2014 - Hsiao - Coherenceproperties of Amplified Slow Light by FWM - Opt Lett 39

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Coherence properties of amplified slow light by four-wave mixing

Ya-Fen Hsiao,1 Pin-Ju Tsai,1 Chi-Ching Lin,1 Yong-Fan Chen,2 Ite A. Yu,3 and Ying-Cheng Chen1,3,*1Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan

2Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan3Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan

*Corresponding author: [email protected]

Received March 21, 2014; revised April 25, 2014; accepted April 26, 2014;posted April 30, 2014 (Doc. ID 208585); published June 3, 2014

We present an experimental study of the coherence properties of amplifiedslow light by four-wave mixing(FWM)ina three-level electromagnetically induced transparency (EIT) system driven by one additional pump field. Highenergy gain (up to 19) is obtained with a weak pump field (a few mW ∕cm2) using optically dense cold atomic gases.A large fraction of the amplified light is found to be phase incoherent to the input signal field. The dependence of theincoherent fraction on pump field intensity and detuning and the control field intensity is systematically studied.With the classical input pulses, our results support a recent theoretical study by Lauk et al. [Phys. Rev. A 88, 013823(2013)], showing that the noise resulting from the atomic dipole fluctuations associated with spontaneous decay issignificant in the high gain regime. This effect has to be taken into consideration in EIT-based applications in thepresence of FWM. © 2014 Optical Society of America

OCIS codes: (190.4223) Nonlinear wave mixing; (270.2500) Fluctuations, relaxations, and noise.http://dx.doi.org/10.1364/OL.39.003394

Slow and stored light associated with the effect of electromagnetically induced transparency (EIT) hasdrawn significant attention because of the many applica-tions they could offer in optical communication andquantum information science. For example, slow lightcan be used as an optical buffer in optical network sys-tems [1], while stored light can be used as a quantummemory to store the quantum state of light [2]. A finiteEIT bandwidth and a finite ground-state decoherencerate can cause pulses to attenuate and broaden, alongwith their propagation or storage. It has been proposed[3] and demonstrated [4–6] that pulse attenuation andbroadening can be compensated through a four-wavemixing (FWM) process. However, it is well known thatthe FWM gain is often accompanied by additional atomicnoises [7–9]. The study of the quantum noise accompany-ing the FWM process is important to evaluating its effectin EIT-based applications.

Recently, Lauk et al. performed just such a theoreticalstudy [10]. Their work shows that the exponential growthof the signal pulse in the high FWM-gain regime is accom-

panied by an equally strong growth in noise. The FWMgain is detrimental to the EIT-based quantum memorywith a single-photon pulse because of the vacuum-gener-ated noise photons. With classical input pulses contain-

ing many photons, the quantum noise resulting fromdipole dephasing by spontaneous emission dominates.Here, we use the beatnote interferometer [11] to exper-imentally evaluate the predictions with classical signal

pulses. The dependence of the incoherent fraction of the amplified slow light on pump intensity and detuningand control intensity is studied. Although the experimentis complicated by the population loss from the FWM sys-tem, the observations agree well with the predictedtrends. Our results show that the quantum noise associ-ated with FWM amplification is significant for opticallydense media and should be taken into consideration in

various applications.

Our energy level scheme with cesium atoms is shownin Fig. 1(a). The weak signal field (a few nanowatts) andthe strong control field (up to 1 mW) drive the j1i) to j3iand j2i to j3i transitions, respectively. The signal andcontrol fields form a three-level Λ-type EIT system.The Rabi frequency of the control field is denoted byΩ. The spontaneous decay rate of the state j3i is denotedby Γ. In the room-temperature-vapor FWM experiments[4,5,12], the off-resonant excitation of the EIT controlfield from state j1i to j3i also acts as the pump field toform an FWM process. For cold atoms with the same

pump parameters, this excitation is reduced by abouttwo orders of magnitude which is the ratio of the effec-tive linewidth of the room temperature vapor to the coldatoms [13,14]. With a 9.2 GHz hyperfine splitting for the6 S 1∕2 state, the effect of FWM from the control field isalmost negligible with our experimental parameters.To introduce a significant FWM, we thus add one addi-tional pump field (with a Rabi frequency denoted by Ω0)to drive the j1i to j3i transition with a detuning Δ∕2π of several tens of MHz. This arrangement allows moreflexible control of the FWM parameters. Specifically,we can independently control the intensity, detuning,

Fig. 1. (a) Relevant energy levels for 133Cs atoms and laser ex-citations. (b) Schematic experimental setup. BS, beam splitter;M, mirror; L, lens; AOM, acousto-optic modulator; FC, fiber cou- pler; P, polarizer; λ∕4, quarter waveplate; PMT, photomultiplier tube.

