Dynamic generation of robust and controlled beating signals in an asymmetric procedure of light storage and retrieval

Qian-Qian Bao; Jin-Wei Gao; Cui-Li Cui; Gang Wang; Yan Xue; Jin-Hui Wu

doi:10.1364/OE.19.011832

1. Introduction

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 [1]. 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) [2] and stimulated Raman adiabatic passage (STIRAP) [3]. 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 [12]. This dynamic process and the associated slow light propagation are usually interpreted in terms of dark-state polariton (DSP) [13] 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 [23]. 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 [26]. 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. [23] 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 ω_c ₁ and amplitude E _c ₁ and the second dipole-allowed transition |b〉 ↔ |e〉 is coupled by another classical field with frequency ω_c ₂ and amplitude E _c ₂. The third dipole-allowed transition |c〉 ↔ |e〉, however, is probed by a weak quantum field described by

{\hat{E}}_{p} (z, t) = ɛ_{p} \sqrt{\frac{\bar{h} ω_{p}}{2 ɛ_{0} V}} E_{p} (z, t) e^{- i ω_{p} t + i k_{p} z}

where ε_p is the polarization vector, ω_p the carrier frequency, V the quantization volume, and E_p(z,t) the slowly-varying dimensionless operator.

Fig. 1 (Color online) Schematic of a four-level tripod system driven by a weak (quantum) probe field of frequency ω_p and two strong (classical) coupling fields of frequencies ω_c ₁ and ω_c ₂, respectively. All stationary atoms under consideration are initially distributed at level |c〉 as denoted by the pink circles.

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The tripod-type stationary atoms, on the other hand, are described by the collective atomic operators

{\hat{σ}}_{μ ν} (z, t) = \frac{1}{N_{z}} \sum_{j = 1}^{N_{z}} {\hat{σ}}_{μ ν}^{j} (z, t)

where N_z denotes the atomic number in a small volume V_z centered at z while

{\hat{σ}}_{μ ν}^{j} (z, t)

is defined as the atomic flip operator |μ_j〉 |ν_j〉 with {μ, ν} ∈ {a, b, c, e}. For simplicity and convenience, we further introduce the slowly-varying operators σ_ce(z, t), σ_cb(z, t), and σ_ca(z, t) for three relevant coherence terms

\begin{array}{l} {\hat{σ}}_{ce} (z, t) & = & σ_{ce} (z, t) e^{- i ω_{p} t} \\ {\hat{σ}}_{cb} (z, t) & = & σ_{cb} (z, t) e^{- i (ω_{p} - ω_{c 2}) t} \\ {\hat{σ}}_{ca} (z, t) & = & σ_{ca} (z, t) e^{- i (ω_{p} - ω_{c 1}) t} \end{array}

whose dynamic evolutions are governed by a set of reduced Heisenberg-Langevin equations

\begin{array}{l} \partial_{t} σ_{ca} & = & - [γ_{ca} + i (Δ_{p} - Δ_{c 1})] σ_{ca} - i Ω_{c 1}^{*} σ_{ce} \\ \partial_{t} σ_{cb} & = & - [γ_{cb} + i (Δ_{p} - Δ_{c 2})] σ_{cb} - i Ω_{c 2}^{*} σ_{ce} \\ \partial_{t} σ_{ce} & = & - (γ_{ce} + i Δ_{p}) σ_{ce} - i Ω_{c 1} σ_{ca} - i Ω_{c 2} σ_{cb} - i g_{p} E_{p} \end{array}

in the weak probe and adiabatic control limits.

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 $\hat{σ}$ _ab(z,t), $\hat{σ}$ _ae(z,t), $\hat{σ}$ _be(z,t), $\hat{σ}$ _aa(z,t), $\hat{σ}$ _bb(z,t), $\hat{σ}$ _cc(z,t), and $\hat{σ}$ _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 F_ca, F_cb, and F_ce in Eqs. (4). In Eqs. (4), γ_ca, γ_cb, and γ_ce represent the coherence decay rates of operators σ_ca, σ_cb, and σ_ce while Δ_p = ω_p – ω_ec, Δ_c ₁ = ω_c ₁ – ω_ea, and Δ_c ₂ = ω_c ₂ – ω_eb are frequency detunings of the three quantum or classical fields. In addition, Ω_c ₁ = E _c ₁ · d _ea/2h̄ and Ω_c ₂ = E _c ₂ · d _eb /2h̄ are complex Rabi frequencies of the first and second (classical) coupling fields while $g_{p} = \sqrt{ω_{p} / 2 \bar{h} ɛ_{0} V} ɛ_{p} \cdot d_{ec}$ 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 G_c ₁ (G_c ₂) and Φ_c ₁ (Φ_c ₂) as the modulus and the argument of the complex Rabi frequency Ω_c ₁ (Ω_c ₂), respectively.

