Propagation of small fluctuations in electromagnetically induced transparency. Influence of Doppler width
PPropagation of small fluctuations in electromagnetically induced transparency.Influence of Doppler width.
P. Barberis-Blostein
Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas. Universidad Nacional Aut´onoma de M´exico,Ciudad Universitaria, 04510 M´exico D.F. M´exico
M. Bienert
Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 48-3, 62251 Cuernavaca, Morelos, M´exico (Dated: October 31, 2018)The propagation of a pair of quantized fields inside a medium of three-level atoms in Λ configu-ration is analyzed. We calculate the stationary quadrature noise spectrum of the field after prop-agating through the medium in the case where the field has a general (but small) noise spectrumand the atoms are in a coherent population trapping state and show electromagnetically inducedtransparency (EIT). Although the mean values of the field remain unaltered as the field propagates,there is an oscillatory interchange of noise properties between the probe and pump fields. Also, asthe field propagates, there is an oscillatory creation and annihilation of correlations between theprobe and pump quadratures. We further study the field propagation of squeezed states when thereis two-photon resonance, but the field has a detuning δ from atomic resonance. We show that thefield propagation is very sensitive to δ . The propagation in this case can be explained as a com-bination of a frequency dependent rotation of maximum squeezed quadrature with an interchangeof noise properties between pump and probe fields. It is also shown that the effect of the Dopplerwidth in a squeezed state propagation is considerable. PACS numbers: 42.50.Gy,42.50.Ar,42.50.Lc
I. INTRODUCTION
Electromagnetically induced transparency (EIT) [1]emerges when coherence between electronic states of anatom suppresses the absorption of incident light. An usu-ally opaque medium consisting of such atoms becomestransparent. Three electronic levels in a Λ-shaped con-figuration are a paradigm for showing EIT when the twolower states are coupled in two-photon resonance via thecommon excited state: In this case destructive interfer-ence between the two excitation paths suppresses absorp-tion of photons. EIT has found many applications incoherent transfer of atoms [2], laser cooling [3, 4], andrecently it was proposed to serve as a quantum memorydevice for applications in quantum information technol-ogy [5, 6].The Λ-configuration is exemplified in Fig. 1, where two(meta-)stable states, | (cid:105) and | (cid:105) are both coupled to anexcited state | e (cid:105) by dipole interaction with the electro-magnetic fields of illuminating laser light. Spontaneousemission rates from the excited state with linewidth γ into | i (cid:105) are denoted by γ i and the detunings of thelasers’ carrier frequency from the atomic transitions arelabeled by δ i . Detuned two-photon resonance is presentif δ = δ = δ (cid:54) = 0.In the usual setup, the so-called pump laser drives onetransition, e.g. | (cid:105) ↔ | e (cid:105) , while the probe laser, interact-ing with | (cid:105) ↔ | e (cid:105) , is tested for transparency [7]. TheRabi frequencies associated with pump and probe aredenoted by Ω and Ω . The linear response of the ab-sorption of the probe by the medium is described by the j e ij i j i g ° g ° ± ± FIG. 1: The atoms have a Λ configuration with stable ormetastable states | (cid:105) and | (cid:105) and common excited state | e (cid:105) .The transitions | j (cid:105) ↔ | e (cid:105) with dipole coupling constants g j underlie spontaneous decay of rates γ j ( j = 1 , | e (cid:105) is γ = γ + γ . imaginary part of the electric susceptibility χ which isproportional to the mean value of the imaginary part ofthe electric dipole. In the case of two-photon resonance, δ = δ , the imaginary part of the electric susceptibility χ vanishes and the medium becomes transparent for theclassical field. In Fig. 2 we plot Im χ as a function of theprobe detuning δ for the cases where the pump is in res-onance ( δ = 0, solid line) and detuned ( δ = γ , dashedline). The maximum absorption frequency of the probefield increases monotonically with the Rabi frequencies.There has been recent interest in understanding thebehavior of the quantum and noise properties of a fieldwhich propagates in EIT media. The studies concentrate a r X i v : . [ qu a n t - ph ] D ec -4 -2 0 2 4 d /g A b s o r p ti on FIG. 2: Absorption spectrum of the probe’s mean value inthe case of a resonant pump field, δ = 0, (solid line) and adetuned pump field δ = γ (dashed line), in arbitrary units.Here, δ = 0 refers to a resonant driving of the probe transi-tion. Parameters: Ω = Ω = γ. on two cases: (i) when the strength of the intensity ofthe pump field if much bigger than the probe field and(ii) when the intensity of the pump field is of the sameorder as the probe field.For case (i), assuming a classical pump field and ne-glecting the small absorption inside the EIT window, cal-culation shows that an incoming quantum state is thesame after propagation in the medium [5]. For the casewhere both fields are quantized, with the resonant pumpfield being a coherent state and the incoming probe statea squeezed vacuum, the absorption of squeezing from thedifferent frequencies of a broadband squeezed vacuumfollows the classical EIT window [8]. Different exper-iments that measure propagation of a probe squeezingvacuum use this fact in order to explain the propagatedfield [9, 10] .When the absorption of the squeezed vacuum followsthe classical EIT window, we can write the effect of themedium on the field in the following way [9]. Let a ( ω )be the annihilation operator for frequency ω . Then, afterpropagation: a (cid:48)(cid:48) ( ω ) = T ( ω ) a ( ω ) + (cid:112) − | T ( ω ) | v ( ω ) , (1)where v ( ω ) is the vacuum contribution and T ( ω ) is thetransmission function from the classical measurement.In some cases it is necessary to include the atomic noisegenerated by the atoms due to decoherence in the baselevels in order to explain the experimental results [11, 12].For case (ii) a theoretical study of general phase noisepropagating in EIT was carried out in references [13, 14],in the stationary regime. In these studies, the spectrumof the difference between the phase noise of both fieldsis treated. They found that as the field propagates thephase noise of both fields correlate and tend to be thesame. Inside the validity domain of the approximation,the length of the formation of this correlation follows theclassical EIT transparency curve. The more transparentthe medium for the mean values, the longer the distancethe field has to propagate in order to get correlated. Notethat this result, although it does not give the noise spec- trum for each field, tells us that the field indeed changesas it propagates and the scale of this change is given bythe scale of the build-up of the