3394 OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014

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and turn-on time of the pump field. The four-photon proc-ess through pump → idler → control → signal providesthe signal gain. All laser beams are nearly copropagating.The control and pump beams nearly overlap with eachother. The control and signal beam intersect by a smallangle (∼1.4°). With this geometry, the phase matchingconfiguration is shown in Fig. 1(b) [4,12].

We use a two-dimensional, temporally dark andcompressed magneto-optical trap (MOT) [15] in addition

to Zeeman-state optical pumping to obtain opticallydense cold cesium gases. The optical depth for the probetransition can be easily varied by tuning the power of thetrapping beams. The maximum optical depth we couldobtain is 235. The measurements are performed at a 10 Hz repetition rate. Near the end of each timing cycle,the MOT quadrupole magnetic field is increased by a factor of 3, and the intensity of the repumping beamof the MOT is reduced by a factor of 110 for 12 ms toimplement the temporally dark and compressed MOT.The MOT magnetic field (trapping beams) is (are) turnedoff 2.5 (0.2) ms before the FWM experiment to minimizethe perturbation. The control beam is turned on 1.2 msbefore the FWM experiment. There is a 1 ms periodfor the trapping and control beams to pump the popula-tion into the F 3 ground state. A small magnetic field of ∼80 mG is applied in the propagation direction of thesignal beam to define the quantization axis. The nearlycircularly polarized, Zeeman-state optical pumping beamwhich drives the j F 3i → j F 0 2i transition of the D2

line, is then on for 30 μs to pump the population into therightmost Zeeman state j F 3;m 3i. It intersects withthe quantization axis by about 3° to induce both σ and π transitions with the Zeeman state j F 3;m 3i beingthe only dark state. The Gaussian signal pulse with anintensity temporal FWHM width of 1.6 μs is then applied.The turn-on time of the pump pulse, which has a square

waveform, can be varied relative to the signal pulse. A master laser is frequency-locked to cesium atomic

transition. Part of the master light injection locks a slavelaser with which the control beam is generated. Partof the master light passes through an electro-opticmodulator (Eospace PM-0K5-10-PFA-PFA-850) drivenat ∼9 GHz. Its 1 sideband injection locks another slavelaser with which both signal and pump beams are gener-ated. In this way, the phase coherences between thecontrol, signal, and pump fields are kept to a good level.To measure the phase coherence properties, the signalbeam is combined with one reference beam, which is80 MHz below the signal transition, to form a beatnoteinterferometer [11]. The signal beam is focused to a waistof 60 μm. The pump and control beams have a diameter of ∼4 mm. The signal and the idler beams are monitoredby two photomultiplier tubes (Hamamatsu H6780-20). A

photodetector (NeFocus 1801) monitors part of the beatsignal before entering the atoms in order to trigger theoscilloscope for the beatnote measurement. Representa-tive power and beatnote data for the FWM experimentare shown in Figs. 2(a) and 2(b). The blue, black, andgreen traces are the input, slow, and amplified pulse, re-spectively. The inset in Fig. 2(a) shows the idler pulseand the inset in Fig. 2(b) shows part of the beatnote near the peak of the amplified signal pulse. Using the numeri-cal calculations based on the Maxwell–Bloch equations

in the three-level EIT system for the input signal pulse,we fit the calculated slow light pulse to the experi-mental one. The optical depth D, Ω, and ground-statedecoherence rate γ can be determined [15].

Because the pump field introduces an AC Stark shift on

state j1i, the optimum signal detuning δ to satisfy the phase matching condition may deviate from zero. TheRabi frequency for the pump field Ω 0 can be determinedby this AC Stark shift. We vary the δ to find the maximumsignal gain. Figure 3(a) shows such an example wherethe energy gain is defined as the ratio of the area of theamplified signal pulse to that of the input signal pulse. Wealso vary the turn-on time of the pump field to maximizethe signal gain. At each turn-on time, δ is set at its opti-mum value. An example is shown in Fig. 3(b). We foundthat the maximum signal gain occurs when the pumpfield is turned on at the time that the signal peak entersthe medium [defined as time zero in Fig. 3(b)]. If the

pump field is turned on too early, even before applying

the signal pulse, the gain goes down and approachesunity. We attribute this trend to the loss of populationfrom the FWM system to state j F 4;m 4i resultingfrom the optical pumping effect of the pump field. Tocheck this idea, we decrease the pump intensity and per-form the same measurements. It is found that the timescale of the variation of the gain versus the pumpturn-on time becomes longer, agreeing with a longer optical pumping time. We have performed a numericalcalculation based on the semiclassical Maxwell–Bloch

