Phase-dependent fluctuations of resonance fluorescence near the coherent population trapping condition
aa r X i v : . [ qu a n t - ph ] F e b Phase-dependent fluctuations of resonance fluorescence near the coherent populationtrapping condition
O. de los Santos-S´anchez ∗ and H. M. Castro-Beltr´an † Escuela de Ingenier´ıa y Ciencias, Instituto Tecnol´ogico y de Estudios Superiores de Monterrey,Avenida San Carlos 100, Campus Santa Fe, Ciudad de M´exico, 01389, M´exico Centro de Investigaci´on en Ingenier´ıa y Ciencias Aplicadas,Instituto de Investigaci´on en Ciencias B´asicas y Aplicadas, Universidad Aut´onoma del Estado de Morelos,Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico (Dated: February 17, 2021)We study phase-dependent fluctuations of the resonance fluorescence of a single Λ-type three-level atom in the regime near coherent population trapping, i.e., alongside the two-photon detuningcondition. To this end, we employ the method of conditional homodyne detection (CHD) whichconsiders squeezing in the weak driving regime, and extends to non-Gaussian fluctuations for satu-rating and strong fields. In this framework, and using estimated parameter settings of the resonancefluorescence of a single trapped Ba + ion, the light scattered from the probe transitions is foundto manifest a non-classical character and conspicuous asymmetric third-order fluctuations in theamplitude-intensity correlation of CHD. Keywords: resonance fluorescence, coherent population trapping, squeezing, non-Gaussian fluctuations.
I. INTRODUCTION
Quantum interference effects in the interaction be-tween matter and light, epitomized by coherent popu-lation trapping (CPT) and electromagnetically inducedtransparency (EIT), have extensively been studied, boththeoretically and experimentally, over the past decades[1, 2]. The most common level structure to enable theseeffects is the three-level system in the Λ configuration (Λ-3LA). Although early research in this regard was primar-ily focused on ensembles of atomic constituents, state-of-the-art experimental developments in atomic spec-troscopy have made it possible to realize EIT with asingle atom in free space [3]. Indeed, these achievementshave paved the way for exploring new avenues of spectro-scopic analyses, besides their potential applications in thethriving field of quantum information, demonstrating, forinstance, the viability of single-atom-based optical logicgates and quantum memories [4].Both CPT and EIT are based on the cancellationof absorption when two lasers are detuned equally onadjacent transitions, thus stopping further fluorescence.Near this two-photon detuning condition, large quantumfluctuations are thus expected. Phase-sensitive fluctua-tions of the electromagnetic field, usually characterizedby the phenomenon of squeezing, are of particular inter-est. Squeezing is the shrinking of a field’s quadraturefluctuations at the expense of increasing those of its con-jugate, and is signaled by negative spectra or variancebelow the shot noise level. For the resonance fluorescenceof a single two-level atom, squeezing was first predictedalmost forty years ago [5, 6], but it was only very recently ∗ Electronic address: [email protected] † Electronic address: [email protected] that squeezing of a two-level quantum dot was observed[7]. This achievement required overcoming the large col-lection losses of resonance fluorescence and the quantumdetection losses of the standard balanced homodyne de-tection (BHD) technique. These issues were addressed,respectively, by the higher photon collection geometry al-lowed by the quantum dot, and by using a method calledhomodyne correlation measurement (HCM) [8–10].The HCM method realizes an intensity-intensity cor-relation of the light of a previously selected quadrature;by measuring for several phases of a weak local oscillator,the method gives access to the variance (squeezing) [7]and a third-order moment of the field. The latter signalsthe evolution of the field after a photon was detected, aswas demonstrated for the resonance fluorescence of a Λ-3LA [11], in a driving regime not weak enough to obtainsqueezing, and far from EIT. The third-order moment isa reachable step above squeezing in the quest for high-order non-classicality [12, 13].Conditional homodyne detection (CHD) is anothermeasurement scheme capable of detecting phase-dependent fluctuations with high efficiency owing to itsconditional character [14–16]. It consists of BHD on thecue of photons recorded in a separate photodetector, giv-ing direct access to the third-order moment of the field.Squeezing is measured if the source is weakly excited (infact, the first motivation for the scheme) since in this casethe third-order fluctuations of the field are small. How-ever, these fluctuations, non-negligible for stronger exci-tation, are no less interesting: CHD goes beyond squeez-ing [17, 18] and reveals the non-Gaussian character of asource.One manifestation of non-Gaussian fluctuations is theasymmetry of the field’s amplitude-intensity correlationwhenever two or more transitions compete [19]; it is notobserved in the resonance fluorescence of a two- or three-level atom driven by a single laser [17, 18, 20]. While this
