Tunable photon-assisted bleaching of three-level systems for noise filtration
Chun-Hsu Su, Andrew D. Greentree, Raymond G. Beausoleil, Lloyd C. L. Hollenberg, William J. Munro, Kae Nemoto, Timothy P. Spiller
aa r X i v : . [ qu a n t - ph ] F e b Tunable photon-assisted bleaching of three-level systems for noise filtration
Chun-Hsu Su, ∗ Andrew D. Greentree, Raymond G. Beausoleil, Lloyd C.L. Hollenberg, William J. Munro, Kae Nemoto, and Timothy P. Spiller Centre for Quantum Computer Technology, School of Physics,The University of Melbourne, Victoria 3010, Australia School of Physics, The University of Melbourne, Victoria 3010, Australia Information and Quantum Systems Lab, Hewlett-Packard Laboratories,1501 Page Mill Road, MS1123, Palo Alto, California 94304, USA NTT Basic Research Laboratories, NTT Corporation,3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa-ken 243-0198, Japan National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Quantum Information Science, School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK (Dated: July 3, 2018)Electromagnetically-induced transparency (EIT) exploits quantum coherence to burn subnaturallinewidth holes within a spectral line. We investigate the less explored properties of EIT to effectabsorptive nonlinear processes without restrictions on the relative intensities of pump and probefields. We show that a three-level medium under imperfect EIT conditions can generate a formof bleaching that is qualitatively similar to two-state saturable absorption. This scheme has theadvantages of greater sensitivity to signal intensity and controllability over its bleaching intensitylevel post-fabrication. Such effects could prove useful for noise filtration at very low light levels.
PACS numbers: 32.30.Jc, 42.50.Gy, 42.65.An
I. INTRODUCTION
An ability to control systems down to the limits im-posed by quantum mechanics is expected by many tousher a new era in technology, the so-called ‘second quan-tum revolution’ [1]. Although often one is interested inharnessing effects not possible, or not practical to achievewith classical physics, quantum processes can also beused to enhance classical processes, and often providelimiting cases in terms of fidelity or power for classical ef-fects. One of the most important and well-studied fieldsshowing quantum control is nonlinear optics. Coherentlyprepared atoms or atomic systems, often under condi-tions of electromagnetically-induced transparency (EIT),can be shown to exhibit large, lossless nonlinearities [2]that can be useful for quantum logic [3–5]. Here we turnour attention to a nonlinearity that is less explored in thequantum space, namely absorptive nonlinearity.Saturable absorption is a well-known nonlinear pro-cess. An optical field impinges on a medium, usuallytreated as an ensemble of two-state systems [6, 7]. Forlow intensities, the field is greatly attenuated, but at highfields, as the medium is saturated, the absorption is rela-tively low. This effect is useful for many practical tasks,including analog-to-digital conversion [8], passive mode-locking of laser radiation [9] and sub-diffraction imagingin confocal fluorescence microscopy [10].In this work, we analyse EIT-based absorptive non-linearities in detail. Usually EIT is used as a means ofgenerating a medium with zero absorption and high dis- ∗ Electronic address: [email protected] persion, which can be the enabler of large lossless opti-cal nonlinearities [2]. EIT employs two coherent fields(e.g., signal and pump) driving a Raman transition be-tween two, long-lived ground states via a shared excitedstate, in the Λ configuration, as depicted in Fig. 1(a).When the ground-state decoherence is absent, quantuminterference is complete and a long-lived dark state is ob-served, which manifests ideally as a perfect transparencywindow at the two-photon resonance point. The associ-ated steep dispersion has recently been shown to allowsub-nanometer-scale resolution for microscopy, surpass-ing the the two-level system based technique [11].Ground-state dephasing plays a critical role in practi-cal EIT schemes. In general, the coherence responsiblefor EIT must compete with decoherence, and this ulti-mately limits the depth and width of the transparencywindow. However, this competition allows for saturableabsorption. Kocharovskaya et al. have shown that theabsorptive nonlinearity can occur even when the radia-tion intensity is much less than that needed for saturationof the transition [12–14]. The effect is termed coherentbleaching.Here we investigate this bleaching effect under con-ditions that are less restrictive than those consideredpreviously in Refs. [12–14]. In particular we show thatthe three-state system provides increased flexibility overthe onset of the nonlinearity with the signal absorptiontunable with the pump field post fabrication, typicallyabsent in two-state schemes. This is distinct from sat-uration and coherent bleaching because photon-assistednonlinearity occurs even when both the coherent and sat-uration nonlinearities are weak. Moreoever the absorp-tion scaling with intensity can be quadratic under certainconditions, and thus can provide a greater sensitivity for
