We present undergraduate-friendly instructions on how to produce light pulses propagating through warm Rubidium vapor with speeds less than 400 m/s, i.e., nearly a million times slower than c. We elucidate the role played by electromagnetically induced transparency (EIT) in producing slow light pulses and discuss how to achieve the required experimental conditions. The optical setup is presented, and details provided for preparation of pump, probe, and reference pulses of the required size, frequency, intensity, temporal width, and polarization purity. EIT-based slow light pulses provide the most widely studied architecture for creating quantum memories. Therefore, the basic concepts presented here are useful for physics and engineering majors who wish to get involved in the development of cutting-edge quantum technologies.
I. INTRODUCTION
It has been two decades since the first demonstration of slow optical pulses propagating through atomic media.1,2 Slow light was revealed as the striking consequence of a quantum mechanical phenomenon in light–matter interaction known as EIT, i.e., electromagnetically induced transparency. EIT arises from the interference between probability amplitudes for absorption pathways that are simultaneously excited by two resonant light fields—one strong, referred to as the pump (or coupling) field, and the other weak, referred to as the probe.3,4 EIT and slow light in warm alkali vapor continue to be central topics of research in quantum information and quantum technology, particularly for building robust quantum memories,5–7 and stable photon-shot-noise-limited electromagnetic field sensors using Rydberg atoms.8 In Ref. 6, for instance, the authors provide an overview of current approaches to quantum memory and state that “although all of these approaches have been studied and demonstrated, EIT remains the most popular scheme for quantum memory,” because, “in comparison to the other approaches, the EIT approach has a long storage time and is a relatively easy-to-implement and inexpensive solution.”
An excellent article on EIT-based experiments for undergraduate laboratories was published just over a decade ago.9 Elegant EIT-based experiments carried out at undergraduate institutions have also been reported.10 However, an undergraduate-friendly experimental description of slow light in atomic vapor does not exist. In this paper, we endeavor to fill this gap. This is important because an increasing number of physics and engineering majors wish to get involved in the development of cutting-edge quantum technologies.
There are several pedagogical advantages to introducing the concepts of EIT and slow light in an optics class for undergraduate seniors and first/second-year graduate students. For instance, in the advanced lecture/lab course “Optics and Laser Physics” that we teach at Miami University, students often ask, “We've read about laser applications in imaging, communications and medicine… can you tell us about quantum applications we haven't heard of?” EIT and slow light can provide a satisfying answer to this question. EIT physics is a natural extension to what these students have already learned about light–matter interaction. Early in the semester, they are introduced to population-rate equations for a laser system, which sets the stage for an EIT-based slow light system, since the simplest model for either system is provided by a three-level atom. Furthermore, students know about group versus phase velocity, and therefore have all the necessary background to learn about how a steep positive gradient of the refractive index within a narrow EIT spectral window can lead to a dramatically slow group velocity.
In this article, we state the relevant theoretical results for EIT and slow light, and describe our experimental setup for producing slow light pulses with group velocities less than 400 m/s. A price list for parts is included in the online supplementary material.11
II. THEORETICAL DESCRIPTION
The theory behind EIT and slow light has been explained in several high-quality undergraduate-friendly articles (see, for example, Refs. 9 and 12–15) including a textbook-treatment in Ref. 16, which is amenable to seniors. Here, we summarize the relevant results. A more complete derivation can be found in the online supplementary material.11
A. Three-level atom model
Following the treatment in Ref. 16, we depict a fictitious three-level atom in Fig. 1, in which transitions are allowed between states 1 and 3 and between states 2 and 3, but not between states 1 and 2. This is the so-called lambda-system because the lower levels 1 and 2 have close-lying energies, resulting in a Λ-shaped configuration. The energy differences between levels 3 and 1, 3 and 2, and 2 and 1 are , and , respectively. The standard EIT experiment consists of a sample of such atoms illuminated by a strong coupling (or pump) laser of frequency ωc and a weak probe laser of frequency ωp, both tuned to near-resonance with from levels and respectively. The laser detunings are and . We assume that the pump beam only addresses the coupling transition and the probe beam only addresses the transition. This may be satisfied by ensuring that the interaction between the pump beam and the probe transition, and vice versa, is forbidden by quantum mechanical selection rules (as in Zeeman EIT, discussed in detail in Sec. III A below). Alternatively, this interaction may be negligible because the two transitions ω31 and ω32 are significantly different (as in hyperfine EIT, see Sec. III A). In either situation, the pump frequency is fixed, while the probe frequency is scanned so that the two-photon detuning , defined as , passes through zero (two-photon resonance). The probe absorption is measured as a function of this scanning probe frequency.
In the supplementary material, Secs. S1A and S1B,11 we write the Hamiltonian for an illuminated three-level atom, assuming that the incident field-induced electric dipole is the predominant light–atom interaction. We calculate the eigenenergies and eigenstates in Eqs. (S9)–(S11), taking the pump and probe beams to be on-resonance for simplicity (i.e., the two-photon detuning is zero). One of the eigenstates is a “dark” state, which is a coherent superposition of the two ground states and . This dark state has no quantum overlap with the excited state . Thus, an atom, once it is transferred to the dark state upon interaction with the light fields, stays there, no longer able to absorb a probe photon. In general, for non-zero detunings Δp and Δc, the steady-state dark state is formed at two-photon resonance . Note that in the absence of the pump, the weak resonant probe would be completely absorbed. Thus, the pump laser “coherently prepares” the atoms to be transparent to the probe—this is EIT.
