Nitric Oxide Beam Intensity Oscillations Induced by the Combined

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Nitric Oxide Beam Intensity Oscillations Induced by the Combined Action of a Static and a Radio Frequency Electric Field† K. Gasmi, A. G. Gonza´lvez, and A. Gonza´lez Urenã* Unidad de La´seres y Haces Moleculares, Instituto Pluridisciplinar, UniVersidad Complutense de Madrid, Madrid 28040, Spain ReceiVed: September 30, 2009; ReVised Manuscript ReceiVed: December 20, 2009

This paper details an experimental and theoretical investigation in which a simplified version of the molecular beam electric resonance technique is employed that requires the use of a C-field only. In the experiment the forward intensity of a NO beam is measured as a function of the frequency of the oscillating electric field over the 900-1460 kHz range. Specifically, the interaction of the NO beam with a radio frequency (rf) field of 1.12 kV/m amplitude and -610 kV/m 2 of gradient at the horizontal plane during 72 µs produces a series of oscillations in the transmitted beam intensity. The theoretical analysis shows how the interaction between a beam of NO molecules and both a static and oscillating rf field produces interferences in the forward beam intensity and how the observed interferences are due to superposition of molecular internal states. Furthermore, the interference model reproduces satisfactorily the observed beam intensity oscillations. The present technique could be useful for the development of new schemes to achieve coherent control of molecular processes using radiowaves. Introduction As is well-known, one of the most important techniques employed in the investigation of the atoms and molecules interaction with external magnetic and electric fields is the molecular beam magnetic or electric resonance spectroscopy.1-6 In the molecular beam electric resonance (MBER) method4 rotational states of polar molecules are prepared or analyzed using the interaction between their electric dipole moment and the nonhomogeneous field.5,6 Initially, the molecule is state prepared in the A-field. Subsequently, it passes through the resonant unit, the so-called (C-field) and, finally, the state analyzer (B-field) is operated in such a way that only molecules in the prepared state reach the detector. Inside the C-field, located between the A and B fields, the polar molecule interacts with two mutually perpendicular fields, a static dc and an oscillating rf field. When the oscillating field is tuned to a frequency resonant with the transition between the Stark substates, a fraction of the molecules are removed from the initially prepared state and the detector signal diminishes. The present work deals with an experimental and theoretical investigation in which a simplified version of the MBER technique is employed, requiring the use of a C-field only. The interaction of a beam of polar molecules with oscillating rf electric fields is a subject currently investigated by our group using different experimental methodologies.7-12 Thus depletion spectra of molecular beams were reported when NO molecules interact with static and resonant rf fields.7-9 Later on, beam splitting of (0.2° toward both positive and negative directions perpendicular to the beam propagation axis of a supersonic NO beam seeded in He was reported10 as a result of its resonant interaction with an rf electric field. More recently,12-14 the observed beam depletion and splitting were rationalized, from a theoretical point of view, using a quantum interference model †

Part of the “Benoît Soep Festschrift”. * Corresponding author. E-mail: [email protected].

that seems to capture the main physics behind all reported observations. This work describes an experiment in which the forward intensity of a NO beam is measured as a function of the frequency of the oscillating electric field interacting with the molecular dipole moment. We anticipate that the main result of the work here described is the observation of clear oscillations in the forward beam intensity when the rf frequency is swept from 900 to 1460 kHz. A theoretical analysis of this observation is also presented, which demonstrates that the observed beam intensity oscillations can be explained by the onset of interferences due to coherent superpositions of internal quantum states of the polar molecule. The applied molecular interference model not only provides a satisfactory description of the experimental results but also confirms that the coherent superposition of states, created by the dc and rf fields, is the main physics required to explain the observed interferences. Although atomic beam interferometry is well developed,15 there are not many examples of molecular beam interferences in the literature, Since the first experimental demonstration of molecule interference by Estermann and Stern16 in 1930, which showed the diffraction of H2 on a LiF crystal surface, it was only during the past decade of the 20th century when additional experiments with diatomic molecules reported advances in molecular interferometry. Thus, different interferometers of the Ramsey-Borde´,17,18 Mach-Zehnder,19 and Talbot-Lau20 types were employed for experiments with I2, K2, Na2, and Li2, respectively. On the other hand, diffraction at nanofabricated gratings not only proved the existence of the weakly bound helium dimer21 but also was used to measure its binding energy.22 Later on, diffraction experiments with these nanofabricated objects were carried out to show the wave-particle duality of molecules as large as, for example, C7023 or C70.24 The de Broglie interferometry using highly polarizable complex molecules suffers from dispersive effects of the interaction between the diffracted molecule and the grating wall. However, a new interferometer was recently developed that