Solving Eqs. (4) in the steady state, we can easily examine the absorption and dispersion spectra, proportional respectively to Im[σ_ce(Δ_p)] and Re[σ_ce(Δ_p)], of a continuous-wave probe. To study the dynamic evolution of the dimensionless operator E_p(z,t), however, we also need the wave propagation equation

(\frac{\partial}{\partial t} + c \frac{\partial}{\partial z}) E_{p} (z, t) = i g_{p} N σ_{ce} (z, t)

which is attained in the slowly-varying envelope approximation. Here N refers to the total number of active atoms in the quantization volume V of the weak probe field while c is defined as the light speed in vacuum. Eq. (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 = γ_ce as the frequency unit, $t_{0} = γ_{ce}^{- 1}$ as the time unit, and $l_{a} = c γ_{ce} / g_{p}^{2} N$ (the absorption length in the absence of EIT) as the position unit. As we can see from Fig. 2, a quantum field E_p(z,t) goes slowly into the atomic sample at the velocity υ_g = 0.25l_a/t ₀ and is totally transformed into stationary spin coherences σ_ca(z,t) and σ_cb(z,t) at the sample center when the two coupling fields with Δ_c ₁ = Δ_c 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 σ_ca(z,t) and σ_cb(z,t) at the same time but with opposite detunings, i.e. Δ_c ₁ = −Δ_c ₂ = 0.02γ in (a, b), Δ_c ₁ = −Δ_c ₂ = 0.05γ in (c, d), and Δ_c ₁ = −Δ_c 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 υ_g = 0.25l_a/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 Δω_beat = Δ_c ₁ − Δ_c ₂ when |Δ_c ₁| = |Δ_c ₂| are large enough [see Fig. 2(e, f)].

Fig. 2 (Color online) Dynamic evolution of a quantum probe field inside a cold atomic sample (a, c, e) and the quantum probe field at the sample exit as a function of time t (b, d, f). The two classical coupling fields are turned off at t = 48.0t ₀ with Δ_c ₁ = Δ_c ₂ = 0.0γ but turned on at t = 72.0t ₀ with Δ_c ₁ = −Δ_c ₂ = 0.02γ (a, b); 0.05γ (c, d); 0.08γ (e, f). Other parameters are set as Δ_p = 0.0γ, γ_ca = γ_cb = 0.3 × 10⁻⁴ γ, $g_{p} \sqrt{N} = 310.0 γ$ , G_c ₁ = G_c ₂ = 0.50γ, and Φ_c ₁ = Φ_c ₂ = 0.0. In (b), (d), and (f), thin grey curves denote the quantum probe field at the sample entrance while thick red curves represent the quantum probe field at the sample exit.

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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 [13], we may adiabatically transform Eqs. (4) and Eqs. (5) into

(\frac{\partial}{\partial t} + c {cos}^{2} θ \frac{\partial}{\partial z}) Ψ (z, t) = 0

which describes a shape-preserving propagation of the two-mode DSP Ψ(z,t) = Ψ_a(z,t) + Ψ_b(z,t) with

\begin{array}{l} Ψ_{a} (z, t) & = & \frac{cos θ}{2} E_{p} (z, t) - e^{i (Φ_{c 1} - Δ_{c 1} t)} sin θ \sqrt{N / 2} σ_{ca} (z, t)] \\ Ψ_{b} (z, t) & = & \frac{cos θ}{2} E_{p} (z, t) - e^{i (Φ_{c 2} - Δ_{c 2} t)} sin θ \sqrt{N / 2} σ_{cb} (z, t)] \end{array}

if Ψ(z,t) originates from the pure quantum field excitation E_p(z,t) or

\begin{array}{l} Ψ_{a} (z, t) & = & e^{i (Δ_{c 1} t - Φ_{c 1})} \frac{cos θ}{2} E_{p} (z, t) - sin θ \sqrt{N / 2} σ_{ca} (z, t)] \\ Ψ_{b} (z, t) & = & e^{i (Δ_{c 2} t - Φ_{c 2})} \frac{cos θ}{2} E_{p} (z, t) - sin θ \sqrt{N / 2} σ_{cb} (z, t)] \end{array}

if Ψ(z,t) originates from the pure atomic coherence excitation

\sqrt{N / 2} [σ_{ca} (z, t) + σ_{cb} (z, t)]