In order to calculate the spectral width of the transparency window, we must take into account the decay of the light-induced dipole moment due to various decohering processes (also called damping, or relaxation, processes). Such processes may arise from atomic motion which may, for example, cause the atom to leave the illuminated region, and the light–atom interaction to cease altogether. Similarly, elastic and inelastic collisions with other atoms and with the container walls, and other processes such as spontaneous emission, affect the phase and amplitude of the light-induced dipole moment. These mechanisms are discussed in more detail in Secs. III B and III C below. We denote all decay processes from and from by the optical decoherence rates γ13 and γ23, respectively. The two ground states are close-lying, so that non-optical processes, such as collisions, may suffice to cause interchange between the two states—we denote this two-way decay by the ground state decoherence rate γ12. In dilute samples where the densities are not so high, non-optical damping processes (e.g., collisions between ground state atoms) begin to dominate over optical damping processes e.g., spontaneous emission or Doppler broadening, γ12 is much smaller than γ13 and γ23. This is an important requirement for EIT which is satisfied in our experiments, see Sec. III C below.
We show in Secs. S1C–S1D of the supplementary material11 that the density matrix formalism allows for simple phenomenological inclusion of decay processes into the calculation of the probe-induced atomic dipole moment and polarizability. This allows us to deduce the complex refractive index . The imaginary part ni yields the absorption spectrum . The complex refractive index is a convenient way to express something well known: Where there is dispersion , there must be absorption α. This is owing to causality-based Kramers–Kronig relations between the real and imaginary parts of the refractive index.17 We calculate the absorption and real refractive index seen by the weak probe (see Eqs. (S22) and (S23)) propagating through a sample of Λ-atoms, which is coherently prepared by a strong pump beam in a dark state, as mentioned above. In Secs. II B and II C below, we discuss some important limiting situations which will be probed by experiments.
B. Absorption and refractive index with pump field off
Here, Ωc is the coupling beam Rabi frequency defined as , where is the transition dipole moment between levels and is the unit polarization vector in the direction of the coupling electric field, and is Planck's constant divided by .18 N is the number of atomic dipoles per unit volume, c is the speed of light in vacuum, ϵ0 is the electric permittivity of vacuum, and μ13 is defined as the projection of the induced probe dipole moment on the direction of the probe field polarization.
Equation (1), plotted in Fig. 2(a), depicts the expected Lorentzian absorptive line shape, with a half width at half maximum (HWHM) γ13 and a maximum at . Equation (2), plotted in Fig. 2(b), depicts the expected dispersive line shape around probe-resonance, again with a HWHM of γ13. In the immediate vicinity of resonance , we see that , i.e., has a negative slope. This is the well-known “anomalous dispersion” effect that occurs close to an atomic resonance and in which nr decreases with increasing optical frequency. By contrast, as we move further away from resonance we see that nr increases with frequency on either side. This is the normal dispersion, consistent with the classic experiment of propagating white light through a glass prism and observing that red light deviates the least and blue the most.
C. Absorption and refractive index with pump field on
D. The EIT “window”
E. The refractive index and slow light
We wish to clarify two issues that often confuse students. First, the dramatic changes in nr near resonance do not much impact the phase velocity of light ( ), because nr never significantly departs from unity for a dilute vapor—this is clearly visible in Figs. 2(d) and 2(f). No physical significance is ascribed to the phase velocity, so that the fact that nr dips below unity, causing the phase velocity to exceed c, is no cause for alarm. Second, the definition of vg above suggests that when is negative, as is the case near-resonance just outside the EIT window, the group velocity may, in principle, exceed the phase velocity and even c. However, outside the EIT window the absorption is high; thus, no signal is actually able to propagate faster than c.15
F. Stored light and EIT-based quantum memory
In effect, a pulse of probe light is slowed and stored in the sample for a time τd,20 and the medium then acts as a quantum memory where information can be stored. Notice that the pulse undergoes a dramatic spatial compression inside the sample: The front end slows down upon entry into the sample and propagates at vg, even as the pulse-rear, which is still outside, propagates at c. The compression factor is given by where Lp is the length of the compressed pulse inside the medium and L0 is the free-space pulse length.20
III. THREE-LEVEL ATOMS: IMPLEMENTATION IN THE LAB
In the lab, Rubidium is a popular choice of alkali, owing to the ready availability of inexpensive single mode diode lasers at the resonance wavelength for transitions from the ground state to the excited state (D1-transition; 795 nm) or the excited state (D2-transition; 780 nm). Figure 3 shows the hyperfine energy-level structure for both stable isotopes 85Rb (nuclear spin I ) and 87Rb (I ).25
Our experiments are performed on the D1 transition in 87Rb atoms. The hyperfine D1 transition for 87Rb involves the ground state and the excited state, which are split by the interaction of the valence electron's total angular momentum (orbital + spin) with the nuclear spin, creating ground energy states (separated by 6.8 GHz), and excited energy states (separated by 815 MHz). Each hyperfine state F has degenerate substates with quantum number mF ranging from to F in increments of 1. Measurement will find the atom with an angular momentum component along a chosen quantization axis—we choose z as our axis of quantization. For the Fg = 2 ground state, for example, five magnetic substates are revealed in the presence of a weak external magnetic field Bz (along z) which lifts the degeneracy by imparting a Zeeman shift to each magnetic sub-level. Here, gf is the Lande g-factor for the hyperfine state ( 1/2 for Fg = 2), μB is the Bohr magneton ( J/T), and is the magnetic sub-level number. The Zeeman shifts between the magnetic sub-levels of the Fg = 2 ground state are Hz/T, resulting in a shift of 0.7 kHz when the field is T (i.e., 1 mG).25
A pump or probe laser tuned near-resonance to a particular transition pumps other close-lying Fe states off-resonantly, creating extraneous channels for atomic population leakage, which diminishes the steady-state fraction of atoms in the dark state. The 87Rb D1 transition is advantageous in that respect because the Fe level-separation of 815 MHz is largest, compared to that of the D1 transition in 85Rb (362 MHz), and of the D2 transitions (Fig. 3). See Sec. IV C for further details.
