Optical Ramsey spectroscopy and coherence measurements of the ...

495

Optical Ramsey spectroscopy and coherence measurements of the clock transition in a single trapped Sr ion L. Marmet and A.A. Madej

Abstract: We report on the observation of coherent excitation and spectroscopy performed using Ramsey’s method of separated excitation fields on a single trapped and laser cooled ion of strontium. Single pulse Rabi excitation studies of the ion have shown linewidths approaching the transform limited value. Studies of transition rate versus pulse duration have shown a quadratic initial increase in transition rate with a duration consistent with coherent excitation and a rapid loss of Rabi oscillations, consistent with the concept of a dephasing of the thermally populated trap oscillator states as the excitation evolves. When two-pulse Ramsey excitation is applied to the single ion, fringe widths down to a linewidth of 840 Hz (FWHM) were observed with measured fringe contrasts consistent with a probe laser linewidth of 500 Hz. PACS Nos.: 32.80Qk, 32.30Jc, 06.30Ft Résumé : Nous présentons nos observations d’excitation cohérente d’un seul ion de strontium, piègé et refroidit, et la spectroscopie de l’ion réalisée en utilisant la méthode des champs d’excitation séparés de Ramsey. Des études faites en excitant l’ion avec une seule impulsion Rabi ont montré des largeurs de raies approchant la valeur limitée par la transformée de Fourier de l’impulsion. Une augmentation quadratique du taux de transition a été observée en fonction de la durée de l’impulsion d’excitation, en accord avec les propriétés d’une excitation cohérente. Un amortissement rapide des oscillations Rabi est expliqué par le déphasage des états d’oscillations thermiques de l’ion dans le piège. En utilisant deux impulsions Ramsey pour exciter l’ion, des franges aussi étroite que 840 Hz (pleine largeur à mi-hauteur) sont observées avec un contraste consistant avec une largeur de raie du laser d’excitation de 500 Hz.

1. Introduction Measurements based on high-resolution spectroscopy have provided the means to test physical theories to an extremely high accuracy [1]. The success of a significant portion of these measurements is attributable to frequency standards based on atomic transitions that provide accurate absolute references. To reduce offsets in frequency measurements and improve the accuracy of these standards, Ramsey proposed the method of separated oscillatory fields [2]. The method used two spatially sepa-

Received November 22, 1999. Accepted March 20, 2000. Published on the NRC Research Press Web site on June 29, 2000. L. Marmet1 and A.A. Madej. Frequency and Time Group, Institute for National Measurement Standards, National Research Council of Canada, Ottawa, ON K1A 0R6, Canada. 1

Corresponding author: Rm. 1106, Bldg. M-36, INMS, National Research Council of Canada, Ottawa, ON K1A 0R6, Canada. Telephone: (613) 998–1317; FAX: (613) 952–1394; e-mail: [email protected]

Can. J. Phys. 78: 495–507 (2000)

© 2000 NRC Canada

496

Can. J. Phys. Vol. 78, 2000

rated interaction regions placed in an atomic beam to coherently excite the atoms and produce narrow interference fringes. Ramsey fringes have also been produced in the time domain using separated pulse excitation [3] of a cloud of stored 3 He+ ions. The Ramsey technique has many advantages, among which are an elimination of the first-order Doppler shift and an improved detection contrast. Moreover, the width of the central fringe is independent of the homogeneity of any stray or bias field present between the interaction regions. The technique has been broadly used in atomic- and molecular-beam spectroscopy and provided the best frequency standards as exemplified in today’s cesium-beam atomic clocks.

The Ramsey technique is more difficult to apply at optical frequencies since the shorter wavelength of the probing radiation increases the phase shift sensitivity caused by a small displacement of the atoms. Such sensitivities can destroy the phase coherence necessary to observe interference fringes. This difficulty has been overcome in a number of different methods. Applications of Ramsey fringe spectroscopy to the excitation of two-photon transitions possess the advantage [4] that the phase is position independent when two photons are absorbed from opposite directions. A recent application of such two-photon Ramsey fringe spectroscopy has been the measurement of the 1S–2S transition frequency in atomic hydrogen using a standing wave for the coherent excitation [5]. Ramsey fringe spectroscopy has been also successfully applied in single-photon absorption using multiple beam configurations of three or more excitation regions, which are insensitive to the velocity spread of atoms in the atomic beam [6–8]. These methods have been particularly directed to the examination of the narrow 1S0 −3P1 intercombination transitions in alkaline-earth elements [9]. With the recent development of laser-trapping and cooling of these neutral atoms, it has been possible to perform Ramsey-fringe spectroscopy on cold atom samples using time-separated, pulsed excitation [10].