FIG. 1: Scheme of the Λ three-level atom with spontaneousdecay rates γ a , γ b , interacting with lasers with Rabi frequen-cies Ω a , Ω b with detunings ∆ a , ∆ b . asymmetry was readily observed for cavity QED systemsboth numerically [14, 19] and experimentally [15], it hasbeen the resonance fluorescence of several 3LA systemsthat have provided clear theoretical access to the under-standing of the asymmetry [21–26]. More recent accountsof asymmetric correlations are found in plasmonics [27]and collective cavity QED [28].In the experiment outlined in [11], squeezing, far fromthe two-photon detuning, was explored in the weak fieldregime. In keeping with the same spirit, quantum fluctu-ations of the light scattered by a coherently driven V-type3LA have thoroughly been analyzed [22]. In this work,near the two-photon detuning, we investigate, within theframework of CHD, the adjoining effect of CPT on thephase-dependent quantum fluctuations of the emittedlight of the probe transition of a Λ-3LA by amplitude-intensity correlations. We follow closely the experimentalconditions of observation of EIT in single Ba + reso-nance fluorescence of Ref. [3], where saturation is present,and we find the fluctuations to be predominantly non-Gaussian.Our work is structured by introducing the atom-lasermodel in Section 2, discussing the role of coherent popu-lation trapping on the state populations and on the emis-sion spectrum; section 3 is devoted to the theory of con-ditional homodyne detection and the analysis of quadra-ture fluctuations via the associated amplitude-intensitycorrelation. We study, in section 4, the quadrature fluc-tuations in the spectral domain, including squeezing andvariance. Finally, in section 5, we present our conclusionsand an appendix shows additional calculations. II. MODELA. Atom-Laser Interaction
Our system, pictorially represented in Fig. 1, consistsof a Λ-type three-level atom (Λ-3LA) with a single ex-cited state | e i coupled by a monochromatic laser withRabi frequency Ω a to the ground state | a i and decayrate γ a , and to a long-lived state | b i by a monochromatic laser with Rabi frequency Ω b and decay rate γ b . Decayfrom | b i to | a i is dipole-forbidden. Henceforth, the fieldsdriving the | a i → | e i and | b i → | e i transitions will bereferred to as the probe and control fields. We define theatomic operators as ˆ σ jk = | j ih k | .Under the above considerations, the system’s evolu-tion, in free space, and in the frame rotating at the laserfrequencies, ν a and ν b , is governed by the master equa-tion ˙˜ ρ ( t ) = − i [ ˆ H, ˜ ρ ] + P j γ j L ˆ σ je [˜ ρ ], in whichˆ H = X j = a,b − ∆ j ˆ σ jj + Ω j σ ej + ˆ σ je ) (1)is the atom-laser Hamiltonian, and ∆ j = ω ej − ν j la-bels the individual atom-laser detunings. Dissipation isaccounted for by the action of the Lindblad generator L ˆ O [˜ ρ ] = 2 ˆ O ˜ ρ ˆ O † − ˆ O † ˆ O ˜ ρ − ˜ ρ ˆ O † ˆ O , with ˆ O = ˆ σ je . Withthe help of the relationship ˆ σ jk ˆ σ lm = ˆ σ jm δ kl , the masterequation can be explicitly recast as˙˜ ρ ( t ) = − i [ ˆ H, ˜ ρ ] + X j = a,b γ j ˜ ρ ee ˆ σ jj − γ j σ ee ˜ ρ + ˜ ρ ˆ σ ee ) . (2)With the relation h ˆ σ jk i = ˜ ρ kj , Eq. (2) allows us to arriveat the following set of linear equations for populations: h ˙ˆ σ aa i = − i Ω a h ˆ σ ae i − h ˆ σ ea i ) + γ a h ˆ σ ee i , (3) h ˙ˆ σ bb i = − i Ω b h ˆ σ be i − h ˆ σ eb i ) + γ b h ˆ σ ee i , (4) h ˙ˆ σ ee i = i Ω a h ˆ σ ae i − h ˆ σ ea i ) + i Ω b h ˆ σ be i − h ˆ σ eb i ) − γ h ˆ σ ee i , (5)with γ = γ a + γ b , and coherences: h ˙ˆ σ ab i = i Ω a h ˆ σ eb i − i Ω b h ˆ σ ae i − i (∆ a − ∆ b ) h ˆ σ ab i , (6) h ˙ˆ σ ae i = i Ω a h ˆ σ ee i − h ˆ σ aa i ) − i Ω b h ˆ σ ab i− (cid:16) γ i ∆ a (cid:17) h ˆ σ ae i , (7) h ˙ˆ σ be i = − i Ω a h ˆ σ ba i + i Ω b h ˆ σ ee i − h ˆ σ bb i ) − (cid:16) γ i ∆ b (cid:17) h ˆ σ be i , (8) h ˙ˆ σ jk i = h ˙ˆ σ kj i ∗ . (9)The solution to these equations (a set of nine Blochequations) is to be obtained numerically and their struc-ture will facilitate the assessment of the sought correla-tion functions via the quantum regression formula, com-bined with the employment of matrix methods. For lateruse, we define the values of the atomic operators in thesteady state as h ˆ σ jk ( t → ∞ ) i = h ˆ σ jk i ss = α jk . (10) FIG. 2: Upper panel: Occupation probability of the excitedstate, h σ ee i ss , versus the scaled detuning ∆ a /γ a and Rabifrequency Ω a /γ a . Lower panel: Populations h σ jj i ss , for j = a, b and e (blue, red and black lines, respectively) as functionsof ∆ a /γ a ; the black curve is the corresponding cross sectionalprofile of the upper figure at Ω a /γ a ≈ .