correlations. Neverthe-less, changing the field strength can make this scale verylong. This result is similar to that in Ref. [8], in the sensethat the distance scale where the field changes follows theclassical EIT transparency curve.There are also several experimental studies of the prop-agation of probe and pump field noise and its correlations[15, 16]. They explain the results using the known effectof transformation of the incoming laser phase-noise to in-tensity noise. This transformation happens due to smallabsorption by the atoms because of decoherence effects.In the case of perfect EIT , this process would not exist,and the correlations would not emerge.Recent calculations, where both fields, probe andpump, are treated quantum mechanically, show whathappens in the case where the probe field is initially asqueezed state. It is shown, in the stationary regime andwhere there is no decoherence between the base levels,that although the medium is transparent for the meanvalues of the field, initial quantum fluctuations are notnecessarily preserved after interaction. In the case of acavity filled with atoms in Λ configuration driven by asqueezed pump field and a coherent pump field, the quan-tum properties of the output field can be very differentfrom the quantum properties of the input field [17, 18].Similar calculations were done in the case of a probesqueezed state and a pump coherent state propagating ina medium showing EIT. When the probe and pump car-rier frequencies drive the atoms in resonance with theirrespective transition, there is an oscillatory interchangeof noise properties between the initially squeezed probefield and the initially coherent pump field as the statepropagates inside the medium. The length of this inter-change of squeezing can be much smaller than the lengthwhere the phase noise of both fields correlate [14] [19].This means that the state measured after interacting withthe EIT medium can be totally different to the incomingstate and that the kind of analysis expressed by Eq. (1) isnot always valid. Note that this interchange of squeezingbetween probe and pump fields is not related to absorp-tion or noise generated by the atoms.In this paper, treating both fields quantum mechani-cally, we study (i) the spectrum of the probe and pumpquadrature and the correlations between them when thepump and probe field have some initial general noise, (ii)the case where there is two-photon resonance, but thecarrier frequencies of the fields are not in resonance withthe corresponding atomic transitions ( δ = δ = δ (cid:54) = 0)[31], and (iii) the effect of the atoms’ Doppler width inthe field propagation. We find that the interchange ofnoise properties between squeezed states and coherentstates described in [19] extends to the case of generalnoise. Thus, for some spectrum frequency and distance,the probe noise becomes equal to the initial pump noise,and vice versa. Also, as the field propagates, there is anoscillatory creation and annihilation of correlations be-tween the pump and probe fields. Our results allow, forexample, to study the propagation of a probe squeezedstate with a pump laser with some phase noise. Althoughthis is usually the case, it has not been treated theoreti-cally before, except for the case where only difference inthe phase noise was studied [14]. Our results allows us topredict the quadrature of each field and the correlationbetween them and not only the spectrum of the phasedifference. Furthermore, we do not make an adiabaticapproximation: Our solution is valid for all frequencies.We find also new qualitative behavior that appears nei-ther in the semiclassical nor in the resonance case. If thecarrier frequencies of the pump and probe fields drivethe atoms with the same detuning from their respectivedipolar transition then, the mean value of the field staysunaltered. Basically this means that as long as thereis two-photon resonance, the photon detuning from theatomic transition does not have a strong effect in thepropagation of classical pulses in EIT. Nevertheless, ouranalysis shows that a detuned (from the atom transition)two-photon resonance of the carrier frequencies have alarge impact on the propagation of the field state.Our results show that the propagation of general statesin EIT media is richer than was usually believed. In par-ticular we show that the maximum squeezed quadratureof an initial broad squeezed vacuum rotates as it propa-gates in the medium, the velocity of rotation depends onthe detuning from the atom transition and the spectrumfrequency of interest. This means that after some propa-gation, the maximum squeezed quadrature of the propa-gated field is different for each frequency. This spectrum-frequency velocity dependent rotation of the quadraturesimplies that the output field is different to the incomingfield for length scales where the mean values of the fieldare almost unchanged.Note that the reduced velocity of pulse propagationin an EIT medium implies a rotation of the maximumsqueezed quadrature. This quadrature rotation is due tothe phase difference between the pulse, after propagat-ing inside the medium, with respect to the local oscil-lator used to measure the quadratures. This phase dif-ference depends on the pulse velocity. Nevertheless thisphase difference is global and because of that the quadra-tures for each spectrum-frequency are rotated the sameamount. The fact that we obtain that the velocity ofrotation depends on the spectrum frequency makes thisresult qualitatively different from the quadrature rotationdue to slow light propagation.We also show that in the case where the probe andpump field have the same Rabi frequency, propagation isa combination of the rotation of the squeezed quadratureplus interchange of noise properties between the probeand pump field.We also present the influence of detuned two-photonresonances on EIT media with some Doppler width. Themovement of the atoms in a medium causes a shift ofthe frequency that the atoms “see” in their referenceframe. Due to this Doppler effect, the atom experiences a velocity-dependent frequency shift relative to the car-rier frequency of the fields measured in the laboratoryframe. For a thermal vapor cell at room temperature,the Doppler width can be several times the decay rate.If the pump and probe fields propagate in the same di-rection, then the Doppler effect consists of detuning thefrequency of both fields by the same amount. This im-plies that the propagation of the mean value of the fieldwould be negligibly disturbed, because, independently ofthe Doppler width, all the atoms would be in two-photonresonance and the media shows EIT [7]. This observationis one reason why conventional (mean value measurementof the field) EIT experiments can be performed at roomtemperature. In contrast, a detuned two-photon reso-nance has a strong effect on the propagation of quantumstates. This implies a significant impact of the atoms’Doppler width on the quadrature spectra measured af-ter propagation. There are many recent experimentswhere the propagation of vacuum squeezed states is stud-ied [9, 20, 21, 22, 23]. Our results impose a restrictionon the possibility of conducting EIT experiments withsqueezed states in denser thermal clouds, even when wecan neglect decoherence between the base levels.The paper is organized as follows: in section II westate the equations for the field and for the atoms, insection III we study the case where the carrier frequenciesof the probe and pump field are in resonance and theincoming field has some general noise, in section IV westudy the case where the carrier frequencies of the probeand pump field are in a detuned two-photon resonance( δ = δ = δ (cid:54) = 0), in section V we study the effect of theDoppler width on the propagation of the field. II. THEORETICAL DESCRIPTION OF THEPROPAGATING FIELDS
