## Abstract

This article describes quantum interference in a lambda system driven by two identical pulses that are each sufficiently broadband to drive both dipole-allowed transitions. The first pulse drives the system into a quantum superposition, making the effect of the second pulse depend critically on its optical phase and resulting in Ramsey-like fringes. This method, using just two pulses of the *same carrier frequency*, is conceptually simpler than in previous Raman–Ramsey studies, which use pump and Stokes pulses in each of two spatially separated regions. The goal here is not efficient population transfer, but to investigate narrow features resulting from quantum interference. I first explore these effects for low-inversion, which illustrates many key features using an easy-to-visualize model. I then use Schrödinger’s equation in a semiclassical model to extend the results to arbitrary inversion. Informative quantum interference features remain when using this simplified scheme.

© 2010 Optical Society of America

## 1. INTRODUCTION

Ramsey-fringe techniques [1, 2, 3] have a long and important history, including applications to time and frequency standards [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and spectroscopy [15, 16, 17, 18, 19, 20, 21, 22, 23]. An important subset of the literature addresses quantum interference effects, including Ramsey-like fringes, in the case of *three-level* systems [6, 7, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. Ramsey-type work, in which a three-level *Λ* system is driven from one lower state to the other via a stimulated Raman transition, is sometimes aptly called “Raman–Ramsey” work. Interference effects in three-level systems driven by pulse pairs also have applications in coherent control [37, 38] and for selecting a narrow velocity distribution of atoms [39, 40]. There are calculations addressing the interaction of a three-level system with a quantized field (e.g., [26, 41, 42]). A subset of this work, e.g., in [32, 35, 36], involves driving the system into a dark state; the advantages of such schemes are discussed therein.

With many variations possible, an archetypical stimulated Raman–Ramsey setup can be pictured as a beam of three-level *Λ* atoms or molecules sent through two interaction regions. In each interaction region, the *Λ* system interacts with a pump and a Stokes electromagnetic field. In zone one, the *Λ* system is driven into a coherent superposition state. The later interaction, in zone two, depends sensitively on the phase relation between the driving fields in zone two and the quantum phase of the *Λ* system, which precessed freely between the zones. The physical reasons for this phase sensitivity are fundamentally similar to the case of traditional Ramsey fringes in a two-level system, as discovered in the Nobel Prize winning work of Ramsey [1, 3].

This article uses a straightforward semiclassical model to study a case that is different from the cases studied in the work cited above: it explores the quantum interference structure that results from a *simplified technique using just one pulse, of one carrier frequency*, in each interaction region instead of using pump and Stokes beams. The amplitude and duration of the pulses are the same in each interaction region. The pulses are modeled as perfectly coherent pulses with a single carrier frequency and square envelopes. They are not few-cycle or “ultrashort” pulses, but are short enough to have sufficient bandwidth to drive both dipole-allowed transitions. Although the calculations below scale, a good exemplar case is the “optical Ramsey” case, with optical dipole-allowed transitions and with hyperfine splitting between the two lower states. The results below (i) show that a Ramsey-like fringe structure is still present and (ii) explore how the fringe structure changes as the relevant parameters change.

## 2. MODEL

Figure 1 shows the system of interest: a *Λ* system with energy levels ${\u03f5}_{i}=\hslash {\omega}_{i}$
$(i=1,2,3)$, where the lowest level is chosen to have ${\u03f5}_{1}=0$. In frequency units, the separation between the lower levels is

*Λ*system is driven by an optical beam having carrier frequency

*ω*. The overall detuning

*Δ*, as shown in Fig. 1, iswhere ${\omega}_{0}$ is the average resonant frequency;

Traditionally, the stimulated Raman–Ramsey effect involves a pump beam, generally nearly resonant with the dipole-allowed |1⟩-|2⟩ transition, and a Stokes beam, generally nearly resonant with the dipole-allowed |2⟩-|3⟩ transition. In this article, however, I consider only one carrier frequency *ω*. It is worth noting that with only one carrier frequency, the system is *never on two-photon resonance*; it is always detuned by the |3⟩-|1⟩ splitting *δ*. From an energy point of view, one might say that energy is conserved because the pulse bandwidth allows the energy absorbed by the |1⟩-|2⟩ transition to be slightly more than the energy emitted into the pulse by the |2⟩-|3⟩ transition.

