We propose an efficient scheme for the robust and controlled generation of beating signals in a sample of stationary atoms driven into the tripod configuration. This scheme relies on an asymmetric procedure of light storage and retrieval where the two classical coupling fields have equal detunings in the storage stage but opposite detunings in the retrieval stage. A quantum probe field, incident upon such an atomic sample, is first transformed into two spin coherence wave-packets and then retrieved with two optical components characterized by different time-dependent phases. Therefore the retrieved quantum probe field exhibits a series of maxima and minima (beating signals) in intensity due to the alternative constructive and destructive interference. This interesting phenomenon involves in fact the coherent manipulation of two dark-state polaritons and may be explored to achieve the fast quantum limited measurement.
© 2011 OSA
Quantum information storage and quantum state engineering are two key elements for the practical implementation of quantum information processing. Photons are believed to be the most fast and robust carriers of quantum information while atomic ensembles may provide the best units for reversible storage and nonlinear processing of weak light signals . To efficiently control the propagation and interaction of weak light signals, there has been a growing interest in the research of laser induced atomic coherence, which is the basis of many interesting phenomena such as electromagnetically induced transparency (EIT)  and stimulated Raman adiabatic passage (STIRAP) . Using the EIT technique, one can either suppress resonant absorption to achieve low-light-level nonlinear optics [4–7] or manipulate light group velocity to attain quantum information storage [8–11]. Typically, the three-level Lambda system with a non-degenerate dark state is adopted to demonstrate the quantum memory of photonic states by transforming a slowly propagating light field into a stationary wave-packet of spin coherence and vice versa . This dynamic process and the associated slow light propagation are usually interpreted in terms of dark-state polariton (DSP)  defined as a joint atom+field excitation. To store a photonic qubit with two basis states, however, we have to consider four-level systems driven into, e.g. the tripod configuration exhibiting multiple degenerate dark states and allowing for two DSP modes [14, 15]. In the tripod system, a pair of light fields can be simultaneously slowed down, respectively stored into two different wave-packets of spin coherence, and then retrieved at the same time or with a controlled time delay [16–18]. The tripod system has also been explored to achieve the enhanced cross-phase modulation between two weak light fields with the ultimate goal to build conditional quantum phase gates for manipulating photonic polarization states [19–22]. Note, in particular, that beating signals due to the interference between two weak probe fields (two DSP modes) have been observed recently in the tripod system as a manifestation of the preserved phase information during light storage . Beating signals can also be observed in the Lambda system when a weak probe field (a DSP mode) interferes with a strong coupling field during light retrieval in the presence of a magnetic field [24, 25]. Last but not least, Heinze et al. demonstrate that the total intensity of a retrieved spatial image may oscillate against the storage time, which is also identified as a beating of DSP modes . Such interferometric beating is claimed to be important in that it has potential applications in the fast quantum limited measurement.
In this paper, we present an alternative scheme for the robust and controlled generation of beating signals via the dynamic EIT technique in a tripod system of stationary atoms. This tripod atomic system is driven by a weak (quantum) probe field and two strong (classical) coupling fields and therefore is different from that in ref.  interacting with two weak probe fields and a strong coupling field. Our numerical calculations show that beating signals can be reliably generated and flexibly controlled in an asymmetric procedure of light storage and retrieval. To be more specific, we first store the probe field into two different wave-packets of spin coherence by switching off the two coupling fields with equal detunings, and then retrieve it after a short storage time by switching on the two coupling fields with unequal detunings instead. In this case, the only retrieved probe field consists of two components characterized by different time-dependent phases so that beating signals (a series of maxima and minima in intensity) arise due to the alternate constructive and destructive interference. The generation of beating signals can be well understood in the polariton picture where a pair of DSP modes are simultaneously excited with their structures totally determined by amplitudes, phases, and detunings of the two coupling fields. Experimental examination of such beating signals may be used to acquire the frequency difference, the relative phase, and their stabilities of two classical coupling fields or to achieve the fast quantum limited measurement of magnetic field amplitudes and atomic transition frequencies between ground state sublevels.