12. The remainingparameters are Ω b /γ a ≈ .
15 and ∆ b /γ a ≈ . Besides, in order for our findings to be possibly putto the test in a given realization, we shall consider thedecays γ a = 14 . γ b = 5 . Ba + ions [3, 11, 29]. Although a more accurate de-scription of Barium resonance fluorescence would entailconsidering its multilevel structure, being composed ofeight energy levels, it suffices for our purposes to dealwith the simplified three-level system as a proxy for spec-ifying the relevant allowed dipole transitions that takepart in the dynamics. Parenthetically, the isolation ofa single three-level configuration can be implementedthrough a proper optical pumping arrangement. B. Role of Coherent Population Trapping
The Λ-type three-level atom is an archetypal systemthat readily fulfills the necessary conditions for coher-ent population trapping (CPT) to take place [1, 2]. Insuch a scenario, the system is known to evolve towards the trapping state | u i = (Ω b | a i − Ω a | b i ) / p Ω a + Ω b thatturns out to be decoupled from the lasers, thereby drop-ping the long-term excited-state population α ee to nearlyzero. The manifestation of this effect is exemplified in theupper panel of Fig. 2, where the steady state populationof the excited state is shown as a function of both thedetuning and Rabi frequency of the probe laser ( e → a transition); the values of the parameters associated withthe control field are, henceforth, taken to be fixed and thesame as those reported in [3], namely, Ω b /γ a ≈ .
15 and∆ b /γ a ≈ .
38. In accord with the well-established pre-scription to determine the frequency region around whichthe atom is essentially transparent to the incoming probefield, the so-called Raman resonance condition, the probedetuning must be such that ∆ a ≈ ∆ b is satisfied; the roleof the probe intensity Ω a is that of slightly modifying thewidth of such a transparent frequency window. Its loca-tion is also depicted in the lower panel of Fig. 2 showingthe cross sectional profile of the upper figure (black line)at Ω a /γ a ≈ .
12 where, in turn, we can observe the com-plete depopulation of the excited state at ∆ a /γ a ≈ . | a i and | b i states are also added as a supplementary view of theirbehavior as functions of the probe detuning.The foregoing was not the actual condition underwhich the experiments [3] were performed, but insteadthe detuning was chosen so as to fit the value of the corre-sponding saturation parameter, Ω j / ( γ j + ∆ j ), and takento be ∼ . a → e transition. This choice gives riseto a detuning of about ∆ a /γ a ≈ .
4, the location of whichbeing also indicated in the figure (dotted-dashed verticalline); the saturation parameter associated with the con-trol field was set to 0 .
8. So, for this particular choice ofprobe and control detunings that drive the a → e transi-tion out of the Raman resonance condition, the completedepopulation of the | e i state can be avoided or delayed,a working situation that will permit us to study the non-classical properties of the scattered light we seek to as-sess. It is worth commenting that if, instead, the a → e transition were driven more strongly than the b → e one,such that Ω a > Ω b , for general detunings, the populationwould end up in the | b i state, with Ω b ≪ γ a . So then,the strong transition would be turned off due to lack ofrecycling population to | e i .So, having established the present configuration oflaser intensities and frequencies, we find it pertinent,at this stage, to depict the stationary power spectrumof the re-emitted light obtained by use of the Wiener-Khintchine formula S ( ω ) = 1 πα ee Re Z ∞ dτ e − iωτ h ˆ σ ea (0)ˆ σ ae ( τ ) i ss , (11)i.e., the Fourier transform of the autocorrelation functionof the dipole field, h ˆ σ ea (0)ˆ σ ae ( τ ) i ss , where ss indicatesthat the process is stationary; the prefactor ( πα ee ) − normalizes the integral over all frequencies. For conve-nience, the spectrum is separated into its coherent and FIG. 3: Incoherent spectrum, S inc ( ω ), of the e → a transitionas a function of the scaled probe laser intensity Ω a /γ a (upperpanel, for ∆ a /γ a ≈ .
