the semiclassical stochastic-field/atom interaction ...

THE SEMICLASSICAL STOCHASTIC-FIELD/ATOM INTERACTION PROBLEM * JAMES CAMPARO Physical Sciences Laboratories, The Aerospace Corporation PO Box 92957, Los Angeles, CA 90009, USA No field in nature truly corresponds to the classical monochromatic ideal. Even the “singlemode” laser can be, in certain regards, a poor approximation: the mode has a width that depends on the photon’s cavity lifetime, and amplified-spontaneous-emission produces fluctuations in the laser’s output. With regard to metrology, the question is twofold: at what level of precision (stability) does a classical field’s stochastic nature become relevant in field/matter interactions, and how does the field’s stochastic nature manifest itself in measurements. Unfortunately, answers to those questions are only known in a few specific cases, and general intuitive models are lacking. Here, we provide a short overview of the semiclassical stochastic-field/atom interaction, illustrating how the stochastic nature of a classical field can have a significant effect on spectroscopic signals, and how those effects are not always intuitively obvious. In concluding, we argue that simple, general, intuitive models of the stochastic-field/atom interaction are needed, so that researchers can be alerted as to when and how stochasticfield effects may manifest themselves in precision measurements.

1. Introduction No field in Nature is truly monochromatic: the most ubiquitous light source on the planet, the sun, gives off thermal radiation, and the singlemode laser’s emission has a finite linewidth and emits photons in cavity modes other than the main lasing mode. Nevertheless, pick up almost any introductory or advanced quantum text; turn to the section on radiative field/matter interactions, and in nearly all cases the discussion will be restricted to monochromatic fields. This is not to say that treating fields as monochromatic in the semiclassical regime is wrong. Rather, it limits what is perhaps a richer and more varied subject. Typically, we account for field stochasticity through the convolution picture. Briefly, a stochastic field will always have some “non-delta function” power spectrum, and we describe the atom’s response to the stochastic field,

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This work was supported under The Aerospace Corporation’s Mission Oriented Investigation and Experimentation program, funded by U.S. Air Force Space and Missile Systems Center under Contract No. FA8802-04-C-0001. 1

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L(ωatom−ωfield) in terms of a convolution between the field spectrum, P(ω−ωfield) and the atom’s absorption cross-section, σ(ω−ωatom): L(ωatom − ωfield ) =

∫ P(ω − ω

field

)σ(ω − ωatom )dω .

(1)

For example, if we have a positively-skewed field spectrum, as might arise for a field with correlated amplitude and frequency fluctuations, it will produce a negatively-skewed atomic-response function, perhaps a fluorescence lineshape. Moreover, the asymmetry will result in a frequency shift of the atom’s response: for a positively-skewed field spectrum we will get a negative frequency shift. The problem with a convolution picture of the stochastic-field/atom interaction is that it has well-defined limits of applicability. In mathematics, convolutions in frequency arise when we consider the Fourier transform of a quantity that is the product of two temporal functions. In other words, the convolution picture requires the atom’s response to the field to be described as the product of an instantaneous absorption cross section and an instantaneous field intensity, which limits it to first-order linear processes. Obviously, the same conclusion is reached when we approach the problem through timedependent perturbation theory. 2. Stochastic-Field Effects and an Atom’s Average Response 2.1. Multi-photon Excitation The first place we might expect the convolution picture to break down is with multiphoton processes: radiative effects that require high intensity and are by definition not first order. If we consider a multimode field, where the modes have uncorrelated phases, then the intensity will show deep fades and extreme spikes all about some average intensity, 〈I〉, as illustrated in the inset to Fig. 1. The rate of N-photon excitation, and hence the multiphoton signal, SN, will be equal to a generalized multiphoton cross section, σN, and the Nth moment of the field intensity (i.e., SN ~ σN〈IN〉) [1]; and for a sufficient number of independent modes (~ 102), the central limit theorem implies that the total amplitude of the field will have a normal probability distribution. Thus, 〈IN〉 = N!〈I〉N, and we find that an N-photon signal in a stochastic field can be N! times larger than the signal from a monochromatic field of the same average intensity. This striking prediction was verified in an experiment by Lecompte et al. a number of years ago [2]: a 20 nsec pulsed Nd:Glass laser that could be operated with one to 100 modes was employed for 11-photon ionization of xenon. Essentially, the number of Xe+ produced was measured as the number of modes