This article addresses the response of a *Λ* system (called an “atom” for convenience) to two optically coherent pulses. The pulses are modeled as having square envelopes. For discussing the quantum interference due to two pulses, we can imagine atoms passing through two interaction zones, or atoms passing through one interaction zone twice, or two consecutive pulses hitting an atomic sample. Depending on the experimental details, the pulses can hit the atom with a zero or with a non-zero phase shift, as illustrated in Figs. 2a, 2b , respectively. Case (a) might, for example, be obtained by pulsed amplification of a coherent cw laser. Case (b), with a stable but non-zero phase shift, could be obtained by splitting a pulse and sending one copy through an optical delay line. For case (b) a very small change in *T* (say by a fraction of an optical cycle) causes a change in the optical phase of the second pulse [16]. Therefore, case (b) requires controlling *T* with interferometric precision, as was done, for example, by Scherer *et al.* [43]. With a stable phase relation, quantum interference can result in either case, and the distinction between cases (a) and (b) has often been discussed, e.g., in [16, 35, 44, 45, 46]. For convenience and tractability, this article takes the phase shift as zero, using

*T*and the detunings, not by any overall phase contained in ${E}_{0}$.

## 3. EQUATIONS

This section gives the equations of motion for the probability amplitudes in the Schrödinger picture. The solutions to these equations will be plotted in Sections 4, 5 to explore the behavior of the system under various conditions. The basis states are the energy eigenstates of the bare atom: $|j\u27e9$, $(j=1,2,3)$. So we have

where ${\widehat{H}}_{0}$ is the Hamiltonian of the unperturbed atom and ${\u03f5}_{j}$ is the energy of state $|j\u27e9$. An atomic superposition state can be written*C*’s when the laser is on.

The time development of the atomic state is given by Schrödinger’s equation,

Using the wave function, Eq. (9), in the Schrödinger Eq. (10) gives the differential equation for the time evolution of the *C*’s:

*C*'s can be extracted from Eq. (12) and is The simplicity of these equations is the result of the approximations mentioned just above and also because ${V}_{11}={V}_{22}={V}_{33}={V}_{13}=0$.

Writing Eqs. (13, 14, 15) together in matrix form yields

where $\mathbf{C}\equiv {[{C}_{1},{C}_{2,}{C}_{3}]}^{\mathrm{T}}$, and## 4. SHORT, WEAK PULSE LIMIT

#### 4A. Solution

The case of short, weak pulses ($\tau \u2aa11\u2215{\Delta}_{i}$ and low inversion) allows for conceptual and mathematical simplification, which facilitates checking some fundamentals. The results in this section are consistent with intuition carried over from traditional Raman–Ramsey studies (using pump and Stokes beams), and also somewhat resemble the simpler case of Ramsey fringes in a two-level system. Good agreement between Sections 4, 5 also shows that the intuition developed in Section 4 can facilitate understanding of the more formal and general results in Section 5.

Consider (throughout this article) an atom starting in the lowest state |1⟩. An optical pulse then drives the atom into a superposition state. In the “weak pulse limit” (this section), the first pulse will leave most of the population in state |1⟩. Some of the population will be driven to state |2⟩, and a small fraction to state |3⟩. But what about the second pulse? A reasonable guess is that the second pulse will drive more population along this two-step path (that is, will cause constructive interference) if the second pulse is in phase with the already oscillating dipoles. By the definition of detunings, the driving field *ω* drifts out of phase with the freely oscillating |1⟩-|2⟩ and |2⟩-|3⟩ dipoles at a rate given by the detunings ${\Delta}_{1}$ and ${\Delta}_{3}$. If the field drifts out of phase by a whole number of cycles, then it is back in phase. That is, the second pulse will arrive in phase with each oscillating dipole transition if

*n*and

*m*are integers. (These pulse delay conditions have a similar significance [51] in frequency comb calculations.) If the pulse is to be in phase with both oscillations, then the oscillations must be in phase with each other. That is, the |1⟩-|2⟩ oscillation and the |2⟩-|3⟩ oscillation should have drifted out of phase by $2p\pi $ where

*p*is an integer:Put differently, Eq. (20) can also be obtained by simply subtracting Eq. (19) from Eq. (18).

