Controlling irreversibility and directionality of light via atomic motion: optical transistor and quantum velocimeter

The Doppler effect of moving atoms can create irreversibility of light. We show that the laser field in an electromagnetic induced transparency (EIT) scheme with atomic motion can control the directional propagation of two counter-propagating output probe fields in an atomic gas. Quantum coherence and the Doppler effect enable the system to function like an optical transistor with two outputs that can generate states analogous to the Bell basis. Interference of the two output fields from the gas provides useful features for determining the mean atomic velocity and can be used as a sensitive quantum velocimeter. Some subtle physics of EIT is also discussed. In particular, the sign of the dispersive phase in EIT is found to have a unique property, which helps to explain certain features in the interference. New Journal of Physics 10 (2008) 123024 1367-2630/08/123024+17$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft


Introduction
In the presence of matter, the reversibility of light can be broken. An example of the broken symmetry is between the absorption and emission processes, due to the presence of spontaneous emission. Atomic motion also breaks the symmetry, particularly between absorption and stimulated emission, which is one of the ingredients for laser cooling [1]. No doubt, irreversibility of light is possible. But how can we control the irreversibility and develop it for useful applications? In a two-level atom, a negative detuned laser would be resonantly absorbed if the atom moves opposite to it and would be transmitted if the atom moves along it, but there is no way to control it.
The absorption of a probe field with frequency ν (on b ↔ a transition) by atoms with a three-level scheme (see figure 1) can be significantly reduced by applying a control laser (on the c ↔ a transition) with frequency ν c that satisfies two-photon or Raman resonance ( = c or ν − ν c = ω c − ω b ), where ω b and ω c are the atomic frequencies of level b and level c, respectively. This effect is called EIT and has been widely studied; mainly in the perspectives of nonlinear processes, slow light and quantum information storage [2]. The physics of EIT is mainly due to destructive interference of two absorption pathways, i.e. from level b to two ac Stark split levels of level a.
In this paper, we consider the EIT effect in an ensemble of atoms with three-level scheme moving with a mean velocity u, sufficiently large such that the Doppler shift is greater than the linewidth of the excited state, ku > . In each atom (figure 1(a)), the same transition (a ↔ b) couples to two counter-propagating probe fields with Rabi frequencies + and − , which are typically weaker than the control field c . The present scheme should be discerned from the polarization gradient cooling scheme [3] and velocity selective coherent population trapping and Doppler induced detuning (DID)) in EIT with and without probe detuning are shown for the case of an atom co-propagating with the probe field. The laser frequency is red-shifted (red arrow downward) by the Doppler frequency ω p = ku = kp/M. A similar diagram can be drawn for a counter-propagating atom, but with blue-shift (blue arrow upward).
(VSCPT) [4] which also has counter-propagating fields, but only one field couples to each transition.
The effect of Doppler broadening (due to moving atoms) on EIT linewidth [5] and slow light [6] has been considered. However, potential applications based on controlling the transmission and irreversibility of two counter-propagating probe fields in moving atoms have not been considered. Under certain controllable conditions, the medium can be made transparent to one of the probe fields while the other field is strongly absorbed. The combined effect of the center of mass (cm) motion and the control field in EIT provides controllable optical directionality or switching of the counter-propagating probe fields 1 . Based on this mechanism, we discuss possible applications as a quantum optical transistor and a quantum velocimeter. Further analysis also yields insights into the difference between detuning from a real level and from a level split by the control laser field.
Before we go through the theory, let us refer to figure 1(b) and try to understand the simple physics (for a resonant control field c = 0). The frequency of a probe field is blue-shifted when absorbed by a counter-propagating atom. Conversely, the frequency is red-shifted when absorbed by a co-propagating atom. The Doppler shift, shown by small arrows and the ac Stark splitting (2˜ ) give rise to three limiting classes of internal mechanisms: (a) DIT occurs when the bare frequency is shifted to the midway between the split levels. (b) DIR-the bare frequency is shifted to resonance with one of the split levels. (c) DIT-the bare frequency is shifted farther away from the nearest ac Stark split level.