A. Zeeman EIT: Spin polarization via optical pumping
Three-level schemes with alkali atoms for EIT-based slow light experiments fall into two categories: Hyperfine EIT and Zeeman EIT. Hyperfine EIT with 87Rb is depicted in Fig. 4(a). A strong pump beam (coupling laser; solid line) is tuned to one transition, while a weak beam (probe; dashed line) is tuned to the other transition, 6.83 GHz apart. The probe is created using an electro-optic modulator to split off a small amount of the coupling beam, or by using two phase-locked lasers, each tuned to either transition. Zeeman EIT, depicted in Fig. 4(b), is less resource-intensive, requiring a single passively-stable laser system and less expensive acousto-optical modulators. Hence, we confine our attention to this method. Zeeman EIT relies upon the optical pumping of the magnetic Zeeman sub-levels created by applying a weak external magnetic field Bz, using a strong coupling beam and a weak probe beam of mutually orthogonal circular-polarization (one beam , the other ). Both beams propagate collinearly with Bz.
Figure 4(b) depicts a strong coupling field (solid arrows) and weak probe field (dashed arrows), tuned to near-resonance with the transition.25,26 The dipole selection rules for transitions induced between magnetic substates by light are for absorption and for stimulated emission, in addition to the selection rule for spontaneous emission transitions.25 This means that after many absorption–emission cycles have occurred, the strong field optically pumps the atoms into the sub-state. From here, the pump excites the atoms to the sub-state, which is followed either by spontaneous decay to the spin sub-states (where they stay) or by emission back into the state (in which case they are re-excited). Eventually, after a time that is long compared to the excited state lifetime of 27.7 ns for the Rb D1 line, the atoms accumulate into the spin sub-states (typically, after pumping on a time-scale of ms). The action of the weak probe is to optically pump some of these atoms toward lower -states. Eventually, in the steady-state, the atomic population is concentrated in the spin sub-states. Thus, the atom is “spin-polarized” via optical pumping.27
Indeed, the ideal three-level lambda system (states , and in Fig. 1) is well-approximated by the ground states and the excited state . In Fig. 4(b), the pump (thick solid arrow) and probe (rightmost dashed arrow) are highlighted. The method for fine-tuning the pump and probe laser frequencies to resonance with these particular Zeeman sub-levels is described in Sec. IV C. These approximate three-level atoms then evolve into the dark state, enabling EIT and slow light. Factors causing spin relaxation that reduce the dark state population are discussed in Sec. III B below.
Note that the assumption made at the beginning of Sec. II, that the coupling beam only addresses levels and , and the probe beam only addresses levels and , is automatically satisfied in Zeeman EIT because pump photons cannot be absorbed by the transition and probe photons cannot be absorbed by the transition. The assumption is also satisfied in the case of hyperfine EIT, where the 6.83 GHz difference between the pump and probe frequencies in Fig. 4(a) is large enough that the pump beam experiences negligible absorption on the probe transition, and vice versa.
B. Spin relaxation mechanisms: Role of buffer gas
There are several spin-depolarization mechanisms that reduce the EIT contrast and affect its linewidth. The spin-polarized Rb atoms prepared in the previous subsection exist within the sample volume that is jointly illuminated by the pump and probe beams. Typically, in a glass cell filled with warm alkali vapor, this volume is the central part of the cell. However, the atoms are moving at thermal speeds, so that the polarized atoms can exit the illuminated volume. Collisions with the glass wall destroy the atoms' polarization,28 and these unpolarized atoms may subsequently re-enter the illuminated region. These transit-time effects reduce the sample polarization.29 Further, there are spin-exchange collisions between spin-polarized Rb atoms which can redistribute the populations in the magnetic sub-states, causing depolarization.28
There are two commonly used approaches to suppress the spin-depolarizing effects described above:27 (i) include an inert buffer gas such as Ne, He, or Ar in the Rb vapor cell or (ii) coat the inside of the glass vapor cell with an anti-relaxation coating such as paraffin.
Coated cells can yield significantly longer spin coherence times than buffer gas cells because the paraffin coating suppresses the Rb spin-depolarization via “softer” atom-wall collisions. However, the light-Rb atom interaction dynamics in coated cells is more complicated, yielding a dual-structured EIT lineshape with a narrow central peak sitting atop a broad pedestal (see, for example, Ref. 5 and references therein). The broad pedestal is due to Rb atoms interacting with light during a one-time pass through the light beams. The narrow peak, on the other hand, results from contributions by polarized Rb atoms that transit in and out repeatedly through the laser beams, interacting coherently each time with the pump and probe fields. These atoms have many wall collisions before eventually becoming depolarized. The linewidth of the narrow peak is significantly affected by factors such as coating quality and cell geometry.5 These effects tend to make the EIT data more difficult to interpret so that, in the work reported here, we do not use coated cells.