Equations (18, 19, 20) can be supported further by solving Eq. (16). For simplicity in the examples below, consider the Rabi frequencies to be equal:

*C*’s, Eqs. (13, 14, 15). Neglecting ${C}_{2}$ compared to ${C}_{1}$, Eq. (13) for ${C}_{1}$ integrates toMoving on to ${C}_{2}$: neglecting decay $(\gamma \approx 0)$, neglecting ${C}_{3}$ compared to ${C}_{1}$, and using Eq. (23), Eq. (14) becomes

The approximate solutions to Eqs. (24, 25) are

*C*'s continuous at the boundaries. I also used the “short-pulse” approximation

The final value of ${C}_{3}$ is given by evaluating Eq. (27) at $t=T+\tau $, yielding

Using Eqs. (4, 5) allows rewriting Eq. (30) in terms of the overall detuning *Δ* as

*Δ*for three different values of

*T*. (Reassuringly, a plot for the same parameters made using the more formal results of Section 5 is indistinguishable from Fig. 3.) Because the variable

*Δ*appears just once in Eq. (31), the curves for various

*T*are simply scaled and shifted cosines. For both the heavy solid curve and the dashed curve in Fig. 3,

*T*is chosen so that $\mathrm{cos}(\delta T\u22152)=1$. Therefore, the term in parentheses varies from zero to 8. The FWHM of each fringe is $\pi \u2215T$. The dashed curve has

*T*three times as long as the heavy solid curve, so the fringes are one-third as wide. All of the fringes in Fig. 3 are much narrower than the bandwidth of a single pulse, which is of order $1\u2215\tau $. ($1\u2215\tau $ equals 20 in the units of Fig. 3.) Having fringe width governed by $1\u2215T$ rather than $1\u2215\tau $ is a hallmark of traditional Ramsey fringes; Fig. 3 shows that this crucial property carries over to the nontraditional case considered in this article.

The light solid curve in Fig. 3 illustrates how the product $\delta T$ can affect the interference amplitude. In Eq. (31), the term $\mathrm{cos}(\delta T\u22152)$ multiplies the oscillating term and therefore sets the scale of the oscillation. Because that cosine can range between $-1$ and $+1$, the oscillation will scale and can change sign as *T* changes. The light solid curve shows a case with $\mathrm{cos}(\delta T\u22152)$ negative, so the oscillations are inverted. Also, because $\mathrm{cos}(\delta T\u22152)$ is not all the way to $-1$, that curve has smaller oscillations.

#### 4B. Frequency Domain Discussion

Equations (18, 19, 20) above, along with the basic structure of Fig. 3, are consistent with intuition about “in-phase timing” for constructive interference. For low inversion, the resonance structure can also be understood from the frequency domain point of view, which is widely used; see, e.g., [1, 2, 52, 53].

Effectively picturing the frequency domain requires keeping in mind that the pulses are broadband. In the time domain, Fig. 2a might have tempted one to say simply that the second pulse is “in phase” with the first. However, each pulse has a frequency *spectrum*, and various frequency components will have various phase relations. The main issue is not necessarily whether the carrier frequency is in phase from pulse to pulse, but whether the frequency components matching the allowed transitions interfere constructively. This issue of pulse- to-pulse interference also arises for a similar reason in frequency-comb spectroscopy. (See, e.g., [9, 22, 23, 51, 54, 55, 56].) Frequency-comb work is in a different parameter regime than the one addressed here; it addresses the more extreme case of ultrashort coherent pulses. Because of their extreme bandwidth, ultrashort pulses can interfere constructively at many frequencies, hence giving rise to the “frequency comb.” Although the parameter regime is different in that work and this, the phase delay concept is of similar importance. For example, [51] (p. 7) describes the pulse-to-pulse interference requirement in their case “by noting that for the laser modes to match both atomic transitions, then the difference between the atomic frequencies must be equal to some multiple of the mode spacing $2\pi \u2215T$.” That point of view agrees with Eq. (20) above. For this article, I will continue the discussion in the time domain, and now turn to the more general case, including arbitrary inversion.