10.1021/jp909398w  2010 American Chemical Society Published on Web 01/08/2010

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solved this problem making possible experiments25,26 with molecules or large as C60F36 and C60F48. As mentioned in previous work from our group,12-14 a model based on molecular interferences was developed to account for the onset of molecular interferences in the transmitted beam intensity. Essentially, the basic interaction considered in our previous work was that of a polar molecule interacting with both a static dc field and an oscillating rf field. Whereas the homogeneous field produces a Stark split, e.g., for a given J into (J, M1) and (J, M2), the rf field produces a coherent superposition of these two states. Because the rf field originates a distinct interaction potential on the different “arms”, i.e., on the distinct substates split by the Stark field, a phase shift is introduced between them. As a result, the interference is frozen and its pattern is observed by scanning the phase difference accumulated when the frequency of the rf field is changed. The present investigation reveals how the MBER technique4 can be converted in a molecular quantum interferometer, which could be used to manipulate polar molecules. It may open the way to investigate quantum phenomena with polar molecules as, for example, quantum reactive scattering or coherent control of molecular processes using radio-waves. One of the key features of quantum physics is the presence of interferences whose manipulation constitutes the basic objective of the modern coherent quantum control field27,28 and where the use of the quantum attributes of light and matter provides an ultimate control over the dynamics of atomic and molecular systems. Here we demonstrate how the interaction between an oscillating rf field with a beam of polar molecules, NO in the present study, produces interferences in the forward beam intensity and therefore how a rf field is employed to control the beam intensity. At the present time no attempt was carried out to investigate how the beam velocity could be affected by the molecule-rf field interaction. Nowadays one of the most successful methods to control the motion of cold molecules29,30 employs the deceleration of molecular beams using external electric,31,32 magnetic,33,34 or optical fields.35 In addition a time varying electric field36 has been used to manipulate atoms released from a magneto-optical trap and, more recently, the possibility of trapping polar molecules in the standing-wave electromagnetic field of a microwave resonant cavity was proposed.37 In this context, since the present technique employs low Stark fields (see further below) it is not suitable for decelerating beam of polar molecules As for the dynamical and stereodynamical applications mentioned above, it should be pointed out that because the present methodology produces a coherent superposition of molecular states, coherent control of chemical reactions could be feasible. Despite the fact that reaction dynamics using oriented and aligned molecules is a well developed field,38,39 the originality of the present approach would rely on the use of radio waves to control the state population of the reagent polar molecule. The paper is organized as follows. Section 2 describes the main features of the experimental setup with emphasis on those conditions relevant to the experimental results. The theoretical basis necessary to interpret and discuss the data is presented in section 3. It comprises some basic formulas for describing the two-state model interaction relevant to the type of experiments under consideration, the main type of interferogram, namely the total beam signal as a function of the frequency of the rf field as well as the calculation of individual state trajectories inside the applied rf field. Section 4 is dedicated to present and

Gasmi et al.

Figure 1. Simplified sketch of the experimental layout. A supersonic beam of NO of 1100 m s -1 peak velocity travels inside a static and an oscillating rf field. With the laser ionization coupled to the time-offlight spectrometer, the transmitted beam intensity is measured as a function of rf field frequency. The nozzle beam diameter is 0.05 cm. The nozzle stagnation pressure was 3 bar. All dimensions in centimeters. The nominal full beam divergence is 2 × 10-3 rad.