In our laboratory, we use a spectroscopic technique based on a laser cooled, trapped single ion to eliminate the phase changes due to motion. By sufficiently reducing the kinetic motion of the ion by laser cooling, the ion becomes confined within a very small volume of space of the trapping potential. Under the condition that the linear dimension of motion is smaller than the wavelength of the light probing the transition of interest, one observes a sharp narrow line at the transition frequency, free from the first-order Doppler effect (Lamb–Dicke regime) [11]. Our optical frequency standard is based on the 674 nm electric-quadrupole-allowed, 5s 2S1/2 – 4d 2D5/2 clock transition in the 88 Sr ion [12] whose natural lifetime is 0.4 s. We present in this paper the results of coherence measurements and Ramsey spectra of the Sr ion. Although these trapped ions do not experience significant Doppler effect, the Ramsey technique brings other improvements to such spectroscopic measurements. Two short pulses separated by a time T produce a central fringe having a width only 56% of the linewidth obtained with a single pulse of duration T . If the probing time exceeds the lifetime of the excited state, the contrast of the Ramsey fringes decreases but their widths become narrower than the natural linewidth of the transition. Another potential advantage is that many ions confined and excited coherently provide an ensemble that improves the frequency measurements [13] via corellated states. High-resolution microwave spectra of Yb ions [14] and Hg ions [15] have already been obtained using Ramsey spectroscopy. In the visible, some indication of coherent excitation was also observed in the Hg ion at 282 nm [16] where the spectrum showed a triplet structure explained by Rabi power broadening. In addition, work in a number of laboratories has recently been directed toward the coherent excitation of single-ion optical transitions for applications such as quantum optics, quantum computation, and the preparation of nonclassical states of motion [17]. To our knowledge, no example of Ramsey spectroscopy in the visible wavelength range have been reported for trapped ions. We present our studies demonstrating coherent excitation and the first Ramsey spectra of an optical transition in the Sr ion. ©2000 NRC Canada

Marmet and Madej

497

2. Theory 2.1. Single-pulse excitation of a trapped ion We consider a laser field described by E(t) = E0 (t) cos(ωt) exciting a two level system in a stationary atom. For weak intensities, the transition probability is proportional to the square of the modulus of the Fourier transform of E(t) [18]. If the exciting field becomes large, the excitation is nonlinear and the population of the atom is described instead by the optical Bloch equations. Since the general solutions for the Bloch equations are complex, we limit ourselves to a few specific examples. For a stationary atom excited by a single square pulse of duration τ , the transition probability is given by the Rabi probability [19]: P (ω, τ ) =

1 2 2 sin2 [τ/2] τ 4 0 [τ/2]2

(1)

where 2 = (ω − ω0 )2 + 20 , ω0 is the resonant frequency of the atomic transition and 0 is the Rabi frequency proportional to the strength of the atomic transition multiplied by E0 . The transition probability as a function of frequency gives a line profile with a full width at half maximum (FWHM) of: 0.89 1ν ∼ = τ

(2)