FIG. 1: (a) Three-level Λ-type atoms as a nonlinear filterwhose bleaching intensity level can be tuned with the pumpfield. The signal (pump) field couples to the | a i−| b i ( | b i−| c i )transition with coupling rate Ω s (Ω p ). The excited state | b i decays to the pair of ground or metastable states at a totalrate 2Γ. The low-frequency coherence dissipates at a dephas-ing rate γ , which is responsible for the absorption in the EITwindow. (b) Proposed configuration in solid-state environ-ment: atomic systems along a single-mode waveguide con-structed from a photonic-bandgap lattice, interacting withthe signal beam and an uniform pump field. (c) An alterna-tive implementation with copropagating fields using atomicvapour cell. applications such as noise filtration and analog-to-digitalconversion applications. Given that our focus is on thisenhanced absorption properties, this work is also distinctfrom the effect of tunable electromagnetically-inducedabsorption in doubly dressed two-level systems [15].This paper is organized as follows: In Sec. II we brieflyreview EIT and describe the absorptive nonlinearities un-der various strength of spontaneous decay, dephasing andcoherent couplings in the steady state. In Sec. III we con-sider propagating fields in optically-thick media in twopossible signal-pump arrangements shown in Fig. 1(b,c).We also explore their application for noise filtration. Fi-nally in Sec. IV we briefly explore some possible imple-mentations of our scheme. II. OPTICAL RESPONSE
The essential features of EIT and its application tomodifying the linear and nonlinear optical properties ofa light field can be quantitatively described using a semi-classical analysis [16]. We consider a dilute ensemble ofidentical Λ-type atomlike systems. Our model Λ-typeatom is depicted schematically in Fig. 1(a). Transitionsbetween the excited state | b i and the two ground states | a i and | c i are dipole allowed. A near-resonant pumpfield of frequency ω p is applied on the | c i − | b i transitionwith detuning ∆ = ( ω bc − ω p ) and Rabi frequency Ω p .The | a i − | b i transition is driven by the signal field, withfrequency ω s , detuning (∆+ δ ) = ( ω ab − ω s ) and Rabi fre-quency Ω s , where δ is the two-photon detuning betweenthe low energy states. We study the absorption of the signal field (Ω s ) ( e.g., for signal processing applications)and allow the signal to be more intense than the couplingfield (Ω p ). Hence our analysis is similar in spirit to thework on strong probe EIT [17], parametric EIT [18], andthe dispersive regime for quantum gates [5].In the rotating-wave approximation, the coherent dy-namics of the semiclassical atom-field system are de-scribed by the Hamiltonian, H / ~ = ∆ σ bb + δσ aa + (Ω p σ cb + Ω s σ ab + h . c . ) , (1)where σ ij = | i ih j | are the atomic projection operators.The evolution equation of the atomic density operator ρ of the system is, dρdt = − i ~ [ H , ρ ]+Γ (cid:16) L [ σ ab ]+ L [ σ cb ] (cid:17) + γ L [ σ aa − σ cc ] , (2)where Γ and γ are the rate of spontaneous emission fromthe excited state and the dephasing rate of the groundstates respectively, and the standard Liouvillian is used L [ B ] = BρB † − (cid:16) B † Bρ + ρB † B (cid:17) , (3)where B is some operator (not necessarily Hermitian) de-scribing the loss channel. The signal field in the mediumresponds to (the expectation value of) the polarizationgenerated by the applied fields or, equivalently the linearoptical susceptibility χ . In terms of the off-diagonal ma-trix elements ρ ij = h i | ρ | j i per atom, the polarization inthe direction of the dipole moment d ab is [16] P = 12 ǫ r ǫ χ ab E s + h . c . = h d i = N (cid:16) ρ ab d ab + h . c . (cid:17) , (4)where E s is the classical electric field amplitude of the sig-nal, ǫ and ǫ r are the permittivity of free space and therelative permittivity respectively, and N is the atomicdensity. The atom-field interactions are taken to beelectric dipole in nature so that the Rabi frequenciesΩ s = d ab E s / ~ and Ω p = d bc E p / ~ , where E p is the fieldamplitude of the pump, d ij is the dipole moment of the | i i → | j i transition and it is related to the radiative de-cay rate by d ij = 