In buffer gas cells, the usual practice for EIT and slow light experiments is to mix noble gas (up to several tens of Torr) with a few microTorr of Rb.5 In our experiments, we use 10 Torr of Ne. Frequent Rb–Ne collisions reduce the Rb atoms' mean free path to values much smaller than the pump beam diameter,30 thus confining the atoms within the illuminated volume. Instead of flying through the laser beam at thermal speeds, the atoms then slowly diffuse through, thereby increasing the laser-atom interaction time by several orders of magnitude.5 Note that Rb–Ne collisions cause the spin-polarized Rb atom's velocity to change, but with negligible spin relaxation. Such collisions that rapidly redistribute the atomic velocities without re-equilibrating the populations of the atomic levels are termed “velocity-changing collisions” in the literature.27,30 At the considered partial pressures, Rb–Rb and Rb-wall collisions are negligible compared to Rb-Ne collisions.
Still, decoherence effects remain that are not addressed by the buffer gas (or by wall coatings). Inhomogeneities in the magnetic field used for Zeeman EIT cause spatial variation of the dark state leading to absorption. In Sec. S2 of the supplementary material,11 we discuss how to suppress these inhomogeneities. Furthermore, an incoherent pumping mechanism, known as radiation trapping, may become significant at high atomic densities. Radiation trapping refers to the reabsorption of spontaneously emitted photons within the illuminated volume, and is expected to become significant inside our probe beam radius at number densities cm−3 (see Sec. IV D).31 Hence, we keep our number density below this value (see Secs. V and S2 D).
C. Slow light conditions; estimate of γ12, γ13
The ground state decoherence rate γ12 arises from Rb–Rb collisions, Rb collisions with the cell walls, and from Rb atoms eventually diffusing out of the illuminated volume despite the buffer gas. Collisions with the inert buffer gas do not contribute significantly to γ12. However, since our experiments use Zeeman hyperfine levels, inhomogeneities in the magnetic field contribute to γ12. Good suppression of stray magnetic fields (Sec. S2, supplementary material)11 results in a value kHz, as reported in the literature.33,34 Clearly, , and since kHz or less (see Figs. 2(e) and 2(f) and 8) our experiments fall in the strong coupling intensity regime, i.e., the condition in Eq. (5) is well-satisfied.
Next, to check that Eq. (11) is also well-satisfied in our experiments, we note that the dipole matrix element for the 87Rb D1 transition in our spin-polarized atom is well-approximated by just the magnetic quantum number-independent value (known as the reduced matrix element) cm.25 The number density N is cm−3 (see Sec. V), yielding GHz, which exceeds the -values by three orders of magnitude.
Finally, we estimate the excited state damping γ13 which arises from the spontaneous emission rate Γ, the dephasing due to Rb collisions with the buffer gas, and the Doppler broadening ΓD. Recall from Eq. (1) and Fig. 2(a) that γ13 is manifested as the half-width of the absorption peak. Our sample consists of a few μTorr of isotopically enriched dilute 87Rb vapor mixed with 10 Torr of Ne buffer gas. To estimate we follow the empirical relation MHz/Torr derived in Ref. 35. We see that, for 10 Torr of Ne, MHz. On the other hand, the Doppler broadening of the absorption profile of Rb vapor, illuminated by a frequency-scanning monochromatic beam, arises due to the thermal motion of the atoms.36 In a simple 1D situation, an atom moving with thermal velocity toward or away from a laser beam of frequency ω0 sees a Doppler-shifted frequency . This means that for a thermal distribution of velocities we may estimate the full width at half maximum (FWHM), ΓD, of the Doppler broadened absorption profile to be . Here, is the magnitude of the incident field wave vector. If we use the most probable velocity , we obtain m/s at K, yielding MHz) at λ = 795 nm. Thus, γ13 is dominated by and (both are significantly larger than Γ). We use their sum as an estimate for γ13.
Note from Eqs. (9) and (13), that, in the high coupling intensity limit, both vg and decrease linearly with the intensity. However, lowering the intensity too far (although still in the high intensity limit) may reduce the steady-state population in the dark state enough that slow light effects start to degrade, causing the probe pulse to distort and appear to speed up again. Thus, we may expect a sweet spot in the pump intensity, where conditions for slow light are optimized (see Sec. VII C).
IV. EXPERIMENTAL SETUP
A. Laser source
We use a commercial external cavity tunable diode laser (ECDL) to provide linearly polarized continuous-wave light at ∼795 nm. A few percent of the output is split off for use in a saturated absorption spectroscopy (SAS) set-up to enable the precise tuning of the laser to the D1 transition, as described below.
The main ECDL output is first passed through a Faraday rotation-based optical isolator (OI) which optically isolates the laser from back reflections (e.g., arising from the optics that couple the light into the fiber). The light then passes through an anamorphic prism pair to circularize the elliptical cross-section of the beam, before coupling into a single-mode polarization-maintaining fiber which preserves the direction of the linear polarization of the input beam (we measured a residual drift of <2% in the polarization direction at the fiber output). The single-mode fiber is typically aligned and operated at a transmission efficiency of only 30%–50%, but produces a clean Gaussian spatial profile. Gaussian spatial profiles are highly desirable for pump-probe experiments where the pump and probe beams must be well-collimated, and must have good spatial overlap that is amenable to quantitative modeling.
B. Creating the pump and frequency-scannable probe
The output from the fiber (∼7 mW) is collimated using an aspheric lens. The collimated light is divided by a polarizing beamsplitter (PBS 1) into two orthogonally linearly polarized beams—the strong coupling (or pump) beam (solid line) and weak probe beam (dashed line).