On resonance, ω−ω0 = 0, the transition probability, follows a sinusoidal oscillation with increasing pulse length, starting with a quadratic increase for short pulses satisfying τ 1. This behavior distinguishes coherent excitation from broadband excitation, which would result instead in a linear increase of the probability. Although the single trapped ion is an extremely close approximation to an isolated atomic system, which is at rest, the fact that it is confined with a three-dimensional harmonic potential significantly alters the population dynamics. As pointed out by Blockley et al. [20], for ion motion having low kinetic temperatures, the quantum mechanical behavior of the ion’s centre-of-mass motion enters into importance. Coupling of this motion with the excitation light field will produce a characteristic Rabioscillation frequency in the population depending on the oscillator energy level quantum number. For an ion that is confined in a harmonic potential with oscillation frequency νtrap and a probe transition having a wavelength λ and a linewidth 0 < νtrap , the system satisfies the Lamb–Dicke condition if ε ≡ Er /(hνtrap ) 1, where Er = h2 /2mλ2 is the classical recoil energy of the atom. The absorption/emission spectrum then consists of a well-resolved Doppler-free line at the transition frequency together with transition features located at the characteristic oscillation or “secular” frequencies of the trap. The trap levels are sufficiently well spaced that for absorption on the central Doppler-free carrier, excitation between the ground and excited state does not change the quantum number of the harmonic oscillator level. Following the procedure presented by ref. 20, one can thus write the equations of motion for a particular ground and excited electronic state occupying the harmonic oscillator state m as |g, m > and |e, m > as an isolated pair and solve for the characteristic Rabi frequency of the oscillator state. The on-resonance (ω − ω0 = 0), one-dimensional solution yields: i h (3) 00 = 0 exp[−ε2 /2]Lm (ε2 ) where Lm (ε2 ) is the mth-order Laguerre polynomial normalized so that Lm (0) = 1. To second order in ε the relation for the mth level Rabi frequency is 1 2 ε (4) 00 = 0 1 − m + 2 ©2000 NRC Canada

498


Fig. 1. Calculated excited state population for a two-level atom in a one-dimensional harmonic oscillator well of ε 2 = 0.005 and with free space Rabi frequency of 0 = 20 000 s−1 . Excitation is tuned to resonance on the Doppler-free carrier line. Calculations are shown as a function of pulse duration for an initial thermal population in the harmonic well having mean quantum number of m = 2, 20, and 200 for the well vibrational level (dotted, broken, and continuous lines, respectively).

Thus, it can clearly be seen that the effective Rabi frequency depends on the occupation quantum number for a particular trap oscillator state and the coupling coefficient ε. For an atomic system placed in a unique trap harmonic oscillator level (Fock state) the transition probability would exhibit Rabi oscillations as shown in (1) modified by the frequency shift given in (4). In our case of an atom laser cooled to a low kinetic temperature, the system will not occupy a single-trap oscillator state but have a probability distribution ρm following a thermal distribution of the form [19]: ρm =

(m)m (1 + m)m+1

(5)

where m is the mean oscillator occupation number for a kinetic temperature T . By summing over all contributions of oscillator states weighted by their probability and noting that each state has a different effective Rabi frequency, one arrives at the important finding that for m sufficiently large, a large number of oscillator states contribute differing Rabi frequencies resulting in a “washing out” of the Rabi oscillation over the first period of excitation. Figure 1 shows the calculated transition probability under conditions of 0 = 20 000 s−1 , ε2 = 0.005 and m = 2, 20, and 200. With our current experimental conditions having ion kinetic temperatures of T = 15 ± 5 mK [12] thus setting m at approximately 150 and ε 2 = 0.005, one can expect that one can observe a rapid decoherence of the Rabi oscillation due to such a spread in effective Rabi nutation frequencies. In addition to the issues of multiple effective Rabi frequencies, one must consider the effect of the finite bandwidth of the coherent source. For a probe laser having a Lorentzian profile with a linewidth 1ν equal to 2π1ν = 2γ , the Rabi oscillations are also effectively damped by the phase fluctuations ©2000 NRC Canada

Marmet and Madej

499

Fig. 2. Energy level diagram of the 88 Sr ion showing the resonance transition at 422 nm used for laser cooling and the detection of quantum jumps, the transition at 1092 nm to maintain the cooling cycle, and the electricquadrupole clock transition at 674 nm. The transition to the 2 D5/2 , m0j = +3/2 state will occur only if the ion is initially in the m00j = +1/2 state.

resulting in oscillations that are smaller in amplitude and tend to a steady-state limit for long pulse lengths. Cirac and co-workers [21] have obtained such results by performing calculations for the case of finite excitation bandwidth and excitation on a secular sideband transition of a single ion. For our case of a laser whose bandwidth is 1ν ≈ 500 Hz thus γ ≈ 1600 s−1 , the loss of coherence is more rapid than the radiative decay rate of 0 = 2.7 s−1 of the upper D state yet will not be a significant contribution to the single-pulse excitation-rate dephasing since the various oscillator levels will lose relative phase over a single Rabi cycle, which, for our experiment, was 0 = 104 s−1 .

