We study the retrieval efficiency of stored light pulses based on electromagnetically induced transparency in multiple simultaneously driven Λ-systems. The light pulses are stored in coherences between different Zeeman states of laser-cooled atoms. When the stray magnetic field from the environment is minimized by compensation coils we observed a smaller retrieved probe pulse amplitude than for a small externally applied magnetic field, i.e., a seemingly shorter coherence time. We identify this effect as a beating of several coherences due to a very small uncompensated dc magnetic stray field. By intentionally applying a small magnetic field larger than this stray field we were able to increase the retrieved probe pulse amplitude up to five-fold to the value determined by the true coherence time of our system.
© 2009 Optical Society of America
Slowing down a light pulse inside a medium prepared by electromagnetically induced transparency (EIT)  and subsequently storing it as a spin excitation of the medium is a well-established technique [2, 3, 4, 5] that has attracted a lot of interest in recent years. One very promising application is its use for future quantum networks as photons are robust and fast carriers of information that can now be stored and retrieved in a coherent process. In the typical Λ-type coupling scheme of EIT two metastable ground states |a〉 and |b〉 are coupled by a coherent two-photon transition via an excited state |c〉. The corresponding energies of the states are ħωx (x = a,b,c). One of the two photons of the two-photon transition is provided by a strong coupling laser field, represented by its Rabi frequency Ωc and the other photon by a weak probe laser field, represented by Ωp. Due to the presence of the strong coupling field the dispersive properties of the medium are modified. If the center frequencies of the probe and coupling fields are tuned such that the two ground states are coupled in two-photon resonance, a probe pulse entering the medium will experience a high transmission. Without the coupling field applied the probe pulse would be completely absorbed by the medium. Along with this high transmission comes a steep variation of the refractive index leading to a strong reduction of the group velocity of the probe pulse [6, 7]. While the probe pulse propagates through the medium it can be described in terms of a quasi-particle, a so-called dark state polariton (DSP) . The DSP has an electromagnetic and an atomic component. The latter one consists of a spin coherence between the two ground states. Once the coupling field is switched off adiabatically while the probe pulse is slowly propagating inside the medium, the electromagnetic component, along with the group velocity of the DSP, is reduced to zero . The DSP now has only a non-moving atomic component, i.e., the probe pulse is stored inside the medium as a spin-wave. This spin-wave evolves temporally with a frequency ωsw = ωb - Ωa determined by the energy difference between states |a〉 and |b〉. Also its spatial modulation period, i.e., its wavelength λsw = 2πc/ωsw is determined by this energy difference for collinear laser beams. By reversing the storage process, i.e., adiabatically switching on the coupling field, the DSP is re-accelerated. The original probe pulse is restored.
In this paper we discuss the effect a very small magnetic field can have on the retrieved probe pulse amplitude for a manifold of simultaneously driven Λ-systems (see Fig. 1(a)). When the stray magnetic field from the environment was minimized by three pairs of compensation coils we experimentally observed a smaller retrieved amplitude than expected from the coherence time. By applying a small magnetic field along the quantization axis defined by the laser polarizations we were able to increase the amplitude by a factor of 5 back to the expected value. Additionally we show how to minimize the magnetic field inside the medium by storage and retrieval of light pulses.
This paper is organized as follows. In Sec. 2 we discuss the theoretical aspects important for the measurements. In Sec. 3 we present the experimental setup. Section 4 presents the experimental results and discusses them before we conclude in Sec. 5 with a summary.