discuss the results of the work: (i) typical model calculations concerning the individual trajectories of each quantum state coupled by the rf field together with the (molecular) wave function phase shift as a function of the interaction time, information necessary to understand the physics behind the observed beam intensity oscillations; (ii) measured frequency interferogram, for a supersonic NO beam and its comparison with the theory developed by our group in previous work. We anticipate that our interference model provides a satisfactory description of the experimental results. Finally, the paper ends with a summary pointing out the most relevant conclusions. 2. Experimental Method The main features of the experimental setup employed in the present investigation have been described elsewhere,7,10,11 and only a brief description is given here. The experiment employs a supersonic NO (20%) beam diluted in He (80%) with a most probable velocity of ca. 1100 m s-1. The beam expansion is produced from a stagnation pressure of 3 bar through a nozzle diameter of 0.5 mm. See Figure 1. The C-field unit11 consists of two parallel Cu-coated glass plates (length 10 cm along the beam, height 6 cm) separated by 0.67 cm. In one plate a 1 mm wide scratch insulates electrically a rectangle of 8 cm by 3 cm; this rectangle and the rest of the plate form the electrodes to which the rf is applied. The static voltage is applied to the rectangular electrode and the opposite plate. As the beam of polar molecules travels inside the unit, the molecule interacts with a static dc field directed along the z-axis and an oscillating rf field of amplitude E1 and frequency ω that is directed along the y-axis. The beam travels parallel to the scratch along the x-axis 0.34 cm away from the plate. At this distance the homogeneous static field (z-axis) is perpendicular to the rf field (y-axis). This nonhomogeneous character of the field, i.e., the non-negligible gradient δE1/δZ * 0 leads to a different and oscillatory trajectory for each initial quantum state coupled by the rf field. (See further below.) As the polar molecule travels inside the rf unit, first the applied dc field splits any rotational state J into its 2J + 1 sublevels characterized by the MJ value; right after, an oscillating rf field is turned on, which can produce coherent superpositions between the (J, M) states. At the final ionization region the beam divergence is determined by the last silt width of 0.15 cm located 48 cm from the nozzle. The present experimental conditions give a full nominal beam divergence e3 × 10-3 rad. Essentially, the NO supersonic beam travels inside an electric rf unit and 21 cm further downstream the beam intensity is measured by means of a laser ionization time-of-flight mass spectrometer.10,11 The

Nitric Oxide Beam Intensity Oscillations

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experiment utilizes a nonquantum-state-selective detection scheme, i.e., a nonresonant two-photon ionization of the NO molecule was achieved by using the fourth harmonic generation (λ ) 266 nm, 7 ns pulse) of our ND:YAG laser. At this excitation energy of ca. 75 187 cm-1 while the (NO)2 + hν f (NO)2+ channel is open, the dissociative ionization channel, i.e., (NO)2 + hν f NO+ + NO is energetically closed.40 The experimental procedure was the following: First a static dc field of 11.92 kV/m was set into the two plates of the electric unit to produce the Stark splitting and to break the 2J + 1 degeneracy. Second, a rf field was established in between the rf plates, with field strength of 1.12 kV/m. The gradient of the rf field at the center of the beam was determined41 to be -0.610 kV/m2. Finally, after optimizing the delay time between the pulsed valve of the gas expansion and the ionization laser pulse, the experimental runs consisted of measuring the transmitted beam intensity as a function of the frequency of the oscillating field for a given static electric field value. For the present experiment the rf field frequency was varied over the 900-1460 kHz range and the magnetic earth’s field was unshielded (see comment on this point in the discussion section). Further details on the experimental method can be found in refs 10 and 11.

Hamiltonian. The nondegenerate two-level system is characterized by the time - dependent wave function Ψ(t) given by

Ψ(t) ) a(t)e-iωatφa + b(t)e-iwbtφb

(4)

where a(t) and b(t) are the coefficients of the lower and higher states φa and φb, respectively. Their expressions using the rotating wave approximation (RWA) are given in Appendix A. Here the energies of the two states are Ea ) pωa and Eb ) pωb so that the energy difference ω0 ) ωb - ωa is the resonant angular frequency, i.e., ω0 ) 2πν0. When the oscillating ε field is applied, the time evolution of the molecular state (and so that of the electric dipole moment) is governed by the generalized Rabi frequency Ω′, which is given by42-45

Ω′ ) √Λ2+ΩR2

(5)

where Λ ) ω - ω0 and ω is the applied rf field frequency and ΩR is given by

pΩR ) µab · E1

3. Theory Model Interaction. For the understanding of the moleculefield interaction here employed we shall recall that when a molecule with a permanent electric dipole moment interacts with a homogeneous Stark field, with E as field strength, the energy of the molecule is perturbed and given by42-45

W ) W0 + WStark

(1)

in which W0 is the energy of the unperturbed system (i.e., when j EE. Here µ j E is the mean component of E ) 0) and WStark ) - µ the electric moment in the field direction. If the molecule is in a Π, ∆, etc., a linear Stark effect arises given by42-45

WStark

ΩM ) -µE J(J + 1)

(2)

Here µ is the permanent electric dipole moment which in the case of the NO molecule equals µ ) 0.158 D,46 J is the total angular momentum, Ω is the magnitude of the projection of J on the internuclear axis, and M is the projection of J on the space fixed axis, i.e., the direction of the static E field. (See comment in the Discussion about the validity of eq 2) As for the interaction of the polar molecule with the oscillating field, we follow previous work from this group13,14 where the well-known two-leVel system interaction has been applied. The interaction of such a model with an electromagnetic field ε ) E1 cos ωt under an electric-dipole approximation is a well-known case whose solution was given by Rabi.42-45 In a semiclassical picture the Hamiltonian can be written as

where µab is the dipole moment of the a f b transition. In addition, under the action of the incident field a dipole moment, P(t), is induced between the two states. This induced dipole moment is given by the expectation value of the dipole moment operator, i.e.