2.2. Ramsey-fringe excitation spectra A Ramsey spectrum is obtained by adding a second pulse of duration τ , identical to the first pulse but delayed by a time 1T with respect to the beginning of the first pulse. The electronic state superposition produced by the first pulse evolves for some time before the second pulse arrives. After the second pulse, the transition probability is proportional to the cosine of the total cumulated phase after the time 1T . In the low-intensity limit 0 < (ω − ω0 )2 and for a stationary atom, the transition probability is ©2000 NRC Canada

500


Fig. 3. Experimental setup to probe the clock transition of the Sr ion. (a) An acousto-optic modulator and two choppers modulate the beams before they are combined on dichroic mirrors. PMT: photomultiplier tube, AOM: acousto-optic modulator. (b) Relative timing of the laser pulses.

given by [18]

P (ω, τ, 1T ) =

1 2 2 sin2 [(ω − ω0 )τ/2] τ {1 + cos[(ω − ω0 )1T ]} 2 0 [(ω − ω0 )τ/2]2

(6) ©2000 NRC Canada

Marmet and Madej

501

Fig. 4. Spectrum of the of the (m0j , m00j ) = (+3/2, +1/2) Zeeman component of the 4d 2 D5/2 –5s 2 S1/2 transition in 88 Sr + with a magnetic field B = 14 µT. The linewidth of 600 Hz is mainly caused by the probe laser linewidth.

The cosine term gives rise to the Ramsey oscillations modulating the Rabi pedestal. The central line of the Ramsey pattern has a FWHM of 1ν = 0.5/1T , only 56% of the width of the Rabi probability given the same available time to probe the transition. Assuming the probe laser has a Lorentzian profile with a linewidth equal to 2γ , the contrast of the Ramsey fringes will decrease with longer times 1T according to [19] C = exp(−γ 1T )

(7)

The intensity of the Ramsey spectrum is then given by (6) with the cosine term multiplied by C. Given the dependence of the excitation rate as a function of the trap oscillator level, as shown in Sect. 2.1, it can be anticipated that the above relations may be perturbed. However, it can be appreciated that the fringe minima observed in Ramsey-type excitation arise from a phase reversal of the applied excitation. This reversal would be individually effective to all individual occupied oscillator states and thus the sum over all probability-weighted oscillator states will also produce a null in the transition probability. For the fringe “maxima,” the dephasing of the individual Rabi excitations in the occupied oscillator levels will reduce the maximum amplitude. Nevertheless, clear cosine type oscillations should be observable with the fringe contrast ultimately limited due to dephasing from the finite laser bandwidth. 2.3. Observation of transition probability via observation of single-ion quantum jumps The transition probability is obtained in our experiment by measuring the rate at which the ion changes state. As is described below, the state of the ion is periodically measured after each excitation with the probe laser pulses, resulting in either a “bright” fluorescing state indicating that the ion was originally in the ground state, or a “dark” state indicating that the ion is shelved in the long-lived 2D5/2 excited state. If the ion is in the bright state, the excitation probability to the dark state is P as presented in the previous section. However, the 2S1/2 ground state of the Sr ion is a doublet so that only one sublevel m00j of the ground state can be excited to the corresponding sublevel m0j of the excited state. Given a probability Pm of finding the ion in the m00j state, the probability of a bright to dark transition is thus Pbright→dark = Pm P . It is assumed throughout the rest of this paper that Pm = 0.5 (as borne ©2000 NRC Canada

502


Fig. 5. Linewidth of the (m0j , m00j ) = (−1/2, −1/2) Zeeman component as a function of the inverse of the pulse duration. The broken line shows the theoretical width.

out in our spectral line intensity measurements). In the case when the ion is found in the dark state, the probability of de-excitation is also given by P . However, the natural decay rate 0 = 2.7 s−1 adds a small probability 0d 1 to the probability of de-excitation during the time d between two measurements of the state of the ion. The Pdark→bright ≈ P + 0d. Our measurements give the average time spent by the ion in the bright state and in the dark state. The inverse of these values give the bright period rate and the dark period rate, respectively. Theoretically, these rates are simply given by: Rbright = Pbright→dark /d = Pm P /d and Rdark = Pdark→bright /d = P /d + 0. We see that for low count rates, the bright period rate is proportional to P while the dark period rate is never smaller than the natural decay rate 0.