Density matrix approach for EIT with moving atoms
The scheme in figure 1(a) is described by the interaction Hamiltonian V = −h[|a c| c e −iν c t + |a b|( + e ikẑ + − e −ikẑ )e −iνt + adj] with the bare Hamiltonian composed of the kinetic energy 4 H kin =P 2 2M and atomic energy H a = l=a,b,ch ω l |l l|. The control laser is taken to be orthogonal to the atomic beam and has no cm effect along the z-axis. Only the counter-propagating probe lasers give the cm effect. The cm position operatorẑ is quantized and this gives rise naturally to the first-order Doppler shift ω p = kp/M and recoil shift ω r =hk 2 /2M in the transition frequency.
We have a set of an infinite number of density matrix equations which contain finite coherence between different momentum families ρ αβ ( p α , p β ) = α, p α |ρ|β, p β that can only be solved numerically. The presence of quantized cm motion makes it impossible to find exact solutions. However, we find that it is possible to obtain analytical results in the weak field limit.
For sufficiently weak probe fields as in the typical EIT case, the population in the excited state is negligible. Thus it is a good approximation to disregard the momentum redistribution as the result of spontaneous emission, which gives rise to integral terms [8] in the equations for the populations. Also, the density matrix equations may be solved analytically by truncating the set of equations based on the approximation of neglecting the coherences between two momentum families with 2hk and larger, i.e. ρ bc ( p ± 3hk, p) ρ bc ( p ± hk, p), ρ aa ( p ± 2hk, p) ρ aa ( p, p) = ρ aa ( p), ρ bb(cc) ( p −hk, p +hk) ρ bb(cc) ( p). We then obtain nine equations in quasi steady state where the slowly varying coherences are The complex decoherences that include the Doppler and recoil shifts are The population inversions are defined as w ac ( p) = ρ aa ( p) − ρ cc ( p) and w ± ab ( p) = ρ aa ( p) − ρ bb ( p ±hk) which can be taken to be the initial values for weak probe fields ρ aa ( p, t) ρ aa ( p, 0) = ρ aa0 f ( p), ρ cc ( p, t) ρ bb ( p, 0) and ρ bb ( p ±hk, t) ρ bb ( p, t) ρ bb ( p, 0) = ρ bb0 f ( p). The populations depend on the momentum distribution of the gas f ( p) with normalization ∞ −∞ f ( p) d p = 1. For a gas with velocity width of p = √ 2Mk B T and a mean momentump = mū (mean velocity towards the right,ū > 0), we represent f ( p) by a Gaussian function 1 The five coupled equations (1)-(3) can be solved exactly, giving where and where κg = N ω ab |℘| 2 2hε 0 c . Please note that we do the momentum integration differently from previous works [5,6]. The present approach is more general, in that it enables the populations in different levels to take different distributions; for example: (7) and (10) show that + is coupled to − due to the nonlinear dependence of equation (7) on I ± and need to be solved numerically when I ± are large. The cross-coupling of the two fields originates from the last two terms in equation (3) and physically corresponds to two probe fields interacting simultaneously with the same atom.
The Doppler effects in nonlinear regime are more complicated and will be reported in the future. Here, we focus on the limit of small probe fields I ± I c , (T ± ab ) 2 where the crossinteraction between the fields is weak and negligible. Thus, we have a linear theory and equation (10) yields the solutions ± (z) = ± (0)e G ± z with the complex 'gain' or response which are related to the linear susceptibilities χ ± = g N ℘ρ ab ( p, p ∓hk, t)/ε 0 ± . If we omit the superscripts ±, we recover the known [5] relationρ ab −i c and is the real double-peak function that primarily determines the EIT features. Note that if ρ cb is neglected (B ± → 0) the probe fields go as (see equation (1)) A ± −i ∓ w ± ab ( p)/T ± ab that gives the usual absorption with Lorentzian resonance profile as in a twolevel system. There would be no EIT (equivalent to setting c = 0) since A ± does not have a denominator with two resonances.

EIT, ac Stark splitting and detunings
Transparency corresponds to a minimum value of Re G ± . For c = 0 this occurs at = 0. However, for finite c there is no simple expression at the minimum. When γ bc = 0 the minimum is at = c .
The beat frequency of emission in EIT is due to ac Stark splitting 2˜ = u − l , where u/l are the detunings of the upper (u) and lower (l) split levels that correspond to two resonant peaks in the absorption spectrum. Expressions for u/l can be found by solving d d (Re G ∓ ) = 0. For c = 0 but finite γ bc we have which reduces to when γ bc γ ab 2 c . Conversely, for γ bc = 0 but finite c we find Thus, the frequencies of the resonant peaks ω ab + u/l depend on the control field detuning in a nonlinear manner. Equation (17) is very useful for physical interpretation of the double resonance found in figure 2 for the total intensity I (z) = | tot (z)| 2 with finite detuning c = , where is the superposition of the two probe fields.