4) and the detuning ∆ a /γ a (lower panel,for Ω a /γ a ≈ . γ a . The remaining parameters are thesame as in Fig. 2. incoherent parts, namely, S ( ω ) = S coh ( ω ) + S inc ( ω ), asa result of considering the dynamics of the atomic vari-ables to be split into their mean and fluctuations, viz.ˆ σ jk ( t ) = α jk + ∆ˆ σ jk ( t ), with h ∆ˆ σ jk ( t ) i = 0. In doing so,we get S coh ( ω ) = | α ea | πα ee Re Z ∞ dτ e − iωτ = | α ea | πα ee δ ( ω ) , (12)where α ea = h ˆ σ ea i ss , and S inc ( ω ) = 1 πα ee Re Z ∞ dτ e − iωτ h ∆ˆ σ ea (0)∆ˆ σ ae ( τ ) i ss , (13)the former being the coherent constituent of the spectrumowing to elastic scattering, and the latter the incoherentpart of the spectrum that is brought about by atomicfluctuations. The main features of the Λ-type three-levelatom spectrum have already been studied from the weakto the strong field limit, both theoretically and experi-mentally [30]. Figure 3 shows a three dimensional viewof the incoherent part of the spectrum associated with the e → a transition, the one of interest to us, as a functionof the probe intensity (upper panel) and the detuning(lower panel); details of the steps involved in the calcula-tions herein via the matrix analysis are included in Ap- FIG. 4: Scheme of conditional homodyne detection (CHD).It features a balance homodyne detection setup to assess thequadrature of the field on the condition of photon detectionvia detector D I . BS and LO stand for beam splitter and localoscillator, respectively. pendix A. The general spectral profile can be understoodin terms of dressed-state configuration that follows fromproperly diagonalizing the atom-field Hamiltonian [31],emphasizing the fact that, above saturation, the spec-trum displays the appearance of Rabi sidebands as theintensity field increases, as one can see in the upper panelof the figure. By setting the Rabi frequency at, say,Ω a /γ a = 1 .
12, such sidebands become sufficiently con-spicuous and the dependency of their profile upon thedetuning is shown in the lower panel.
III. AMPLITUDE-INTENSITY CORRELATION
In this section we present the theory of conditional ho-modyne detection (CHD) in order to assess and discussthe time-asymmetry, the non-Gaussianity and the non-classicality of the light scattered from the atomic systemunder study. In the next section we move to the fre-quency domain. The CHD setup is sketched in Fig. 4.In one arm of the setup a quadrature of the source light, E φ ∝ ˆ σ φ = (ˆ σ ea e − iφ + ˆ σ ae e iφ ) /
2, is analyzed in bal-anced homodyne detection (BHD), where φ is the phaseof the local oscillator (LO). This signal has a delay τ withrespect to the measurement of the source’s intensity inanother arm, proportional to the excited-state popula-tion, I ∝ h ˆ σ ea ˆ σ ae i = h ˆ σ ee i . Thus, the outcome is anamplitude-intensity correlation that reads h φ ( τ ) = h : ˆ σ ea (0)ˆ σ ae (0)ˆ σ φ ( τ ) : i ss α ee α φ , (14)where the dots :: indicate normal and time operator or-dering, and α φ = ( α ea e − iφ + α ae e iφ ) / - - - Γ a Τ h Φ H Τ L FIG. 5: Amplitude-intensity correlations of light from the e → a transition, as a function of the scaled time γ a τ , for φ = 0 (continuous line) and φ = π/ a ≈ . γ a , Ω b ≈ . γ a ∆ a = 3 . γ a ,∆ b = 2 . γ a . A. Time-asymmetry and Non-Gaussianity
Resonance fluorescence is a highly non-linear process,preventing its description in terms of quasi-probabilitydistributions, i.e., it does not admit a Fokker-Plancktype of equation. The non-linearity leads to non-Gaussian fluctuations, thus giving rise to non-vanishingodd-order moments. Autocorrelation functions suchas that for the spectrum, h σ ea (0) σ ae ( τ ) i ; squeezing, h ∆ σ φ (0)∆ σ φ ( τ ) i ; and photon-photon correlation, h σ ea (0) σ ea ( τ ) σ ae ( τ ) σ ae (0) i , are of even-order and, assuch, time-symmetric [19]. These functions do notaddress the non-Gaussianity of the field’s fluctuations.In amplitude-intensity correlations, Eq.(14), on theother hand, such a symmetry is not guaranteed: beingdifferent observables, the outcome will be dependenton the time order of measurements. For instance, thequadrature is measured (preselected) for τ ≥ τ ≤ h φ ( τ ≥
0) = h ˆ σ ea (0)ˆ σ φ ( τ )ˆ σ ae (0) i ss α ee α φ , (15) h φ ( τ ≤