3. Experiment The experimental setup used for trapping, cooling, and probing the ion is described in more detail elsewhere [22]. An oven produces 88 Sr atoms that are ionized and captured by an electrodynamic trap. A single Sr ion is confined within the rf or Paul-type trap, which consists of a ring electrode with an inner radius r0 = 0.7 mm and two endcaps separated by 2z0 = 1.0 mm [22,23]. An electric potential with an amplitude of 530 V peak-to-peak at a frequency of 12.0 MHz is applied to the endcap electrodes to produce a time-averaged trapping potential. Figure 2 shows the energy levels of the Sr ion relevant to this experiment. A laser beam at 422 nm cools the ion on the 5s 2S1/2 − 5p 2P1/2 transition. The radiation at 422 nm is produced by frequency doubling the output of an 844 nm diode laser in a potassium niobate crystal within a triangular resonant cavity. The frequency of the 422 nm output is stabilized using a passive lock to a resonance in Rb [24]. Since a decay to the 4d 2D3/2 state would stop the fluorescence/cooling cycle, a repumping laser, ©2000 NRC Canada

Marmet and Madej

503

Fig. 6. (a) Bright period rate as a function of pulse duration of the probe laser for the (m0j , m00j ) = (+3/2, +1/2) Zeeman component with a probe laser power of 30 µW. A detailed view for short excitation times with a quadratic fit (broken line) for the same transition. (b) Bright rate for longer excitation times. The broken line shows a theoretical fit using the model described in the text.

tuned to the 4d 2D3/2 − 5p 2P1/2 transition at 1092 nm, is used to maintain the cycle. This radiation is generated in a diode-pumped Nd-fiber laser and its polarization is rapidly modulated with an electrooptic modulator (EOM), at a frequency of 15 MHz [25], to reduce the effects of optical pumping into the Zeeman sublevels of the 2D3/2 state. The beams at 422 and 1092 nm are combined on a dichroic mirror and focused into the trap as shown in Fig. 3a. The 422 nm S − P fluorescence of the ion is monitored with a photomultiplier giving a signal that is processed through a pulse shaper, a single channel analyzer, and a linear rate meter converting it to a voltage representing the fluorescence intensity. Typical rates of 6000 photons/s with a background of 300 photons/s are observed. A diode laser generates the probing radiation at 674 nm that is optically narrowed with optical feedback from a folded FP cavity, then further narrowed using a reference ultra-stable Fabry–Perot resonator [23]. Depending on the specific diode laser, linewidths of 250 to 700 Hz are obtained as is inferred from the measured width of a resonance line in a spectrum of the ion. An acousto-optic modulator (AOM), used in a double-pass configuration, shifts the fixed frequency of the stabilized laser to the frequency of the 2S1/2 −2D5/2 transition. The beam is brought to the ion trap with an optical fiber, attenuated with neutral density filters, and then focused to a 2ω = 60 µm diameter spot inside the trap. The fluorescence at 422 nm allows the observation of transitions to the 4d 2 D5/2 state with almost 100% efficiency [26]. When the ion electronic state is shelved into the metastable D level, the ion becomes decoupled from the resonant 422 nm light and ceases to produce the strong fluorescence. A spectrum of the Zeeman components can be obtained from this observed quantum jump transition rate as a function of the probing frequency. Three pairs of Helmholtz coils generate a stable 10–20 µT magnetic field to separate these Zeeman components. To avoid power broadening of the reference S − D transition, the cooling beam and the probe beam alternately illuminate the ion via the use of two light choppers and one AOM. The timing of the laser beams is shown in Fig. 3b. The chopper wheels, synchronized with each other, modulate the cooling and the repumping beam at a frequency of 70 Hz, producing 6 ms pulses for the cooling beam and 7 ms pulses for the repumping beam. Intensity control of the probe 674 nm beam is obtained with a pulse generator modulating the output of the frequency synthesizer used to drive the AOM. The response time of the modulation is 10 µs. One or two pulses are generated, synchronized with the choppers. Since the relative optical phase of the two pulses is important to obtain good Ramsey fringes, the phase variations ©2000 NRC Canada