3 π ~ ǫ ǫ r c Γ / ( ω ij η ) in the medium withrefractive index η . Following Eq. 4, the signal field seesthe first-order complex susceptibility χ ab = 2 N d ab ~ ǫ r ǫ ρ ab Ω s , (5)and is therefore attenuated according to the absorptioncoefficient in the units of inverse length, α = ω s nc Im[ χ ab ] , (6)where n is the bulk refractive index. It is related to thereal part of the susceptibility by n = 1 + Re[ χ ab ], andcontributes to dispersions and phase shift.Before proceeding to investigate the absorption nonlin-earity of the three-state system, it is instructive to briefly - I (cid:144) I sat A b s o r p ti on c o e ff i c i e n t Α (cid:144) Α FIG. 2: Comparison of the absorption coefficient of the two-state and three-state systems. Dash-dot curve: two-state α (2) .Dashed: three-state α s with I coh /I sat = 50, I p /I sat = 0 . α s with I coh = I p = I sat . Beyond the saturation orcoherent points, the scaling with the intensity is very differ-ent for the two systems with the the two-state (three-state)system being α/α ∼ I − ( ∼ I − ). review the well-known result of the two-state absorber.The corresponding absorption coefficient can be derivedusing the above prescription with a two-state Hamilto-nian H / ~ = δσ bb + (Ω s σ ab + h . c . ). We calculate thesteady-state solution of the density matrix (Eq. 2) andsubstitute the associated off-diagonal matrix element intoEqs. 5 and 6. Finally considering near-resonant opera-tion, we expand the coefficient about δ = 0 to arrive at, α (2) = 2 ξ/ Γ1 + 8Ω s / Γ + O ( δ ) , (7)where we have introduced a medium-dependent constant ξ = 2 N d ab ω s / ( ~ ǫ r ǫ nc ), and noted that Re[ χ ab ] = 0 and n = 1 at resonance.We rewrite this expression as α (2) = α / (1 + I/I sat ),where α is the small signal or linear absorption co-efficient, and the input field intensity I = ζ Ω s where ζ = ~ cǫ ǫ r / (2 d ). The medium containing multiple two-level system will become optically saturated when thesignal intensity is strong, leading to reduced absorption.This turn-on of transmittance occurs at the saturationintensity I sat = ζ Γ / s exceeds the decay rate.This is shown by dash-dot curve in Fig. 2. This form ofscaling with I in Eq. 7 is also found in three- and four-level schemes of saturable absorbers where a coherentdriving of a transition with a second field is not con-sidered [6, 19]. By contrast, the ground-state coherenceincluded in our model is responsible for an absorptionscaling different from Eq. 7.We now turn to absorptive properties in three-statesystems. The emergence of EIT can be understood inthe dressed-state picture of the Hamiltonian Eq. 1 [20].At two-photon resonance ( δ = 0), the Hamiltonian hasan eigenstate that is completely decoupled from the ex-cited state: |Di = (Ω s + Ω p ) − / (Ω p | a i − Ω s | c i ). Bybeing in state |Di under the application of the two fieldssatisfying δ = 0, system population in the dissipative - Detuning ∆ (cid:144) W Α s (cid:144) Α FIG. 3: Absorption spectra of the three-state system withdifferent decay rate Γ in the absence (dashed) and presence(solid) of dephasing ( γ = Γ / p = Ω s = Ω, andΓ / Ω = 1 (blue), 5 (yellow). When γ = 0, increasing Γ effectsthe narrowing EIT window but this diminishes with γ . - - Detuning ∆ (cid:144) G Α s (cid:144) Α FIG. 4: Absorption spectra of the three-state system withincreasing dephasing rate γ/ Γ = 0 (solid), 0.5 (dashed), 1(dash-dot), 5 (dotted). The other parameters are Ω p = Ω s =Γ. state | b i is zero, so that there is no spontaneous emissionand hence absorption of both fields cannot occur. Thistransparency window is positioned between two symmet-rical absorption peaks (Autler-Townes (AT) splitting),which corresponds to transitions from |Di to the othertwo dressed states. In the typical perturbative limit ofΩ s ≪ Ω p , the peak separation is 2Ω p . This separation in-creases with Ω s , and we solve dα s /dδ = 0 (with γ, ∆ = 0)to find that the peak maxima are centred at, δ ± = ± (Ω p + Ω s ) / p Ω p . (8)where α s denotes the absorption coefficient of the three-state scheme in the steady state. Notably while increas-ing spontaneous emission has the effect of broadeningeach peak (linearly as 2Γ, when Γ ≫ Ω p , Ω s ) and nar-rowing the EIT window, the peak positions remain un-changed, as shown in Fig. 3.In the presence of dephasing, field absorption betweenthe two peaks becomes finite. While this has so far rep-resented a fundamental limit for many of the EIT appli-cations, this effect is exploited here. As the dephasingrate increases, the peak-to-peak distance of the AT split-ting decreases and the overall absorption increases, asillustrated in Fig. 4. Solving for the peak maxima yields, δ ± = ± s A h Ω p (5 γ + 2Γ)Γ ′ + Ω s Ω p i − γA Γ ′ (9)where A = ( γ Γ ′ +Ω p +Ω s ) / and Γ ′ = γ +2Γ is the totaldecoherence rate. When the absorption peak is centredat zero detuning, Eq. 9 becomes imaginary. In generalprovided that Ω p , Ω s ≫ γ Γ, all the important features ofEIT remain observable.If we restrict our analysis to the case of single andnear two-photon resonances with ∆ = 0 and δ ≈