The dynamics of the composite system of atoms andpump and probe fields are described by Heisenberg’sequation of motion [14], (cid:18) ∂∂t + c ∂∂z (cid:19) a j = − ig j N σ je , (2a)and ∂∂t (cid:36) = 13 ( − γ − γ )(1 + (cid:36) + (cid:36) ) − ig ( σ e a − a † σ e ) − ig ( σ e a − a † σ e ) + f (cid:36) ,∂∂t (cid:36) = 13 ( − γ − γ )(1 + (cid:36) + (cid:36) ) − ig ( σ e a − a † σ e ) − ig ( σ e a − a † σ e ) + f (cid:36) ,∂∂t σ e =( − γ iδ ) σ e + ig (cid:36) a − ig σ a + f e ,∂∂t σ e =( − γ iδ ) σ e + ig (cid:36) a − ig σ a + f e ,∂∂t σ =( − γ − i [ δ − δ ]) σ − ig a † σ e + ig σ e a + f . (2b)where j = 1 ,
2. This description relies on a multi-mode decomposition of the electromagnetic fields (cid:126)E j = (cid:126) E j a j ( z, t ) exp[ ik j z − iω j t ] + h . c . around the carrier fre-quencies ω j = ck j , where | (cid:126) E j | is the corresponding vac-uum electric field. The detuning with respect to theatomic transition ω je is given by δ i = ω i − ω je . In thisnotation, the field envelope operators a j ( z, t ) are slowlyvarying in space and time, allowing us to write Maxwell’sequation in the form (2a), where the atomic polarizationproportional to σ je acts as a source.The atomic operators σ µν ( z ) =lim ∆ z → LN ∆ z (cid:80) z ( j ) ∈ ∆ z σ ( j ) µν and (cid:36) j = σ ee − σ jj arewritten in the continuum limit, where σ ( j ) µν = | µ (cid:105) ( j ) (cid:104) ν | is the atomic operator of atom j at position z j . Here N is the number of atoms, L the length of the mediumand ∆ z a space region around z . This approximation isjustified if the inter-atomic distance is smaller than thelength scale introduced by the wavelength of the carrierfields.The coupling between atoms and field relies on a dipoleinteraction with coupling constant g j = (cid:126)℘ · (cid:126) E j / (cid:126) , where (cid:126)℘ is the atomic dipole moment. In order to arriveat the form of equations (2a) and (2b), the rotatingwave approximation was performed and all the opera-tors are in a reference frame rotating with the corre-sponding carrier frequencies. The laboratory frame no-tation can be obtained by the transformations σ (cid:48) ej = σ ej exp [ − i ( δ j + ω je )] and σ (cid:48) = σ exp [ − i ( δ − δ )].To account for the noise introduced by the couplingof the atomic system to the free radiation field, delta-correlated, collective Langevin operators, f j , were intro-duced. They have vanishing mean values and correla-tion functions of the form (cid:104) f x ( z, t ) f y ( z, t (cid:48) ) (cid:105) = LN D xy δ ( t − t (cid:48) ) δ ( z − z (cid:48) ). The diffusion coefficients D xy can beobtained from the generalized Einstein equations [24].Their explicit form can be found, for example, in [25].In order to obtain the spectrum of the quadrature fluc-tuations, the system of equations is solved in the small-noise approximation using the standard technique oftransforming them into a system of c-number stochasticdifferential equations [26]. The initial conditions for ouranalysis are: The probe and pump field has (cid:104) a j (cid:105) = α j ,thus driving the atomic transitions with Rabi frequencyΩ j = | g j α j | .Denoting the fluctuation of the θ quadrature of field j = 1 , δY θj ( z, t ) = δa j ( z, t ) exp( − iθ ) + δa † j ( z, t ) exp( iθ ) , (3)where δo = o − (cid:104) o (cid:105) , the θ -quadrature noise spectrum, inthe stationary regime, is given by S j ( z, ω ) = ∞ (cid:90) −∞ e − iωt (cid:104) δY θj ( z, t ) δY θj ( z, (cid:105) dt , (4)and the correlations noise spectrum between, the probe θ -quadrature and the pump θ -quadrature, is given by S c ( z, ω ) = ∞ (cid:90) −∞ e − iωt (cid:104) δY θ ( z, t ) δY θ ( z, (cid:105) dt , (5)where ω = 0 corresponds to the carrier frequency of thefield in accordance with the co-rotating reference framewe use. The initial conditions for the fluctuations can bewritten as S ( z = 0 , ω ) = 1 + 2 g ( ω ) cos(2 θ ) + 2 f ( ω ) , S ( z = 0 , ω ) = 1 + 2 g ( ω ) cos(2 θ ) + 2 f ( ω ) , S c ( z = 0 , ω ) = 0 , (6)where f j ( ω ) and g j ( ω ), j = 1 , III. CARRIER FREQUENCIES IN RESONANCEWITH ATOMIC TRANSITIONS ( δ = δ = 0 ) We solve equations (2) with δ = δ = 0 and noisespectrum given by (6), following the treatment used in[19]. We list in appendix A the analytical expressions forthe quadrature noise spectra of the pump ( j = 1) andprobe ( j = 2) field, and the correlations between them.The behavior of the spectrum is characterized by twoquantities: as the field propagates Q ( i ) characterizes anexponential decay of noise properties and Q ( r ) charac-terizes the coherent propagation of the field which corre-sponds to an oscillatory evolution of noise properties.The behavior of Q ( i ) as a function of the parametersresembles the behavior of the classical transmission spec-trum ( T ( ω ) in Eq. (1)). The absorption of the initialproperties of the field is then characterized by this quan-tity and allow us to define a length scale, z abs , wherethese absorptions are important, z abs ≈ / ( | Q ( i ) | ) . (7) A. Propagation of the field quadratures
When Q ( i ) z (cid:28)
1, that is, when we consider positions z where the exponential absorption of the fluctuationscan be neglected, we can easily understand the effect ofthe oscillatory behavior characterized by Q ( r ) . Let usassume that α = α , then, as the field propagates, thereis a complete oscillatory interchange of noise properties: S ( Q ( r ) z = kπ ) = S ( Q ( r ) z = ( k + 1) π ) , (8) S j ( Q ( r ) z = kπ + π/
2) = ( S ( z = 0) + S ( z = 0)) / , (9)where k is an integer an j = 1 , Q r z i a) Q i 0 p 2 p 3 p 4p 5p 6p p 2 p 3 p 4p 5p 6p Q r z i b) zabs zosc 2.02 FIG. 3: Quantum properties of probe and pump fields propagating through an EIT medium. (a) The fluctuation spectrum ofthe probe (solid), initially (at z = 0) in a squeezed state with squeezing parameter ξ = 1 and of the pump (dashed), initiallyin a coherent state, as a function of position for θ = 0 and no absorption ( γ i = 0). (b) Same as (a) but with absorption. Theamplitude of the oscillations decays 1 /e after z abs / | z osc | = 2 .