In pump-probe spectroscopy, the probe is swept in frequency around the fixed pump frequency, and the probe transmission spectrum is detected. The sweeping is accomplished with an acousto-optic modulator (AOM), which is a device that uses a sound wave propagating through a crystal to form a diffraction grating for incoming light waves, producing frequency-upshifted (+1) and downshifted (–1) orders, in addition to the 0 order at the incident frequency.17,37 The ±1 orders emanate at a slight angle to the direction of the incident beam, symmetrically on either side, while the 0 order continues along the incident beam direction. By carefully aligning the angle of incidence of the beam on the AOM, we achieve about of the incident power in the first order of choice, with most of the remaining power in the 0 order, while a small amount leaks into the undesired first order. The AOM performs a frequency-scan by introducing a fixed offset (80 MHz in our case) and sweeping back and forth about that fixed offset value by some desired amount (chosen by the user to be any value from zero up to a maximum sweep ±20 MHz for our AOM). For the purpose of measuring the EIT linewidth in Figs. 2(c) and 2(e), a sweep of ±100 kHz suffices (see Fig. 8). Because of the fixed frequency-offset of 80 MHz that the AOM introduces, we must insert an identically configured AOM in the path of the pump beam as well, so that both the pump and probe beams are offset in frequency by the same amount. To drive the AOMs, a dual-output waveform generator which creates twin phase-locked identical 80 MHz 1 V signals is used. Each signal is amplified using a standard RF amplifier before being fed to the AOM. Planoconvex lenses of focal length 30 cm are placed on either side of each AOM, with the AOM crystal located at the common focal spot.
One problem we encountered is that the angle of the diffracted orders changes when the output frequency is varied, causing a spatial shift in the beam at the vapor cell located a few feet downstream. This shift is a problem because the probe beam moves off the center of the vapor cell during the course of its frequency scan. To overcome this problem, the desired diffracted order is retroreflected back through the crystal (see top right hand corner of Fig. 5), such that the double-shifted order is aligned with the incident beam but in the counter-propagating direction.37 For this double-shifted beam the angular deflection from the second pass cancels the deflection from the first pass. The cancellation is not ideal, due to unavoidable imperfections in alignment, but the spatial shifting of the scanning probe at the site of the vapor cell is highly suppressed in this double-pass AOM configuration. The presence of the quarter waveplate just before the retroreflecting mirror in Fig. 5 serves to orthogonally polarize the double-shifted order with respect to the incident beam, enabling the separation of the double-passed and incident beams at a polarizing beamsplitter (PBS 2 for the pump, PBS 3 for the probe).
C. Frequency tuning pump and probe to EIT window
The frequency-tuning of the laser for implementing Zeeman EIT is achieved in two steps. First, the laser is tuned to the 87Rb D1 transition, using the method of saturated absorption spectroscopy (SAS). Next, the pump and probe beams are fine-tuned to the and Zeeman sub-levels with the AOMs (see Fig. 4(b)).
SAS is performed in a second vapor cell, that is not explicitly shown in Fig. 5. This cell is at room temperature, is filled with natural abundance Rb vapor (72% 85Rb, 28% 87Rb), has no buffer gas, and the walls are uncoated. Figure 6 shows the transmission spectrum of a weak beam diverted from the ECDL into the SAS cell, where the D1 transitions for the two Rb isotopes are displayed in one continuous scan. In plot (a), the Doppler-broadened D1 transitions are shown, before SAS is performed. Only for the 87Rb D1 Fe states do the level separation of 815 MHz exceed the Doppler broadening for Rb vapor (see Sec. III C), enabling the two excited states to be resolved. The 85Rb D1 transitions, with their smaller Fe separations, are totally smeared out by Doppler broadening. However, when SAS is performed, by introducing in the cell a strong beam that is spatially overlapping with the weak beam but counter-propagates, the absorption is saturated for the velocity-class of atoms that travels in a direction perpendicular to both the strong and weak beams. The hyperfine transitions are revealed as narrow “holes” that are “burnt” into the Doppler-broadened absorption spectra—these holes are manifested as “bumps” in the transmission spectrum in Fig. 6(b). In plot (c), the Doppler component is subtracted away and the vertical scale is magnified. The laser is tuned to the 87Rb D1 peak in plot (c) by reducing the scan to zero, while staying centered on this particular feature, and the goal of SAS is achieved. Note that atoms that have a velocity component, say, along one of the laser beams, may be down-shifted into resonance with that beam on a particular D1 transition, while being simultaneously up-shifted into resonance with the counter-propagating beam on the higher adjacent D1 transition—this results in “crossover” peaks midway between adjacent D1 transitions. See Ref. 38 for further details on SAS.
In order to obtain EIT transmission spectra, such as those shown in Fig. 8, the frequencies of the pump and probe beams need to be further fine-tuned into resonance with the Zeeman sub-levels and , respectively (see Fig. 4(b)). This fine-tuning is achieved inside the magnetically shielded vapor cell in Fig. 5 described in Secs. III B and III C. The magnetic field Bz for Zeeman splitting is created by a solenoid. Details on the Rb–Ne vapor cell are provided in Sec. V. Details on the magnetic shielding and solenoid are given in Sec. S2 of the supplementary material.11 Note that we must take into account the Zeeman shifts of 0.7 kHz/mG between the magnetic sub-levels of the Fg = 2 ground state.25 In our experiments, Bz = 50 mG, yielding a Zeeman splitting between adjacent ground sub-states of 35 kHz. Because the SAS procedure tunes the ECDL frequency before the laser is split into pump and probe, this means that if the pump happens to be in resonance with the transition, the probe (which is at the same frequency) is detuned from the probe transition by kHz (position A in Fig. 7). Scanning the probe AOM frequency (A ↔ C) symmetrically about position B in Fig. 7, where the probe detuning (or more precisely, where the two-photon detuning is zero; see Sec. II D), yields EIT resonance spectra as displayed in Fig. 8. For the slow light experiments, the scan is reduced to zero while staying centered at B.