0) = Re[ e − iφ h ˆ σ ea (0)ˆ σ ee ( τ − ) i ss ] α ee α φ . (16)The asymmetry in time revealed by CHD, as shown inFig. 5, is an indicative of non-Gaussian noise. The corre-lation (14) contains a product of three dipole operatorsor, more generally, three field amplitude operators. This means that h φ ( τ ) provides access up to third order fluc-tuations; since these are non-Gaussian, this third-ordercorrelation does not vanish. To better distinguish theasymmetry and the size of these fluctuations, we proceed,as we did with the spectrum, to split the dipole dynamicsinto its mean plus fluctuations, ˆ σ jk = α φ + ∆ˆ σ jk [17], h φ ( τ ) = 1 + h (2) φ ( τ ) + h (3) φ ( τ ) , (17)where h (2) φ ( τ ) = h : [ α ea ∆ˆ σ ae (0) + α ae ∆ˆ σ ea (0)]∆ˆ σ φ ( τ ) : i ss α ee α φ , (18)and h (3) φ ( τ ) = h : ∆ˆ σ ea (0)∆ˆ σ ae (0)∆ˆ σ φ ( τ ) : i ss α ee α φ , (19)are the components of, respectively, second- and third-order in the dipole fluctuations of h φ ( τ ), where ∆ˆ σ φ =(∆ˆ σ ea e − iφ +∆ˆ σ ae e iφ ) / h (3) φ ( τ ) →
0, which can occur when the transition is weakly driven.For positive time intervals between photon and quadra-ture detections, we get h (2) φ ( τ ≥
0) = 2Re[ α ae h ∆ˆ σ ea (0)∆ˆ σ φ ( τ ) i st ] α ee α φ , (20) h (3) φ ( τ ≥
0) = h ∆ˆ σ ea (0)∆ˆ σ φ ( τ )∆ˆ σ ae (0) i st α ee α φ . (21)We show in Fig. 6 the foregoing second- and third-ordercorrelations. For both quadratures, the third-order con-stituent (blue lines) represents the main contribution, al-most that of the total h φ ( τ ) (black lines). This is under-standable from the fact that we are above the saturationthreshold [22], a regime where non-Gaussian fluctuationsbecome significant. In this regime the dipole α ea , indica-tive of the coherence induced by the laser, is small; mostof the total emission is incoherent. This observation canbe quantitatively revealed from the fact that h φ (0) = 0(just as it occurs for photon correlations in resonancefluorescence), which leads to the relation [22] h (3) φ (0) = − h h (2) φ (0) i = 2( | α ea | − α ee ) α ee . (22)For strong fields, | α ea | ≪ α ee , h (3) φ (0) reaches its extremalvalue -2, thereby making the dipole factor in Eq.(20)small compared to the third-order term.Even though the splitting itself cannot be directly re-alizable from the experimental viewpoint via the mea-surement scheme, it provides us with valuable theoreticalinformation to be able to discern the actual contributionto the system’s fluctuations.For τ <
0, we want to stress the fact that the outcomeof CHD correlation should be taken with special care: itis to be interpreted as the measurement of the intensity (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:6)(cid:2) (cid:6)(cid:3) - (cid:6)(cid:7)(cid:1) - (cid:1)(cid:7)(cid:8)(cid:1)(cid:7)(cid:1)(cid:1)(cid:7)(cid:8) γ (cid:1) τ (cid:1) (cid:1) ( (cid:2) ) ( τ ) (cid:1) (cid:1) (cid:1) ( (cid:3) ) ( τ ) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:6)(cid:2) (cid:6)(cid:3) - (cid:6)(cid:7)(cid:8) - (cid:6)(cid:7)(cid:1) - (cid:1)(cid:7)(cid:8)(cid:1)(cid:7)(cid:1)(cid:1)(cid:7)(cid:8)(cid:6)(cid:7)(cid:1)(cid:6)(cid:7)(cid:8)(cid:2)(cid:7)(cid:1) γ (cid:1) τ (cid:1) π / (cid:1) ( (cid:1) ) ( τ ) (cid:1) (cid:1) π / (cid:1) ( (cid:2) ) ( τ ) FIG. 6: Splitting of the intensity-field correlations shown inFig. 5 into their second- ( h (2) φ ( τ ), brown line) and third-order( h (3) φ ( τ ), blue line) constituents, for φ = 0 (upper panel) and φ = π/ h (2) φ ( τ ) + h (3) φ ( τ ). The param-eters are the same as those of Fig. 5. after the detection of the amplitude. Thus, as previouslyunderlined, the asymmetry results from the different fluc-tuations of the light’s amplitude and intensity. Time andnormal operator ordering leads to h φ ( τ ≤
0) = 1 + Re[ e − iφ h ∆ˆ σ ea (0)∆ˆ σ ee ( | τ | ) i st ] α ee α φ , (23)i. e., the correlation is only of second order in the dipolefluctuations, albeit with ∆ˆ σ ee instead of the quadratureamplitude ∆ˆ σ φ fluctuation operator. B. Non-classicality
The initial motivation for CHD was to detect squeezingfrom weak sources, such as cavity QED [14, 15]. Reso-nance fluorescence is also a producer of weakly squeezedlight [5, 6]. In order to produce light in a squeezed state,a non-classical property of light, these sources must beweakly driven, so that the third-order fluctuations dis-cussed above are small. As we will see later, the re-maining second-order signal is related to the spectrumof squeezing. CHD, hence, gives non-classical criteria in the time domain as resulting of violation of the classicalinequalities [14, 15] 0 ≤ h φ ( τ ) − ≤ , (24) | h (2) φ ( τ ) | ≤ | h (2) φ (0) | ≤ , (25)where the second relation is derived for Gaussian fluctua-tions. More recently, it was found that light in a coherentstate obeys [22] − ≤ h φ ( τ ) ≤