02 periods. coherent pump (dashed), as a function of propagationlength z . For a complete discussion when the probe fieldis a squeezed state, see [19].Equations (8) and (9) clearly display a spectrum-frequency dependent oscillatory transfer of the initialnoise properties of the probe to the pump and back whiletraveling through the medium, and it extends the resultof [19] to any initial noise. The length scale of the oscil-latory transfer z osc = 2 π/Q ( r ) can be much smaller thanthe absorption length scale z abs . In fact z abs | z osc | = | Ω − ω | πγω . In the case where ωγ (cid:28) Ω (corresponding to obser-vations frequencies inside the transparency window) wehave z abs (cid:29) | z osc | .In Fig. 3b), the interplay of both scales, oscillatory andabsorption, can be clearly observed.The oscillatory behavior implies that the outgoing fieldcan be completely different from the incoming field, al-though the mean values stay exactly the same. This effectcannot be explained with the usual approach expressedby Eq.(1). Moreover, it is qualitatively different from theabsorption since the noise properties, for each frequency,is recovered after some propagation. B. Propagation of correlations
A previous study of propagation of noise correlationbetween the pump and probe field was done in [14]. They studied the particular case of phase difference fluc-tuations. We first obtain a generalization (for the no-decoherence case) of the known result of the spectrumof phase difference fluctuations given in [14]. We presenthere the result for no decoherence in the base level, butvalid for all frequencies (no adiabatic approximation).Let φ j represent the phase of field j . We use the factthat for small noise the phase quadrature noise is pro-portional to the phase noise [26]. From Eqs. (A1), (A2)and (A3), we obtain ( α = α = α ): S φ ( ω, z ) = ( δφ − δφ ) ≈ α {S θ = π/ ( ω, z ) + S θ = π/ ( ω, z ) − S θ = θ = π/ c ( ω, z ) } = 1 α e − Q ( i ) ( ω ) z S φ ( ω, z = 0) . (10)Observe that as the field propagates, the difference be-tween the phase noise of the fields disappears, which im-plies that correlations between the phase noise of bothfields builds up. For the case of no decoherence, Eq. (10)is an extension of Eq.(12) in Ref. [13], in the sense thatit is valid for all ω . It is easy to see that the length ofthe fading of the phase difference between the fields goeslike z abs , Eq. (7). For the phase noise difference, we donot observe any oscillatory behavior, as in the case of thepropagation of the field quadratures.Nevertheless, the oscillatory transfer of initial noiseproperties between the pump and probe field generatesalso an oscillatory creation and annihilation of correla-tions (defined in Eq. (5)). When z (cid:28) z abs , from Eq. (A3)we obtain S c ( z, ω ) = sin (cid:16) Q (r) z (cid:17) [sin ( θ − θ ) ( f ( ω ) − f ( ω ))+ sin ( θ + θ ) ( g ( ω ) − g ( ω ))] . (11)Equation (11) shows that if the noise properties of thepump and probe field are different, then correlations be-tween the fields oscillates for length scales small withrespect to the absorption scale. Due to Q (r) frequency-dependence, these correlations are different for each spec-trum frequency, showing a rich coherent interaction be-tween the probe and pump fields. Note that for thespectra of phase noise difference, nothing happens for z (cid:28) z abs , implying that the correlation oscillations shownin Eq. (11) are qualitatively different from the correla-tions implied in Ref. [14] and Eq.(10).For z (cid:29) z abs , the correlations reach a constant valuegiven by: S c ( z, ω ) = 12 (cid:0) cos ( θ − θ ) ( f ( ω ) + f ( ω ))+ cos ( θ + θ ) ( g ( ω ) + g ( ω )) (cid:1) . (12)Note that, except for initial coherent states ( f i = g i = 0),there is always a formation of correlations.We can interpret the generation of the z > z abs con-stant correlations as a signature of initial noise in thefields, and the oscillatory correlations between the fieldsas a signature that the initial noise spectrum is differentfor each field.The interchange of noise properties and the oscillatorycreation and annhilation of correlations can be explainedby the fact that the two fields are coupled via the atomicmedium. Due to EIT this coupling is such that the fieldmean values are unaltered and only the fluctuations getcoupled. The atoms base level coherence plays a key rolein this behavior, the oscillatory interchange of noise iscomplete and the generated correlations reach a maxi-mum when the base level coherence of the atoms, (cid:104) σ (cid:105) ,is maximal. When (cid:104) σ (cid:105) goes to zero there is neither in-terchange of noise nor generation of correlations betweenthe fields.When α = α the oscillatory interchange of noiseproperties seems to resemble the noise interchanges thatoccurs when two oscillators, with the same frequency,are coupled (see Ref. [27] for an example with squeezedstates). Differently to the case we are studying, in the twocoupled oscillators the transfer of noise does not dependon the mean value of the fields: it is always complete.Also in this case the field mean value is altered duringthe interaction. IV. CARRIER FREQUENCIES IN A DETUNEDTWO-PHOTON RESONANCE ( δ = δ (cid:54) = 0 ) In this section we study the case where the carrier fre-quencies of the pump and probe field are in a detuned two-photon resonance δ = δ = δ (cid:54) = 0. The pump fieldis initially in a coherent state and the probe field hasinitially some general noise.We solve equations (2) with δ = δ = δ and f ( ω ) = g ( ω ) = 0 in the initial conditions Eq.(6) (i.e., a coher-ent pump field) following the treatment used in [19]. Inappendix A 2 we list the analytical expressions for thequadrature noise spectra of the pump ( j = 1) and probe( j = 2) field, and the correlations between them. A. Asymptotic behavior As Q ( i ) ± is always negative, for large propagationlengths, z → ∞ , we find S ( z, ω ) ≈ S ( z = 0 , ω ) − α α ( α + α ) , (13) S ( z, ω ) ≈ S ( z = 0 , ω ) − α ( α + α ) , (14)which are reminiscent to similar correlations known fromcavity EIT [28] and the effect of pulse-matching [1, 13].The asymptotic behavior does not depend on δ . Thedistance where this asymptotic behavior is dominant isgoverned by the exponentials in Eqs. (A5) and (A6), andis of the order of z abs ≈ / (Max( | Q ( i )+ | , | Q ( i ) − | )) . (15)When α (cid:54) = 0, the asymptotic quadrature fluctuationsof the pump field shows a fraction of the noise proper-ties of the probe field. Also the noise of the probe fielddiminishes if S ( z = 0 , ω ) > α = α , both fields haveasymptotically the same fluctuations. B. The probe field is a vacuum squeezed state
For a broadband vacuum squeezed state ( α = 0)theinitial conditions (6) reduce to S ( z = 0 , ω ) = 1 , (16) S ( z = 0 , ω ) = e − ξ cos θ + e ξ sin θ , (17)where ξ is a real number characterizing the amount ofsqueezing.From Eq. (A5) can be concluded that the pumpfield quadratures remains as in the initial condition, S ( θ, ω ) = 1, as the field propagates. We also knowthat in the case where both fields are in resonance themedium is transparent for the squeezed vacuum, exceptfor the expected absorption for frequencies detuned fromresonance (see section III and Ref. [29]). Nevertheless,when the fields are detuned but in two-photon resonance,although there is no change for the mean values, the vac-uum squeezed state is altered. We can see this by sub-stituting α = 0 in Eq. (A6): S ( z, ω ) = 1 + (cid:16) e Q ( i ) − z + e Q ( i )+ z (cid:17) sinh ( ξ ) − e Q ( i ) − z + Q ( i )+ z cos( Q ( r ) − z − Q ( r )+ z + 2 θ ) sinh(2 ξ ) . (18)The propagation of a squeezed vacuum in an EITmedium for the case of detuned two-photon resonance ischaracterized by a frequency dependent quadrature rota-tion and absorption, characterized by Q ( r ) ± and Q ( i ) ± , re-spectively. The spectrum absorption of the squeezing isthus different from the spectrum absorption of the meanvalue.
1. Quadrature rotation
To better understand the behavior of the coherent part(i.e., neglecting the absorption part) of the propagation,let us suppose that z (cid:28) z abs . In that case we can replacethe exponentials in Eq. (A6) with 1. Comparing the re-sult with the incoming field, Eq. (17), we can write: S ( z, ω, θ = 0) = S ( z = 0 , ω, θ = ( Q ( r ) − − Q ( r )+ ) z/ . (19)This last equation shows that the propagation of asqueezed probe vacuum in EIT, is equivalent to the rota-tion of the angle of maximum squeezing. The velocity ofthis rotation is given by Q ( r ) ± and is a function of the de-tuned two-photon resonance parameter δ and spectrumfrequency ω . Due to the dependence of the velocity ofrotation on ω , after propagation the maximum squeezedprobe quadrature would be different for each spectrumfrequency. Note that this behavior can not be modeledusing Eq.(1). Since the θ = 0 quadrature fluctuation ofthe propagated field is different for each frequency, wecan say that the media would not be transparent for theincoming broadband θ = 0 vacuum squeezed state.Note that this result is qualitatively different from theknown case [30], where as a result of the pump detuningall the quadratures are rotated the same amount, inde-pendently of ω . The difference between our results andthe results in [30] are due to the approximations used. Inparticular we do not make any adiabatic approximation.Note also that the frequency-dependence on the veloc-ity of quadrature rotation cannot be explained only by aglobal phase rotation due to slow light propagation.