It is worth noting that, before the strong (co-propagating) pump is turned on, the weak probe experiences an absorption profile in the Rb–Ne vapor cell that resembles Fig. 6(a). In Sec. III C we estimated the linewidth γ13, arising from Doppler and collisional broadening in the Rb–Ne cell, for the D1 transitions to be comparable to those shown in Fig. 6(a). Turning on the strong co-propagating pump activates the narrow EIT window predicted by Fig. 2(b), and observed in Fig. 8. It is clear from Fig. 6(a) that even though the 87Rb D1 transition is advantageous for slow light experiments because this particular transition has the least spectral overlap with neighboring D1 transitions, some residual overlap of the levels is visible. This suggests it is likely there is leakage of atoms from the Fe = 1 to the Fe = 2 level in the Rb–Ne vapor cell, resulting in a diminished dark state population and a reduced EIT contrast.
D. Pump and probe beam size and pump intensity
The probe beam is Gaussian with a -radius of 1.13 mm (the distance from the center of the beam where the intensity drops to of its value at the center). To approximate an ideal plane wave (see Sec. 1 in supplementary material Notes), the pump beam, which is also Gaussian, is expanded to a -radius of 2.26 mm using a telescope comprising two simple plano–convex lenses mounted in a cage assembly (see Fig. 5). The pump intensity was mostly varied between 1.25 and 5.6 mW/cm2.
E. Probe and reference pulses and probe intensity
In our experiments, the probe beam is in the form of a short (temporal) Gaussian pulse. Slow light demonstrations consist of measurements of the delay of the probe pulse propagating through the sample, relative to an identical reference pulse propagating along a similar path-length outside the sample. This reference pulse, depicted by the dotted lines in Fig. 5, is split off from the probe pulse by inserting a non-polarizing beamsplitter NPBS 4 into the probe path, as shown. The probe and reference pulses are created by amplitude modulation of the probe AOM: The probe RF amplifier output is pulsed on and off by pulsing the input from its waveform generator. It is then straight-forward to measure time-delays between the centers of temporal Gaussian pulses. The pulse repetition rate is chosen so that the spacing between consecutive probe pulses far exceeds the achievable time-delay between a probe—reference pulse pair. These delays are typically tens of μs (see Sec. VII). Thus, we set the time period for the amplitude modulation of the probe AOM at a few ms (3.4 ms in our case).
Matching the reference pulse path-length to the probe path is a loose requirement because light takes only about a nanosecond to travel 30 cm in air—which is negligible compared to the delay induced by the slowing down of the light in our experiment. The reference and probe pulses are made incident on the same photodetector, and the delay in their arrival times is measured on an oscilloscope by blocking one pulse (with a card) while detecting the other.
From Sec. II F, the durations of the Gaussian probe and reference pulses must be long enough that the pulse frequency-bandwidth (estimated as the inverse of the -pulse-duration τp) fits inside the EIT spectral window . According to Eq. (9) the EIT linewidth broadens linearly with pump intensity, necessitating probe pulses of progressively longer duration as the pump intensity decreases. For convenience, we keep the probe pulse duration constant, and long enough so that the frequency bandwidth fits inside the EIT linewidth for the full range of pump intensities (see Secs. VI and VII B).
The probe intensity must be kept significantly less than the pump, in order to satisfy the weak probe assumption. However, the probe pulse is too short to register on a typical power meter. Therefore, we illuminated a fast photodiode with continuous-wave light of known intensity, and calibrated the response in volts (as measured on an oscilloscope) per mW. Next, the probe pulse was shone upon the diode and the shape of the voltage response recorded. In our case the pulses have a -width of 170 μs, and the peak of the probe Gaussian temporal waveform is set at 0.3 mW/cm2 (the average intensity of the probe pulse is 0.12 mW/cm2).
F. Leakage of pump beam into the detected probe mode
The pump and probe beams are recombined at polarizing beamsplitter PBS 5, so that the pump is reflected while the probe is transmitted toward the Rb–Ne vapor cell. A half-wave plate, placed in each beam before PBS 5, adjusts the relative intensity of the pump and probe beams. Because of the large pump beam size, the corresponding half-wave plate (10 mm diameter) is located before the beam expander. PBS 5 has the property that 99.5% of s-polarization (light polarization normal to plane of incidence) is reflected whereas only 90% of p-polarized light (light polarization parallel to plane of incidence) is transmitted. In our setup, s is vertical polarization (perpendicular to the optics table surface) and p is horizontal (parallel to the table surface). For this reason, the pump beam, which must be strong, is chosen to be s-polarized, and the probe is p-polarized.
The combined beams, which are orthogonal-linearly-polarized, are passed through a quarter-wave plate (QP 1 in Fig. 5) and converted to orthogonal-circularly-polarized before entering the vapor cell, as is required for the Zeeman EIT Λ-scheme. Irises are inserted to assist in day-to-day alignments. A second quarter-wave plate (QP 2) placed after the vapor cell converts the and polarizations back to linear polarization so that the pump can be separated from the probe at a polarizing beamsplitter (PBS 6). A half-wave plate placed just after QP 2 is adjusted so that the probe transmits through PBS 6 and is focused onto a photodiode, while the pump is reflected away. The probe transmission spectrum is recorded via a fast photodiode connected to a digital oscilloscope. The impedance of the detector is kept low (10 kΩ) in order to reduce electrical reflections in the BNC cable between the detector and oscilloscope, at the cost of reduced overall voltage signal.