504


Fig. 7. Ramsey-fringe absorption spectra of (a) the (m0j , m00j ) = (+3/2, +1/2) Zeeman component using a pair of t = 0.06 ms pulses delayed by 1T = 0.15 ms; (b) the (m0j , m00j ) = (+3/2, +1/2) Zeeman component using a pair of τ = 0.05 ms pulses delayed by 1T = 0.15 ms; and (c) the (m0j , m00j ) = (−1/2, −1/2) Zeeman component using a pair of τ = 0.10 ms pulses delayed by 1T = 0.60 ms. The continuous line shows the theoretical fit.

were carefully monitored by observing the beat between the AOM-shifted beam and the unshifted laser beam. A random-phase variation not exceeding 10◦ was observed between pulses separated by 1 ms, this variation being mostly due to the noise induced in the optical fiber carrying the light to the ion trap. ©2000 NRC Canada

Marmet and Madej

505

The pulse generator can be configured to output one square pulse of duration τ for Rabi spectroscopy or two identical pulses of duration τ , with the second pulse delayed by 1T after the beginning of the first pulse, for Ramsey spectroscopy.

4. Results The spectrum of a Zeeman component was first measured to find which magnetic field would isolate it as much as possible from other neighbouring lines. A magnetic field of 14 µT was chosen to separate the central Zeeman components by 158 kHz and giving the spectrum of the (m0j , m00j ) = (+3/2, +1/2) Zeeman component shown in Fig. 4. Since a long probe pulse duration of τ = 3.0 ms was used, the component linewidth of 550 ± 50 Hz FWHM reflects the linewidth of the probing laser. At current experimental conditions, the ion exists in a background pressure of 100 nPa composed of mostly hydrogen where the mean classical collision rate is 0.05 s−1 . Thus, collisional dephasing and quenching effects on the linewidth can be ignored. We verified that the nearest neighbouring lines were not closer than 32 kHz, these lines being sidebands produced by the secular motion of the ion. The proximity of the neighbouring lines imposes a lower limit on the probe-pulse duration that can be used for Ramsey spectra. The probe pulse should not be shorter than τ ∼ 0.05 ms since the Rabi pedestal of the neighbouring lines will then start to overlap with the central line being examined. The bright rate obtained at line center reached 5 s−1 with a power of 180 nW. Strong saturation was observed at larger laser powers, which limited the maximum bright rate to about 9 s−1 at 5 µW and resulted in a broader apparent linewidth. This saturation is discussed in greater detail below. To verify the coherent properties of excitation, we first measured the width of the spectral line obtained for shorter pulse durations. Figure 5 shows the measured linewidth of the (m0j , m00j ) = (−1/2, −1/2) Zeeman component as a function of the inverse of the pulse duration 1/τ . The width follows the theoretical dependence given by (2), but with a minimum value of 380 ± 140 Hz, consistent with the width of the probe laser, which becomes the largest contribution for long pulses. The calculated width, shown as a broken line, slightly underestimates the measured values. This is thought to be caused by some saturation still present at small intensities, which has the effect of increasing the apparent linewidth. Figure 6 shows the measured bright rate as a function of pulse duration. The probe laser excited the (m0j , m00j ) = (+3/2, +1/2) Zeeman component with 30 µW of power and the pulse length τ was varied from 0 to 0.5 ms. For short pulses, a quadratic increase of the bright rate as a function of the pulse duration was seen as shown in Fig. 6a following the expected behavior obtained with coherent excitation. For longer pulses, an increase of the excitation rate was observed up to a saturation point of about 8 s−1 . For pulses longer than 0.3 ms, the signal became erratic, sometimes showing an abnormally small rate. Measurements of the dark rate showed a very small change with pulse duration, being always near the natural decay rate 0 = 2.7 s−1 . This stark contrast in the observed excitation and stimulated emission rates into and out of the D state are believed to arise owing to possible variations in the transition probability for the ion starting from the S state or D state. Since bright-period durations have laser cooling applied every chopper cycle while the dark periods corresponding to residence times in the D state are necessarily decoupled from laser cooling, it is considered that the ion may be suffering from a rapid heating in the D state that quickly brings the ion out of the Lamb–Dicke condition of Doppler-free probing. In the case where the ion’s motional amplitude exceeds the scale length of the wavelength of light probing the ion, the transition probability quickly drops in the Doppler-free central component. These results indicate a loss of coupling to the D − S transition occurring when the cooling laser is off or when the ion is in a dark state. The bright-rate measurements provide better results since the ion is observed under conditions of lower motional amplitude. Further experiments will be required to clarify the cause of this asymmetry in the transition rate. Assuming that this effect decreased the measured bright rates by a constant factor f , we fitted the results of Fig. 6b with the formula for Rabi oscillation given by (1) modified for each effective oscillator level by (3) and summed over the thermal probability distribution. The fitted curve yields the Rabi frequency 0 = 20 000 s−1 , and f = 0.4 ©2000 NRC Canada