0, wecan express the absorption of the signal field compactlyunder the condition ˙ ρ = 0, α s = 2 ξ/ Γ ′ Ω s Ω p + s ΓΓ ′ + (Ω p +Ω s ) Ω p γ Γ ′ + O ( δ ) . (10)Note that because we have assumed near two-photon res-onance, ρ ab is almost purely imaginary and the phaseshift is near zero. Obviously this condition is strictly onlytrue for the regime that the signal is a true continuous-wave (cw) field, but because we expect the non-zero ab-sorption to always dominate the phase shifts, this ap-proximation has little effect.For the simplified case of equal dipole moments d ab = d bc = d , the expression becomes α s = α I (cid:16) I p + I sat + I coh (cid:17) + I p I coh + I I p I coh (11)with the intensity of the pump field I p = ζ Ω p and nowthe saturation intensity is I sat = ζ ΓΓ ′ /
12. We can define I coh = ζγ Γ ′ as the coherence intensity. ConsequentlyΩ p and Ω s can only be varied through adjusting theirrespective field intensities.It should be noted that this result is distinct from,but compatible with the previous works examining thespecial case of a single monochromatic field interactingresonantly with both optical transitions in the spectrally-broad pulse limit [12, 13] and the case of two input fieldsof equal intensity [14]. Specifically to retrieve the atten-uation coefficient part of Eq. 3.5 in Ref. [14], we equate I p = I to yield α s = α I/I sat + I/I coh , (12)and the saturation and coherence intensities are definedin the same way. Similar to the two-state systems, bothof the systems in Eqs. 11 and 12 can become saturatedwhen the signal field intensity is greater than saturationintensity. In addition, field absorption diminishes when I > I coh in the effect known as coherent bleaching, whichis different from saturation bleaching because it occurseven when the usual saturation condition is not met –i.e. I sat > I > I coh [12, 14]. - - - I (cid:144) I sat Α s (cid:144) Α FIG. 5: Tuning the absorption coefficient of the three-statesystem with the pump field intensity I p . The parameter val-ues are I coh = I sat , and I p /I sat is 0.01 (solid), 0.1 (dashed),and 1 (dash-dot). Closer inspection of Eq. 11 reveals two additional cir-cumstances under which the absorption of the three-statescheme changes with signal intensity. The first is when I sat , I coh , p I p I coh > I > I p . The coefficient turns from α ≈ α ( I < I p ) to become inversely proportional to I ,i.e., I p α /I , when the incident field exceeds I p . Secondly,when the intensity exceeds I coh + 2 I p , the system shows aquadratic reduction in absorption with α ≈ I p I coh α /I .This is illustrated in Fig. 2. This form of absorptionnonlinearity represents an alternative to the conventionaltwo-state saturable absorbers as a dynamically tunablenonlinear filter with a potentially greater intensity sen-sitivity. We refer to this effect as photon-assisted bleach-ing . The bleaching point for rapid transmittance changecan be adjusted by varying the pump field strength (seeFig. 5) over a wide range I p ∈ (0 , min { I sat , I coh } ] wheresaturation and coherent bleaching begin to dominate atthe upper limit. Beyond this regime, we note that for I > I p I coh + I sat I coh + 2 I p I sat I sat , (13)the coefficient changes ∼ C/I (for some constant C ) tothe quadratic scaling, depicted by the lower end of thedashed curve in Fig. 2.Our analyses and observations are valid for a signalfield with a small spectral width compared to the size ofthe EIT features at δ = 0, because we have only used thezeroth-order term of α s . This bandwidth requirementfundamentally limits the operational speed of the three-state scheme. For instance, the FWHM of the two-state α (2) is given by ∆ ω (2) = Γ p I/I sat and the responsetime of the system t r commensurates with 2 π/ ∆ ω (2) ,which is necessarily faster than the excited-state lifetime.For the EIT-based scheme, we use the result in Eq. 9 toestimate the allowed bandwidth ∆ ω ≈ ( δ + − δ − ) / γ ≪ Γ , Ω p , Ω s tofind,∆ ω ≈ | δ ± | n γ [Γ + 2Ω p (Ω p − q Ω p + Ω s )]2Γ(Ω p + Ω s ) o , (14)which increases as Ω p p I/I p to first order, and thusthe bandwidth is tunable with the pump field. In theregime of dominant dephasing, we retrieve a single-peakabsorption curve that simplifies to α s ≈ / (1 + 4 δ / Γ ′ ) (15)with FWHM of ∆ ω = Γ ′ .Our results and discussion so far provide the founda-tion of the proposed scheme for optically-thin absorber.However, to determine the overall response of an ex-tended medium, we need to consider the evolution of thepropagating fields. This is the subject of the next section. III. OPTICALLY-THICK MEDIA