2. Spectrum of squeezing absorption
The effect of the atom decay rate is, as the field prop-agates, to dampen the squeezing of the quadrature withmaximum squeezing of the probe field. For the resonance case, the absorption of the maximum squeezed quadra-ture follows the classical absorption spectrum. This isnot the case when there is two photon resonance but eachfield is detuned from the atomic transition. In Eq. (18)it can be observed that the absorption is proportional to | Q ( i ) ± | and the sum | Q ( i )+ + Q ( i ) − | . The two maxima ω ± ofeach of the Q ± are located at ω ±± = δ + 12 ( ± δ ± (cid:112) δ + 4Ω ) . (20)Figure 4 shows the absorption spectrum. The absorp-tion spectrum of the noise differs qualitatively from theabsorption spectrum of the mean values Fig. 2, only for ω = δ there is no absorption in either case. We denotethe region around ω = δ where absorption is negligible as∆ ω δ . The length of this spectral region, ∆ ω δ , decreasesas δ increases. When δ increases, the difference between ω ++ − ω + − = ω − + − ω −− increases. When this difference ismuch bigger than γ , a spectral region appears between ω ++ and ω − + , and between ω −− and ω + − where absorption isnegligible. This is shown as the dotted line in Fig. 4(b).Then, when δ (cid:29) γ, Ω, instead of the three regions oftransparency that appears in the spectrum absorptionfor the mean values (see Fig. 2), there are five regionsof transparency: (i) around ω = δ ; (ii) between ω + − and ω −− ; (iii) between ω − + and ω ++ ; (iv) when ω (cid:29) ω − + and (v) ω (cid:28) ω + − . The width of the transparency window for case(i) is of order ∆ ω δ ≈ /δ , for case (ii) and (iii) is oforder δ .Note that in Ref. [8, section II-C] , it is concludedthat maximum absorption takes place at ω = δ , and thatthere is an absorption of squeezed states around ω = δ ,and transparency elsewhere. This contradiction betweenthe results presented here and those from the referenceis due to the fact that they make the δ (cid:29) γ approxi-mation before solving the equation. As a result of thatapproximation, they do not obtain the transparency re-gion around ω = δ .An example of the interplay between the rotation ofthe angle of maximum squeezing and the absorption ofthe initial squeezing due to the linewidth of the excitedlevel, given by Eq. (18), is plotted in Fig. 5. C. The probe field is a squeezed state with α = α We discuss now the propagation of a squeezed probefield in the case where α = α . We separate this dis-cussion into three parts. From the form of the Q ( r,i )+ ,Eq. (A8), we can identify three regions of qualitativelydifferent behavior, denoted by the Roman numbers I, II,and III in Fig. 6. We will therefore divide the studyof the propagation of squeezed fields in EIT media intothese three parts for the case δ >
0, the other case beingsymmetric. dww wd w/g A b s o r p ti on a) w/g A b s o r p ti on -4 -2 0 2 4 6 8 b) FIG. 4: Absorption spectrum for squeezed states for thecase where there is a detuned two-photon resonance. a) | Q ( i )+ | (solid line), | Q ( i ) − | (dotted line). b) | Q ( i )+ + Q ( i ) − | for δ = 1 γ (solid line), δ = 3 γ (dotted line). Note how five regions oftransparency start to appear in figure b), see text for details.Parameter: Ω = 0 . γ . p 2 p 3 p 4 p 5 p 6 p 7 p Q r Q r z FIG. 5: Propagation of the θ = 0 quadrature spectrumof the probe field, initially (at z = 0) in a broadbandvacuum squeezed state with squeezing parameter ξ = 1,along the z -direction in the case of a pump detuning suchthat Q ( r )+ /Q ( r ) − = 1 .
106 (solid line), and no pump detun-ing Q ( r )+ /Q ( r ) − = 1 (dashed line). The length scale is givenby the parameters of the solid line curve. Parameters: Q ( i )+ /Q ( i ) − = 1 . | Q ( r )+ /Q ( i )+ | = 60 π (solid line), Q ( i )+ /Q ( i ) − =1, | Q ( r )+ /Q ( i )+ | = 63 π (dashed line).
1. Near-resonant case (Region I)
We analyze here the propagation of the field in thenear-resonant case, | ωδ | (cid:28) Ω . A qualitatively differentbehavior from the case δ = 0 appears. d/g a) I II III Q (r) wW g g WW -w+WW +w- d/g Q (i) b) I II III
FIG. 6: The oscillating parts of the quadrature fluctuationspectrum are proportional to Q ( r ) ± . In a) we show Q ( r )+ (solidline) and Q ( r ) − (dashed line) as a function of δ . The extremaare obtained for δ ± = − γ ± Ω − ω ω . The absorbing parts of thequadrature fluctuation spectrum oscillations are proportionalto Q ( i ) ± . In b) we show Q ( i )+ (solid line) and Q ( i ) − (dashedline) as a function of δ . For Q ( i ) ± the extrema are obtainedfor δ ± = ± Ω − ω ω . The maximum of Q ( i )+ ( Q ( i ) − ) is centeredbetween the two extrema of Q ( r )+ ( Q ( r ) − ). Figs. 7 and 8(a) show a typical example of how the θ = 0 quadrature of the field propagates, according toEqs. (A5),(A6), for the case of negligible absorption. InFig. 8 we show the effect of the decay rate γ . The blackarea in the plots represents the fast oscillations. Thereare three scales that can be seen in Figs. 7 and 8: (i)a quick scale z osc , where we have oscillating transfer ofsqueezing between pump and probe fields for some re-gions, (ii) an intermediate oscillating scale, z int , given bythe oscillation of the envelope of the fast oscillations; and(iii) the absorption scale z abs which damps the oscillationbehavior, given by Eq.(15). On the intermediate scale,the probe θ = 0 quadrature goes from a squeezed value toa maximum excess noise ( S ( θ = 0) >
1) and then backto the squeezed value. It is not difficult to show that for e ξ (cid:29) θ = 0 probe quadrature is S ( θ = 0 , z max ) ≈ e ξ / e − ξ .We proceed now to obtain analytical expressions forthe scales. When Ω > | ω ( ω ± δ ) | and Q ( r ) − , Q ( r )+ >
0, wecan extract fast oscillating terms (given by Q ( r ) − , Q ( r )+ )and slow oscillating terms (given by Q ( r ) − − Q ( r )+ ) fromEqs.(A5) and (A6). z osc is given by the fast oscillatingterms: when ωδ (cid:28) Ω , z osc ≈ π/Q ( r ) − ≈ π/Q ( r )+ . (21)The intermediate oscillating scale, z int , is given by theslow oscillating terms: z int = 2 π/ ( Q ( r ) − − Q ( r )+ ) . (22)To gain more insight into the propagation of an ini-tially θ = 0 quadrature squeezed state in EIT media, itis necessary to study all the quadratures. Excess noisein the θ = 0 quadrature after some propagation does notnecessarily imply that the state is no longer a squeezedstate. The squeezed quadrature could have been rotated(see section IV B). It is worth then, instead of studyingthe propagation of a specific quadrature, to study thepropagation of the quadrature where the minimal andmaximal fluctuations are achieved. To calculate thesequadratures we minimize Eqs. (A5) and (A6) for θ . Wewill assume that z (cid:28) z abs . Simple expressions can be ob-tained for the angle θ min of the probe quadrature wherethe minimum value is reached and the angle θ max wherethe maximum value is reached: θ min , n = πn + ( Q ( r )+ − Q ( r ) − ) z , (23) θ max , n = θ min , n + π/ , (24)where n is an integer. Similarly to the case where theprobe field is a vacuum squeezed state (see section IV B),the quadrature of minimal fluctuations rotates with prop-agation.In Fig. 9 we compare the θ min quadrature with the θ max