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Fig. 2. (a) Representative power data. The blue trace is theinput signal pulse. The black trace is a slow light pulse withoutFWM. The green trace is the amplified signal pulse and is shifted vertically for clarity. The inset shows the idler pulse. (b) Thecorresponding beatnotes for the same conditions as in (a).The inset shows part of the beatnote of the amplified signal pulse around its peak. The parameters { D, Ω, γ , Ω0, Δ∕2π }are {102, 0.60Γ, 0.001Γ, 0.68Γ, 49 MHz}, respectively.

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Fig. 3. (a) Energy gain versus signal detuning. The parameters{ D, Ω, γ , Ω0, Δ∕2π } are {95, 0.73Γ, 0.001Γ, 0.60Γ, 49 MHz}, respec-tively. (b) Energy gain versus turn-on time of the pump pulse.Time zero is the time when the signal peak enters the medium.The parameters { D, Ω, γ , Δ∕2π } are {127, 0.85Γ, 0.001Γ, 49 MHz},respectively. The red circles and blue squares indicate data withΩ0 of 1.08 and 0.76Γ, respectively.

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equation with the four-level model system. The popula-tion loss due to the optical pumping effect is modeledby an effective loss rate in the total population. Withcomparable parameters to those in the experiment, thecalculations show the same behavior and support theabove explanation. For all data shown later, the turn-on time of the pump field is fixed at time zero. At a veryhigh pump intensity, it is found that the temporal width of the amplified signal pulse become very short, even

shorter than that of the input pulse. A similar observationis also found in the room-temperature vapor experiment[16]. We also attribute this to the population loss. At a high pump intensity, populations are lost in a very short

period of time when the pump field is turned on. TheFWM-gain builds up and at the same time the populationis depleted such that the gain drops quickly which resultsin a short, amplified signal pulse. Our numerical simula-tion also supports this explanation.

Although achieving a high FWM gain is not the main purpose of this study, we also attempt to determinethe greatest gain we could obtain and under what param-eters this could be achieved. The highest energy gain is19 (not shown in the figure) with an optical depth of 235.The intensity of the pump field is only 4.3 mw∕cm2 whichis at least two orders of magnitude lower than the valuein room-temperature vapor experiments [4,12,17]. Thereason is the narrower excited state linewidth for thecold atoms such that the pump detuning can be chosensmall (49 MHz in this case) and the required intensity isthus relatively low. Examples of the systematical varia-tion of the FWM gain versus the pump detuning, pumpintensity, and control intensity are shown in Figs. 4(a),4(c), and 4(e). The FWM gain is monotonically propor-tional to the pump intensity. There is an optimum pumpdetuning Δ that maximizes the gain. When Δ is too large,there is a decrease in the cross coupling between the

pump and the idler field which is responsible for theFWM gain. Setting a too small Δ will destroy the EITand depletes the population quickly. There is also an op-timum control intensity that maximizes the gain. A weakcontrol intensity causes a significant signal attenuationdue to a narrower EIT spectral width. At a higher controlintensity, the state j1i is closer to an ideal dark state suchthat the pump excitation is more difficult and thus theFWM gain is reduced [10,12].

The classical behaviors of the FWM have been dis-cussed in many literatures [4,6,10,12,18], either for a three-level or four-level system. Under the perturbativeand far pump detuning approximations, the steady-statesignal gain follows a parabolic relation versus the FWMcoupling strength x DΩ0∕ΩΓ∕2Δ in the low gain re-gime and an exponential relation with x in the high gainregime. At given laser parameters, the FWM gain is mono-tonically proportional to the optical depth. A more rigor-ous analysis without assuming the far pump detuningapproximation can be referred to [19]. For a given opticaldepth and pump intensity, there is an optimum pump de-tuning and control intensity that maximizes the signalgain. This optimum gain is proportional to the opticaldepth. High optical depth is the decisive factor inobtaining a high FWM gain.

