1. Introduction

Great advances have been made in quantum optics over the past decade, driven by improved control of coherent interactions between light and atomic systems. To date the most investigated phenomenon is electromagnetically induced transparency (EIT) [1]. This refers to an effect that, in a medium driven by a control laser, a probe laser whose frequency is close to an otherwise absorbing transition will experience a narrow window of transparency at the center of the absorption profile. The effect is based on coherent population trapping [2], a process where the combination of two laser fields excites a 3-level system (e.g., |1〉, |2〉 and |3〉 in Λ configuration) into a so-called coherent superposition state of the two lower energy-states (|1〉 and |3〉). In such a case, the quantum system can simultaneously occupy both states in a phase-locked fashion and the two possible light pathways (i.e. the transitions |1〉 → |2〉 and |3〉 → |2〉) can interfere and cancel each other. The net result of this destructive quantum interference is that none of the atoms or molecules are promoted to the excited state, leading to vanishing small optical absorption [1]. In addition, the transparency is accompanied with a very sharp change in dispersion. These features find compelling applications in topics as various and varied as ultraslow light [3] and light storage [4], laser cooling [5], nonlinear optics [6] and atomic clocks [7]. They are indeed the driving motives behind the extensive study of EIT since its first experimental demonstration [8] and the growing endeavors and proposals presently undertaken aimed at finding ways to implement it in all-optical switching and signal processing in optical communication as well as in building blocks for quantum computing and teleportation.

Despite the consensus on the potential of EIT mentioned above, experiments have been mainly restricted to atomic vapors (e.g. Rb), and few studies have addressed the occurrence of EIT in solid-state materials [9–11] and gaseous molecular systems, e.g., [12]. Gaseous molecular systems have a number of distinctive features which could not only broaden our fundamental understanding of EIT-related phenomena by offering new test grounds but would also open new technological prospects. For instance, many molecular systems exhibit quantized and spectrally resolvable vibrational and regularly spaced rotational-vibrational transitions which cover the whole VIS-IR spectrum. An appropriate combination of two transitions could form a three-level system in the Λ, V or cascade configurations where electromagnetically induced transparencies could in principle occur. Figure 1 schematically illustrates this in the case of rotational transitions between two vibrational states in a parallel band of a linear molecule [13]. The figure clearly shows that for a control laser which is at resonance with an absorption line (e.g. P(J+1), J being the rotational quantum number), one could observe transparencies with a probe laser tuned around either the line R(J-1), thus forming a Λ interaction configuration, or the R(J+1) line, forming a V interaction scheme. Consequently, the R-branch could be used as a comb of transparencies spanning several THz for future devices such as all-optical routers in telecommunications. In this case, molecules such as acetylene and hydrogen cyanide are the natural choice, since their v ₁ + v ₃ and 2v ₁ bands respectively offer a comb of stable and regularly spaced ro-vibrational overtone transitions covering the whole telecommunications wavelength range and are already used as grids of frequency standards [14]. Using the very same absorption lines as a grid of “transparency windows” at which light could propagate with controllable group velocity is a highly attractive prospect.

 

Fig. 1. Schematic illustration of the energy level diagram of rotational transitions between two vibrational states (v and v’). The rotational transitions in the P-branch correspond to transitions accompanied by an excess of one unit of angular momentum, i.e. ΔJ = -1 , and in R-branch to transitions with ΔJ = +1 With this system a combination of control-probe lasers resonant with P(J+1)-R(J-1) forms a Λ interaction and P(J+1)-R(J+1) forms a V interaction producing a comb of transparencies over the spectrum covered by the R-branch.

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There are two hurdles, thus far, to the observation of EIT in gas-phase molecules. On one hand, the strength of the typical molecular transition dipole moment is orders of magnitude smaller than their atomic vapor counterparts (e.g., in Rb the effective dipole moment is ~60 Debye while it is only a few milli-Debye for the v ₁ + v ₃ band of acetylene). On the other hand low intensities in coherent interactions are often required to avoid power broadening and frequency shifts which result in an unfavorably low signal-to-noise ratio. This drawback is particularly felt in molecular systems where even the simplest molecules are open systems in that every excited molecular ro-vibrational level is radiatively coupled to many more energy levels than any atomic excited state thus necessitating extremely long interaction lengths. Fortunately this requirement can be satisfied in hollow-core photonic crystal fiber (HC-PCF) [15].