P(t) ) e〈Ψ(t)|r|Ψ(t)〉

(3) where µab is the transition dipole moment, ω the angular frequency of the oscillating field and H0 is the unperturbed

(7)

Thus, replacing (4) into (7) and after some algebra, one obtains

P(t) ) µab{a · be-iw0t + b · aeiw0t}

(8)

Now the interaction energy ∆W(t) between the rf field and the induced dipole moment is given by

∆W(t) ) -P(t)E1 cos ωt ) -P(t)E1 ·

1 iωt {e + e-iωt} 2 (9)

Substituting (8) in (9), one finds for ∆W(t)

1 ∆W(t) ) µabE1{a · b[1 + e-2iωt] + ab · [1 + e2iωt]} 2 (10) In ref 13 it was shown that for a system in which a(0) ) 1 and b(0) ) 0, ∆W0(t) is given by

∆W0(t) ) -µabE1

E1 H ) H0 - µabE1 cos ωt ) H0 - µab (eiωt + e-iωt) 2

(6)

(

)

sin ΩRt · sin 2ω0t 2

(11)

On the contrary, if b(0) ) 1 and a(0) ) 0, one then gets

∆Wb(t) ) +µabE1

(

)

sin ΩRt · sin 2ω0t 2

(12)

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Individual Path Trajectories. In this section the individual path trajectories for the a and b states are evaluated. The given example is for the z-axis; however, a similar procedure can be employed to deduce that of y-axis. The initial step is to evaluate the imparted momentum PZ under the action of the force FZ during the time period from 0 to t. PZ is given by

Pz ) m∆Vz )

∫0t Fz dt

(13)

where m is the particle mass, ∆Vz the velocity increment from 0 to t, and Fz is given by Fz ) -δW/δz; the distance ∆z is then obtained by

∆z )

∫0 ∆Vz dt

( )

p δΩR sin ω0t{sin ΩRt + (ΩRt) cos ΩRt} 2 δz

(15)

Replacing this expression in the integral, one gets for the displacement in the z-axis ∆za the following expression

A B ∆za ) {G (t) - G-(t)} + {H (t) + H-(t)} 2m + 2m + (16) where

G((t) )

{

cos(ΩR ( 2ω0)t (ΩR ( 2ω0)

2

+

t sin(ΩR ( 2ω0)t (ΩR ( 2ω0) 1 (ΩR ( 2ω0)2

{

}

(17)

2t H((t) ) sin(ΩR ( 2ω0)t + (ΩR ( 2ω0)2

[

]

( )

and

B)

)

Λ2 2 2Λ q cos(Λt + φab) + pq sin(Λt + 2 Ω′ Ω′ ΩR2 2 q cos(Λt - φab) φab) + Ω′2

As explained in a previous work from our group,12 the perturbation associated with dipole-molecule interaction with the rf field is equal to or less than a few megahertz, i.e., equal to or less than 10-9 eV. This energy is far less than the molecule’s kinetic energy, in our case on the order of 0.1 eV. Thus, the only action of the potential is restricted to an action on the internal component part of the wave function. In this picture, also supported by the experimental fact that when the rf field is off no interference effect is manifested, we calculate the phase shift using only the interaction potential between the induced dipole and the rf field. Therefore, the phase shift at an interaction time is given by13

∆φab )

}

( )

δΩR p ΩR 2 δz

∆Wab t p

(21)

where ∆Wab ) ∆Wb - ∆Wa. The quantities ∆Wb and ∆Wa represent the interaction energy between the rf field and the induced dipole when the system initial state population is in a or b, respectively. Using the same methodology employed in ref 13 but not restricted to resonant conditions gives the phase shift as13,14

[

∆φab ) t ·

ΩR2 sin Ω′t · sin 2ωt Ω′ 4ΩR2 · Λ

· sin2

]

Ω′t · cos2 ωt 2

(22)

4. Results and Discussion

(18)

and A and B stand for

p δΩR 2 δz

(

u2 ) p2 -

Ω′

2 (ΩR ( 2ω0)3

(20)

where C is a constant that depends on the particle density and other experimental factors like the ionization and detection efficiency14 and 〈u2〉 stands for the velocity average value of u2 given by

2

2 t2 cos(ΩR + 2ω0)t 3 (ΩR ( 2ω0) (ΩR ( 2ω0)

A)

S ≈ C(2 + 2〈u2〉)