2. Model and equations
We consider here a medium of length L consisting of an ensemble of stationary atoms coherently driven into the four-level tripod configuration (see Fig. 1). The first dipole-allowed transition |a〉 ↔ |e〉 is coupled by one classical field with frequency ωc 1 and amplitude E c 1 and the second dipole-allowed transition |b〉 ↔ |e〉 is coupled by another classical field with frequency ωc 2 and amplitude E c 2. The third dipole-allowed transition |c〉 ↔ |e〉, however, is probed by a weak quantum field described by
The tripod-type stationary atoms, on the other hand, are described by the collective atomic operators
The weak probe assumption allows us to perturbatively solve the original Heisenberg-Langevin equations to the first order in the probe field so that we don’t need to consider the dynamic evolutions of operators ab(z,t), ae(z,t), be(z,t), aa(z,t), bb(z,t), cc(z,t), and ee(z,t). The adiabatic control assumption, on the other hand, is the solid foundation for us to safely neglect the δ-correlated Langevin noise operators Fca, Fcb, and Fce in Eqs. (4). In Eqs. (4), γca, γcb, and γce represent the coherence decay rates of operators σca, σcb, and σce while Δp = ωp – ωec, Δc 1 = ωc 1 – ωea, and Δc 2 = ωc 2 – ωeb are frequency detunings of the three quantum or classical fields. In addition, Ωc 1 = E c 1 · d ea/2h̄ and Ωc 2 = E c 2 · d eb /2h̄ are complex Rabi frequencies of the first and second (classical) coupling fields while is the real coupling constant of the (quantum) probe field with d μν being the dipole matrix element on transition |μ〉 ↔ |ν〉. To be more specific, we may further set Gc 1 (Gc 2) and Φc 1 (Φc 2) as the modulus and the argument of the complex Rabi frequency Ωc 1 (Ωc 2), respectively.
Solving Eqs. (4) in the steady state, we can easily examine the absorption and dispersion spectra, proportional respectively to Im[σce(Δp)] and Re[σce(Δp)], of a continuous-wave probe. To study the dynamic evolution of the dimensionless operator Ep(z,t), however, we also need the wave propagation equationEq. (5) coupled with Eqs. (4) will be numerically solved in the next section to examine an interesting light propagation dynamics concerning the generation and control of quantum limited beating signals imposed on a probe field.
3. Results and discussions
In what follows, we show by numerical calculations how to generate and control robust beating signals in an asymmetric procedure of light storage and retrieval via the dynamic EIT technique. Without loss of generality, we will set γ = γce as the frequency unit, as the time unit, and (the absorption length in the absence of EIT) as the position unit. As we can see from Fig. 2, a quantum field Ep(z,t) goes slowly into the atomic sample at the velocity υg = 0.25la/t 0 and is totally transformed into stationary spin coherences σca(z,t) and σcb(z,t) at the sample center when the two coupling fields with Δc 1 = Δc 2 = 0.0γ are simultaneously turned off at t = 48.0t 0. After a short storage time of Δt = 24.0t 0, we switch on the two coupling fields to retrieve the quantum field from σca(z,t) and σcb(z,t) at the same time but with opposite detunings, i.e. Δc 1 = −Δc 2 = 0.02γ in (a, b), Δc 1 = −Δc 2 = 0.05γ in (c, d), and Δc 1 = −Δc 2 = 0.08γ in (e, f). As expected, the retrieved quantum field appears and vanishes in a periodic pattern when it propagates inside the atomic sample once again at the velocity υg = 0.25la/t 0. That is, the retrieved quantum field exhibits a series of maxima and minima (beating signals) in its average intensity, which oscillate at the frequency of Δωbeat = Δc 1 − Δc 2 when |Δc 1| = |Δc 2| are large enough [see Fig. 2(e, f)].
The interesting beating signals in Fig. 2 can be qualitatively understood in terms of DSP as a coherent mixture of quantum field and spin coherence. Adopting the standard procedure , we may adiabatically transform Eqs. (4) and Eqs. (5) intoEqs. (7) and Eqs. (8), we have set and Gc 1 = Gc 2 for simplicity so that both DSP modes Ψa(z,t) and Ψb(z,t) can attain the same propagating velocity υg = c cos2 θ at the retrieval stage.
Now we begin to explain the main results shown in Fig. 2 with Eqs. (7) and Eqs. (8) in the limit of Φc 1 = Φc 2 = 0.0. At t = 0.0t 0, a fast probe field Ep(z,t) enters the sample of stationary atoms and evolves into two slowly-moving DSP modes described by Eqs. (7) through its two-photon resonant interaction with both coupling fields Ωc 1 and Ωc 2. In this case, the field component in Ψa(z,t) and that in Ψb(z,t) have the same vanishing phase so that no beating signals are observed in the probe intensity before t = 48.0t 0. At t = 48.0t 0, when both coupling fields Ωc 1 and Ωc 2 are switched off, the two slowly-moving DSP modes described by Eqs. (7) turn into a pair of stationary atomic excitations and limited by Δc 1 = Δc 2 = 0.0γ. This is why the probe intensity is exactly zero during the period of t = 48.0t 0 ∼ 72.0t 0. At t = 72.0t 0, when the two coupling fields are switched on with Δc 1 ≠ Δc 2 instead, the pair of stationary atomic excitations become the origin of two slowly-moving DSP modes described by Eqs. (8). In this case, the field component in Ψa(z,t) and that in Ψb(z,t) attain, respectively, time-dependent phases Δc 1 t and Δc 2 t so that they interfere with each other to produce a series of beating signals in the probe intensity after t = 72.0t 0. At the sample exit, the two DSP modes described by Eqs. (8) finally turn into a pair of fast field components e iΔc1t Ep(z,t)/2 and e iΔc2t Ep(z,t)/2 with beating signals perfectly reserved. It is clear that the field components of both DSP modes experience little absorptive loss even if we set Δc 1 ≠ Δc 2 after t = 72.0t 0 because they are well contained in two separate EIT windows located, respectively, at Δp = Δc 1 and Δp = Δc 2 .