In an optically-thick system, the intensity of the trans-mitted fields must be treated as a function of positionwithin the absorbing medium. Under the cw illumina-tion or when the pulse has a broad and smooth distribu-tion, the Beer-Lambert law can be used to determine thepropagation of the pulse, ddz I ( z ) = − α ( z ) I ( z ) , (16)where z is the depth in the medium. This can alsobe regarded as a short-memory Markov-like conditionon the time evolution. There are at least two possi-ble implementations, as depicted in Figs. 1(b) and (c),namely uniform-pump and copropagating-pump arrange-ments, respectively. We discuss the physics of each ar-rangement in details as follows. A. Uniform-pump scheme
In the uniform-pump arrangement, the pump fieldstrength is uniform across the entire medium and onlythe signal field sees an optically-thick medium. The im-plementation therefore may involve directing the pumpfield at some angle to the propagation axis of the signalfield, and thus this is most suitable in solid-state or coldatomic vapour environment where Doppler broadening isnegligible. Since I p is a constant, we only need to inte-grate the propagation differential equation (Eq. 16) forthe signal field using Eq. 11 to arrive at the general lawof intensity transfer for the three-state scheme, α z = (cid:16) I p I coh (cid:17) ln h I I ( z ) i + I − I ( z ) I p I coh + [ I − I ( z )] (cid:16) I p + 1 I sat + 2 I coh (cid:17) (17) Α z I H z L (cid:144) I FIG. 6: Different decay rates of the intensity of the signal fieldas it propagates in a three-state medium in a uniform-pumparrangement. For all curves, I sat = I coh . Dotted curve: 2 I p = I / I sat . Bold: I p = I /
10 = I sat . Solid: I p / I / I sat . Dash-dot: I p = I / I sat . Dashed: I p = 5 I = I sat .The first four curves show the tunability of the medium withthe pump field. To solve for the characteristic penetration depth, weconsider the different regimes separately. For the firstforementioned case where bleaching occurs I ≫ I p , thepenetration depth of the input field has extended from L = 1 /α to ∼ I / ( α I p ) at nonlinearity, compara-ble with the usual case of coherent bleaching (where L ≈ I / ( α I coh )) and saturation ( L ≈ I / ( α I sat )). Onthe other hand in the second case, this length is given by L ≈ I / ( α I p I coh ).If I ≪ max { I p , I sat , I coh } , the absorptive nonlinear-ity is not manifested and the field decays exponentiallyin accordance with I ( z ) ≈ I exp( − α z ). However if I ≫ min { I p , I sat , I coh } = ¯ I , a linear decay occurs as I ( z ) ≈ I − α ¯ Iz until the field intensity reaches I ( z ) < ¯ I that the decay becomes exponential. Finally when thecondition in Eq. 13 is true, we find the slowest decay as I ( z ) ≈ I p − α I p I coh z/I that attenuates the signalfield enough to become a linear decay, followed by an ex-ponential drop off. These different bleaching behavioursare depicted in Fig. 6 for variable pump field and incidentfield strengths.To plot the transmittance of the medium of length l , we rewrite Eq. 17 in terms of the small-signal filtertransmission T = exp( − α l ) and overall transmittance T = I ( l ) /I , ln (cid:16) T T (cid:17) + I (1 − T ) (cid:16) I p + 1 I sat + 2 I coh (cid:17) + (1 − T ) I I p I coh = 0 . (18) Treating T as an independent parameter, we plot thesolutions of this equation with T = 0 .
01 (without lossof generality) in Fig. 7 for different signal intensities. Byvarying the pump field strength, the transmission of thethree-state-based filter is tunable and approaches unitywhen I is greater than I p . Importantly T can approacheunity at I ≈ I p more abruptly than in the two-levelsystem because of the contribution of the quadratic term.Group velocity reduction and spatial compression ofthe propagating fields occurs under the conditions of EIT. I (cid:144) I sat T r a n s m itt a n ce T FIG. 7: (color online) Transmittance of an optically-thick(solid curves) three-state and (dashed) two-state media in anuniform-pump setup for different input signal strengths. Forthe three-state systems, we use I coh /I sat = 1 , T = 0 .
01 andvariable pump field strength to adjust the bleaching level, inparticular (from left to right) I p /I sat = 0 .
01 (blue), 0.1 (red),1 (violet), and 10 (orange). Observe that the turn on of thetransmittance varies with pump field intensity and can occurwith a sharper gradient than the two-state medium.