(14)

One can differentiate eq 11 or 12 to obtain the respective force Fza or Fzb and, subsequently, integrate it to calculate the a or b displacement depending upon which trajectory a or b is selected. In general, at each time Fza ) -Fzb and therefore ∆za ) -∆zb. After some algebra one obtains for Faz the following equation

Fza )

angle distribution, etc. It can be shown14 that under our experimental conditions the beam signal can be expressed as

(19)

Phase Shift and Beam Signal Intensity. If we consider a single molecule interacting with the rf field, the measured signal results from the average of a single molecule squared wave function over all statistically distributed parameter like velocity,

As mentioned in the Experimental Method section, because the oscillatory character of the rf field and since the E1 amplitude is z-dependent (i.e., there is a nonzero field gradient), the time dependent force along the z-direction, perpendicular to the beam propagation axis, should originate oscillatory trajectories for each quantum states split by the dc field. State-selected trajectories calculated using eq 14 of the text are shown in Figure 2. The red or blue line corresponds to the case when the initial state population is a(0) ) 1, b(0) ) 0 or a(0) ) 0, b(0) ) 1, respectively. The simulation was done for a Rabi frequency νR ) 6 kHz a rf resonant frequency of ν0) 30 kHz and a rf gradient given by δνR/δz ) -2 kHz/mm, which is a typical gradient employed in our experiments. Notice the oscillatory behavior of each trajectory where the displacements are identical except for the sign and how both paths merge after


Figure 2. Individual trajectories calculated using eq 14 of the text. The red or blue line corresponds to the case when the initial population state is a(0) ) 1, b(0) ) 0 or a(0) ) 0, b(0) ) 1, respectively. The simulation was done for a Rabi frequency νR ) 6 kHz, a rf resonant frequency of ν0) 30 kHz, and a rf gradient given by δνR/δz ) -2 kHz/mm, which is a typical gradient employed in our experiments. Notice the oscillatory behavior of each trajectory where the displacements are identical except for the sign and how both paths reach the zero value after a given period. The horizontal dot lines represent the displacement corresponding to the de Broglie wavelength. See text for comments.

Figure 3. Top panel: individual trajectories calculated for two quantum states coupled by the rf field. Bottom panel: molecular wave function phase shift calculated using eq 22 of the text as a function of the molecule-filed interaction time. Both calculations were made for a Rabi frequency of 6 kHz and a resonant frequency of 30 kHz and a filed gradient of 2 kHz mm-1. See text for details.

a given period. The horizontal dot lines represent the displacement corresponding to the de Broglie wavelength estimated to be 12 pm for the employed experimental conditions. Notice how the trajectories are symmetric and due to their oscillatory behavior they periodically approach within the de Broglie wavelength. To illustrate the occurrence of a non-negligible phase shift that leads to molecular interferences, a model calculation is also depicted in Figure 3. Here the phase shift (bottom panel) calculated by eq 22 is depicted as a function of the molecule-field interaction time. The calculation was made using the same experimental parameters as those employed in the trajectory calculations shown in the top panel. The latter is part of the trajectory results displayed in Figure 2 that has been reproduced here for a better comparison. The oscillatory behavior of the phase shift value as a function of the interaction time is clearly manifested. As mentioned further below, it is interesting to notice a non-negligible phase shift especially when both trajectories either merge or approach each other within a shorter distance than the de Broglie wavelength. The latter features not only will originate constructive and nondestructive interferences

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Figure 4. Same type of presentation as in Figure 2 except for the rf field. The same conditions are employed for the individual trajectories. No oscillations are seen due to the absence of the rf field. However, the presence of the same z-gradient induces a typical Stern-Gerlach splitting, which goes beyond the de Broglie wavelength value for interaction times longer than 0.03 ms. See text for comments.