quadrature. We observe that for some values of Q ( r )+ z ,the minimum quadrature and the maximum quadratureof the probe field reach the same value. This means thatfor these particular values of Q ( r )+ z , all the quadratureshave the same value. These values of Q ( r )+ z can be cal-culated analytically by minimizing Eqs. (A5) and (A6)with respect to z such that they are independent of θ .We obtain Q ( r )+ z , + ( m ) = 4 πm + π ,Q ( r ) − z , − ( m ) = 4 πm + π , (25)where m is an integer.The previous results tell us that for an initiallysqueezed probe field, the propagation has the effect ofdistributing the noise between the quadratures until, fordistances defined by Eq. (25), the noise is equally dis-tributed between all the quadratures. Then, as the fieldcontinues to propagate, the inverse process starts untila particular mode quadrature is squeezed. This processrepeats until the initial condition of the mode (a θ = 0quadrature squeezed state for the probe field, a coher-ent state for the pump field) is reached. We understand this dynamics as a combination of the rotation of thequadrature of minimal fluctuations due to detuning (seeEq. (23)) and the interchange of noise between the probeand pump fields characterized by z osc .A similar process occurs for the pump field but withthe distances z , ± ( m ) where all the quadratures have thesame value given by, Q ( r )+ z , + ( m ) = 4 πm ,Q ( r ) − z , − ( m ) = 4 πm . (26)The value of all the quadratures for these distancesoscillates and is given by, S ( z , ± ( m )) = S ( z , ± ( m )) =1 + 12 (cos( Q ( r ) ∓ z , ± ( m )) + 1) sinh ( ξ ) .
2. Intermediate detuned two-photon resonance (region II)
In region II (see Fig. 6(a)), it is not difficult to seethat exponential absorption, in the vicinity of the maxi-mum of Q ( i )+ ( δ ), makes all the properties of propagationdescribed in the previous section fade away for distances z < z osc , z int .
3. Large detuned two-photon resonance (region III)
We now study the case of a large detuned two-photonresonance | δω | (cid:29) Ω , ω (region III). For that case, Q ( r )+ ≈ − Q ( r ) − ≈ − C δω γ ω + δ ω , (27) Q ( i )+ ≈ Q ( i ) − ≈ − C γω (cid:16) γ ω + δ ω (cid:17) . If δ (cid:29) γ then | Q ( r )+ | (cid:29) | Q ( i )+ | and a simple expression canbe found from Eqs. (A6) and (A5) for z < z abs . S ( z, ω ) = S ( z + π/Q ( r )+ , ω ) ≈ e − ξ cos (cid:32) Q ( r )+ z (cid:33) + sin (cid:32) Q ( r )+ z (cid:33) + 14 e ξ sin ( Q ( r )+ z ) . From the former equation, it can be concluded thatinterchange of quantum fluctuation between pump andprobe happens when Q ( r )+ z = nπ , n being an integer.Differently to the near-resonant case, described in sec-tion IV C 1, the distances where both fields have the sameexcess noise is reached before the interchange of noise be-tween the fields occurs. In Fig. 10 an example of propa-gation for large detuned two-photon resonance is shown.0
20 p 40 p 60 p 80 p Q r z i FIG. 7: Quantum properties of probe and pump fields propagating through an EIT medium with a detuned two-photonresonance. It is shown how the fluctuation spectrum of the probe field, initially (at z = 0) in a squeezed state with squeezingparameter ξ = 1 propagates along the z -direction in the case where z (cid:28) z abs . The oscillatory transfer of squeezing betweenprobe and pump can be clearly seen to be transformed into excess noise until both fields reach the same value. Parameters: Q ( i )+ = Q ( i ) − = 0, Q ( r )+ /Q ( r ) − = 1 . Q ( i )+ /Q ( i ) − = 1 .
200 p 400 p 600 p Q r z a Q i Q r z b Q r Q i
449 p
200 p 400 p 600 p Q r z c Q r Q i
224 p
200 p 400 p 600 p Q r z d Q r Q i
45 p
200 p 400 p 600 p
FIG. 8: Quantum properties of probe and pump fields prop-agating through an EIT medium with a detuned two-photonresonance. It is shown how the fluctuation spectrum of theprobe field, initially (at z = 0) in a squeezed state withsqueezing parameter ξ = 1 propagates along the z -directionfor different decay rates. The black area of the plots rep-resents the fast oscillations. a) No decay rate ( Q ( i )+ = 0),b)-c) | Q ( r )+ /Q ( i )+ | = 449 π, π, π ( Q ( i )+ is proportional tothe decay rate). The z int scale from minimum e − ξ to e ξ / Q ( r )+ /Q ( r ) − = 1 . Q ( i )+ /Q ( i ) − = 1 . α = α . V. MOVING ATOMS AND TWO-PHOTONRESONANCE: DOPPLER EFFECT
In this section we study numerically the effect of theatoms’ Doppler width in the spectrum of quadrature fluc-tuation.As we studied in the previous section, if the pump fieldis not in resonance with the dipole transition, then thepropagation of the state is hugely altered and the finalpropagated state can be very different from the initialstate. This means that for moving atoms, due to the Q r z q min , q max FIG. 9: Propagation of the probe’s θ max quadrature spectrumcompared with the probe’s θ min quadrature spectrum for z (cid:28) z abs . Here θ max maximize the quadrature and θ min minimizethe quadrature. It is clearly seen that there are distanceswhere the θ max and θ min quadratures are equal. Parameters: Q ( r )+ /Q ( r ) − = 1 . Q ( i )+ = Q ( i ) − = 0, α = α , ξ = 2 p 2 p 3 p 4 p zQ r i FIG. 10: The θ = 0 quadrature spectrum of the probe field(solid line) initially (at z = 0) in a squeezed state with squeez-ing parameter ξ = 1 and the pump field (dashed line) initiallyin a coherent state, propagates along the z -direction in thecase of large detuning | δ | (cid:29) γ , | δω | (cid:29) Ω , ω , α = α . Doppler width, groups of atoms with different directionsand velocities would see different detuned two photonresonance, and each group would have a different influ-ence on the propagation of the initial state. Using thesuperposition principle of the field and the fact that weare working in a linear approximation, the propagated fi-nal field would be a sum of the propagated field for each1group of atoms a ( t ) = (cid:82) ∞−∞ ρ ( δ ) a δ ( t ) dδ , where a δ ( t ) is thesolution of equation (2a) when the atoms’ velocity is suchthat the detuned two-photon resonance is δ , and ρ ( δ ) isthe density of atoms with velocities and directions suchthat the detuned two-photon resonance is δ . The finalquadrature spectrum is then given by S ( θ, ω ) ∆ δ = (cid:90) ∞−∞ (cid:90) ∞−∞ ρ ( δ ) ρ ( δ ) S ( θ, ω, δ , δ ) dδ dδ , (28)where S ( θ, ω, δ , δ ) = ∞ (cid:90) −∞ e − iωt (cid:104) δY θδ ( t ) δY θδ (0) (cid:105) dt , (29)and δY θδ i ( t ) = δa δ i ( t ) exp( − iθ ) + δa † δ i ( t ) exp( iθ ). We willassume that ρ ( δ ) is a Gaussian with variance ∆ δ .To obtain the effect of the Doppler width in thequadrature spectrum we calculate, from Eqs. (2), S ( θ, ω, δ , δ ) following a method similar to the one usedin [19]. Then we numerically integrate Eq. (28).The frequency ω obtained from Fourier transform-ing Eqs. (2) is the spectrum frequency measured fromthe field carrier frequencies. For fixed ω , changing δ also changes the spectrum frequency with respect to thedipole transition of the atom. In order to correctly obtainthe Doppler effect in the quadrature, before integratingEq. (28) we substituted ω by ω − δ . After this sub-stitution, and in this section, ω represents the spectrumfrequency with respect to the atomic transition. A. The probe field is squeezed vacuum