506


and mean vibrational quantum number of m = 175 with ε2 = 0.005. Although the fitted curve applies for a one-dimensional model of vibration, the near equal oscillation frequencies of the three canonical trap directions employed in the current studies result in the fit yielding a good approximation to the observed results. The rapid increase at short excitation times is reproduced followed by the dephasing and limited transition rate for times τ > 0.2 ms. The fitted mean oscillator quantum number m = 175 yields an ion kinetic temperature of T = 18 mK, which is agreement with previous measurements of the equilibrium trapped ion kinetic temperature of T = 15 mK ± 5 mK [12] obtained via measurement of the spectral line intensities of the Doppler-free carrier transition relative to the secular frequency sideband transitions. Figure 7a shows a Ramsey spectrum obtained with the (m0j , m00j ) = (+3/2, +1/2) Zeeman component. The spectrum was obtained with a pair of τ = 0.06 ms pulses delayed by 1T = 0.15 ms, with a measurement interval of 800 Hz. The fringes, observed in the bright rate, have a contrast of 80% and reasonably follow the theoretical fit of (6). No Ramsey fringes were observed in the dark rate. Figure 7b shows a spectrum taken with a smaller measurement interval of 500 Hz and a shorter pulse length of τ = 0.05 ms. The data points follow the stationary atom Ramsey pattern within the measurement errors. Figure 7c shows narrower fringes of 840 Hz linewidth full width at half maximum (FWHM) were observed with a pair of τ = 0.10 ms pulses delayed by 1T = 0.60 ms, tuned to the (m0j , m00j ) = (−1/2, −1/2) Zeeman component. Using the contrast C = 40% observed in this measurement, (5) gives a linewidth of 500 Hz for the probing radiation, in agreement with previously derived values.

5. Conclusions We have demonstrated for the first time in a Sr single-ion system the possibility of coherent excitation and Ramsey-fringe spectroscopy with visible radiation. For single-pulse excitation, linewidths approaching the transform-limited value were obtained. A quadratic increase of the excitation probability was observed with increasing pulse duration, in agreement with coherent excitation. Based on the coupling of the trap oscillator states with the coherent excitation, a dephasing of the individual trap vibrational level excitations were observed for the thermal distribution of population having a high mean quantum number m. We believe that this is the first experimental observation of this effect. Based on a simple one-dimensional model, the fitted dependence yielded a value for the ion kinetic temperature consistent with previous measurements. Strong saturation of the count rate was observed, limiting the bright rate to 9 s−1 and the dark-period rates near the natural radiative lifetime. Further improvements of the ion-trap and laser-cooling sources are planned to help reduce the asymmetry observed in the ion absorption and stimulated emission rates. This would allow greater total transition rates and a better signal-to-noise ratio for a specific averaging time. For two-pulse excitation, Ramsey interference fringes were observed with pulse separation of up to 0.6 ms giving a fringe width of 840 Hz FWHM. At the present time, the probing laser linewidth is much wider than the natural linewidth of 0.4 Hz of the transition. The Ramsey method can be employed to improve the symmetry of the line for a more accurate measurement of the line center. However, lasers with narrower linewidth would make possible more accurate measurements using coherent properties that could be applied to higher precision optical frequency measurements.

