IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) S345–S358 doi:10.1088/0953-4075/40/11/S08

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal

Jens Klein, Fabian Beil and Thomas Halfmann

Fachbereich Physik, Universitat Kaiserslautern, Erwin-Schrodinger-Strasse,D-67663 Kaiserslautern, Germany

E-mail: jeklein@physik.uni-kl.de

Received 27 October 2006Published 16 May 2007Online at stacks.iop.org/JPhysB/40/S345

AbstractWe report on rapid adiabatic passage (RAP) in a Pr3+:Y2SiO5 crystal, cooledto cryogenic temperatures. The medium is prepared by optical pumping andspectral hole burning, creating a spectrally isolated two-level system withinthe inhomogeneous bandwidth of the 3H4 →1D2 transition of the Pr3+ ions.A chirped laser pulse drives a RAP process in the medium, i.e. inverts theinitial population distribution. We study the properties and dynamics of RAPby means of fluorescence detection, absorption spectroscopy and amplifiedspontaneous emission. Time-resolved absorption measurements serve tomonitor the adiabatic population dynamics during the excitation process.In addition, we compare the results with coherent excitation at fixed laserfrequency detuned from resonance, i.e. coherent population return (CPR).

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Manipulation of population distributions as well as the linear and nonlinear optical response incoherently driven quantum systems is a major topic in quantum optics at present. Adiabatic,coherent processes, e.g. electromagnetically induced transparency (EIT) (see [1] and referencestherein), rapid adiabatic passage (RAP), Stark-chirped rapid adiabatic passage (SCRAP),stimulated Raman adiabatic passage (STIRAP) and coherent population return (CPR) (see [2]and references therein) have been studied extensively, but predominantly in the gas phase. Incontrast, solid state media are very attractive for applications. Due to their high density andscalability, solid media offer significant advantages for, e.g. optical data storage or quantuminformation processing. Usually ultra-fast decoherence processes in solids are an obstacle,as they prohibit successful implementation of coherent excitations. However, these problemscan be overcome in some special solid state media, e.g. quantum dots and rare earth ion dopedinorganic crystals. Such media permit the combination of the advantages of solids and thecoherence properties of atoms. Therefore, a growing number of studies on coherent excitationsin quantum dots and rare earth ion doped inorganic crystals is performed at present.

0953-4075/07/110345+14$30.00 © 2007 IOP Publishing Ltd Printed in the UK S345

S346 J Klein et al

In rare earth ion doped crystals (RE materials) the coherent properties of the dopant ionsare preserved since the optically active electrons (4f) are shielded from the environment byouter electrons. Homogeneous linewidths in the order of a few kHz are typical at cryogenictemperatures. However, as the crystal field varies for the different dopant sites, the crystallinehost causes a inhomogeneous broadening of the optical transition. This broadening is typicallyin the order of a few GHz. Therefore, a large number of different ensembles of ions canbe addressed individually using different laser frequencies with narrow bandwidth, e.g. byspectral hole burning. The latter offers attractive possibilities for data storage and quantumcomputing. In fact, RE materials were proposed as quantum computer hardware [3–5] and theimplementation of qubit distillation and qubit operations in these materials is subject of currentresearch [6–8]. Moreover, fundamental coherent effects have been studied in RE materials.EIT and the enhancement of four-wave mixing and phase conjugation have been demonstratedby Ham et al [9–12]. Also, the storage of light pulses in these solid state materials has beenachieved using EIT [13]. Storage times of more than one second were obtained applyingdynamic decoherence control [14].

Adiabatic population transfer using RAP was studied by de Seze et al in a Tm3+:YAGcrystal [15]. These authors observed population transfer by absorption measurementssubsequent to the excitation process. First results showed a population transfer efficiencyonly slightly above the limit of incoherent excitation. However, in an experiment involving amodified repetitive RAP process to prepare a narrow absorption line within the inhomogeneousbandwidth of the crystal, the coherent nature of the process was proven. In subsequentpublications [16, 17] improved population transfer efficiency close to 100% was reported. Thetransfer was driven by a highly stabilized diode laser and laser pulses of complex hyperbolicsecant (in the following we refer to them as CHS pulses) rather than a gaussian temporalprofile.

In [17] it was demonstrated, that the insensitivity of population transfer by CHS pulseswith regard to fluctuations in the pulse amplitude (above a critical threshold) is due to theadiabaticity of the process. CHS pulses, known from nuclear magnetic resonance (NMR)[18], were proposed for qubit manipulation in quantum computing with RE materials by Roosand Mølmer [19]. This was experimentally demonstrated by Rippe et al [8] in a Pr3+:Y2SiO5

crystal. These authors present convincing data, showing population inversion due to excitationwith CHS pulses, observed by absorption spectroscopy. In the experiment a dye laser, highlystabilized in frequency (jitter ∼30 kHz) and intensity, was used. The population transferefficiency was found to be better than 90%.

Moreover, the technique was applied to demonstrate qubit distillation, i.e. the preparationof the medium to facilitate selective interaction with different mutually interacting qubits. Theexcitation of an ion influences the resonance frequencies of neighbouring ions by dipole–dipoleinteraction. This allows qubit–qubit interaction, which is necessary to implement quantumgates. Therefore, efficient and robust transfer to an excited state, e.g. by RAP, is interestingfor qubit manipulation in quantum computer hardware based on RE materials.