1; (26)light outside these bounds violates Poissonian statistics.According to these criteria, we see in Figs. 5 and 6 thatboth the in-phase ( φ = 0, continuous line) and out-of-phase ( φ = π/
2, dashed line) quadratures of the fielddisplay a non-classical character, violating one or moreinequalities. The fact that h φ (0) = 0 already shows anon-classical feature, akin to antibunching in the inten-sity fluctuations. Also, moderately strong fields easilydrive h φ ( τ ) out of the classical bounds.We see, then, that CHD clearly reveals non-classicalityof quadratures in the time domain. Let us now proceedto scrutinize the spectral profile of amplitude-intensitycorrelations in the frequency domain. IV. QUADRATURE SPECTRA
Since in CHD the signal is time-asymmetric, carryingdifferent information for positive and negative intervals,the spectra of quadratures measured from the amplitude-intensity correlation should be calculated separately [22]: S ( τ ≥ φ ( ω ) = 4 γ a α ee Z ∞ dτ cos ωτ [ h φ ( τ ≥ − , (27) S ( τ ≤ φ = 4 γ a α ee Z −∞ dτ cos( ωτ )[ h φ ( τ ≤ − , (28)for positive and negative time intervals, respectively. Theprefactor γ a α ee is the photon emission rate in the probetransition. In Fig. 7 we show the spectra calculated fromEqs. (27) and (28) for both quadratures and the same pa-rameter values of Fig. 5. From the CHD viewpoint, nega-tive values of the spectrum are signature of non-classicalscattered light beyond squeezing, which is confirmed forboth quadratures, with the π/ π/ h φ ( τ ≥ S ( N ) φ ( ω ) = 4 γ a α ee Z ∞ dτ cos ωτ h ( N ) φ ( τ ) , (29) - (cid:1) (cid:2) (cid:1) - (cid:3)(cid:4)(cid:2) - (cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:2)(cid:2)(cid:4)(cid:1) ω/ γ (cid:1) (cid:1) ϕ =( (cid:1)(cid:2) (cid:2) / (cid:3) )( τ≥ (cid:1) ) ( ω ) - (cid:1) (cid:2) (cid:1) - (cid:3)(cid:4)(cid:2) - (cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:2)(cid:2)(cid:4)(cid:1)(cid:3)(cid:4)(cid:2) ω/ γ (cid:1) (cid:1) ϕ =( (cid:1)(cid:2) π / (cid:3) )( τ≤ (cid:1) ) ( ω ) FIG. 7: Spectra, Eqs. (27) and (28), upper and lower panels,respectively, for φ = 0 (continuous line) and π/ for N = 2 ,
3, so that S ( τ ≥ φ ( ω ) = S (2) φ ( ω ) + S (3) φ ( ω ).These are shown in Fig. 9 for both quadratures, corre-sponding to the CHD signals of Fig. 6. We find that thesecond-order spectra are mostly positive, while the third-order spectrum is negative for φ = 0, there are negativebands for φ = π/
2. In Fig. 10 the dependence of S ( N ) π/ onthe detuning of the probe laser is shown. A quite similarspectral landscape (not shown) was found in the second-order correlation for τ ≤
0, Eq. (28). The lineshapes arevery complicated, but the dispersive features at the sidesreveal the non-Gaussianity of the field [20, 22].The above spectra clearly deviate from the more con-ventional measure of non-classical phase-dependent fluc-tuations, squeezing, due to the non-linearity induced bythe strong lasers. Understood operationally as the re-duction of quantum fluctuations below the shot noiselimit, squeezing can be obtained in the spectral domainas the Fourier transform of symmetric photocurrent fluc-tuations in homodyne detection [33]. For our source, S φ ( ω ) = 8 γ a η Z ∞ dτ cos ωτ h : ∆ˆ σ φ (0)∆ˆ σ φ ( τ ) : i ss , = 8 γ a η Z ∞ dτ cos ωτ × Re (cid:2) e − iφ h ∆ˆ σ ea (0)∆ˆ σ φ ( τ ) i ss (cid:3) , (30)where η is a combined collection and detection efficiency, FIG. 8: Fourier cosine transform of h ( N ) π/ ( τ ), for τ ≥ τ ≤ a /γ a , for Ω b ≈ . γ a ∆ a = 3 . γ a and ∆ b =2 . γ a . and the dots :: state that the operators must follow timeand normal orderings. It was shown in [14, 15] that,in the weak-field limit, when third-order fluctuations canbe neglected, the second-order spectrum from CHD, fromEqs. (29) and (20), is indeed the spectrum of squeezing,but unafected by detector losses, i.e., S (2) φ ( ω ) = S φ ( ω ) /η ,owing to the conditional character of CHD.Figure 11 displays a 3D plot of the spectrum of squeez-ing, given by Eq. (30) with η = 1, as a function of theprobe laser intensity Ω a /γ a (upper panel) and detuning∆ a /γ a (lower panel), for the φ = π/ a /γ a = 2 .