2. Theoretical background
The following discussion applies in principle to three-level systems |a〉, |b〉, |c〉, each level again consisting of a manifold of Zeeman states. In view of the experiment discussed later and to simplify the discussion we explicitly consider here the level scheme of 87Rb with |a) = |52S1/2,F = 1〉, |b〉 = |52S1/2,F= 2〉, and |c〉 = |52P3/2,F′ = 2〉 and couplings as depicted in Fig. 1(a). The Zeeman shifts of the two hyperfine ground states are of equal magnitude but opposite, which applies also to all other alkali atoms. All population is initially equally distributed in state |a〉. When, e.g., a Gaussian-shaped probe pulse is stored in the medium, its information, i.e., its amplitude, frequency and phase is encoded in coherences ρmaba between states |a〉 and |b〉, where ma,b is the magnetic quantum number of the projection of the total atomic angular momentum F of states |a〉 and |b〉 onto the quantization axis. For the coupling shown in Fig. 1(a) we always have ma = mb. This scheme applies to laser fields of σ+ -polarization when either no magnetic field is present, or the magnetic field direction is parallel to the propagation direction of the laser beams. Three independent Λ-systems exist for this coupling scheme. If there is no magnetic field present, all Zeeman states within one hyperfine level are degenerate and the three coherences ρmama(t) = ρmama(0) exp (ma = -1,0,+1) evolve likewise with frequency ωsw 0 = ωb - 〉a. Measuring the retrieved probe pulse energy, i.e., the pulse area as a function of storage time t will show an exponential decay of the energy with a time constant given by the coherence time τ of the system. For Gaussian shaped probe pulses - as in the experiment discussed later - the pulse area is proportional to the pulse amplitude. Therefore, we will from now on only discuss the retrieved probe pulse amplitude A(t), which in the case of no magnetic field present is given by
where A(0) is the initial probe pulse amplitude at t = 0. This behavior is illustrated in Fig. 3(a) (dashed line).
If, however, a magnetic field of strength B is present, the amplitude A(t) evolves differently due to two reasons: (i) Other coherences ρmamb with ma ≠ mb, and |∆m|max = |ma - mb|max = 2 can be excited if the quantization axis defined by the laser polarizations ẑL and the one defined by the magnetic field direction ẑB (compare Fig. 2) is different. (ii) The degeneracy is lifted, i.e., the frequency difference ωsw(ma,mb) = ωsw 0 + ∆ωZ(ma,mb) between the two ground states |a〉 and |b〉 depends for each Λ-system on the quantum numbers ma and mb.
∆ωZ(ma,mb) = (gbmb - gama)μB B/ħ is the relative Zeeman shift with ga,gb being the hyper-fine Landé g-factors of the two ground states and μB being the Bohr magneton. If ẑL∥ẑB the laser fields only drive transitions according to their polarization. For σ + -polarized laser fields, only states with ∆m = 0 are coupled (Fig. 1 (b)). If ẑL and ẑB have different directions, however, the coupling will be different. The coordinate system defined by the laser polarizations has to be rotated to obtain the polarization as seen in the atomic frame of reference. Generally speaking, an angle α ≠ 0 between ẑL and ẑB will lead to an acquisition of other polarizations in the atomic frame of reference. For example, if the laser fields have σ + polarization the atoms will experience coupling by a mixture of σ +-, σ -- and π-polarized light. Due to these multiple polarization components in the atomic frame of reference the coupling between the states becomes quite complicated. However, the result of these complicated couplings can be summarized in quite an easy way: the change of the quantum number ∆m between initial and final state of each driven Λ-system is not limited to ∆m = 0 anymore, but has multiple values |∆m| = 0,1,2. For transitions with ∆m = 0 the probe and coupling photons have the same polarization. |∆m| = 1 corresponds to one photon having π- and the other having σ ±-polarization. |∆m| = 2 finally corresponds to one of the photons having σ +- and the other having σ - -polarization. This leads to an excitation of additional coherences which can have different oscillation frequencies ωsw(ma,mb) = ωsw 0 + ∆ωZ(ma,mb) = ωsw 0 + nωZ, where nωZ are harmonics of the fundamental oscillation frequency determined by the Zeeman shift ωZ = gFμBB/ħ. As can be seen easily from the possible coupling schemes, n can have values of n = ±1,±2,±3. It was shown in Ref.  that these oscillations of the retrieved probe pulse amplitude correspond to oscillations between a DSP and a bright-state polariton, where only the DSP contributes to the retrieved probe pulse.