HC-PCF offers the unique opportunity to propagate light over km length-scale [16] in any gas-phase material in a diffractionless fashion, contrasting thus with the intrinsic diffractive nature of light propagation, which prevents most μm-wide laser beams from traveling even a few cm (i.e., twice the Rayleigh range). This means, for applications such as nonlinear optics and laser-induced particle or atom guidance, a dramatic increase in the effective interaction length [17–19], radically changing our approach to experiments involving interactions between laser light and gas-phase materials. Furthermore, we recently demonstrated that a HC-PCF filled with gas can be hermetically sealed and spliced to conventional optical fibers [20]; when filled with acetylene this offers an excellent alternative for an efficient, compact and integrable laser frequency stabilisation system covering a large spectral range of telecommunication wavelengths.

In continuation of this work, we report here on experimental observations of EIT in both Λ and V configurations covering a large part of the R-branch of acetylene v ₁ + v ₃ band, using HC-PCF. We report, on observation of transparencies in different regime of pressure. A steady-state theoretical model was used to fit our experimental results and account for the different sources of decoherence. The recent paper by Ghosh et al. [21], which has just appeared, demonstrates EIT in acetylene-filled HC-PCF; however, the authors have demonstrated it only for one transition using the Λ configuration and their transparency was limited to 40%.

2. Experiments

The experiments were performed using acetylene gas (¹²C₂H₂) at pressures ranging from 0.001 to 1 mbar. The gas is kept at room temperature and is contained in a HC-PCF. The fibre was preliminarily spliced at one end to a single-mode fiber (SMF) using the procedure described in [20] (splice loss ~1 dB) and evacuated before being filled with acetylene through its second end which is hermetically attached to a gas chamber [17]. The pressure in the acetylene-filled HC-PCF was then brought to equilibrium at pressures of 0.1-1 mbar (using a mechanical vacuum pump), and 0.1-0.001 mbar (using an adsorption vacuum pump). The fiber has a guidance band centered at 1550 nm, exhibits a linear loss of 5-10 dB/km in the range 1450-1700 nm and has a 20 μm core diameter (see inset of Fig. 1). The choice of a larger core in comparison with the fiber used in [20] is motivated by needing to reduce the collision rate of the gas with the core wall, which is likely to be one of the dominant sources of decoherence given the confined geometry of the interaction zone. The control beam is tuned around the P(J+1) line corresponding to the transition 0(${\sum }_g^{+}$)(J + 1) → (v ₁ + v ₃)(${\sum }_u^{+}$)(J) where J is the rotational quantum number of the vibrational state and physically can take any value from 0 to 27, covering thus all the strong absorption lines of the P-branch of the overtone v ₁ + v ₃ band. For each laser control coupled to a P(J+1) line, the probe is swept around R(J-1) (i.e. 0(${\sum }_g^{+}$)(J-1) → (v ₁ + v ₃)(${\sum }_u^{+}$)(J)transition) for Λ interaction scheme, and R(J+1) (i.e. 0(${\sum }_g^{+}$)(J + 1) →(v ₁ + v ₃)(${\sum }_u^{+}$)(J + 2) transition) for V-type interaction.

Our experimental arrangement is shown schematically in Fig. 2. The probe beam is delivered from a commercial tunable external cavity diode laser (ECDL) with wavelength tuning range between 1508 and 1575 nm. This beam is coupled to a fiber system consisting of an isolator and the 10% port of a 90/10 coupler. The control beam, from a second ECDL, is amplified by a ~1W erbium-doped fiber amplifier (EDFA). It passes through a polarization controller and is coupled to the 90% port of the 90/10 coupler where it is combined with the probe beam. The combined beams pass through a circulator and finally enter the acetylene-filled HC-PCF. The role of the circulator is first to avoid feedback to the laser sources and secondly to monitor the input beams via their residual reflection at the HC-PCF/SMF splice. At the output of the HC-PCF, the control beam is filtered out using interference filters, leaving mainly the probe beam to be transmitted and detected. The measurements were carried out for three different fiber lengths (~1 m, ~2 m and ~5 m) and under different pressures. For each length and pressure, we first fixed the control to be resonant with a given absorption line from the P-branch. Then we recorded the transmission profile of the corresponding probe. This was generated by sweeping its frequency around the desired absorption by driving the ECDL PZT, giving a span bandwidth of ~1 GHz. The transmission spectrum of the probe is recorded for different power levels in the control beam.