. In deriving Eqs. (7) and Eqs. (8), we have set

tan θ = g \sqrt{N / (G_{c 1}^{2} + G_{c 2}^{2})}

and G_c ₁ = G_c ₂ for simplicity so that both DSP modes Ψ_a(z,t) and Ψ_b(z,t) can attain the same propagating velocity υ_g = 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 Φ_c ₁ = Φ_c ₂ = 0.0. At t = 0.0t ₀, a fast probe field E_p(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 Ω_c ₁ and Ω_c ₂. In this case, the field component in Ψ_a(z,t) and that in Ψ_b(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 Ω_c ₁ and Ω_c ₂ are switched off, the two slowly-moving DSP modes described by Eqs. (7) turn into a pair of stationary atomic excitations $\sqrt{N / 2} σ_{ca} (z, t)$ and $\sqrt{N / 2} σ_{cb} (z, t)$ limited by Δ_c ₁ = Δ_c 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 Δ_c ₁ ≠ Δ_c 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 Ψ_a(z,t) and that in Ψ_b(z,t) attain, respectively, time-dependent phases Δ_c ₁ t and Δ_c 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 ^iΔ_c1t E_p(z,t)/2 and e ^iΔ_c2t E_p(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 Δ_c ₁ ≠ Δ_c ₂ after t = 72.0t ₀ because they are well contained in two separate EIT windows located, respectively, at Δ_p = Δ_c ₁ and Δ_p = Δ_c ₂ [27].

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 |Δ_c ₂ – Δ_c ₁| in very small steps. As we can see there is only a single maximum when |Δ_c ₂ – Δ_c ₁| < 0.003γ while a second maximum tends to arise when |Δ_c ₂ – Δ_c 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 [27]. From Eqs. (7) and Eqs. (8) we also can see that the two DSP modes are sensitive to arguments Φ_c ₁ and Φ_c ₂ of the two complex Rabi frequencies Ω_c ₁ and Ω_c ₂. So one may simply modulate the relative phase ΔΦ = Φ_c ₂ – Φ_c 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 ^{(Δ_c1t−Φ_c1)} E_p(z,t)/2 and eⁱ(^{Δ_c2t − Φ_c2)} E_p(z,t)/2 that describe the pair of field components at the sample exit.

Fig. 3 (Color online) The quantum probe field at the sample exit as a function of time t with Δ_c ₁ = −Δ_c ₂ = 0.0γ (black-solid), 0.0015γ (red-dashed), 0.003γ (blue-dotted) in (a) and Δ_c ₁ = −Δ_c ₂ = 0.005γ (black-solid), 0.0065γ (red-dashed), 0.008γ (blue-dotted) in (b). Other parameters are the same as in Fig. 2.

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Fig. 4 (Color online) The quantum probe field at the sample exit as a function of time t with ΔΦ = 0.0 (black-solid), π (red-dashed) in (a) and ΔΦ = π/2 (black-solid), 3π/2 (red-dashed) in (b). Other parameters are the same as in Fig. 2 except Δ_c ₁ = −Δ_c ₂ = 0.03γ.

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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 γ_ca, γ_cb, and γ_ce. 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, m_F = +1〉, |5² S _1/2, F = 2, m_F = −1〉, |5² S _1/2, F = 1, m = +1〉, and |5² P _1/2, F′ = 1, m = 0〉, respectively [30].

4. Conclusions

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 ω_c ₂ − ω_c ₁, the relative phase Φ_c ₂ − Φ_c ₁, and their stabilities can be easily inferred from the beating signals imposed on a weak probe field. If the detuning difference |Δ_c ₁ − Δ_c ₂| is induced by shifting hyperfine state sublevels with a magnetic field B, we may have h̄ |Δ_c ₁ − Δ_c ₂| = 2gμ_B · B with g being the Lander factor and μ_B 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 |Δ_c ₁ − Δ_c ₂| = |ω_ab| 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).

Acknowledgments

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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Dynamic generation of robust and controlled beating signals in an asymmetric procedure of light storage and retrieval

Abstract

1. Introduction

2. Model and equations

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (8)

Optics Express