in the probability density F ) |Ψ2| and, consequently, in the measurement of the beam intensity but also will demonstrate the molecular interferences are due to a superposition not of external but of internal states. The oscillatory character of the rf field plays a crucial role in the onset of the observed molecular interferences. Figure 4 depicts the individual trajectories for the same two Stark sublevels and z-gradient used in the calculations shown in Figure 2 but now under the action of a nonoscillating and a nonhomogeneous electric field. In contrast to the results shown in Figure 2, no oscillations are now seen. Indeed, the presence of the same z-gradient induces a typical Stern-Gerlach beam splitting, which for interaction times longer than 0.03 ms is greater than the de Broglie wavelength. The underlying Stern-Gerlach splitting opens the question about the expected role of such splitting in the observed interferences. Intuitively, one could attribute some variation of the forward beam intensity to deflection of molecules out of the beam. However, this contribution due to Stern-Gerlach deflection is negligible as already explained in refs 7 and 12 and it is discussed in the following. Since in our experiments the resonant rf field has a gradient of δE/δz ) -610 kV/m2, one could think that Stern-Gerlach transverse beam deflection is relevant for the present observation. The calculated beam deflection by the rf gradient during the ca. 72 µs the NO molecule spends inside the rf field taking into consideration that its dipole moment is µ ) 0.158 D46 is about 0.4 × 10-6 cm, which is 6 orders of magnitude smaller than the last collimator diameter equal to 0.15 cm. Therefore, the present observations cannot be attributed to beam deflection from the rf field gradient. In other words, the beam deflection due to the z-gradient of the nonhomogeneous rf field, although present, is not relevant for the observed beam intensity oscillations. Solid blue circles in Figure 5 show the transmitted beam signal measured as a function of the rf field frequency over the 900-1460 kHz range. Fifty laser shots were averaged for each frequency. The displayed spectrum termed frequency interferogram is the result of averaging 30 runs. The gray line is just drawn through the points to guide the eye. The red line shown in the middle of the figure is the frequency spectrum measured at the same experimental conditions but with the dc Stark field off.

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Figure 5. Solid blue circles: transmitted NO beam signal as a function of the rf field frequency. The solid dark gray line goes though the points to guide the eye. The middle red line is the transmitted beam spectrum taken at the same experimental conditions but with the dc field off. Notice how the latter shows no oscillations and runs through the middle part of the on spectrum.

Looking at Figure 5 the transmitted beam signal, for example, at 1360 kHz, increases from 0.8 to 1 when the dc field is turned on (compare the red line with the blue points). This is a signal enhancement of 25%. On the contrary, at 1074 kHz the signal decreases from 0.82 down to 0.64, which is a signal depletion of 22%. It is clear that while at some frequencies a beam signal enhancement is observed, at other frequencies a signal depletion is seen. This shows the constructive and destructive interferences of the observed interferogram, and it is even clearer in Figure 7 (see further below). An oscillatory structure is clearly manifested in the beam transmission spectrum depicted by the solid blue points, which, however, is absent in the middle (red line) spectrum, i.e., the one measured when the dc field is off that seems to show a very small oscillation, which, nevertheless, is below the experimental error estimated to be less than 5%. The structure of the experimental spectrum taken when both dc and rf fields are on is a bit complicated. However, it clearly shows a main dip centered at 1208 kHz and oscillations more noticeable over the side bands toward the red and blue shift regions of the central main dip. For example, over the 1390-1490 kHz range the oscillations show a frequency spacing of 17 kHz and a visibility of

V ) (Smax-Smin)/(Smax+Smin) ) 17% where Smax and Smin stand for the maximum and minimum signal, respectively. Switching off the dc field eliminates the Stark splitting, and therefore the remaining rf field has no possibility to induce the coherent superposition of internal states responsible for the observed quantum interferences Due to the lack of information about the population for every selected rotational state, the comparison between experimental and simulated transmission spectra in absolute values is not possible. Simulating the transmitted spectra at distinct rotational temperatures, we observed that the only change was the total signal intensity of the transmitted beam but not the detailed spectral shape. In this view, the comparison between experimental and simulated data was aimed to investigate whether or not the present theory is capable of reproducing the main features of the observed transmission spectra.

Gasmi et al.

Figure 6. Simulated NO transmission spectrum calculated using eqs 20 and 22 of the text. The parameters employed were peak velocity ν0 ) 1200 m s-1, rf resonant frequency ν0 ) 1208 kHz, and Rabi frequency νR ) 27 kHz. See text for comments.