We now suppose that the probe field is in a broad-band squeezed vacuum. This means that α = 0. Inthis case, for atoms at rest, and a coherent pump field inresonance with its corresponding transition, the mediumis transparent for the squeezed vacuum (see section IIIand Ref. [29]). In Fig. 11 we show the numericallycalculated probe θ = 0 quadrature spectrum for an ini-tially squeezed vacuum propagating in a medium com-posed of atoms with Doppler distribution with width ∆ δ .For ∆ δ = . γ , the detuned two-photon resonance dueto the Doppler effect is small and does not affect thepropagation. As ∆ δ increase, the effect of the differentquadrature rotation on the incoming field, due to atomswith different detuned two-photon resonance, begins tobe important. When ∆ δ (cid:39) . γ we even have excessnoise in the θ = 0 quadratures for some distances. Theexcess noise is larger the larger ∆ δ . In the limit z → ∞ itcan be calculated from Eq.(14) that the θ = 0 quadraturewill be 1. z p .01 p.1p.25p.5p FIG. 11: Propagation of the θ = 0 quadrature spectrumof the probe field initially (at z = 0) in a vacuum squeezedstate with squeezing parameter ξ = 2 along the z -directionin a medium composed of atoms with Doppler width ∆ δ = . γ, . γ, . γ, . γ . It is clearly seen how the Doppler ef-fect affect the propagation of a vacuum squeezed state. Pa-rameters: γ = γ , g = g = γ/
10, Ω = 1 γ , Ω = 0, ω = γ/ N = 10 . B. The probe field is squeezed state
We study now how the Doppler effect affects the prop-agation of a squeezed state as a probe initial condition.We will suppose that the mean value of the probe fieldis equal to the mean value of the pump field, α = α .We will also suppose that the carrier frequencies of bothfields are in resonance with their respective dipole tran-sitions of the atoms at rest.Numerical results are presented in Fig. 12 for g = g = γ/
10, Ω = Ω = 1 γ , ω = γ/ N = 10 , ξ =2. In Fig.12(a), for a Doppler width of ∆ δ = 0 . γ ,the behavior of the state propagation is very similar tothe studied in section III. From Fig.12(b) to Fig.12(c),it can be seen how the increase of the Doppler widthstarts to destroy the interchange of squeezing propertiesbetween the probe and pump field and the differencesbetween both fields vanish. In Fig.12(d) both fields havepractically the same θ = 0 quadrature spectra for z C /γ > VI. CONCLUSIONS
We have shown that in the propagation of pump andprobe fields through an EIT medium, the initial noiseproperties of the field are not conserved, except for thecarrier frequencies which drive the atoms on two-photonresonance. We found a series of novel behavior in thepropagations.The results reported in Ref. [19] extend to any initialspectrum of noise. The effect of coherent propagation(i.e., neglecting the exponential decay terms) is to inter-2
50 100 150 200 250 300 z g i Dd=.01g
50 100 150 200 250 300 z g i Dd=.1g
50 100 150 200 250 300 z g i Dd=.25g
50 100 150 200 250 300 z g i Dd=.5g
FIG. 12: Propagation of the θ = 0 quadrature spectrum of theprobe field initially (at z = 0) in a squeezed state with squeez-ing parameter ξ = 2 along the z -direction in a medium com-posed of atoms with Doppler width ∆ δ = . γ, . γ, . γ, . γ .It is clearly seen how the Doppler effect can destroy the os-cillatory transfer of squeezing between the probe and pumpfield. Parameters: γ = γ , g = g = γ/
10, Ω = Ω = γ , ω = γ/ N = 10 . change the noise properties between the probe and pumpfield as the field propagates. The interchange is maximalwhen both fields have comparable intensities. The fre-quency of the interchange of noise properties depends onthe spectrum frequency. This implies that the noise spec-trum of the outcoming field can be completely differentto the noise spectrum of the incoming field.When the initial noise spectrum is different betweenthe fields, as the field propagates there is an oscilla-tory creation and annihilation of correlations between thepump and probe fields.The oscillatory interchange of noise properties betweenthe two fields is reminiscent to the case of two cou-pled harmonic oscillators [27]. However, in the case pre-sented here, the coupling is realized by a quantum sys-tem, namely the atomic media, which shows some spe-cial properties due to EIT: In terms of mean values,there is no excited state population, and thus the fieldsand the atomic media are uncoupled. Consequently, themean values are stationary, in contrast to the case ofthe two oscillators. Coupling is enabled by the quan-tum fluctuations of the atomic dipole moments, and theground state coherence conveys the interaction betweenthe two fields. Indeed, the generated correlations aremost strongly present when (cid:104) σ (cid:105) is maximal. Apartfrom the coherent dynamics, the atomic media intro-duces noise due to the finite lifetime of the excited statelevel. This is reflected in the decay of the oscillatoryinterchange of noise properties between the fields at acertain time scale.The effect of a detuned two-photon resonance in thepropagation of an initially broad band vacuum squeezedstate as the probe field and a coherent state as the pumpfield, is to rotate the quadrature of maximum squeezing as the field propagates. The velocity of the rotation is afunction of the detuned two-photon resonance and spec-trum frequency. With a frequency-dependent velocity ofrotation, the outcoming field can be completely differentto the incoming field. This frequency-dependent phaseis different to the expected global phase (the same phasefor all frequencies) due to slow velocity propagation in-side the medium.When both fields have comparable Rabi frequencies,with the probe field initially in a squeezed state andthe pump field in a coherent state, the effect of a de-tuned two photon resonance in the propagation can bedescribed as a combination of a rotation of the maxi-mum squeezed quadrature with an interchange of squeez-ing properties between the pump and probe fields. Theeffect of this combination is to redistribute the noise insuch a way that gives rise to mode-dependent propaga-tion distances where all the quadratures have the samenoise. As the field continues to propagate we recover themode squeezed state.Besides the fields’ coherent propagation, the squeezingproperties of the quadratures are absorbed as the fieldpropagates. The absorption spectrum for the squeezingproperties is different to the absorption spectrum for themean values. It is interesting to note that the trans-parency window around spectrum frequency ω = δ isinversely proportional to the detuned two-photon reso-nance and can be very narrow.An important result, is the influence of the atoms’Doppler width on the propagation of a squeezed probefield. Differently to the propagation of the mean values,where the Doppler effect has a small influence, and dueto the considerable effect that detuned two-photon reso-nance has for states propagating in an EIT medium, theatoms’ Doppler width can destroys the initial squeezingproperties of the field as it propagates. Acknowledgments
PBB gratefully acknowledges support from DGAPAand CONACYT. MB gratefully acknowledges sup-port from the Alexander-von-Humboldt foundation.We thank David Sanders for carefully reading themanuscript.