## 5. GENERAL CASE, INCLUDING ARBITRARY INVERSION

#### 5A. General Solution

The approximate solution for the population in state |3⟩, Eq. (31), is useful because it reveals the physics effectively; it explicitly displays how each parameter enters. Even when discussing the exact solution below, I will sometimes refer back to Eq. (31), because it can make the physics transparent.

To obtain a more general solution that holds for arbitrary inversion, we can directly integrate Eq. (16), giving the formal solution

Equation (16) could also be addressed by finding the eigenvalues and eigenvectors of**M**, but that algebra yielded no further physical insight, so I will leave the general solution written compactly as Eq. (32). For plotting the various cases below, I used a power series expansion of the exponential in Eq. (32), while keeping enough terms to insure that each plot represents the actual solution. The number of terms needed depends on the parameters.

Finding the final population in state |3⟩ requires stepping through the “on-off-on” pulse sequence, much like in the previous section. $\mathbf{C}\left(\tau \right)$ is given by evaluating Eq. (32) at $t=\tau $, and with the atom starting in the ground state ${\mathbf{C}}^{\mathrm{T}}\left(0\right)=(1,0,0)$. The “pulse-off” part of the evolution is simple: putting both Rabi frequencies equal to 0 in **M** shows that the atom simply precesses and decays. Therefore at the end of the “off-time” we have

#### 5B. Agreement in the Low Inversion Case

In any situation to which the assumptions of Section 4 apply, Eq. (31) and Eq. (35) agree. For example, for the three curves plotted in Fig. 3, Eqs. (31, 35) give indistinguishable results. The reason for presenting both equations is this: Eq. (35) applies more generally, because neither the low inversion approximation [Eqs. (22)] nor the short-pulse approximation [Eq. (28)] was used. However, the more restricted result, Eq. (31), is useful because it explicitly displays the parameter dependence, allowing for a more intuitive understanding of the physics.

#### 5C. Velocity Averaging

For the case of an atomic beam traveling through two interaction regions, different atoms have different speeds, and hence have different travel times; that is, different atoms have different *T*. This is a Doppler broadening effect that is considered disadvantageous in some circumstances, but in Ramsey experiments it offers the important *advantage* of identifying the main peak at $\Delta =0$. This advantage is important enough that averaging over *T* is often artificially introduced into two-pulse experiments with trapped atoms by using a range of optical pulse delays and then averaging the results (e.g., in [2, 18, 57, 58]). The point of averaging over *T* in a two-pulse experiment using *two-level* atoms is this: the constructive interference peak with zero detuning remains, because a resonant drive stays resonant regardless of the dark time. However, constructive interference peaks due to a relative phase shift between pulses of $2\pi $, $4\pi $, $6\pi $, etc., are washed out, because having *those* phase delays is *T*-dependent. Because the side peaks wash out, the central peak can be unambiguously identified. The results in this subsection show that for *three-level* atoms, averaging over *T* depends on the averaging interval in a more complex way.

A great deal can (and has) been said about the details of velocity averaging. The details depend on many factors, including whether the variation in *T* is due to a spread of atomic velocities in a beam or due to controlled variation of two pulses interacting with a sample of trapped atoms. This article does not address a myriad of averaging schemes; rather, it addresses two core issues: (1) Do the fringes remain after velocity averaging? (They do.) (2) How does changing the range of delay times *T* included in the averaging affect the signal?

The *T* averaged results shown as bold curves in Fig. 4 are obtained by a simple boxcar average,

*T*. That is why the $\Delta =0$ peak does not wash out in the averaging for a

*two-level*atom. However, some additional complexity of a three-level atom is revealed by the $\mathrm{cos}(\delta T\u22152)$ term that multiplies $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\Delta T$ in Eq. (31). The $\mathrm{cos}(\delta T\u22152)$ term changes sign if

*T*is varied greatly, eventually causing the $\Delta =0$ peak to wash out. Figure 4 shows an example of such a progression. In Fig. 4a, the range of velocity integration is $T=4\pi \pm 1$, in units with $\delta =1$. The purpose of this choice is to keep $\mathrm{cos}(\delta T\u22152)$ always positive. This makes the last term in Eq. (31), evaluated at $\Delta =0$, $4\times 1\times \left(\mathrm{a}\phantom{\rule{0.3em}{0ex}}\text{positive}\right)$. Averaging over just positive numbers retains the peak at $\Delta =0$. In Fig. 4b, the solid curve is averaged over $T=4\pi \pm 4$, which includes some negative values of $\mathrm{cos}(\delta T\u22152)$, thereby reducing the height of the main peak. This broader range of

*T*is sufficiently large that the

*non-central*peaks are averaged over enough positive and negative values to wash out. This is a hallmark of the effect of velocity averaging in Ramsey fringes.