Inhomogeneous broadening and Doppler width
The effects of finite velocity width of the gas on EIT have been well studied in optically thick medium [9]. Under the approximation that the Doppler width is much greater than the EIT window, it was found [5] to have little effect on EIT linewidth at low control field, but leads to linewidth narrowing at high field. However, no analytical expression exists for the general case. Although all results are obtained using the Gaussian distribution, here we use Lorentzian as in a previous analysis [5]. For simplicity, takē ω = kū = 0 and from we can see that G + = G − by defining the variable as ω p → −ω p . The Doppler width is ω = k u.
When + ω r = 0, c = 0, the response function is real, expressible as The S and the f functions are symmetric upon reflection at ω p = 0. The imaginary part of G is zero because the imaginary part of the integrand in equation (19) is an antisymmetric function.
In the case of stationary atoms (no Doppler effect), the real part of G is zero if γ bc = 0 and c = . However, the real part of G in equation (20) is finite because of the overlap between the integrand S(ω p ) and f due to the finite momentum width, even if γ bc = 0. Thus, the resulting signal depends on the overlap of the transparency window W EIT = 2˜ and the Doppler width W D = 2k p/M = 2k √ 2k B T /M. When the Doppler width W D increases, the transparency decreases since more atoms coincide with the absorption peaks. In general, the medium is essentially transparent when W D < W EIT . Figure 3 shows how the total intensity I varies with increasing Doppler width. As the temperature increases, W D increases and only the regions with sufficiently large control field 2˜ W D show large signal.

Real and imaginary parts
The real parts of G ± (corresponding to the imaginary parts of χ (1)± ) give the absorption coefficients (if negative) and gain (if positive). Conversely, the imaginary part of G ± (corresponding to the real parts of χ (1)± ) give the refractive index beyond unity that affects the wavevector. The polar plots in figure 4 illustrate the spatial evolution of the amplitudes | ± (z)| = 0 e ReG ± z (radial lengths) and the phases θ ± = Im G ± z (angles). In the absence of EIT ( c = 0) the resonant ( = 0) counter-propagating probe fields are Doppler shifted equally above and below the resonance (neglecting the much smaller recoil shift). Figure 4  that the counter-propagating probe fields ± acquire phases with opposite signs. However, in the presence of EIT, we find a new subtle effect. The polar plot in figure 6(a) shows that the two fields acquire the same phase with the same sign despite being shifted above and below the split level. This is in contrast to the case without EIT ( figure 4(a)) where the phases of the fields have opposite signs.

Directional flow of light
Several potential applications of quantum coherence effects of EIT in systems with moving atoms are discussed below.

Directional propagation
We start by analyzing the simplest case-without EIT ( c = 0), as shown in figure 4(b). If the probe is detuned = −ωp(< 0) with ωp = kū = kp/M, the field that propagates opposite to the atoms would be absorbed since G − κgw ab0 GL is real and negative value where GL = ∞ −∞ e −x γ 2 +x 2 dx, x = ω p − ωp, whereas the co-propagating field is transmitted since G + −iκgw ab0 f ( p) d p ωp+ω p is primarily imaginary for γ ωp. The counter-propagating probe fields undergo optical directionality, i.e. the medium is transparent to one of the probes while opaque to the other probe field. Thus, an appropriate probe field detuning with respect to the mean velocity of a gas jet can serve as an optical diode.
The mechanism may also be used as a directional emitter, if an emitting source like an electrically driven quantum dot is embedded inside the channel of a hollowed waveguide or fiber containing a gas flow. The source emits light in all directions but would be guided only along two opposite directions. The light in one direction propagates with little damping whereas the light in the opposite direction is heavily absorbed.