To account for the different oscillation frequencies, Eq. (1) can be rewritten as
The terms oscillating with frequency ωsw 0 contribute a constant to the amplitude that decays exponentially with the coherence time τ as in Eq. (1). The terms oscillating with frequencies ωsw 0 + nωZ give rise to an oscillatory behavior of the retrieved probe pulse amplitude. The factors pn depend on the sum of all coherences ρmamb that oscillate with frequency ωsw 0 + nωZ and, in turn, determine the contribution of each frequency to the total amplitude. The retrieved probe pulse amplitude A(t) depends therefore on the relative phases of the coherences ρmamb(t) at the time of retrieval in addition to the decay due to decoherence. The beating of the coherences leads to an oscillation of the retrieved probe pulse amplitude with respect to the storage time (Fig. 3(a), solid line). Such beating of coherences has already been discussed and demonstrated before [9, 10, 11], with a detailed analysis of the amplitude A(t) in [9, 10]. Here, however, we are only interested in the combined contribution of all coherences oscillating at a certain frequency. Therefore we do not discuss the exact form of the factors pn. We can further simplify Eq. (2) by setting pn = p -n. Numerical simulations show that Eq. (2) is rather insensitive to the difference (pn - p -n) and depends mainly on the sum (pn + p -n). This approximation has also been verified by fitting experimental data shown in Sec. 4 by Eqs. (2) & (3). No difference between both fits was visible. We obtain the simple equation
where the factors qn now determine the contribution of each frequency to the amplitude and we replaced ωZ by 2π/T with the oscillation period T. The fundamental oscillation period of the retrieved pulse amplitude is solely determined by the magnetic field strength B, i.e., the Zeeman shift ωZ. The actual oscillation behavior, i.e., the number n of driven harmonics of the fundamental oscillation frequency ωZ, however, is determined by the relative orientation of the quantization axes ẑL and ẑB. Also the relative contribution qn of each oscillation frequency to the overall amplitude depends on this orientation. This is illustrated in Fig. 3(a) & 3(b) for α = 0° and α = 90°, respectively. The Zeeman shift, i.e., the magnetic field strength is the same in Fig. 3(a) & 3(b). Figure 3(a) corresponds to the coupling shown in Fig. 1(b), i.e., only the frequency components 2ωZ are present (compare Fig. 1(c)). In Fig. 3(b), more frequency components nωZ with n = 1,2,3 are present.
We now discuss how the previous findings can be used to characterize an experimental setup and improve the retrieval amplitude of a stored probe pulse. By placing three orthogonal pairs of coils in Helmholtz configuration around a medium, it is possible to adjust the magnetic field Beff = Bcoil + Bstray, i.e., the sum of the magnetic field of the coils and the stray field from the environment, along any direction ẑB within the medium. As stated before, the oscillation frequency of the retrieved probe pulse amplitude depends solely on the magnetic field strength. Measuring this frequency for various magnetic field strengths B eff in the three orthogonal directions therefore allows one to determine the settings for which B eff reaches its minimum. However, minimizing B eff is not always preferable as we will show next: The Zeeman shift of the ground state of, e.g., 87Rb is 700 Hz for a magnetic field strength of 1 mG. This will cause the retrieved probe pulse amplitude to oscillate with a fundamental period of T = 1.4 ms. The oscillation influences the retrieved amplitude for times much shorter than its period T as can be seen from Fig. 3. Therefore, the retrieved amplitude is already reduced for storage times below 100 μs and will be observable if the coherence time is of same order of magnitude which is typically the case for MOTs. Even for a magnetically shielded Rb cell the effect of a small magnetic field of 50 μG has been observed recently on a timescale of milliseconds . There, the authors discussed a technique to measure directly the coherence time, based also on oscillations between dark- and bright states. In Fig. 4(a) we plot the retrieved amplitude A(t) versus the storage time for different magnetic field strengths to demonstrate the effect a small magnetic field has on the retrieved probe pulse amplitude. The calculation is based on Eq. (3). The parameters were obtained empirically from experimental data shown later in this paper (see Sec. 4). The black dotted line corresponds to the case of no magnetic fields present B eff = 0. The amplitude decays exponentially on a timescale given by the coherence time. For magnetic field strengths of either B′∥ = 2.3 mG or B′⊥ = 4.0 mG (blue dashed line, the prime indicates from now on small magnetic fields that can not be further reduced) parallel or perpendicular, respectively, to ẑL the amplitude decreases faster than for B eff = 0, as A(t) approaches its first minimum. At a first glance, it looks like the coherence time is shorter. This, however, is a deception due to the oscillatory behavior. By applying a field B ∥ parallel to ẑL in addition to Beff ≃ B′∥⊥ which is still rather weak but stronger than B′∥⊥ = B′∥⊥, the oscillation behavior changes. This is shown in Fig. 4(a) (red solid line) for Beff = 27 mG≃B ∥.A(t) has now two maxima whose values are limited by the coherence time only. In Fig. 4(b) we plot the ratio of the retrieved amplitudes for Beff ≃ B ∥ and Beff = B′∥⊥ for the curves shown in Fig. 4(a). A maximum enhancement of about a factor of 5 is reached within the coherence time of τ = 50 μs. The maxima of the oscillations can be adjusted to any desired storage time simply by changing B ∥. This is shown in Fig. 4(c) where the maximum achievable enhancement for each storage time is plotted for several values of B′⊥. The larger the small uncompensated magnetic field B′ is, the larger the enhancement will be. The degree to which the magnetic field B ∥ can be adjusted is limited by the Zeeman splitting that should not exceed the bandwidth of the EIT window to avoid absorption of the probe pulse. However, the limited tuning range of the magnetic field strength was not a limiting factor in our experimental system discussed next.