APPENDIX A: ANALYTICAL EXPRESSIONSFOR THE NOISE AND CORRELATIONSPECTRA.
We solve equations (2) with δ = δ = 0 and noisespectrum given by (6), following the treatment used in[19]. We obtain the following analytical results.3
1. The resonance case
For the resonance case, δ = δ = 0, and noise spec-trum given by (6) we obtain the following expressions for the pump S and probe S quadrature noise spectra: S ( z, ω ) = 1( α + α ) (cid:40) e − Q (i) z cos (cid:16) Q ( r ) z (cid:17) α ( f ( ω ) − f ( ω ) + cos(2 θ ) ( g ( ω ) − g ( ω ))) α +2 e − Q ( i ) z α (cid:0) f ( ω ) α + α f ( ω ) + cos(2 θ ) (cid:0) g ( ω ) α + α g ( ω ) (cid:1)(cid:1) +2 (cid:0) f ( ω ) α + α f ( ω ) + cos(2 θ ) (cid:0) g ( ω ) α + α g ( ω ) (cid:1)(cid:1) α (cid:41) + 1 , (A1) S ( z, ω ) = 1( α + α ) (cid:40) e − Q ( i ) z cos (cid:16) Q ( r ) z (cid:17) α α ( − f ( ω ) + f ( ω ) + cos(2 θ ) ( g ( ω ) − g ( ω )))+2 e − Q ( i ) z α (cid:0) f ( ω ) α + α f ( ω ) + cos(2 θ ) (cid:0) g ( ω ) α + α g ( ω ) (cid:1)(cid:1) +2 α (cid:0) f ( ω ) α + α f ( ω ) + cos(2 θ ) (cid:0) g ( ω ) α + α g ( ω ) (cid:1)(cid:1) (cid:41) + 1 . (A2)For the case, g = g and α = α we have for the correlation spectrum S c ( z, ω ) = 12 (cid:0) cos ( θ − θ ) ( f ( ω ) + f ( ω )) + cos ( θ + θ ) ( g ( ω ) + g ( ω ))+2 e − Q (i) z sin (cid:16) Q (r) z (cid:17) (sin ( θ − θ ) ( f ( ω ) − f ( ω )) + sin ( θ + θ ) ( g ( ω ) − g ( ω ))) − e − Q (i) z (cos ( θ − θ ) ( f ( ω ) + f ( ω )) + cos ( θ + θ ) ( g ( ω ) + g ( ω ))) (cid:1) , (A3)with Q = ω C − ω + iωγ/ , (A4a) Q ( r ) ≡ Re Q = ω C Ω − ω [Ω − ω ] + ω γ / , (A4b) Q ( i ) ≡ Im Q = C − ω γ/ − ω ] + ω γ / , (A4c) C ≡ N ( g Ω + g Ω Ω c ) , Ω ≡ (cid:113) Ω + Ω .
2. Detuned two-photon resonance case
For the detuned two-photon resonance case, δ = δ = δ , and assuming that f = g = 0 in Eqs. (6), we obtain S ( z, ω ) = α α ( α + α ) (cid:110) f ( ω ) (cid:104) e Q ( i ) − z + e Q ( i )+ z − e Q ( i ) − z cos Q ( r ) − z − e Q ( i )+ z cos Q ( r )+ z (cid:105) + 2 g ( ω ) (cid:104) cos 2 θ + e Q ( i ) − z + Q ( i )+ z cos( Q ( r ) − z − Q ( r )+ z + 2 θ ) − e Q ( i )+ z cos( Q ( r )+ z − θ ) − e Q ( i ) − z cos( Q ( r ) − z + 2 θ ) (cid:105)(cid:111) , (A5)4 S ( z, ω ) = 1( α + α ) (cid:110) α (cid:104) f ( ω ) (cid:16) e Q ( i ) − z + e Q ( i )+ z (cid:17) + 2 g ( ω ) e Q ( i ) − z + Q ( i )+ z cos( Q ( r ) − z − Q ( r )+ z + 2 θ ) (cid:105) + α α (cid:104) f ( ω ) (cid:16) e Q ( i ) − z cos Q ( r ) − z + e Q ( i )+ z cos Q ( r )+ z (cid:17) + 2 g ( ω ) (cid:16) e Q ( i )+ z cos( Q ( r )+ z − θ ) + e Q ( i ) − z cos( Q ( r ) − z + 2 θ ) (cid:17) (cid:105) + 2 α ( f ( ω ) + g ( ω ) cos 2 θ ) (cid:111) . (A6)For the correlations we have, assuming α = α and g = g S c ( z, ω, θ , θ ) = − e Q ( i ) − z (cid:16) cos ( θ − θ ) ( f ( ω ) + f ( ω )) + 2 sin( Q ( r ) − z ) sin ( θ − θ ) ( f ( ω ) − f ( ω ))+ cos (cid:16) Q ( r ) − z + θ + θ (cid:17) ( g ( ω ) − g ( ω )) (cid:17) + 12 e Q ( i )+ z cos (cid:16) Q ( r )+ z − θ − θ (cid:17) ( g ( ω ) − g ( ω )) − e Q ( i ) − z + Q ( i )+ z cos (cid:16) Q ( r ) − z − Q ( r )+ z + θ + θ (cid:17) ( g ( ω ) + g ( ω ))+ 12 (cos ( θ − θ ) ( f ( ω ) + f ( ω )) + cos ( θ + θ ) ( g ( ω ) + g ( ω ))) (A7)with Q ± = ω C − ω ( ω ± δ ) + iωγ/ Q ( r ) ± ≡ Re Q ± = ω C Ω − ω ( ω ± δ )[Ω − ω ( ω ± δ )] + ω γ / Q ( i ) ± ≡ Im Q ± = C − ω γ/ − ω ( ω ± δ )] + ω γ / C ≡ N ( g Ω + g Ω Ω c )Ω ≡ (cid:113) Ω + Ω [1] S. Harris, Phys. Today , 36 (1997).[2] W. Merkel, H. Mack, M. Freyberger, V. V. Kozlov, W. P.Schleich, and B. W. 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