 0 . 28, | ^{M( x A )} | ⁼ 0 . 60 ) .

is essentially a secular e€ ect. Figure 5 shows the population q 11 for the same parameters as in ® gure 4 but for a narrow-bandwidth squeezed vacuum with g c / g = 10. There is no population trapping for the narrow-bandwidth squeezed vacuum.

4. Flu ore sc e n c e spe c tru m

The stationary spectrumof the resonance ¯ uorescence fromatwo-level atomis given by the Fourier transform of the two-time atomic correlation function as [4, 36]

F ( x ) = 1 p Re ò ¥

0 k s + ( 0 ) s - ( ¿ ) l ^{ss exp} [ ^{i ( x - x L ) ¿} ] ^{d ¿}

{ } ^{, ( 45 )}

where Re denotes the real part of the integral. The two-time correlation function appearing in (45 ) can be found from the Bloch equations (37 ) by applying the quantum regression theorem [41]. The equations of motion for the two-time correlation functions can be written as

¶ ¶ ^¿ k s + ( 0 ^{) s} _- ^{( ¿ )} l ^ss

k s + ( 0 ) s + ( ¿ ) l ^ss

k s + ( 0 ) s z ( ¿ ) l ^ss

æ çç è

ö ÷÷

ø ^{= A} k s + ( 0 ^{) s} _- ^{( ¿ )} l ^ss

k s + ( 0 ) s + ( ¿ ) l ^ss

k s + ( 0 ) s z ( ¿ ) l ^ss

æ çç è

ö ÷÷

ø ⁺ k s + l ^ss

0 0

æ çç è - g

ö ÷÷

ø ^{, ( 46 )}

where A is the 3 ´ 3 matrix

A =

- g ^{( 1} ₂ + ~ N - ^{i d )} - g ^{M ~} _{2 X} ⁱ

- g ^{M ~ *} - g ^{( 1} ₂ ^{+ ~ N +} i d ) - 2 X ⁱ

i ( X + b ^* ) - ^{i ( X + b )} - g ⁽ 1 + 2 N ^~ )

æ ç è ö _÷

ø ^{, ( 47 )}

and the initial values for the correlation functions are s

0

0.5 1

1.5 2

0 2 4 6 8 10 0.6 0.65 0.7 0.75 0.8 0.85

W /g D /g r

Figure 3. Population of the dressed state | 1 l as a function of ^X / g and ^¢ / g for a narrow-

bandwidth squeezed vacuum with g c / g = 10, ^² / g c = 0 . 5 and ^u = p / 2.

k s + s - l ^{ss = 1 2 ( 1 +} k s z l ^{ss )} ,

k s + s + l ^{ss = 0} ,

k s + s z l ^{ss =} - _k ^{s +} _l ss .

( 48 ⁾

Taking the Laplace transform of (46 ) we obtain the system of algebraic equations for the transformed variables which can be easily solved. The solution gives us the following formula for the Laplace transform of the correlation function

k s + ( 0 ^{) s} _- ^{( ¿ )} l ^ss

0 2

4 6

8 10

0 10 20 30 40 50 0.5 0.6 0.7 0.8 0.9 1

W /g D /g r

Figure 4. Dressed state population q 11 for a broadband squeezed vacuum with g c / g = 100000, ^² / g c = 0 . 5 ( N( x A ) = 1 . 78, | M( x A ) | ⁼ 2 . 22 ) , and ^u = p .

0 2

4 6

8 10

0 10 20 30 40 50 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

W /g D /g r

Figure 5. The same as in ® gure 4, but for a narrow-bandwidth squeezed vacuum with

g c / g = 10.