3. Experimental setup
We performed the experiments in laser-cooled 87Rb atoms. We trapped about 109 atoms in a MOT with rectangular coil geometry . This geometry led to a cigar-shaped atom cloud of dimensions 10.0 · 2.5 × 2.5 mm3. The temperature of the atom cloud was less than 300 μK . Three pairs of coils in Helmholtz configuration were placed around the MOT to minimize static magnetic stray fields. These coils also allowed us to apply a nearly homogeneous magnetic field of arbitrary direction and field strengths up to B = 500 mG over the atom cloud. The probe and coupling fields were derived from the same external cavity diode laser (ECDL) by injection locking two laser diodes with a frequency offset of 6.8 GHz induced by an electro-optic modulator, to ensure phase coherence of both fields. The ECDL was locked to the cross-over transition between |52S1/2,F = 2〉 ↔ |52P3/2,F′ = 2〉 and |52S1/2,F = 2〉 ↔ |52P3/2,F′ = 3〉 by saturation absorption spectroscopy. All atoms were optically pumped into the unpolarized hyperfine state |52S1/2,F = 1). The probe field drove the transition |52S1/2,F = 1〉 ↔ |52P3/2,F′ = 2〉. The coupling field drove the transition |52S1/2,F = 2〉 ↔ |52P3/2,F′ = 2〉. Both fields had circular polarization (σ +). The output of the probe and coupling lasers was sent through acousto-optic modulators (AOMs) to temporally shape the pulses needed for the experiment. The probe pulse had a Gaussian temporal shape with a 1/e full width of the intensity of τp = 0.6 μs. The coupling Rabi frequency was Ωc = 0.7Γ, where Γ = 2π × 6 MHz is the decay rate of the excited state. A computer controlled the experimental sequence while the coupling field sequence (on/off) was produced by a delay generator that modulated the AOM output. The probe field was focused onto the atom cloud to a beam diameter of about 270 μm and propagated along the main axis of the atom cloud to ensure the best interaction with the atoms that resulted in an optical density of 36. The coupling field was collimated with a beam diameter of about 6 mm and intersected the probe beam with an angle of about 0.5° to avoid saturation of the avalanche photo detector (APD, Hamamatsu C5460, rise time 36 ns) detecting the probe pulses. The output of the APD was averaged 256 times by an oscilloscope and then transferred to a computer. The magnetic field of the MOT was turned off 2.5 ms before the probe pulses entered the medium such that the magnetic quadrupole field had already decayed during the measurement. The MOT trapping beams were turned off 5 μs before the arrival of the probe pulses. Further details of our setup can be found in .
4. Results and discussion
Figure 5 shows typical experimental data of an input (magenta diamonds), delayed (black squares), and retrieved probe pulse (red circles). We first determined the retrieved probe pulse amplitude for different storage times and magnetic fields strengths and orientations from data as those shown in Fig. 5 (black squares and red circles). Figure 6 shows such experimental data of the retrieved probe pulse amplitude for a magnetic field applied parallel (a) and perpendicular to the laser quantization axis ẑL (b). The data is normalized with respect to the amplitude without storage (black squares in Fig. 5). The red solid line corresponds to the best fit of the experimental data with Eq. (3). We estimated the oscillation frequency ωZ for the initial fit parameters by using the relation fZ = 700 kHz/G×B(Icoil) according to the Zeeman shift. The approximately applied magnetic field B(Icoil) was determined by calibrating the Helmholtz coils with a Hall sensor probe for different currents Icoil. By performing measurements as those shown in Fig. 6(a) & 6(b) for several magnetic field strengths Bcoil along a fixed direction and fitting each data set to obtain the oscillation frequency we obtained data as shown in Fig. 6(c). Here, the oscillation frequency is plotted versus the current Icoil (bottom scale) and the effective magnetic field strength B eff (top scale). The effective magnetic field strength was calculated from the oscillation frequency obtained for each measurement as in Fig. 6(a) & 6(b). Negative values of Icoil correspond to a reversed magnetic field direction of the compensation coils. Negative values of 5eff correspond to a reversed effective magnetic field direction. The red solid lines represent the best linear fits of the experimental data. By extrapolating both linear fits to a oscillation frequency of fZ = 1/T = 0 the current Icoil for the smallest magnetic field along this direction can be determined. Repeating these measurements and analysis for all three axes of the compensation coils it is therefore possible to determine the lowest achievable magnetic field B eff of the system. In our measurements we were able to determine this point with a precision of about 1 mG, i.e., about 1 % of the applied magnetic field Bcoil.