It is obviously important to suppress as much as possible leakage of the strong pump beam into the detector at polarizing beamsplitter PBS 6. Polarizing beamsplitter cubes typically provide an extinction of 103:1. For the highest pump powers used in our experiment, we found that a 0.1% pump leakage at PBS 6 more than doubles the power in the detected probe mode. Further, pump leakage may distort the shape of the probe pulse because the pump-profile is not entirely flat. A cleaner probe signal is obtained by the use of a Glan–Thompson polarizer (extinction ratio 105:1) as PBS 6 for pump-probe separation. Despite careful attempts to minimize the pump light from leaking into the probe detector, we observed some residual pump leakage. We suspect this is due to the use of low-order wave plates (as opposed to more expensive zero-order wave plates) and to residual errors in the alignment of QP1 and QP2's optical axes.
V. MAGNETICALLY SHIELDED WARM VAPOR CELL
For our slow light experiments, we use a pyrex glass vapor cell, that is a sealed cylinder of 25 mm diameter, containing a small amount of isotopically pure solid 87Rb metal, along with 10 Torr of Ne buffer gas. Heating the cell is a convenient way to controllably vary the Rb atomic concentration in the vapor phase, yielding a few μTorr of 87Rb vapor. A heater assembly is used to vary the cell temperature from 55 to 65 °C during the experiment, thus varying the Rb atomic number density N participating in EIT between and cm−3 (see the supplementary material, Sec. S2).11 The length of the cell (28 mm) includes 1.6 mm windows (not anti-reflection coated) on either end. Therefore, the length of the vapor sample L is taken to be 25 mm. The cell is placed inside a solenoid as shown in Fig. 5. To enable Zeeman EIT, the solenoid applies a small magnetic field Bz = 50 mG, co-linear with the laser beam propagation. The cell, solenoid, and heater assembly are placed inside a magnetic shield in order to reduce stray magnetic fields incident on the sample below 0.2 mG, i.e., < 0.5% of Bz (see the supplementary material, Sec. S2).11
VI. SETTING THE PROBE PULSE BANDWIDTH
In order to observe slow light, we need to ensure that the probe (and reference) pulse bandwidth is less than the EIT spectral window . The EIT spectrum is obtained by the procedure described in Sec. IV C.
Figure 8 shows EIT lineshapes (and in the inset, EIT linewidths) obtained by measuring the probe transmitted power as a function of the probe detuning Δp for pump intensities ranging from 1.3 to 5.5 mW/cm2, which correspond to -values of 1.9 and 3.9 MHz, respectively (see Sec. III C). The maximum contrast measured (for the highest intensity of 5.5 mW/cm2) is only ∼25%, for reasons explained in Secs. II D and III B; the contrast decreases as the pump intensity is lowered. In order to enable a visual linewidth comparison between the lineshapes obtained at different pump intensities, we normalized the lineshapes and displayed them on the same vertical scale, as is shown for (a) the highest and (b) the lowest intensities. We fit each lineshape to a Lorentzian (thin black curves), for which the FWHM yields the value for . In the inset, the datapoints represent EIT linewidths extracted from these fits for five different pump intensities, and the line represents the prediction for from Eq. (9). We note that a close look at the zero probe detuning point in the main figure reveals that the data are spectrally narrower at the center of the Lorentzian owing to coherent contributions from atoms that diffuse in and out of the laser beam multiple times without decohering (this is known as Ramsey-narrowing).5,33
We chose a -width of 170 μs for our Gaussian probe and reference pulses (see Sec. IV E). The corresponding bandwidth of kHz fits inside the EIT window, which, from Eq. (9), is at least kHz wide even for the smallest pump intensities.
Generating a EIT lineshape measurement as in Fig. 8 typically takes 40 ms for the higher pump intensities, and 160 ms for the lower intensities (we set the AOM for a ±100 kHz frequency-sweep in 10 ms; each spectrum in Fig. 8 is an average of either 4 or 16 such sweeps). Our laser system's passive stabilization yields a frequency drift of <10 MHz/h, resulting in a drift of less than 0.5 kHz while generating these EIT data-profiles.
VII. RESULTS AND DISCUSSION
In order to create short pulses, the probe frequency scan is now turned off, the probe AOM offset voltage is tuned to the EIT peak, and the amplitude modulation is turned on, as described in Sec. IV E. Once both slow and reference pulses are detected on an oscilloscope, the temporal waveform for each is fit to a Gaussian curve, and the relative delay time τd is extracted. Typical measurements of τd are presented in Fig. 9, where the data (fuzzy lines) are shown along with Gaussian fits (smooth lines).
In Figs. 9(a) and 9(b), the reference pulse arrives at the detector earlier than the probe by and 68 μs, respectively, which corresponds to a slowed probe group velocity vg of 610 and 368 m/s (using . In each case, the peaks of the fits for reference and probe are scaled to the same value for easier comparison. The reference and probe pulse intensities are similar at the non-polarizing beamsplitter (NPBS4, Fig. 5), but the probe pulse suffers some absorption in the vapor (recall that our EIT contrast never exceeds 25%), which explains the increased noise on the slowed pulses. The slower group velocity was achieved by lowering the pump intensity which further reduces the overall signal-to-noise ratio.
A. Observed , τd, and vg versus theoretical prediction
To predict the EIT linewidth , we insert in Eq. (9) parameter values that are relevant to our experiment, as discussed in Sec. III C: kHz, MHz. The optical depth is estimated using cm−3 and L = 2.5 cm (see Sec. V), yielding . The inset in Fig. 8 shows that the datapoints for are in reasonable agreement with the prediction (line) from Eq. (9), without the use of any fitting parameters.