The blue line of Figure 6 displays the calculated transmission beam signal using eqs 20 and 22 of the text and the experimental parameters described earlier except for the rf field strength. In this calculation the velocity distribution of the molecular beam was also taken into account. Notice how the calculated transmission beam intensity is able to reproduce the main appearance of the frequency interferogram, i.e., the main dip, relative intensity of the bands, and especially, the frequency spacing of the oscillations as discussed further below. The simulated frequency interferogram shows a central symmetry that is not so evident in the experimental spectrum. We attribute this small difference to the equal state population used in the theoretical calculation. In other words the calculated spectrum employed an equal initial state population, i.e., a(0)2 ) b(0)2, which, in principle, has not been proved experimentally. Preliminary simulations using a distinct state population indicate the lack of the spectral symmetry when the initial state population is not the same. This matter is now a subject of current investigation by our group and will be reported in a future publication. A detailed comparison between the measured and calculated spectrum over the 1390-1460 kHz frequency range is displayed in Figure 7. Blue points are the experimental results from Figure 5, and the red line is the simulated spectrum using the same parameters employed in the Figure 6 simulation. It is clear how the molecular quantum interferences model can reproduce satisfactorily the observed spectrum. Both Figures 6 and 7 simulated frequency interferograms were calculated using a peak velocity of 1200 m s-1, a full-width at half-maximum (fwhm) value of ∆νM ) 80 m · s-1, a resonant frequency of 1208 kHz, and a Rabi frequency of ΩR ) 27 kHz. Except for the Rabi frequency value, these best fit parameter values are very close to the experimental ones. Thus, the experimentally measured peak velocity was 1100 m s-1. The resonant transition for the nearest molecular transition J ) 3/2, Ω ) 1/2, and ∆M ) (1, calculated by eq 2 with the experimental µ value cited above, is 1208 kHz. The significant broadening introduced by the opening pulse of the nozzle valve (∼150 µs) did not allow us to perform a detailed investigation on whether the rf field reduced the width of the beam velocity distribution or not. This possibility will be the subject of a future investigation. At present we have no explanation why the best fit is obtained with the resonant transition calculated with eq 2. It should be


J. Phys. Chem. A, Vol. 114, No. 9, 2010 3235 5. Summary and Conclusions The present work dealt with an experimental and theoretical investigation in which a very simplified version of the MBER technique was employed requiring the use of a C-field only. It described an experiment in which the forward beam intensity of a supersonic NO (peak velocity of 1100 m s-1) was measured as a function of the frequency of the oscillating electric field interacting with the molecular dipole moment.

Figure 7. Solid blue circles: transmitted beam signal as a function of the rf field frequency. The solid red line is the calculated best fit frequency interferogram using eqs 20 and 22 of the text. See text for details. Notice how an oscillatory structure is clearly manifested in the beam transmission spectrum depicted in the bottom panel which, however, is absent in the top spectrum.

recalled that eq 2 is valid only in the high field regime, i.e., when the Stark splitting is higher than the Λ-doubling split for the 2Π1/2 J ) 3/2 state and if one neglects hyperfine interaction. These approximations are too crude to be valid in the present case. Indeed, whereas the employed Stark splitting is ca. 1.2 MHz, the Λ-doubling splitting for the J ) 3/2 state is several orders of magnitude higher. New experiments are planned with CO and 15N16O that have no hyperfine interaction or a different nuclear spin coupling, respectively, to clarify these important points. The best fit Rabi frequency value is significantly smaller than the nominal (inside the rf unit) ΩR ) 119 kHz one.12 The only explanation for such a difference we can think of is the action of decoherence, since the ionization region is situated 21 cm away from the rf unit, as mentioned earlier. After the molecular beam exits the rf unit, the rf field is no longer interacting with the dipole moment and quantum decoherence is taking place. Finally, since NO is paramagnetic, a comment about the magnetic Earth’s field effects on the observed spectra follows. As is well-known, the NO 2Π1/2 state is well represented by Hund’s case (a) for which the Zeeman interaction energy ∆WB in the weak field limit47 is given by

∆WB ) µBB ·

(∆ + 2.002Σ)MΩ J(J + 1)

(23)

Here µB is the Bohr magneton, B is the magnetic field, and ∆ and Σ stand for the projection of the electronic orbital and spin angular momentum on the molecular axis, respectively. M is the projection of J on the B direction axis. Since the |Λ + 2.002Σ| value for 2Π1/2 is 0.001, eq 23 gives ∆WB values lower than 20 Hz for the magnetic Earth’s field value of our laboratory (B < 10-4 T). This splitting is below our spectral resolution and 5 orders of magnitude lower than the Stark splitting used in our experiments. The suggested hypothesis of quantum decoherence opens the way for future investigations aimed at studying decoherence mechanisms of these molecular beams by monitoring the evolution of fringe visibility as a function of (i) the molecule-field interaction time or (ii) the pressure of a scattering gas previously introduced in the molecular beam chamber.48