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in the pulse varied, and this was compared to the number of Xe+ produced for a single-mode pulse of the same average intensity. The results are shown in Fig. 1: as the number of modes increased, the ionization signal was enhanced by a factor that asymptotically approached 11! = 4×107… a 7 orders-of-magnitude increase in ionization efficiency due to the field’s stochastic nature. 11! = 4.0x107

107 106 105

100 Mode Field

Instantaneous Intensity

Enhancement in Number of Xe+ formed

108

104 103 102

〈Ι〉

101 Time

100 0

10

20

30

40

50

60

70

80

90

100

110

Number of Modes in Laser Pulse Figure 1: Enhancement of 11-photon ionization of Xe with an amplitude-fluctuating stochastic field; simulated realization of intensity variations shown in inset for illustrative purposes.

2.2. AC-Stark or Autler-Townes Splitting The stochastic nature of a field can also play a profound role in the qualitative nature of the field/atom interaction. As illustrated in Fig. 2, when states |1〉 and |2〉 of a three level system are coupled by a strong field, a weak probe tuned around the |2〉 → |3〉 transition produces a doublet (i.e., the Autler-Townes doublet). The origin of this doublet is best understood through the dressed atom energy level structure of the atom [3]: For ωs < ω21, the eigenfunctions of the dressed atom consist of an upper state (with predominant |2〉 character) and a lower state (with predominant |1〉 character), so that we can refer to the lower energy resonance as a “sequential” excitation resonance (i.e., |1〉 → |2〉 → |3〉) and the upper energy resonance as a direct two-photon resonance (i.e., |1〉 → |3〉). Though the 2-photon process has an “intrinsically” smaller probability of occurrence than the sequential process, the sequential resonance is nonetheless smaller because there are very few atoms in |2〉 when the laser is off resonance.

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The situation is completely different for a field suffering phase fluctuations. In that case, the instantaneous frequency fluctuations of the strong field yield a much larger probability for real transitions into |2〉 even when the strong field is (on-average) off resonance. As a consequence, the asymmetry of the Autler-Townes doublet reverses, and the sequential excitation resonance becomes larger than the direct 2-photon resonance. This stochastic-field effect was first observed and explained by Hogan et al. [4] and latter studied in detail [5]. 2-photon

ωs < ω21

|3〉 ωp

Sequential

ωs < ω21

Sequential

-3

-2

-1

2-photon

0

1

2

3

-3

-2

-1

0

1

Probe Field Detuning

Probe Field Detuning

Monochromatic Strong Field

Stochastic Strong Field

2

3

|2〉

ωs

|1〉

Dressed Atom Energy

E3 2-photon

E1+(N+1)hωf Sequential

E2+Nhωf ωs = ω21 Strong Field Frequency Strong Field Frequency

Figure 2: A 3-level system excited by a strong field & weak probe, and the resulting Autler-Townes doublet. The doublet asymmetry reverses when the strong field suffers white frequency fluctuations.