F ( z ) = 1

z d ( z ) - ⁱ _k ^{s +} _l ss X

2 [ ^{g 2 ( 1} 2 + ~ N + ~ M + i d ) + g ^{( 3} ₂ ^{+ ~ N + ~ M +} i d ) z + z ² ]

{

+ ¹ ₂ ( 1 + k s z l ^{ss )} [[ ^{g 2 ( 1 + 2 N ~ )} ( ¹ 2 + ~ N + i d ) ^{+ 1} 2 X ( X + b ) ] ^z

+ g ^{( 3} ₂ + 3 N ^~ + i d ) z ² + z ³ ] } ^{, ( 49 )}

where

d ( z ) = d + [ ⁵ 4 + 5 N ^~ ( N ^~ + 1 ⁾ - | ^{M ~} | ^{2 + d 2 + X ( X +} Re ^{b )} ] ^{z + 2 ( 1 + 2 N ~ ) z 2 + z 3}

( 50 ⁾ with d given by (41 ) , and

k s + l ^{ss =} k s x l ^ss - ⁱ _k ^{s y} _l ss = i ^X 2 d g ^{2 ( 1} ₂ ^{+ ~ N + ~ M *} - ^{i d ) . ( 51 )}

The Laplace transform (49 ) contains both the coherent and incoherent contribu- tions to the spectrum [4]. The coherent part of the spectrum is the delta function d ( x - ^{x L )} centred on the laser frequency, the amplitude of which is de® ned by the residuum for z = 0

F coh = lim _{z ® 0} zF ( z ) = 4 ^{X 2} d ² | g ^{2 ( 1} ₂ ^{+ ~ N + ~ M +} i d ) | ^{2 . (} 52 ⁾ The incoherent part of the spectrum is then given by

F inc ( x ) = 1

p Re { ^{F inc ( z )} | z = - i ⁽ x - ^x L ) }, ^{( 53 )}

where

F inc ( z ) = F ( z ) - ¹ _z ^{F coh =} _d ¹ _{( z )} - ⁱ _k ^{s +} _l ss X

2 g [ ^{( 3 2 + ~ N + ~ M +} i d ) ^{+ z} ]

{

+ ¹ ₂ ( 1 + k s z l ^{ss )} [ ^{g 2 ( 1 + 2 N ~ ) ( 1} 2 + ~ N + i d ) + ¹ _{2 X} ( X + b )

+ g ^{( 3} ₂ ⁺ 3 N ^~ + i d ) z + z ² ] ^{- F coh d ( z )} _z ^{- d} } ^{. ( 54 )}

We can relate the incoherent part of the resonance ¯ uorescence spectrumtothe quadrature-noise spectrum (squeezing spectrum ) as [42]

F inc ( x + x L ) = S X ( x ) _{+ S} Y ( x ) _{+ S A} ( x ) , ^{( 55 )}

where

S X ( x ) = 1 2 p Re ò ¥

0 cos ⁽ x ^¿ ) [ k s + ( 0 ⁾ , ^s - ( ¿ ) l ^{ss +} k s + ( 0 ⁾ , ^{s + ( ¿ )} _l ss ] ^{d ¿ , ( 56 )}

S Y ( x ) = 1 2 p Re ò ¥

0 cos ⁽ x ^¿ ) [ k s + ( 0 ⁾ , ^s - ( ¿ ) l ^ss - _k ^{s + ( 0 )} , ^{s + ( ¿ )} _l ss ] ^{d ¿ , ( 57 )}

are, respectively, in-phase and out-of-phase quadrature components of the noise spectrum, and

S A ( x ) = - _p ¹ ò ¥

0 sin ⁽ x ^¿ ) Im k s + ( 0 ⁾ , ^s - ( ¿ ) l ^{ss d ¿ ( 58 )}

s

is the asymmetric contribution tothe spectrum. In (56 ) ± (58 ) , k ^a , ^b _l ^º _k ^ab _l - _k ^a _{l k} ^b _l

denotes the covariance.