Next, we come to the main point of this paper. We observed, that when precisely minimizing the stray magnetic field inside the MOT as described in the previous paragraph, the retrieved probe pulse amplitude was always smaller for any storage time as when a magnetic field along the quantization axis ẑL was applied. Figure 7(a) shows normalized retrieved probe pulse amplitudes versus the storage time for several magnetic field configurations. The data represented by black squares was obtained for a minimized magnetic field B eff. From this data one would estimate the coherence time to be around τ ≃ 25 μs. However, fitting the data with an exponential decay function yields a bad fit (blue dashed line). Fitting the data by Eq. (3) (black solid line) for B eff = B′⊥ or B eff = B′∥, in contrast, yields a good fit with a coherence time of τ ∥ = (49 ± 5)μs or τ ⊥ = (52 ± 6) μs, respectively. We fitted the data with parameters for a magnetic field B eff parallel and perpendicular to ẑL because the exact orientation is difficult to determine when the oscillation period is much longer than the coherence time. According to oscillation periods of T ⊥ = 631 μs or T ∥ = 356 μs obtained from the fits, a small effective magnetic field of strength B eff ≃ 2.3 - 4.0 mG is present - even when the compensation coils are best adjusted. Attempts to further minimize this stray field were not successful. We note again that the precision for minimizing B eff was about 1 mG along each axis of the compensation coils. This led to an total uncertainty for B eff of 1-2 mG which is close to the value when assuming B eff = B′⊥.
We demonstrate now, that the retrieved probe pulse amplitude can be increased for a certain storage time by applying a magnetic field along ẑL as discussed in Sec. 2. The data represented by red dots in Fig. 7(a) was obtained for a magnetic field applied parallel to ẑL with B ∥ = 27 mG. The red solid line corresponds to the best fit with Eq. (3). The coherence time obtained from the fit is τ = (54 ± 3) μs in agreement with the time obtained for the data in black. Plotting the retrieved amplitude enhancement, i.e., the ratio of the data in red and the one in black in Fig. 7(a) (compare Fig. 4(b)), shows an enhancement of about a factor of 5 (Fig. 7(b)) at a storage time equal to the coherence time of τ ≃ 50 μs. The storage time for which the enhancement has a local maximum can be easily changed by slightly varying the magnetic field strength to ensure a maximum pulse amplitude for any desired storage time (see Fig. 4(c)). By applying a magnetic field and thereby removing the degeneracy of the three Λ-systems, the transmission of the probe pulse is reduced. Figure 8 shows that this decrease is rather small compared to the enhancement obtained at the same time. For a magnetic field of B ∥ = 27 mG the overall amplitude is reduced by less than 5 % while the retrieved amplitude is enhanced five times.
We have presented a way to enhance the retrieved amplitude of stored probe pulses based on the effect of electromagnetically induced transparency (EIT). The method is suitable for coupling schemes where multiple EIT systems of Zeeman states are driven simultaneously. By employing the oscillatory behavior of retrieved probe pulses from a medium subjected to a homogeneous magnetic field we were able to enhance the retrieved probe pulse amplitude by a factor of 5 within the coherence time of the system. This method can be useful for all kinds of media as, e.g., laser-cooled atoms, solids and hot atomic gases as small magnetic fields that can not be completely eliminated are always present. This effect not only allows one to enhance the amplitude for better experimental detection, but also might be of interest for precise coherence time measurements, e.g., in solids.
This project was funded by the National Science Council of Taiwan under Grants No. 95-2112-M-007-039-MY3 and No. 97-2628-M-007-018.
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