 

Fig. 2. Schematic representation of the experimental set-up. PD: photodetector, IF: interference filters, S: splice between the HC-PCF and a solid SMF, FPC: fiber polarization controller, EDFA: erbium-doped fiber amplifier, ECDL: external cavity diode laser. The inset on the left-hand side is a SEM of the HC-PCF used as the acetylene cell.

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3. Results and discussion

Figure 3(a) shows the normalized transmission spectrum of light from an LED source through ~1 m long acetylene filled HC-PCF (pressure ~1mbar). Also, it shows the different absorption lines with their corresponding labeling. The insets illustrate the combination control-probe for the two interaction schemes we investigated, P(J+1)-R(J-1) for the Λ-type configuration and P(J+1)-R(J+1) for the V-type. Figure 3(B) shows the typical EIT traces recorded (black line) at the R-branch line of a selection of P-R combinations for a control-laser with coupled power in the range of 400-500 mW, gas pressure in the range of 0.1-1 mbar and a fiber length of ~2 m. The upper row corresponds to a Λ-interaction scheme formed by the control-probe combinations P(15)-R(13), P(16)-R(14) and P(17)-R(15) while the lower row corresponds to the V-type interaction and shows the combinations P(15)-R(15), P(16)-R(16) and P(17)-R(17). It is noteworthy that transparencies were observed at around 15 R-branch lines (from R(5) to R(19)) when the gas is driven at the appropriate P-line; we were limited by the gain-bandwidth of our EDFA. Moreover, even when the molecules were driven at lines with smaller absorbance such as P(16), EIT was routinely observed in our control-laser power range.

The transparency height was measured by taking the ratio of the signal at zero-detuning of the probe (the control laser being on) to the same signal in the absence of the control laser. We found it to vary between 10 and 30% for the experimental parameters mentioned above, with a corresponding FWHM in the range of ~40 to ~70 MHz. Moreover, we carried out a simple theoretical fit using the density matrix formalism [22], following the approach described in [23]. The results are represented by the grey curve in each of the graphs in Fig. 3(b) and show good agreement with the experimental data by appropriate choice of fitting parameters.

 

Fig. 3. (a) The transmission spectrum of a ~1m long acetylene-filled HC-PCF at a pressure of ~0.1-1 mbar. Inset: schematics of Λ and V interactions. (b) Measured (black line) and theoretical (grey line) transmission spectra for coupled power in the range of 400-500 mW, gas pressure in the range of 0.1-1 mbar and a fiber length of ~2 m. The asymmetry in some of the transparency traces are due to the probe-frequency detuning from the absorption transition.

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In this theoretical model we use the following assumptions: (i) that spontaneous decay from the upper level transfers the population equally between the two ground-states (i.e., Γ₂₁ = Γ₂₃ = Γ₂/2 = Γ_sp/2) and is zero for the ground-states; (ii) that the system is closed (this means the Clebsch-Gordan weighted portion of spontaneous decay into the spectral reservoir formed by the rest of the acetylene energy-states is neglected); (iii) that the effect of the probe is ignored as its power is very weak (<100 μW) and a first-order analysis of the density matrix motion and its steady-state solution is valid, i.e., all the molecules are in the vibrational ground state and equally populate the two lower states |1〉 and |3〉 (ρ ₁₁ ≈ ρ ₃₃ ≈ 0.5,ρ ₂₂ ≈ 0); (iv) finally, that saturation effects and optical pumping are ignored for simplicity. With these assumptions, our detected signal is fitted to an expression proportional to exp(-4πk_pL_fibχ″(ω_p )), where k_p is the wavevector of the probe laser, L_fib is the fiber length and χ″(ω_p ) is the imaginary part of the susceptibility. The susceptibility χ is deduced from the off-diagonal elements of the density matrix and averaged-out over the velocity distribution which is taken to be a Maxwellian, and is given by [23]:

where z is given by:

where μ is the dipole moment for the probe transition, ω_P the angular frequency of the probe, N the molecular density, c the speed of light in a vacuum and u/√2 the root-mean-square of the thermal velocity of the molecules. Δp and Δc are the detuning of the probe and control laser respectively from their corresponding molecular transition. Ω_c is the control-laser Rabi frequency defined as Ω_c = μE_c /ħ. Expressed as a function of the power coupled into the fiber P_c , this gives $\overrightarrow{k}$ ${\Omega }_C=(\mu /\hslash )\sqrt{2P_C/c{\epsilon }_0{A}_{eff}}$ with A_eff = (4/9)${\pi R}_{\mathit{\text{fib}}}^2$ where R_fib is the fiber core radius.

In Eq. (2) γ_gr and γ_od are related to the ground-state coherence relaxation rate and the optical decay rate of the upper state respectively, and are treated as fitting parameters. They sum up contributions due to diffusion (time-of-flight through the laser beam and collisional relaxation of the ground-state coherence), spontaneous decay and phase noise between the laser fields and laser linewidths. With the above assumptions and for a Λ-type system, the expressions for the two decay rates are given by [24,25]:

where Γ_sp is the natural linewidth of the upper levels (spontaneous emission rate). For acetylene overtone transitions this is of the order of a few Hz [26]. δω_C and δω_P are the laser linewidths of the control and probe fields respectively and they have been measured to be ~20 MHz for the control laser and ~3 MHz for the probe laser. γ_tf is the decay rate due to time-of-flight through the laser beam (escape rate). For simplicity, we neglect this decay given that the guided-mode fills almost all the hollow-core physical area. γ_coll is the dephasing rate due to inter-molecular collisions and collisions with the inner wall of the fiber core. The pressure broadening is typically 12 MHz/mbar [27], which for operating pressures of less than 1 mbar is negligible compared with the dephasing rate due to collisions with the walls. The latter is given in the hard sphere limit by [28] ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ 2π = 2.4052 [D_s /$R_{\mathit{\text{fib}}}^2$ (1 + 6.8(λ_fp /R_fib ))), where D_s and λ_fp are the self-diffusion coefficient and the mean-free-path of the acetylene respectively and are both pressure dependent. Using the Chapman-Enskog equation for D_s [29], ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ takes the form 212/(√p + 3.76/√p) MHz where p is the acetylene pressure expressed in mbar (the square-root reflects the fact that we have treated acetylene as an ideal gas). Consequently, if one keeps only the dominant elements, the expression of the two decay coefficients is reduced to a function of ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ and δω_c :

From the above equations, one sees that the main sources of decoherence in our system are due to the control-laser linewidth and collisions with the fiber wall. Furthermore, the value of γ_gr /γ_od deduced from the above expressions agrees qualitatively with the fitting parameters used for the different experimental results (see the grey curves in Fig. 3(b)).

 

Fig. 4. Evolution with the control-laser Rabi frequency of the transparency height (A) and full-width-at-half-maximum (B). The operating pressure range is 0.1-1 mbar and the fiber length is ~2 m.

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Fig. 5. (49 kB) Animation of the evolution of the electromagnetically induced transparency with control-laser power launched into the HC-PCF for P(15)-R(13) combination.

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In order to deduce γ_gr and γ_od independently, we examined experimentally and theoretically the evolution of the height and the FWHM of the EIT with control-laser power, H_EIT and Γ_EIT respectively. The theoretical values were estimated numerically from the expression for the susceptibility (Eq. (1)) using an iterative method.

Figure 4 shows the evolution of H_EIT and Γ_EIT with increasing Ω_c for the configuration P(15)-R(13). The experimental data show a trend that is in good agreement with theoretical fit using Eq. (1). The discrepancies at higher Ω_c are due to uncertainties in the coupled control-laser power, and also to the fact that our theoretical model does not take into account optical pumping. The numerical fit for both curves gives a value of ~20 MHz for ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ , corresponding to a pressure of 0.13 mbar which is in excellent agreement with our experimental pressure range using the mechanical vacuum pump.