To see how beating signals are formed, we plot in Fig. 3 the pair of field components at the sample exit by increasing the coupling detuning difference |Δc 2 – Δc 1| in very small steps. As we can see there is only a single maximum when |Δc 2 – Δc 1| < 0.003γ while a second maximum tends to arise when |Δc 2 – Δc 1| > 0.003γ. In addition, a part of light energy is redistributed into a later time region and the destructive interference occurs between the former and later time regions when a wider EIT window is gradually split into two narrower ones by an absorption line . From Eqs. (7) and Eqs. (8) we also can see that the two DSP modes are sensitive to arguments Φc 1 and Φc 2 of the two complex Rabi frequencies Ωc 1 and Ωc 2. So one may simply modulate the relative phase ΔΦ = Φc 2 – Φc 1 to control the beating signals attained in an asymmetric procedure of light storage and retrieval, which is illustrated in Fig. 4. It is found that beating signals with ΔΦ = 0.0 (π/2) and beating signals with ΔΦ = π (3π/2) are exactly staggered by a half period, i.e. a maximum in the black-solid curves corresponds to a minimum in the red-dashed curves. In this case, it is ei (Δc1t−Φc1) Ep(z,t)/2 and ei(Δc2t − Φc2) Ep(z,t)/2 that describe the pair of field components at the sample exit.
We would like to note that the stationary atoms mentioned above can be either stray impurities doped in a solid materials or cold atoms confined in a magneto-optical trap (MOT). For the former, the Pr3+: Y2SiO4 crystal working at the cryogenic temperature [28, 29] should be a good candidate for attaining beating signals via the dynamic EIT technique. But the inhomogeneous broadening of absorption lines due to random crystalline fields should be suitably included into the coherence decay rates γca, γcb, and γce. For the latter, we may choose the D1 line of cold 87Rb atoms to construct the tripod system with |a〉, |b〉, |c〉, and |e〉 referring to |52 S 1/2, F = 2, mF = +1〉, |52 S 1/2, F = 2, mF = −1〉, |52 S 1/2, F = 1, m = +1〉, and |52 P 1/2, F′ = 1, m = 0〉, respectively .
In summary, we have demonstrated by numerical calculations and analyzed in the polariton picture an efficient scheme for the dynamic generation and flexible control of beating signals via an asymmetric light storage and retrieval technique. In experiment, the frequency difference ωc 2 − ωc 1, the relative phase Φc 2 − Φc 1, and their stabilities can be easily inferred from the beating signals imposed on a weak probe field. If the detuning difference |Δc 1 − Δc 2| is induced by shifting hyperfine state sublevels with a magnetic field B, we may have h̄ |Δc 1 − Δc 2| = 2gμB · B with g being the Lander factor and μB the Bohr magneton. In this case such interferometric beating signals can be used to measure magnetic field amplitudes. If the two coupling fields have the identical frequency and differ only by σ + and σ − polarizations, however, we may have |Δc 1 − Δc 2| = |ωab| so that atomic transition frequencies can be determined from such interferometric beating signals. In particular, when a nonclassical squeezed light is used, one can achieve the quantum limited measurement with a sub-shot noise precision because quantum properties are well conserved during light storage and retrieval [10, 11]. Moreover, this novel method for measuring atomic transition frequencies is much faster than the standard spectroscopy method because a light storage experiment can be implemented in a fraction of the time required for the acquisition of a complete EIT spectrum.
Finally it is worth emphasizing what are special in our four-level tripod scheme involving an asymmetric procedure of light storage and retrieval. First, we use only one probe field to generate two DSPs, whose optical components of different frequencies interfere to generate beating signals. In refs. [23–26], however, beating signals are resulted from the interference between two probe fields or between a probe field and a coupling field. Second, it is easy for us to let the two DSPs have the same amplitude and experience the same loss and diffusion by simply manipulating the two coupling fields so that beating signals imposed on the only probe field can be kept perfect (i.e. always have vanishing minima in intensity).
This work is supported by NBRP of China (No. 2011CB921603), NSFC of China (No. 10874057 and No. 10904047), BSRF of Jilin University (No. 200905019), and the Graduate Innovation Fund of Jilin University (No. 20101051).
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