One of most significant results of the work was the observation of clear oscillations in the forward beam intensity when the rf frequency is swept from 900 to 1460 kHz. A theoretical analysis was also presented that rationalizes the observed oscillations to be due to molecular interferences originated by the superposition of internal quantum states of the polar molecule. Although a full trajectory calculation is now possible for every NO (J, Ω, M) state traveling inside the C-field, the present investigation uses a simpler two-state model interacting with both dc and rf electric fields. In spite of its simplicity, the employed interference model seems to describe the observed oscillations in the forward beam intensity. It was shown how every quantum state coupled by a rf field follows a different but symmetrical trajectory inside the nonhomogeneous rf field. However, due to the oscillatory behavior of the field both trajectories merge periodically, showing a relative phase shift exclusively originated by the interaction energy between the oscillating electric field and the different quantum states of the polar molecule. The phase shift depends on several parameters as, for example, the interaction time, the rf field strength or frequency. The present study exploits the latter dependence which undoubtedly results of a great experimental simplicity. While the observation of the oscillations in the forward beam intensity is out of question and the possibility they are originated by beam deflection has been shown to be very unlikely, a full and complete interpretation of the data presented here is still needed. Despite the fact the two-level model employed in the data analysis seems to describe satisfactorily the observed oscillation, an important question still remains open about the validity of the model parameters deduced from the best data fit. If from the outcome of new experiments the explanation of molecular interferences is fully confirmed, it should be remarked how the MBER technique could be converted in a molecular quantum interferometer perhaps useful to investigate quantum phenomena using polar molecules as, for example, quantum decoherence,48 quantum reactive scattering, or coherent control of molecular processes using radio waves. Acknowledgment. This work was partially supported by the Ministerio de Educacioń y Ciencia of Spain under Projects CTQ2007-61749/BQU. A.G.G. acknowledges a FPI fellowship from the MICINN of Spain. Financial support from the Universidad Complutense as well as from the Madrid Comunidad Autonoma is gratefully acknowledged. Appendix A The a(t) and b(t) coefficients are given, under the so-called rotating wave approximation (RWA), by the expressions44

3236


{ [ ( )

a(t) ) a(0) cos

Ωt Ωt iΛ + sin 2 Ω 2 ΩR Ωt iΛt/2 i b(0) sin e Ω 2

( )]

( )}

Gasmi et al.

(A1)

and

{ [ ( )

b(t) ) b(0) cos

Ωt Ωt iΛ + + sin 2 Ω 2 ΩR Ωt -iΛt/2 i a(0) sin e Ω 2

( )]

( )}

(A2)

where a(0) and b(0) are the initial t ) 0 values of the coefficients. Note Added after ASAP Publication. This article posted ASAP on January 8, 2010. The name of the first author has been revised. The correct version posted on January 21, 2010. References and Notes (1) Scoles, G. Atomic and Molecular Beam Methods; Oxford University Press: Oxford, U.K., 1992; Vols. 1 and 2. (2) Pauly, H.; Toennies, J. P. Methods of Experimental Physics; Academic Press: New York, 1968; Vol. 7, p 227. (3) Muenter, J. S. In Magnetic and Electric Resonance Spectroscopy in Atomic and Molecular Beam Methods; Scoles, G., Ed.; Oxford University Press: Oxford, U.K., 1992; Vol. 2, Chapter 2. (4) Hughes, H. K. Phys. ReV. 1947, 72, 614. (5) Stolte, S. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 413. (6) Reuss, J. State Selection by Non Optical Methods in Atomic and Molecular Beam Methods; Scoles, G., Ed.; Oxford University Press: Oxford, U.K., 1988; Vol. 1, p 276. (7) Montero, C.; Gonza´lez Urenã, A.; Caceres, J. O.; Morato, M.; Najera, J.; Loesch, H. J. Eur. Phys. J. D 2003, 26, 261. (8) Morato, M.; Gasmi, K.; Montero, M.; Gonza´lez Urenã, A. Chem. Phys. Lett. 2004, 392, 255. (9) Caćeres, J. O.; Montero, C.; Morato, M.; Gonza´lez Urenã, A. Chem. Phys. Lett. 2006, 426, 214. (10) Morato, M.; Caćeres, J. O.; Gonza´lez Urenã, A. Eur. Phys. J. D 2006, 38, 215. (11) Morato, M.; Caćeres, J. O.; Gonza´lvez, A. G.; Gonza´lez Urenã, A. J. Phys. Chem. A 2009, 113, 14291. (12) Gonza´lez Urenã, A.; Caćeres, J. O.; Morato, M. Chem. Phys. 2006, 328, 156. (13) Gonza´lez Urenã, A.; Requena, A.; Bastida, A.; Zuń˜iga, J. Eur. Phys. J. D 2008, 49, 297. (14) Gonza´lez Urenã, A.; Gonza´lvez, A. G.; Requena, A.; Bastida, A.; Zuń˜iga, J. Manuscript submitted for publication. (15) Baudon, J.; Mathevet, R.; Robert, J. J. Phys. B: At. Mol. Opt. Phys. 1999, 32, R173.

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