Acknowledgements The authors thank Professor R. Blatt of the Universität Innsbruck and Dr. Christian Tamm of the Physikalisch Technische Bundesanstalt for helpful discussions into single-ion dynamics and excitation phenomena. The technical expertise of R. Pelletier and B. Hoger in the support of the ion-trap experiments is gratefully acknowledged. ©2000 NRC Canada

Marmet and Madej

507

References 1. J.C. Bergquist (Editor). Proceedings of the 5th Symposium on Frequency Standards and Metrology. World Scientific, Singapore, 1996, and references therein. 2. N.F. Ramsey. Rev. Mod. Phys. 62, 541 (1990) and references therein. 3. H.A. Schuessler, E.N. Fortson, and H.G. Dehmelt. Phys. Rev. 187, 5 (1969). 4. Ye.V. Baklanov, V.P. Chebotayev, and B.Ya. Dubetsky. Appl. Phys. 11, 201 (1976). 5. A. Huber, B. Gross, M. Weitz, and T.W. Hänsch. Phys. Rev. A: At. Mol. Opt. Phys. 58, R2631 (1998). 6. Ye.V. Baklanov, B.Ya. Dubetsky, and V.P. Chebotayev. Appl. Phys. 9, 171 (1976). 7. Ch.J. Bordé, Ch. Salomon, S. Avriller, A. Van Lerberghe, Ch. Bréant, D. Bassi, and G. Scoles. Phys. Rev. A: Gen. Phys. 30, 1836 (1984). 8. J.C. Bergquist, S.A. Lee, and J.L. Hall. Phys. Rev. Lett. 38, 760 (1977). 9. R.L. Barger, J.C. Bergquist, T.C. English, and D.J. Glaze. Appl. Phys. Lett. 34, 850 (1979). 10. F. Riehle, H. Schnatz, B. Lipphardt, G. Zinner, T. Trebst, and J. Helmcke. IEEE Trans. Instrum. Meas. 48, 613 (1999). 11. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt. Phys. Rev. Lett. 41, 233 (1978). 12. J.E. Bernard, A.A. Madej, L. Marmet, B.G. Whitford, K.J. Siemsen, and S. Cundy. Phys. Rev. Lett. 82, 3228 (1999). 13. J.J. Bollinger, W.M. Itano, D.J. Wineland, and D.J. Heinzen. Phys. Rev. A: At. Mol. Opt. Phys. 54, R4649 (1996). 14. P.T.H. Fisk, M.J. Sellars, M.A. Lawn, C. Coles, A.G. Mann, and D.G. Blair. IEEE Trans. Instrum. Meas. 44, 113 (1995). 15. D.J. Berkeland, J.D. Miller, J.C. Bergquist, W.M. Itano, and D.J. Wineland. Phys. Rev. Lett. 80, 2089 (1998). 16. J.C. Bergquist, W.M. Itano, F. Elsner, M.G. Raizen, and D.J. Wineland. In Light induced kinetic effects on atoms, ions and molecules. Edited by L. Moi, S. Gozzini, C. Gabbanini, C. Arimondo, and F. Strumia. ETS Editrice, Pisa. 1991. p. 291. 17. Ch. Roos, Th. Zeiger, H. Rohde, H.C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt. Phys. Rev. Lett. 83, 4713 (1999). 18. J. Vanier and C. Audoin. The quantum physics of atomic frequency standards. Adam Hilger, Bristol and Philadelphia, IOP Publishing Ltd. 1989. p. 628. 19. R. Loudon. The quantum theory of light. 2nd ed. Clarendon Press, Oxford. 1983. 20. C.A. Blockley, D.F. Walls, and H. Risken. Europhys. Lett. 17, 509 (1992). 21. J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller. Phys. Rev. A: At. Mol. Opt. Phys. 49, 1202 (1994). 22. A.A. Madej and K.J. Siemsen. Opt. Lett. 21, 824 (1996). 23. L. Marmet, A.A. Madej, K.J. Siemsen, J.E. Bernard, and B.G. Whitford. IEEE Trans. Instrum. Meas. 46, 169 (1997). 24. A.A. Madej, L. Marmet, and J.E. Bernard. Appl. Phys. B, 67, 229 (1998). 25. G.P. Barwood, P. Gill, G. Huang, H.A. Klein, and W.R.C. Rowley. Opt. Commun. 151, 50 (1998). 26. W. Nagourney, J. Sandberg, and H. Dehmelt. Phys. Rev. Lett. 56, 2797 (1986).

©2000 NRC Canada