The group velocity of the signal field is related to the realpart of the complex susceptibility (Eq. 5), v g s = cn + ω s ∂Re [ χ ab ] ∂δ , (19)where the parital derivative for ∆ = 0 , δ = 0 is ∂Re [ χ ab ] ∂δ = 2 N d ~ ǫ ǫ r α s α I coh + I ) / Γ ′ − ( I p + I ) / ( γ Γ ′ ) I coh + I + I p . (20) Therefore the group velocity of the signal field can varyconsiderably depending on field intensities. In the regimeof
I, I p ≪ I sat , I coh , we find that v g s ≈ cn + N d ω s ~ ǫ ǫ r Γ ′ (21)so that in the limit of fast spontaneous decay and/ordephasing the velocity reduction is minimized. On theother hand in the limit of small dephasing, v g s ≈ cn + N d ω s ~ ǫ ǫ r Ω s I p /I + I/I p ) , (22)where the velocity is also maximized with strong pumpand/or signal field intensities. However the advantageof velocity reduction in EIT systems is that the mediuminteraction length v g t r required for the atoms to reachthe steady state can be reduced. B. Copropagating-pump scheme
In this setup, the signal and pump fields are coprop-agating and both see an optically-thick medium. Whilethis arrangement can be implemented with solid-state I (cid:144) I sat T FIG. 8: Transmittance of an optically-thick three-statemedium in the copropagating-pump arrangement with I sat = I coh . We vary the bleaching level with the pump field, specifi-cally (from left to right) ( I p /I sat , α l ) = (0 . ,
5) (blue), (0.1,7) (red), (1, 12) (violet), and (10, 70) (orange). systems, it is also suitable for media such as hot atomicgases, where one must work in a Doppler-free config-uration to overcome the effect of Doppler broadening.Consequently, the absorption of the pump must be ad-dressed and here we consider the absorption coefficient α p of the pump field to investigate the field evolutionwithin and transmittance of the medium in the similarway. By the symmetry of the system in the two-photonresonance regime, the associated coefficient has the sameform as Eq. 11 except I p is replaced with I and vice versa, α p = α I p (cid:16) I + I sat + I coh (cid:17) + II coh + I p II coh . (23)When one of the fields is completely absorbed, themedium becomes transparent to the other field. Thisis because, e.g. , when the pump field is absorbed, theatoms are in the state | c i due to spontaneous decay andthus the absorption of the signal field can no longer oc-cur. This happens when the ratio α s /α p = I p /I at entryto the system is less than unity, or I p < I . As a resultthe pump field is attenuated at a much faster rate thanthe signal field and the values of the ratio and α p in-crease with the distance. Conversely when I p > I and α s /α p >
1, the signal field is attenuated rapidly with in-creasing α s . Consequently the transmission response ofthe medium in this arrangement is more sensitive to thesignal field intensities than the uniform-pump case.Under the assumption that the fields are travelling atthe same group velocity, we solve Eq. 16 as coupled dif-ferential equations with α s and α p in the same movingframe z to plot T versus I in Fig. 8. The sharp increasein transmittance of the medium to the signal field oc-curs at I = I p , and comparing with Fig. 7, this gradientis larger than the uniform-pump counterpart when thedephasing is weak.Group velocity reduction under the conditions of EITmay diminish the observed properties of this copropa-gating arrangement. The mismatch of group velocity be-tween the signal and pump fields can lead to reductionin effective two-beam interaction and different absorp-tion behaviours akin to a two-state scheme. The solid-state based implementation can however take advantageof available techniques to engineer the group velocity ofone of the fields to match the other, e.g., by tailoringthe dispersion properties of the optical channel such asin photonic bandgap materials [22] or coupled-resonatoroptical waveguides [23]. This control is not availablein atomic vapour systems, but we note that there is arestricted operating conditions where the effect of mis-match can be minimized, as discussed below.The group velocity v g p of the pump field can be writtendown in an analogous way to Eqs. 19 and 20, v g p = c/ (cid:16) n + ω p ∂Re [ χ bc ] ∂ ∆ (cid:17) , (24)where ∂Re [ χ bc ] ∂ ∆ = 2 N d ~ ǫ ǫ r α p α I coh + 2 I p ) / Γ ′ I coh + I + I p (25)such that v g s and v g p are different in general. Howeverin the same operating condition as Eq. 21, we find that v g p ≈ v g s if ω s ≈ ω p , and the mismatch with the signalfield velocity is minimized. C. Noise filtration