To gain some physical intuition and predictive insight into how the ground-state coherence relaxation and optical decays affect the transparency resonance and evaluate them at least approximately, we deduced an expression for H_EIT by expanding the susceptibility (Eq. (1)) around zero probe-detuning. For large Doppler broadening, Δw_D , this gives H_EIT ~${\mathrm{\Omega }}_C^2$/2Δw_D γ_gr . Hence, the product of the ground-state decay with the Doppler linewidth sets the Rabi frequency “threshold” required for the EIT occurrence. This statement qualitatively agrees with our experimental observation where significant EIT occurrence starts for Rabi frequencies around 50 MHz if one takes the value of γ_gr deduced from above. Similarly, an empirical form of the transparency linewidth (FWHM) is given in the absence of the control-laser one-photon detuning by Γ_EIT ≈ 2γ_gr + ${\mathrm{\Omega }}_c^2$/${\gamma }_{\mathit{\text{od}}}^{*}$ [24,25]. A simple examination of this expression shows that the ground-state coherence decay determines the minimum transparency width one could hope for. ${\gamma }_{\mathit{\text{od}}}^{*}$ is the effective optical decay and is equal to γ_od/V; it controls the rate of the power broadening of the EIT (V represents the weight due to thermal velocity distribution [24]). A fit of the above expressions with our experimental data gives a value of ~25 MHz for γ_gr and ~1.2 GHz for ${\gamma }_{\mathit{\text{od}}}^{*}$, corresponding to a value of ~10 MHz for γ_od . Including these values in equations (4) gives a value between 15 and 20 MHz for ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ , which represents an excellent agreement with the ones deduced from our numerical fit.

 

Fig. 6. Measured and theoretical transmission trace of R(11) line in the absence of control beam (A) and in the presence of ~800 mW control power (B). The operating pressure range is 0.001-0.01 mbar and length of fiber ~2m.

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In order to minimise the decoherence effect due to the fiber wall we evacuated the fiber to pressure around 0.001-0.01 mbar using an adsorption vacuum pump. As we have seen from the simple analysis above, this has the effect of increasing the transparency height whilst lowering the minimum EIT linewidth. At this range of pressure, collisional relaxation due to the fiber wall drops to between 5.6 and 1.7 MHz and consequently is not as dominant as for the pressure range we have with the mechanical pump. Figure 6 shows the absorption line R(11) when only the probe beam is coupled to HC-PCF (Fig. 6(a)) and in the presence of the control-laser with a power of ~800 mW (Ω_c ≈ 198 MHz) (Fig. 6(a)). The transparency height reaches ~70% for a transparency linewidth of ~57 MHz. The numerical fit gives a value of 15 MHz for γ_gr , corresponding to ${\gamma }_{\mathit{\text{col}}}^{\mathit{\text{wall}}}$ ≈ 5 MHz, which in turn corresponds to a pressure of ~8 × 10^-3 γ_gr , which is within our pressure range.

It is noteworthy that despite the present promising results, we believe that with a further decrease in the gas pressure and better frequency stability of the lasers, one could further improve the present system to expect unity transparency and sub-MHz linewidth at relatively moderate control-laser powers (<1W). Indeed, for a pressure between 10^-5 and 10^-6 mbar the total collisional decay is below 100 kHz. Consequently, taking a value for the absorption coefficient of the v ₁ + v ₃ acetylene overtone transitions of 3 cm^-1 mbar^-1 [30], one could in principle achieve with detectable contrast a resonance with total transparency and ~200 kHz linewidth using a fiber length of ~50 m.

4. Conclusion

In conclusion, we have demonstrated generation of EIT in acetylene-filled HC-PCF in both Λ and V interaction schemes over several lines of the R-branch – to our knowledge the first time this has been demonstrated. At a pressure below 10 µbar, we have achieved a transparency as high as 70%. The linewidth of the transparency was around 60 MHz. Such a large linewidth was attributed to collisions of the gas with the inner wall of the HC-PCF core and to the poor quality of the pump laser lines. Moreover, we have showed that near-unity transparency along with sub-MHz linewidth is possible using an appropriate length of HC-PCF filled at suitable pressure, making it an excellent candidate in areas such as compact atomic clocks and slow light applications.