To predict the delay time τd and the slowed probe group velocity vg, we refer to Eqs. (13) and (14), which use just the power-broadened component of . For intensities I = 3 and 1.2 mW/cm2 as in Fig. 9, the calculated delays τd are 72 μs (yielding a predicted vg of 347 m/s) and 181 μs (predicted 138 m/s), respectively. The predicted vg-values are a factor two to three slower than the observed vg-values in Figs. 9(a) and 9(b). To make more accurate predictions, we must move away from an idealized three-level atom model and include the full hyperfine structure (Fig. 3), which is beyond the scope of this article.19
B. Role of EIT transparency window
Figure 9(a) highlights the importance of the EIT process. The squat data-waveform is the transmitted probe pulse when the pump beam is blocked causing the EIT window to cease to exist: Large absorption and no slowing is observed. The transmitted power of the slow pulse for the unblocked pump is of the incident probe power, while for the blocked pump is less than 5%.
To further highlight the important role played by EIT, we show in Fig. 9(c) what happens if we select a pulse duration for which the probe bandwidth does not fit inside the EIT transparency window. The input Gaussian pulse in this case, s (which yields a -pulse bandwidth of kHz), is nearly an order of magnitude shorter than the 170 μs pulse employed in Figs. 9(a) and 9(b). Thus, in the case of Fig. 9(c) the pulse bandwidth is comparable to the EIT window which ranges from a few kHz to 30 kHz for our experiments. The frequency components of this pulse that do not fit within the EIT window are not slowed and are instead significantly absorbed. This leads to significant temporal stretching and distortion of the pulse (we scaled the vertical size of the reference pulse to resemble the transmitted probe).
In fact, some probe pulse broadening is visible even in the case of the slowest group velocity measured in Fig. 9(b). In this case, higher frequency components of the 170 μs Gaussian probe pulse ( -frequency bandwidth kHz) extrude past the 10.8 kHz Lorentzian EIT transparency window, and are strongly absorbed. This causes an effective narrowing of the probe bandwidth, which leads to a broadening of the pulse duration.
C. Slow group velocity versus pump intensity
In Fig. 10, we plot the experimentally observed slow light group velocity for several pump intensities at three different vapor temperatures. The observed temperature dependence of vg is in accordance with what we expect from Eq. (13): N increases with T (see Sec. S2 D in supplementary material11) causing vg to decrease.
At a fixed temperature, we surmise from Fig. 10 that two competing slow light effects occur when we vary the pump intensity. The linear increase in vg with increasing pump intensity I is expected from Eq. (13). Recall that this originates from the power-broadening of the EIT transparency window as described by Eq. (9), thereby reducing the “tightness of the pinch” of the “wrinkle” in the real refractive index nr at in Figs. 2(d) and 2(f).
However, it is obvious that one cannot keep reducing the pump intensity to achieve lower and lower group velocities as there is no slow light in the absence of a pump.
At really low pump intensities, the weak probe assumption starts to break down and, as mentioned in Sec. III C already, the population of atoms being pumped to the dark state may decline to the point that the probe pulse appears to start speeding up again.
Group velocities of several thousand m/s suffice for most cutting-edge experiments on quantum memory and image storage in warm vapor.5,6 At higher temperatures, the density increases, but so do spin depolarization mechanisms such as Rb–Rb spin exchange collisions and radiation trapping, as indicated in Sec. III B. In vapor cells that use paraffin-based anti-spin relaxation coatings instead of buffer gas, temperatures exceeding 80 °C may cause coating breakdown. Coatings, such as OTS (octadecyltrichlorosilane), permit higher temperatures.39,40
VIII. CONCLUSION
We have presented detailed theoretical and experimental undergraduate-friendly instructions on how to produce light pulses propagating through warm alkali vapor with speeds as low as few hundred m/s, emphasizing the role played by EIT in the production of slow light.
The experimental setup described here is remarkably versatile. It can be used for investigations into slow and stored light, including detailed measurements of the pulse delay τd as the pulse width τp is varied, and further measurement of the subtle Ramsey narrowing seen at the EIT line-center in Fig. 8 as the pump and probe beam sizes are varied.33 By slightly varying the angle between the pump and probe beams, one can study EIT linewidth-narrowing due to the spatial localization of alkali atoms from frequent velocity-changing collisions with the buffer gas.41 Furthermore, one may study how atomic diffusion degrades storage times for slow light pulses42 of various transverse profiles, e.g., Laguerre–Gaussian43 and Bessel beams,44 which possess topological phase features that permit significantly more robust storage in comparison to the usual Gaussian beams. The setup may be adapted for innovative magnetometry with potential applications in magnetic induction tomography and the detection of concealed objects.45,46
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant No. W911NF2110120. The authors appreciate insightful discussions with Drs. Irina Novikova, David Phillips, Ofer Firstenberg, and Ran Finkelstein, and invaluable detailed feedback from the three anonymous referees and from the editor. The authors thank Hong Cai, Kaleb Campbell, Richard Jackson, Dillon deMedeiros, Bradley Worth, Amanda Day, Somaya Madkhaly, Yuhong (Iris) Zhang, Peter Harnish, and Jason Barkeloo for help during the initial setup. The authors thank the Miami University Instrumentation Laboratory for crucial help in machining and electronics.
AUTHOR DECLARATIONS
The authors have no conflicts to disclose.
Author Contributions
Kenneth DeRose and Kefeng Jiang contributed equally to this work.