Any absorbing material can be used to build a sat-urable absorber for signal-noise discrimination and a widerange of systems ranging from dyes, synthetic crystallinematerials, to semiconductor quantum wells and dots arewidely used [9]. Our EIT-based scheme may also be use-ful as a dynamically-tunable absorber for similar appli-cations. As an illustration, we demonstrate noise filter-ing by simulating a transmission of a corrupted signalpulse through the three-state medium in the uniform-pump setup. In particular, we lift the restrictions for theuse of the Beer-Lambert law and consider the Maxwell-Bloch (MB) equation to study the field propagation.We begin by invoking the slowly-varying amplitudeand phase envelope approximation to transform thesecond-order Maxwell’s wave equation into a first-orderMB equation for the real field amplitude, (cid:16) ∂∂z + 1 v g ∂∂t (cid:17) E s ( z, t ) = N dω s ǫ ǫ r Im[ ρ ab ] . (26)It is convenient to solve the propagation equation interms of the Rabi frequency Ω s ( z, t ) that reads (cid:16) ∂∂z + 1 v g ∂∂t (cid:17) Ω s ( z, t ) = − α s ( z, t ) . (27)As the medium polarization is affected by the field thatis generates, we solve this equation by discretizing the FIG. 9: (color online) Simulation of noise filtration of thesignal field Ω( z, t ) according to Eq. 28 in two steps (a)Ω p = γ = Γ = 10 rad/s, ζ = 10 m − s − , and (c) Ω p = Γ / γ = Γ = 10 rad/s, ζ = 10 m − s − . The Gaussian signalpulse with pink noise is initialized at z = 0 and the noise com-ponents are absorbed as the signal propagates in the medium.The power spectral densities of the signals are shown in (b,d)respectively. Dashed curve: spectral density of the originalsignal. Bold blue: original corrupted signal. Solid red: fil-tered signal. expression, z Ω nj +1 + 12 v ng j +1 ∆ t (Ω n +1 j +1 − Ω n − j +1 ) + α nj +1 nj +1 =2∆ z Ω nj − v ng j ∆ t (Ω n +1 j − Ω n − j ) − α nj nj (28) where the subscript s for Ω s is dropped, n = 1 , , ..., N is the time step of size ∆ t = T /N and T is the simula-tion time. The index j = 1 , , ..., J is the sampling indexfor z where a length unit in the medium of length l is∆ z = l/J . The expression can be recast in the matrixform A j +1 Ω j +1 = B j Ω j . At each step j in space, the ex-pression Ω j +1 = [ A j +1 ] − B j Ω j is iteratively evaluatedto reach convergence. The inverse matrix [ A j +1 ] − iscalculated with initial trial values of Ω j +1 given by theprevious step Ω j , and the matrices A j +1 and Ω j +1 aresolved for self-consistency.In the simulation the atoms in the medium ( z > | a i (i.e. the steadystate when the signal field is switched off) and the signalfield Ω(0 , t ) is a Gaussian pulse corrupted with 1 /f pinknoise components. The signal is initialized at entry tothe medium for simplicity so that we do not treat thein-coupling of the field to the medium. The distributionand the spectral density of the input signal are shown inFig. 9. In Fig. 9(a,b), we consider the case where thepump intensity is commensurate with the saturation andcoherent bleaching intensities. In Fig. 9(c,d), the caseof relatively weaker signal and pump fields is simulated.In both cases, the high-frequency noise components areabsorbed, showing the retrieval of the Gaussian signal.A temporally broad pulse is required for the second casebecause the system takes a much longer time to reachsteady state to produce the absorptive behaviour pre-dicted by Eq. 10. IV. IMPLEMENTATIONS
The proposed scheme for tunable absorption nonlin-earity can be implemented in a variety of solid-state andvapour phase media. We concentrate our attention ondiamond defects for the solid-state and rubidium (Rb)vapour for the gas phase, but we note in passing thatquantum dots [24, 25], and rare-earth crystals [26] pro-vide equally promising solid-state implementations.In the solid state, we envisage one practical realizationusing a doped diamond waveguide implemented as a step-index fibre [27, 28], a line defect in the photonic-bandgapstructure [22, 29] [Fig. 1(a)], or in coupled-resonator con-figuration [23, 30]. Diamond hosts numerous optically-active impurity- and/or vacancy-based defects, and aprominent centre featured in numerous quantum-opticaldevice designs is the negatively-charged nitrogen-vacancy(NV) centre [31]. Coherent population trapping dueto the same mechanism responsible for EIT has beendemonstrated with the centre [32, 33].By operating near the zero-phonon line resonance ofthe centre λ = 638 nm, the NV can be treated as a Λ-type system where we identify the required energy levelswith the available spin states, | b i = | E, m = 0 i beingthe excited spin triplet state whereas states | a i and | c i correspond to the m = 0 and m = +1 (or m = − | A i [34].The dipole moment between transitions | a i−| b i and | c i−| b i is ∼ − Cm [35]. To avoid optical cross couplingbetween the NVs, they should be spatially separated by ≥ λ/n ≈
250 nm in the single-mode waveguides with across-section area of 200 ×
200 nm , limiting the atomnumber density to N ≤ m − . This density levelcan also controlled and achieved using ion implantationin synthetic diamond [36]. Consequently, with ǫ r = 10,the characteristic constant ξ = 7 × m − s − .The NV has a radiative lifetime of 11.6 ns or Γ /π =86 MHz and a dephasing lifetime up to 1 µ s [37], so thatthe small-signal attenuation coefficient is α = 2 ξ/ Γ ′ =244 m − and saturation and coherence intensities areof order 1 MW/m . For T = 0 .