Figure 4c uses an even broader range of *T* in averaging, and reveals an effect with no exact equivalent in Ramsey fringes for two-level systems. Figure 4c can be thought of algebraically or physically. In algebraic terms, the product of cosines in Eq. (31) will include both positive and negative numbers, and will therefore average to near zero if an extremely wide range of *T* is sampled. The exception to that is when $\Delta =\pm \delta \u22152$; for that case, the last term in Eq. (31) is a cosine squared term, and will therefore never average to zero. In physical terms, $\Delta =\pm \delta \u22152$ is where the carrier is resonant with one of the dipole-allowed transitions, as shown by Eqs. (4, 5). Hence, the peaks at $\Delta =\pm 1\u22152$ (in units where $\delta =1$) in Fig. 4c show the allowed transition frequencies. This is an interesting shift—from identifying the central fringe at $\Delta =0$, to identifying the two dipole-allowed resonances.

Figure 5a further illustrates how Eq. (31) can afford some simple understanding of population inversion. First, since the important $\mathrm{cos}\left(\Delta T\right)$ term contains *Δ* and *T* only as a product, the graph shows basically hyperbolic contours, because the product is constant along hyperbolas. Second, Eq. (31) clearly reveals why the contours flip from black to white along horizontal bands at $T=\pi $ and $3\pi $. This is because the $\mathrm{cos}\left(\Delta T\right)$ term causing the hyperbolas is multiplied by $\mathrm{cos}(\delta T\u22152)$, which changes sign at $T=\pi $ and $3\pi $.

Comparing Figs. 5a, 5b shows that using Eqs. (18, 19) to get a good sense of inversion is justified even for nontrivial inversion. Figure 5a shows the actual inversion as a function of *Δ* and *T*, while Fig. 5b shows where one might have casually guessed inversion would be large. The guess in Fig. 5b is simply a plot of $\mathrm{cos}\left({\Delta}_{1}T\right)+\mathrm{cos}\left({\Delta}_{3}T\right)$, chosen because that expression gives a crude map of where Eqs. (18, 19) are true. More precisely, $\mathrm{cos}\left({\Delta}_{1}T\right)+\mathrm{cos}\left({\Delta}_{3}T\right)$ is contained in the middle term in Eq. (30), which reveals the same physics.

#### 5D. Longer Pulses

There is another way (besides averaging over *T*) to identify the central interference peak. Figure 6 illustrates this. The dashed curve in Fig. 6 is identical to the dashed curves in Fig. 4. The heavy solid curve in Fig. 6 shows the effect of increasing *τ*. There is no velocity averaging whatsoever in Fig. 6; rather, increasing *τ* simply gives the pulses a narrower bandwidth and identifies the central peak. The lighter solid curve in Fig. 6 shows that decay $(\gamma \ne 0)$ has the expected effect of lowering the signal.

## 6. CONCLUSIONS

A pulse of sufficient bandwidth can drive both dipole-allowed transitions in a *Λ* system. Two optically-coherent pulses can cause quantum interference effects: the first pulse leaves the atom in a superposition state, and the effect of the second pulse depends critically on its phase relation with the already-excited system. This article demonstrates that such stimulated Raman–Ramsey quantum interference effects can be studied in a simplified manner—using a total of just two pulses of one carrier frequency, rather than using the traditional pump and Stokes fields in each interaction. The physics underlying the resulting quantum interference effects has been explored and illustrated in detail here using a straightforward semiclassical model.

## ACKNOWLEDGMENTS

Drew University and Stevens Institute of Technology provided support for this research. I appreciate Marty Ligare, Robert Murawski, and Chris Search making insightful comments on a draft of this article. Paul Berman, Wally Buell, Dan Gauthier, A. “Kumar” Kumarakrishnan, Rainer Martini, Ron Walsworth, Ed Whittaker, and two reviewers also made very helpful suggestions.

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