Control of directionality
Now, by applying a laser field c in EIT configuration, we can control the directional emission by controlling the absorption strength of the two fields. There are basically three control parameters: (a) the field c that causes ac Stark splitting of the upper levels, (b) the probe frequency or detuning and (c) the Doppler shift ωp. Consider the result with positive detuning = kū shown in figure 5. For˜ < ωp (or 2˜ < 2kū ) in figure 5(a), the co-propagating field (experiences DIT) is damped more rapidly than the counter-propagating field (experiences DID) although it experiences an EIT. Both fields experience different degrees of transparency, but the transparency due to EIT is less than that of a largely detuned (non-resonant) field. In fact, upon the Doppler shift, the co-propagating frequency becomes closer to one of the split levels than the counter-propagating frequency. We should realize that EIT does not necessarily provide the best transparency. Rather, its essence lies in making a medium more transparent to a resonant probe field that would be otherwise heavily absorbed. The above explanation also applies to the case˜ > ωp ( figure 5(b)) which is more obvious, i.e. the medium becomes more transparent to the counter-propagating field through DIT. When˜ = ωp, the two fields are equally detuned from the ac Stark shifted upper level, so both fields are equally damped.

Optical transistor
In principle, the EIT scheme with stationary atoms works like an optical valve, with the capability to control the direction and intensity of the transmitted probe signal. Typical control parameters are the probe detuning and the strength of the control field c . In the absence of atomic motion or Doppler effect, both counter-propagating probe fields are either equally damped or transmitted and there is no rectification. The presence of atomic motion can also make the two probe fields equally transmitted (figure 6(a)) or blocked (absorbed) ( figure 6(b)). More importantly, the presence of atomic motion can create rectification of the combined probe signal, by making the medium transparent to one of the counter-propagating fields but opaque to the other field (figures 6(c) and (d)).
The ability to coherently control the resulting direction of the two counter-propagating fields makes the device function like an optical transistor, shown in figure 6(e). Since the probe fields can be weak compared to the control field, the present optical transistor does not involve nonlinear optical processes, but uses quantum interference in atomic dynamics and the Doppler effect as the switching mechanisms. Despite the linear mechanism, the total signal can vary in a nonlinear manner with the control field as shown in figure 6(f) for switching from state |1 L |1 R to state |0 L |1 R as the Rabi frequency varies from˜ = kū (case figure 6(a) to 2kū (case figure 6c). The present optical transistor is different from existing optical transistors that use (classical) nonlinear processes as the underlying mechanisms; such as the use of nonlinear refractive index in Fabry-Perot [10] and engineered discontinuity in the electromagnetic density of states using photonic crystals [11]. The recently proposed single photon transistor [12] uses surface plasmons to establish strong nonlinear coupling with an atom for use as a single photon gate.

Connection with quantum states
The four limiting cases in figures 6(a)-(d) show a possible connection or analogy between the classical directional states with quantum states. Forū = 0, a resonant field with c = 0 is damped, giving no-field state |0 L |0 R . However, a resonant field with finite c corresponds to transparency (EIT), thus the state of light is |1 L |1 R . For positiveū, when = kū the co-propagating field experiences transparency whereas the counter-propagating field is absorbed, thus the state is |0 L |1 R , referred to as 'right-field'. For = −kū and˜ = 2kū or (for negativeū and = k|ū|) the situation is reversed giving the 'left-field' state |1 L |0 R . The states are subjected to physical factors, namely the driving field and the direction of the fluid.
If these four states (shown in figure 6(e)) can be used to construct the well-known Bell basis the scheme could be useful in quantum information. For example, the superposition of 'on' and 'off' control field can be described by a macroscopic entangled state Similarly the superposition of 'right-field' and 'left-field' states is described by which represents an indefinite mean velocity associated with the case of chaotic flow. Thus, the field system can be described by four Bell basis states.