38, around which CPT takes place, even forlaser intensities above saturation. A slightly higher de-gree of squeezing comes about within certain regions ofthe spectrum by fixing the laser intensity, at Ω a /γ a = 0 . V φ = h : (∆ σ φ ) : i st = Re (cid:2) e − iφ h ∆ σ eg ∆ σ φ i ss (cid:3) , (31)related to the integrated spectrum as R ∞−∞ S φ ( ω ) dω =4 πγ a ηV φ . This quantity is depicted in Fig. 12 for φ = π/ a /γ a ≈ .
38, is found to reduce - (cid:1) (cid:2) (cid:1) - (cid:2)(cid:3)(cid:4) - (cid:2)(cid:3)(cid:5) - (cid:2)(cid:3)(cid:6)(cid:2)(cid:3)(cid:2)(cid:2)(cid:3)(cid:6)(cid:2)(cid:3)(cid:5)(cid:2)(cid:3)(cid:4)(cid:2)(cid:3)(cid:7) ω/ γ (cid:1) (cid:1) (cid:1) ( (cid:2)(cid:3)(cid:4) ) ( ω ) - (cid:1) (cid:2) (cid:1) - (cid:3)(cid:4)(cid:2) - (cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:2)(cid:2)(cid:4)(cid:1) ω/ γ (cid:1) (cid:1) π / (cid:1) ( (cid:1)(cid:2)(cid:3) ) ( ω ) FIG. 9: Fourier cosine transform of h ( N ) φ ( τ ≥ N = 2 (brown) and N = 3 (blue). Upper and lower panelscorrespond, respectively, to the cases φ = 0 and π/
2. Theparameters are the same as those of Fig. 5. fluctuations, approximately, to the extent of a coherentstate. It was also verified that the in-phase quadrature(not shown) did not feature squeezed fluctuations in anyparameter regime of the aforesaid transition.
V. CONCLUSIONS
Using the framework of conditional homodyne detec-tion, we have analyzed the nearby effect of coherent pop-ulation trapping on the phase-dependent quantum fluc-tuations, in both time and frequency domains, of thelight fluoresced in the probe transition from a coher-ently driven Λ-type three-level atom. Given the feasibil-ity of implementing the outlined optical system, a single Ba + ion [3, 11], our findings are expected to bolsterfurther experimental investigations to be benchmarkedagainst CHD-based theoretical predictions.It is worth underlying that the CHD framework provesto be a versatile tool to discern the contribution of phase-dependent fluctuations of different orders, concludingthat the light scattered under the aforesaid conditionsis essentially non-Gaussian; i.e., the correlation of third-order in the dipole fluctuation operators prevails. Non-Gaussianity, notably, manifests in two main ways. Onthe one hand, the amplitude-intensity correlation is, ingeneral, time-asymmetric, indicating that amplitude and FIG. 10: Fourier transform of h (2 , π/ ( τ ≥ a /γ a , for Ω a /γ a ≈ .
12. The set ofparameters is the same as in Fig. 5.FIG. 11: Spectrum of squeezing of the φ = π/ a /γ a = 2 .
38) and detuning (lower panel, for Ω a /γ a = 0 . b ≈ . γ a and ∆ b = 2 . γ a . intensity of the radiated field have different noise proper-ties. On the other, the non-linearity imposed by a satu-rating excitation regime leads to fluctuations away fromthe ideal weak-field squeezing regime.The role of CPT in CHD is explored with particular FIG. 12: Variance for φ = π/
2, as a function of the scaledRabi frequency Ω a /γ a . For both, Ω b ≈ . γ a and ∆ b =2 . γ a . focus on the spectra of quadratures. In this regard, asa function of the probe detuning, the spectral contentconfirms once again the prevailing contribution of third-order fluctuations to the outcome of the measurements,for both quadratures. This fact is also reinforced by ex-amining the variance (the integrated spectrum) of fluo-rescence. VI. ACKNOWLEDGMENTS
The authors thank Dr. Ir´an Ramos-Prieto for usefulcoversations and help with the figures.