01 in Fig. 7, the re-quired length l of medium to demonstrate the tunable ab-sorption properties for the uniform-pump setup is 2 cm.Similarly for the copropagating-pump implementation inFig. 8, we estimate l = 2 cm (for I p /I sat = 0 .
01) and aninteraction length of 29 cm (for 10) for the same valueof absorption coefficient. This can be achieved by fold-ing the waveguide back onto itself for multiple times in a standard diamond wafer. The dephasing time is relatedto the concentration of electron paramagnetic impuritiesin the diamond lattice [38] and hence α can be in partcontrolled at fabrication.The Λ-type configuration also appears naturally in thehyperfine structures of the atomic Rb vapour. Usingthe D transition, the pump and signal fields of wave-length λ = 795 nm couple pairs of Zeeman sublevels ofthe atomic 5 S / , F = 2 state via the common upperstate 5 P / , F = 1 [39, 40]. The overall rate of sponta-neous emission into the ground states is Γ /π = 37 MHzso that the dipole moment is d ∼ − Cm and the sat-uration intensity is ∼
350 W/m . The usual conditionof weak ground-state dephasing of γ/ π = 117 Hz [40] inan typical EIT experiment is not required in our schemebecause the coherent bleaching would become the dom-inant effect. The atomic density can vary from 10 to 10 cm − in a typical 5 cm cell at room tempera-ture [39, 41]. Suppose that a density level N = 10 m − and ξ = 5 × m − s − . In a velocity-matched,copropagating-pump or Doppler-free uniform pump se-tups, we expect a short cell of length of 10 − µ mshould be sufficient for operations with I p /I sat = 0 . α ≈ × m − . V. CONCLUSION
We have analysed nonlinear absorption in an ensembleof three-level atoms in a signal-pump EIT configuration.In the presence of ground-state dephasing, EIT is notcomplete at two-photon resonance and bleaching occurs.While this is comparable with conventional saturable ab-sorption in an ensemble of two-level atoms, the effectiveabsorptive nonlinearity associated with the signal field inthe EIT system can be stronger, and can scale quadrati-cally with signal intensity. Moreover as the EIT featuresdepend on the pump beam intensity, the bleaching in-tensity of the absorber can be modified post-fabrication.We considered two different signal-pump arrangementswith uniform pump and copropagating pump beams, andshow that nonlinear change in the transmission propertyof an optically-thick three-state medium is sharper thanthe two-state scheme. This effect may be useful for noisefiltration and microscopy, especially in the weak signalregime, and perhaps in the future, for manipulating thequantum, rather than classical, properties of fields.
Acknowledgments
We thank A. M. Martin, J. Q. Quach, and A. Hay-ward for helpful discussions. This project is supportedby the Australian Research Council under the DiscoveryScheme (DP0880466 and DP0770715), the Centre of theExcellence Scheme (CE0348250), and in part by MEXTand FIRST in Japan. CHS acknowledges the support of the Albert Shimmins Memorial Fund. [1] J. P. Dowling and G. J. Milburn, Phil. Trans. R. Soc. A
947 (1967).[7] D. Atkinson, W. H. Loh, V. V. Afanasjev, A. B. Gru-dinin, A. J. Seeds, and D. N. Payne, Opt. Lett.
Handbook of Optoelectronics , editedby J. Dakin and R. Brown (Taylor & Francis, New York,2006).[10] K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, andS. Kawata, Phys. Rev. Lett.
945 (1986).[13] Y. I. Khanin and O. A. Kocharovskaya, J. Opt. Soc. Am.B
523 (1990).[15] H. Ian, Y. X. Liu, and F. Nori, Phys. Rev. A
267 (2001).[20] M. Fleischhauer, A. Imamoˇglu, and J. P. Marangos, Rev.Mod. Phys.
633 (2005).[21] M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robin-son, V. L. Velichansky, L. Hollberg, and M. O. Scully,Phys. Rev. Lett.
692 (2008).[26] B. S. Ham, M. S. Shahriar, and P. R. Hemmer, Opt. Lett.
741 (2008).[30] N. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.Ryu, Opt. Express
21 (2008).[32] C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D.Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D.Greentree, S. Prawer, F. Jelezko, and P. Hemmer, Phys.Rev. Lett.86,