Next, we turn to the phase coherence properties of theamplified slow light pulses. As shown in Fig. 2(b), the

green trace is the beatnote for the amplified signal pulse.The contrast of the beatnote reflects the coherent part of the amplitude of the electric field. Compared to the slowlight trace without the FWM, it is easy to see that the co-herent part of the electric field is amplified. However, the

peak contrast of the amplified pulse is not larger than

that of the input pulse in this case. From the power data as shown in Fig. 2(a), the peak power and area for theamplified pulse is larger than that of the input pulsethough. That is, a significant portion of the amplified

pulse is phase incoherent to the input pulse. This situa-tion is observed over all amplified pulses under variouslaser parameters. To quantify the incoherent percentage,we model the electric field of the amplified signal to con-tain two components; one has a definite phase relationand the other has a random phase relation to the refer-ence field. From the difference between the top and thebottom envelope of the beatnote and the known ampli-tude of the reference pulse, the coherent part of the elec-tric field can be determined. Based on this and the total

power data, the incoherent energy fraction of theamplified signal pulses can be determined.

Figures 4(a), 4(c), and 4(e) depict the total energy gainand energy gain of the coherent part versus the pump

power, pump detuning, and control power, respectively.Figures 4(b), 4(d), and 4(f) show the correspondingincoherent fractions in the amplified pulses. In [10],the authors analyzed the quantum noises associated withFWM. They identified two noise sources: one is vacuum-generated noise and the other is the noise resulting fromthe dipole fluctuation in spontaneous emission. The vac-uum-generated noise photon is independent of the pho-ton number of the input signal pulse and is dominated

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Fig. 4. (a), (c), and (e) Total energy gain (red circle) and thecoherent part energy gain (blue square) of the signal pulse ver-sus the pump power, pump detuning, and control power, re-spectively. (b), (d), and (f) The fraction of the incoherent part energy corresponding to the data in (a), (c), and (e).

The parameters { D, Ω, γ , Δ∕2π } for (a) and (b) are {153, 0.79Γ,0.001Γ, 49 MHz}, respectively. Ω 0 is 0.41Γ for a pump power of 0.5 mW. The parameters { D, Ω, γ , Ω0} for (c) and (d) are {102,0.60Γ, 0.001Γ, 0.68Γ}, respectively. The parameters { D, γ , Ω0,Δ∕2π } for (e) and (f) are {22, 0.001Γ, 0.71Γ, 49 MHz}, respec-tively. Ω is 0.55Γ for a control power of 0.16 mW.

3396 OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014



with a single-photon signal pulse. This noise source re-duces the fidelity of the quantum memory. The noise

photon number from the second source is proportionalto the excited-state population and is thus dominatedwith a classical input pulse containing many photons. Be-cause we use classical signal pulses with a large photonnumber, typically ∼1 × 105, the second noise contributiondominates and we are only concerned with this noisesource.

To include the effect of quantum noise, one has to con-sider the Bloch–Langevin equations which include theatomic dipole fluctuations. Under certain approxima-tions, Lauk et al. [10] derived an expression for the num-ber of noise photons which is N SE D∕2Δω2

0∕Ω2

1 x 2∕8 Ω02∕Δ21∕2 x 2 in the low gain re-gime and N SE D∕16e2 x Δω2

0∕Ω25∕2 x 2Ω02∕Δ2

in the high gain regime, where Δω0 is the pulse band-width. For a given pulse width, it is evident that byincreasing the control power and the pump detuningand by decreasing the pump power, the number of noise

photons decreases. These three trends correspond to de-creasing the population in the excited state. The data inFigs. 4(b), 4(d), and 4(f) support these predictions. In col-lecting these data, we kept the gain relatively low(1-5). If it is too high, the incoherent fractions are all quitehigh with little variation versus the laser parameters. Wehave to point out that a quantitative comparison of thetheoretical predictions with the experimental results iscomplicated by the issue of population loss. In the theo-retical model, the FWM is considered in a closed system.In the actual system, the population can be depleteddue to the optical pumping by the pump field. However,the population loss limits the FWM gain as well as thenumber of noise photons simultaneously. We expectthe qualitative trends on the incoherent fraction versuslaser parameters to stay the same.

In conclusion, we performed an experimental study onthe coherence properties of FWM-amplified slow light us-ing classical pulses. It is found that the amplified slowlight is partially coherent, especially in the high gain re-gime. This effect should be taken into consideration inthe applications utilizing FWM-amplified slow light.

This work was supported by the National ScienceCouncil of Taiwan under grant number 101-2112-M-001-014-MY2.

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2014 - Hsiao - Coherenceproperties of Amplified Slow Light by FWM - Opt Lett 39

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