Quantum velocimeter
An interferometric setup with one arm in the quantum coherence medium has been proposed as magnetometer [13]. The present setup (figure 7(a)) is different as both outputs emerge from the same phaseonium (EIT) medium but their linear responses can be substantially different due to the Doppler effect. The interference of the counter-propagating output fields can be used to detect atomic motion in a gas. The use of the Doppler effect to measure the atomic velocity reminds us of the existing technique of laser Doppler velocimetry [14] which uses crossed laser beams and the Doppler effect to measure the flow velocity, a concept entirely based on classical physics.
Here, we introduce quantum velocimetry that incorporates a different underlying mechanism, based on a quantum coherence effect through the EIT with a laser as a control knob to create a sensitive velocimeter. By combining the two probe fields as shown in figure 7(a), useful results can be obtained from the total field tot (z) = 0 (e G + z + e G − z ). The presence of atomic motion in the gas gives G + = G − . Figure 7(b) illustrates the physics behind the damped and oscillatory nature of the signal I (z). The real parts of G ± determine the degree of absorption or transparency of the fields. The imaginary parts superimpose to give oscillations or beating in I (z) = | tot (z)| 2 , as shown in figure 7(c). The key point is that gross atomic motion or mean velocityū can be detected through the presence of oscillations in I (z) versus z or c . There are no oscillations whenū = 0. A small change inū can be detected from rapid oscillations of I (z) versusū (see figure 8) and will be discussed in the subsequent section.
The existence of the oscillatory feature in figure 7(c) at c = 0 might suggest that a velocimeter could even be contrived in the absence of EIT. However, it is the exponentially decaying region around c ω D = kū (due to EIT resonance) in figure 7(c) that provides information aboutū. From figure 4(a), one can immediately deduce that a velocimeter without EIT may not be feasible since it would produce very weak signal for smallū because the Doppler induced detuning is small. This conclusion is supported by figure 8 for = 0 and small L. For large L, the propagation effect amplifies the small phase difference between the two Doppler shifted probe fields, giving rise to small oscillations at c = 0.
The region around c kū corresponds to the EIT resonance which arises from DIR as shown by the level schematic (on the right) of figure 7(c). Here, the two fields are shifted to resonance and the imaginary part of G ± essentially vanishes, so I (z) shows no oscillations. The probe fields add as (e −bz + e −bz ) 2 ∼ e −2bz . The channel divides the profile into two regions with oscillations. The oscillations in c < kū are more rapid than the region c > kū. The fields in these regions interfere as (e iaz e −bz + e −iaz e −bz ) 2 ∼ e −2bz cos 2 az. For = 0 (finite detuning), figure 7(d) shows the ridge at c ω D corresponding to equal detuning (but opposite signs) of both probe fields from the ac Stark shifted level. One field experiences EIT whereas the other is just normal detuning. We learn from figure 6(a) that their phases would be of the same sign. So the fields interfere as |e −bz e iaz + e −bz e iaz | 2 → e −2bz , which explains the exponential decay. However, at c 2ω D , we have one EIT and one resonant with the shifted upper level. Here, the fields interfere as |e −bz e iaz + e −(b+c)z | 2 = e −2bz 4e −cz cos 2 az 2 where c > 0 which means that the damping due to absorption is larger than the damping in EIT. The rate of oscillations is less rapid by half. The rising ridge can be explained as due to the term (e −cz − 1) 2 in equation (24). The effect of Doppler broadening on the total signal has been shown in figure 3 and section 3.3. At T = 1 K, the Doppler width is W D 0.7γ ac . Thus, the effect of Doppler broadening is important only at the field around c ∼ γ ac and not visible in figure 7. Effects of other noises such as finite control laser linewidth and dephasings would be similar to the effects of increasing the Doppler width and decoherences, respectively. Typical semiconductor laser linewidth of MHz corresponds to a width of less than 1 K, so it would not affect the above results. Detailed theory using the quantum noise approach will be reported elsewhere.

Sensitivity to mean velocity
In the presence of EIT, the quantum velocimetry can be a sensitive device. There is no unique way to determine the mean velocity. The sensitivity depends on the method used. Consider that the total detected intensity I is converted to the detection currentj. Then, in analogy to [13] I ∝j = n in , is δū ∼ 10 −7 ms −1 , which is comparable to the sensitivity of the former method. We have used typical values: γ ac = 10 7 s −1 , A = 10 −5 and λ c = 500 nm. Although the sensitivity is higher ( dū d c is smaller) for smaller c (which is compatible with the narrow EIT lines), the signal is much weaker and less detectable. The gradient dū d c of the features in figure 8 decreases with L, in agreement with the conclusion of equation (28).

Conclusions
We have presented the physics behind EIT for moving particles and discussed potential applications of controlled irreversibility and directional flow of light. In particular, we explore the mechanisms for controlled switching of the probe fields that can function as a two-output optical transistor and as the basis for quantum states. We also showed that the interference of the two counter-propagating fields produces oscillations that can be used to detect a small change in the mean velocity, a quantum version of laser Doppler velocimetry.
Finally, we note that the full potentials of optical directional control and quantum velocimetry could be realized through miniaturization and integration into microoptical systems. Since EIT has been shown in liquid [16] containing active components, it would be possible to integrate the quantum optical transistor and velocimeter concepts into existing microfluidic technology [17], creating a new class of quantum optical-microfluidic sensors for chemical and biomedical sensing. The study of this exciting possibility would involve fluid dynamics with large inhomogeneous broadening and decoherence rates. It is beyond the present scope and will be the subject of future publications.