In order to calculate the spectra of the normally ordered quadrature com- ponents of the ¯ uorescent light [43] we need to evaluate the correlation function

k s ₊ ( 0 ⁾ , ^{s + ( ¿ )} _l ss which, on the other hand, can be found from equations (46 ) and (40 ) . The Laplace transform for the function k s + ( 0 ^{) s} + ( ¿ ) l ^ss has the form

S ( z ) = 1

z d ( z ) i k s + l ^ss X

2 g [ ^{2 ( 1 2 + ~ N + ~ M *} - ^{i d} ) ⁺ g ^{( 3} ₂ + ~ N + ~ M ^* - ^{i d ) z + z 2} ]

{

- ¹ 2 ( 1 + k s z l ^{ss )} [[ ^{g 2 ( 1 + 2 N ~ ) M ~ * - 1 2 X ( X + b * )} ] ^{z + g M ~ * z 2} ] } ^{. ( 59 )}

The components of the squeezing spectra (56 ) ± (58 ) are related to the functions

F ( z ) and S ( z ) , given by (49 ) and (59 ) , in the following way

S X ( x ) = ¹ ₂ Re { F ( - ^{i x ) + F ( i x ) + S (} - ^{i x ) + S ( i x )} },

S Y ( x ) = ¹ ₂ Re { F ( - ^{i x ) + F ( i x )} - ^{S (} - ^{i x )} - ^{S ( i x )} }, ^{( 60 )}

S _A ( x ) = Re { F ( - ^{i x )} - ^{F ( i x )} }

In ® gure 6( a ) , we plot the ¯ uorescence spectrum for ^¢ = 0, ^X = 10g , and the squeezed vacuum produced by a DPO with g c / g = 10 and ^² / g = 0 . 5 ( N ( x A ) = 1 . 78, | ^{M ( x} A ) | = 2 . 22 ) . The spectrum in the squeezed vacuum is sym- metric and contains three peaks, just as does the Mollow spectrum in the normal vacuum[4]. The linewidths of the spectral features, however, are di€ erent and can be narrower than those in the normal vacuum. Moreover, the intensities of the spectral lines are di€ erent and we observe a shift of the Rabi sidebands from their resonant positions. The shift is due to the presence of the parameters d N and d M

which are di€ erent from zero only in a narrow-bandwidth squeezed vacuum. The shift vanishes when the bandwidth goes toin® nity. Therefore, the shift of the Rabi sidebands, seen in ® gure 6( a ) , is a signature of a narrow-bandwidth squeezed vacuum. In order todiscuss linewidths and frequencies of the spectral components it is enough to® nd eigenvalues of the matrix A given by (47 ) . It is easy toshowthat for ^¢ = 0 and u = 0 the eigenvalues are

z 1 = - g ^{( 1} ₂ + ~ N + | ^{M ~} | ⁾

z 2 = - ¹ 2 g ₍ ³ ₂ + 3 N ^~ - | ^{M ~} | ₎ + i [ ^{( X + 1 2 b ) 2 - 1 4 g 2 ( 1 2 + ~ N + | M ~ | ) 2 - 1 4 b 2} ] ^{1 / 2 ,}

z 3 = - ¹ 2 g ₍ ³ ₂ + 3 N ^~ - | ^{M ~} | _{) -} i [ ^{( X + 1 2 b ) 2 - 1 4 g 2 ( 1 2 + ~ N + | M ~ | ) 2 - 1 4 b 2} ] ^{1 / 2 .}

( 61 ⁾

The eigenvalues for ^u = p are obtained by changing | ^{M ~} | ^® - | ^{M ~} | . It is seen from (61 ) that the spectral linewidths can be narrower than those in the normal vacuum and for a strong driving ® eld the Rabi sidebands are shifted from 6 X positions to

6 ( X + ¹ ₂ b ) . According to(22 ) the shift ^b depends on the parameters d N and d M . It

is interesting to explain the mechanism which leads to these narrow spectral lines

and the role of the squeezed vacuum in this process. In ® gure 6( b ) , we plot the

components S X ( x ) and S Y ( x ) of the noise spectrum and the asymmetric con-

tribution S A ( x ) .The latter is zero in this case. Both components of the noise

spectrum are positive indicating that there is no squeezing in the emitted

¯ uorescence ® eld. Comparing the noise and ¯ uorescence spectra we see that the Rabi sidebands correspond to S Y ( x ) , whereas the central component of the