3. Stochastic-Field Effects and an Atom’s Dynamic Response While the mean response of an atom to a stochastic field is important in determining a spectroscopic signal’s amplitude and linewidth, it is necessary to remember for precision spectroscopy that stochastic fields will induce stochastic atomic responses, and that these affect signal-to-noise ratios. The importance of the atom’s dynamic response to a stochastic field was brought home forcefully to the frequency standards community in the 1990s, during attempts to replace the rf-discharge lamp in Rb vapor-cell clocks with diode lasers [6]. In one experiment [7], the signal amplitude and noise for a commercial lamp-pumped clock were measured. Then, the lamp was removed, replaced

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with a diode laser (Δνlaser ~ 50 MHz), and the signal amplitude and noise remeasured. As anticipated, laser pumping was much more efficient than lamp pumping for clock signal generation, with the clock signal increasing by more than an order of magnitude. (The linewidth of the hyperfine resonance was essentially unchanged.) Surprisingly, however, the noise for the laser-pumped clock signal was more than two orders of magnitude larger, so that the clock’s signal-to-noise ratio was actually poorer with the laser. The origin of the increased noise was eventually traced to the influence of laser phase noise on the atomic vapor’s absorption cross section [8]. Anytime two quantum states are coupled by a field, a lower-state/upper-state superposition is created, and one can show that fluctuations in laser phase yield fluctuations in the superposition state’s expansion coefficients. The expansion coefficients, however, define the atom’s absorption cross section. If the coefficients vary randomly, so too does the absorption cross section, and this in turn produces stochastic fluctuations in the transmitted light intensity. Consequently, the absorption process yields laser phase-noise (PM) to transmitted intensity-noise (AM) conversion. This process is intrinsically atomic: it cannot be avoided; it can only be mitigated. Researchers have had success mitigating PM-to-AM conversion either by reducing the magnitude of the laser phase fluctuations (i.e., reducing the laser linewidth) [9], or by making the atoms less sensitive to phase fluctuations through pressure broadening [10]. 4. Intuitive Models of the Stochastic-Field/Atom Interaction 4.1. Rabi Resonances All of the isolated examples of the stochastic field’s influence on radiative interactions discussed above are exactly that: isolated examples. What is sorely lacking in the area of semiclassical stochastic-field/atom interactions is a cohesive, intuitive picture that experimentalists can employ to guide their thinking, identify potential problems in particular experiments, and illuminate areas of research that might broaden our general understanding of radiative interactions. As an example of the type of experiment that may eventually lead to that intuitive picture, investigations of Rabi resonances appear promising. Rabi resonances are generally described as enhancements in the amplitude of population oscillations when an atom interacts with a time-varying electromagnetic perturbation, and when the perturbation’s period matches the Rabi period, 2π/Ω [11]. To illustrate how these may lead to a better understanding of the stochastic-field/atom interaction, Camparo et al. [12]

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employed a diode laser (Fig. 3) to optically pump a vapor of Rb87 atoms in a TE011 microwave cavity (contained with 10 torr of N2). Optical pumping created a population imbalance between the two ground-state hyperfine levels, so that when microwaves (resonant with the 0-0 hyperfine transition) were applied to the sample, atoms returned to the absorbing state with a consequent decrease in the transmitted light intensity, Itrans. Thus, the atoms’ response to a fluctuating microwave field could be studied by examining Itrans with a spectrum analyzer.

Figure 3: Experiment examining the effect of white frequency fluctuations on a 2-level quantum system: the Rb87 0-0 transition was a surrogate for all two-level systems. Though the frequency fluctuations were white, the atom’s response exhibited a “bright line” at Fourier frequencies equal to the Rabi nutational frequency. (VCXO = Voltage-Controlled Crystal-Oscillator).

In the experiment, the Rb87 0-0 hyperfine resonance was used as an illustrative 2-level atom interacting with a stochastic field. Consequently, in