Appendix A: Correlations and Spectra
Here, we succinctly describe the evaluation of the ex-pectation values of two-time correlations and spectra used throughout this work.From the equations of motion of the atomic op-erators, Eqs. (3) to (9), which can be put intothe concise form ˙ s ( t ) = Ms ( t ), with s = { σ ee , σ ae , σ be , σ ea , σ aa , σ ba , σ eb , σ ab , σ bb } T and M the pa-rameter matrix (to be specified), together with the use ofthe quantum regression formula [32], we seek the generalsolution to the equation ∂ τ g ( τ ) = Mg ( τ ) , (A1)where g ( τ ) = h ∆ σ ea (0)∆ s ( τ )∆ A − (0) i ss is the corre-sponding vector of correlation functions. For the second-order correlations, ∆ A − = ; for the third-order ones,∆ A − = ∆ σ ae . Its solution can be written in the form g ( τ ) = e M τ g (0), where the initial condition g (0) = g ss ,given in terms of the steady state solution of populationsand coherences, is solved numerically.The incoherent spectrum requires, for instance, han-dling the time dependence of the correlation { g ( τ ) } m = h ∆ σ ea (0)∆ σ ae ( τ ) i ss , where the subindex m -th denotesthe element of the vector to be taken. The present matrixanalysis saves the work of solving the correlation explic-itly followed by time integration, namely, for ∆ A − = 1, S inc ( ω ) = 1 πα ee Re (cid:26)Z ∞ dτ e − ( iω − M ) τ h ∆ σ ea ∆ s i ss (cid:27) m = 1 πα ee Re n − ( iω − M ) − e − ( iω − M ) τ | ∞ ×h ∆ σ ea ∆ s i ss } m = 1 πα ee Re (cid:8) ( iω − M ) − h ∆ σ ea ∆ s i ss (cid:9) m , where is the n × n identity matrix. The spectra corre-sponding to the CHD correlations are calculated in thesame manner, thus giving us the sought results S (2) φ ( ω ) = 2 γ a α φ Re (cid:8) α ae e − iφ (cid:2) ( iω − M ) − − ( iω + M ) − ) g (0) (cid:3) m (cid:9) + 2 γ a α φ Re (cid:8) α ae e iφ (cid:2) ( iω − M ) − − ( iω + M ) − ) g (0) (cid:3) n (cid:9) ,S (3) φ ( ω ) = 2 γ a α φ Re { e − iφ (cid:2) ( iω − M ) − − ( iω + M ) − ) g (0) (cid:3) p } , for the second- and third-order fluctuations, respectively, and S ( τ ≤ j,φ ( ω ) = 2 γ j α φ Re { e − iφ [(( iω − M ) − − ( iω + M ) − ) g (0)] q } for fluctuations associated with negative time intervals. The elements of the vectors, denoted by subindexes m, n, p and q , have to be chosen appropriately to match the corresponding correlation it seeks to assess.For the sake of completeness, the initial conditions of the correlations ( τ = 0) are encapsulated by using the0fluctuation operator approach as h ∆ σ ij ∆ σ kl i ss = α il δ jk − α ij α kl , h ∆ σ ig ∆ σ jk ∆ σ gi i ss = 2 | α ig | α jk + α ii ( δ gj δ kg − α jk ) − α ik α gi δ gj − α ig α ji δ kg , for the second- and third-order fluctuations, respectively. More explicitly, for the e → a transition, they become h ∆ σ ea ∆ s i ss = (cid:8) − α ea α ee , α ee − α ea α ae , − α ea α be , − α ea , α ea (1 − α aa ) , − α ea α ba − α ea α eb , α eb − α ea α ab , − α ea α bb (cid:9) T , and h ∆ σ ea ∆ σ ee ∆ σ ae i ss h ∆ σ ea ∆ σ ae ∆ σ ae i ss h ∆ σ ea ∆ σ be ∆ σ ae i ss h ∆ σ ea ∆ σ ea ∆ σ ae i ss h ∆ σ ea ∆ σ aa ∆ σ ae i ss h ∆ σ ea ∆ σ ba ∆ σ ae i ss h ∆ σ ea ∆ σ eb ∆ σ ae i ss h ∆ σ ea ∆ σ ab ∆ σ ae i ss h ∆ σ ea ∆ σ bb ∆ σ ae i ss = α ee (2 | α ea | − α ee ) − α ae ( α ee − | α ea | ) α be (2 | α ea | − α ee )2 α ea ( | α ea | − α ee )(2 | α ea | − α ee )( α aa − α ba (2 | α ea | − α ee ) − α ea α be α eb (2 | α ea | − α ee ) α ab (2 | α ea | − α ee ) − α eb α ae α bb (2 | α ea | − α ee ) . [1] E. Arimondo, Progress in Optics, , 257 (1996).[2] M. Fleischhauer, A. Imamoglu and J. P. Marangos, Rev.Mod. Phys.
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