¯ uorescence spectrum corresponds to S X ( x ) . It follows that the ¯ uorescence spectrum exactly reveals the noise spectrum. This result can be understood in terms of the two noise quadratures S X ( x ) and S Y ( x ) of the atom acting as

s

20 0 15 10 5 0 5 10 15 20

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

(w - w

)/g F

(w )

( a )

20 15 10 5 0 5 10 15 20

0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

S

,S

,S

(w - w

)/g

( b )

Figure 6. ( a ) The ¯ uorescence spectrum for a narrow-bandwidth squeezed vacuum (solid line ) for g c / g = 10, ^² / g c = 0 . 5 ( N = 1 . 78, | M | ⁼ 2 . 22 ) , ^X / g = 10, ^¢ / g = 0 and

u = 0. The dotted line is the Mollow triplet in the absence of the squeezed vacuum.

( b ) The quadrature-noise spectrum for the same parameters as in ® gure 6( a ) ; S X ( x )

(dashed line ) , S Y ( x ) (dashed-dotted line ) . The asymmetric part S A ( x ) is equal to

zero in this case.

independent scatterers of the input squeezed noise. Each quadrature acts as a bandpass ® lter of the input noise of di€ erent linewidth. Changing the phase of the input squeezed vacuum merely alters the bandwidths of the noise quadratures of the atom. Therefore, the narrowing of the spectral lines is not due to squeezing in the ¯ uorescence ® eld but arises from the noise reduction in the input squeezed vacuum ® eld.

An interesting deviation of the ¯ uorescence spectrum from the noise spectrum can appear when the detuning ^¢ di€ ers from zero. In ® gure 7, we plot the

¯ uorescence spectrum together with the quadrature-noise components S X ( x ) , and S Y ( x ) and with the asymmetric part S _A ( x ) for ^X / g = 2 . 5, ^¢ / g = - ^{5, u =} p and the DPOparameters g c / g = 10, ^² / g = 0 . 07. The spectrumis composed of two narrow lines symmetrically located about the central frequency. For the para- meters employed in ® gure 7, the approximate eigenvalues of the matrix A are

z 1 = - ¹ 5 g ( ⁹ ₂ + 9 N ^~ + | ^{M ~} | ) ,

z 2 = - 10 ¹ g ( ¹¹ ₂ + 11 N ^~ - | ^{M ~} | ) - ^{i5 1 / 2X} ,

z 3 = - 10 ¹ g ₍ ¹¹ ₂ + 11 N ^~ - | ^{M ~} | ₎ + i5 ^{1 / 2X} .

( 62 ⁾

It is evident from (62 ) that the linewidths predicted from the real parts of the eigenvalues cannot be signi® cantly narrowed. This indicates that for a detuned driving ® eld the squeezed vacuumdoes not act as a® lter of the noise. However, the linewidths seen in ® gure 7 are quite narrow indicating that there is another mechanism which can lead to the narrowing of the spectral lines. As we see from ® gure 7, the noise component S X ( x ) is negative at the frequencies 6 5 ^{1 / 2X} . The fact that S X ( x ) is negative indicates that the X quadrature is squeezed.

According to(55 ) the ¯ uorescence spectrumis asumof the twoquadratures S X ( x ) and S Y ( x ) . Since S X ( x ) is negative the sum of S X ( x ) and S Y ( x ) gives e€ ectively

10 8 6 4 2 0 2 4 6 8 10

0.015 0.01 0.005 0 0.005 0.01 0.015

S

,S

,S

(w - w

)/g

Figure 7. The ¯ uorescence spectrum (dotted line ) and the quadrature-noise components S X ( x ) (dashed line ) , S Y ( x ) (dashed-dotted line ) for a narrow- bandwidth squeezed vacuum with g c / g = 10, ^² / g c = 0 . 07, ^¢ / g = - ^{5, X} / g = 2 . 5 and

u = p . The solid line is the asymmetric part S A ( x ) .