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addition to the dc voltage, Vc, which controlled the microwave detuning from resonance, Δ, the detuning was randomly modulated by adding a “white” noise voltage to Vc. The atoms’ response to the stochastic field was then studied via a spectrum analyzer. Interestingly, the atoms exhibited enhanced “noise” at the Rabi nutational frequency: [ Ω2+Δ2]1/2 as shown in the inset of Fig. 3. These results are consistent with a linearized theory of the stochastic-field/atom interaction, which shows that the atom responds to the field like a damped, driven, harmonic oscillator with the Rabi frequency playing the role of the oscillator’s resonant frequency [13,14]. 4.2. Bloch-Vector Trajectory Entropy In addition to novel experimental investigations of the semiclassical stochasticfield/atom interaction, new theoretical tools will also be necessary if we are to develop an intuitive understanding of the problem. In this vein, Fig. 4 shows theoretical results examining an atom’s Bloch-vector trajectory when interacting with a field whose frequency suffers a 2-frequency quasi-periodic variation. In particular, the atom’s response was studied in Avan and Cohen-Tannoudji’s “instantaneous” frame of Bloch-vector trajectories [15]. This reference frame is essentially the standard rotating frame of resonance problems, but with an additional rotation that keeps the frame’s x-axis aligned with the effective field. In the instantaneous frame, trivial aspects of the field’s random variations are removed from consideration, and what is left is the atom’s intrinsic response to the stochastic field. The interesting feature of these trajectories is that in general they take on essentially two very specific geometrical patterns: a weak-field pattern and a strong-field pattern. Insight into these trajectories can be gained by calculating a Bloch-vector trajectory entropy, SB: SB = Σpiln(pi), where pi is the probability of the Bloch-vector trajectory falling within the ith infinitesimal volume element of (instantaneous-frame) Bloch space [16]. The results of this Bloch-vector trajectory entropy analysis are also shown in Fig. 4, where SB and the amplitude of the atom’s population oscillations are plotted as a function of Ω. Note that in the transition from weak to strong stochastic fields SB reaches a maximum. Thus, there is what amounts to an “entropy barrier” separating the behavior of Bloch-vector trajectories in weak fields from their behavior in strong fields. (Does this imply that by somehow “clamping” the Bloch vector trajectory’s entropy, an atom’s stochastic response to a field can be limited?) Moreover, increases in entropy correspond to increases in the amplitude of population variations, so that these amplitude enhancements must come at the B

B

B

B

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expense of what can only be described as randomizations of the instantaneous frame’s Bloch-vector trajectory. Therefore, at least with regard to phasefluctuating/amplitude-stable fields, we may have arrived at a fairly general and relatively simple picture of the stochastic-field/atom interaction, a picture in which the Rabi nutational frequency plays a primary role [13,14]. Strong Field

Weak Field

Figure 4: Left - Instantaneous frame Bloch-vector trajectories for an atom interacting with a field suffering 2-frequency quasi-periodic phase variations; Right - Bloch-vector trajectory entropy showing that the transition from weak field to strong field is accompanied by an entropy increase.

5. Summary & Conclusions In the preceding sections, I have tried to show that the stochastic-field/atom interaction can play an important role in precision spectroscopy, and that stochastic field effects cannot always be adequately captured by a convolution picture of the problem. This is not to say that just because a field is nonmonochromatic we must discard our well-worn quantum texts: many lasers come exceeding close to the monochromatic ideal, and not all radiative problems will be sensitive to the field’s stochastic nature. Nevertheless, as we push to unprecedented levels of spectroscopic precision, it may be important to keep in mind that the stochastic nature of a field can change both the quantitative and qualitative aspects of radiative interactions.

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Moreover, there is a real need for general intuitive models of the stochastic-field/atom interaction. Experimentalists need to know when and where a “cavalier” treatment of the problem is fair, and if not fair then what effects are anticipated; and they need answers to these questions without launching their own research effort into the stochastic-field/atom interaction problem. At present, it appears that for phase-fluctuating/amplitude-stable fields the Rabi frequency provides a key parameter: the atom responds to the field as if it were a damped driven harmonic oscillator, with its resonant frequency equal to Ω, and Bloch-vector trajectories in the instantaneous frame have easily categorized patterns quantified by the trajectory’s entropy. What we don’t intuitively understand well is the atom’s response to an amplitudefluctuating field. In particular, how do we define a Rabi frequency for such a field that will allow us to categorize the atom’s dynamics as non-adiabatic (i.e., weak field) or adiabatic (i.e., strong field)? References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12 13. 14. 15. 16.

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