In the work presented in the following, we report on results on RAP in a Pr3+:Y2SiO5

crystal with linearly chirped laser pulses derived from a less complex laser system, withoutany additional frequency stabilization involved (frequency jitter ∼2 MHz). We observe andstudy adiabatic population transfer by different and independent techniques, i.e. laser-inducedfluorescence, absorption spectroscopy and amplified spontaneous emission. Moreover,the population dynamics during the excitation process were investigated by time-resolvedabsorption measurements. For excitation with laser pulses at fixed frequency, i.e. withoutchirp, our data reveal the transient adiabatic population transfer dynamics of a CPR process.In the case of chirped excitation, we observe rapid and efficient population transfer by RAP.

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S347

2. Dynamics of a coherently driven two-level system

Consider a two-level quantum system, driven by a radiation field at a detuning � fromresonance. The coupling strength is defined by the Rabi frequency �(t) = µE(t)/h, with theelectric field E of the laser and the transition dipole moment µ. The population dynamics inthis system is described by the time-dependent Schrodinger equation

ihd

dt��(t) = H(t) ��(t), (1)

where ��(t) = [c1(t), c2(t)]T is the state vector consisting of the probability amplitudes ck(t)

of the two states |k〉 (k = 1, 2). In the rotating wave approximation (RWA), the HamiltonianH is given by

H = h

2

(−� �

� �

). (2)

For � = 0 the solution of (1) shows Rabi oscillations, i.e. the population of state |2〉oscillates between 0 and 100%. At a certain time, e.g. for finite excitation time, the systemends up in state |1〉 or |2〉 or a coherent superposition of both, depending on the pulse area.However, this transfer depends critically on the experimental parameters. If variations ofthe Rabi frequency over the spatial laser profile, or fluctuation of the laser intensity occur,the transfer efficiency averages to the limit of incoherent excitation, i.e. 50%. In contrast,adiabatic excitation, e.g. by RAP, permits complete and robust population transfer [2]. Thetransfer efficiency does not depend upon variations in the experimental parameters, providedsome limits are maintained. In the following section, we will discuss these features.

2.1. Rapid adiabatic passage

To derive the properties of RAP, we consider now the adiabatic states |�±〉. These states arethe instantaneous eigenvectors of (2) and can be expressed as coherent superpositions of thebare states |1〉 and |2〉 by

|�+(t)〉 = cos θ(t) · |1〉 + sin θ(t) · |2〉 (3)

|�−(t)〉 = sin θ(t) · |1〉 − cos θ(t) · |2〉 (4)

with the mixing angle θ (t) given by

tan θ(t) = �(t)

�(t)+

√1 +

�(t)2

�(t)2. (5)

From equations (3)–(5) we see, that for � < 0 and � → 0 the mixing angle θ is equal to0 and therefore the adiabatic state |�+〉 is identical to state |1〉. However, the same adiabaticstate is identical to state |2〉, if � > 0 and � → 0. We assume conditions for adiabaticevolution, i.e. the system stays in the initial adiabatic state during the interaction. Thencomplete population transfer is achieved, if the laser frequency is chirped across resonanceduring the interaction. The process is known as rapid adiabatic passage (RAP). It does notdepend upon the sign of the chirp.

Figure 1(a) shows the population dynamics for RAP obtained from the analyticalexpressions (3)–(5) by calculating |〈�±(t)|k〉|2 (k = 1, 2). The calculation shows, thatall population flows smoothly, i.e. adiabatically, from the initial state to the target state.

We note that an extension of RAP, involving a fixed frequency pump pulse to drivea transition and an additional off-resonant laser pulse to induce dynamic Stark shifts, also

S348 J Klein et al

(a) (b)

Figure 1. Population dynamics of (a) rapid adiabatic passage and (b) coherent population return.The figure shows the population of the bare states, i.e. |〈�±(t)|k〉|2 (k = 1, 2), deduced from theanalytical expressions (3)–(5) for a pump pulse of gaussian temporal shape (duration 10 µs (FWHMof intensity)) with a Rabi frequency of � = 2π × 500 kHz and a linear chirp of dω/dt = 2π ×200 kHz/µs (a) or fixed detuning of � = 2π × 50 kHz (b).

permits complete population transfer. This technique is called Stark-chirped rapid adiabaticpassage (SCRAP). In the SCRAP process, the transition frequency is varied rather than thelaser frequency [20–22].

2.2. Coherent population return

RAP provides a way to drive complete persistent adiabatic population transfer between aground and an excited state. In addition, the dynamics, governed by equations (3)–(5) alsopermit the observation of a transient adiabatic population transfer process.

For � = const �= 0 one of the adiabatic states is identical to the ground state, before andafter the interaction. Therefore, no net population transfer takes place, although the excitedstate |2〉 is populated during the interaction. However, this population completely returns tothe ground state towards the end of the interaction. This effect is known as coherent populationreturn (CPR). To avoid diabatic couplings and loss of adiabaticity, the detuning � must belarger than the bandwidth of the Fourier-limited laser pulse (� > 1/τPulse). CPR can beexploited to suppress power broadening and support applications in coherent spectroscopy,e.g. in trace isotope detection and analysis [23–25].

Figure 1(b) illustrates the population dynamics in the case of CPR. The calculation usesparameters similar to the case of RAP. While in the case of RAP a time-varying detuning, i.e.a chirp, is used, in CPR the detuning is fixed. Population is transferred adiabatically from theground to the excited state and back again.

3. Experiment

3.1. Spectroscopic properties of Pr3+:Y2SiO5

For our studies on rapid adiabatic passage a Pr3+:Y2SiO5 crystal (in the following abbreviatedas Pr:YSO) serves as the medium. The manipulation of the Pr3+ ions by coherent radiation ata wavelength of λ = 605.977 nm involves the substructure of the transition from the groundstate 3H4 to the excited state 1D2 (see figure 2). As the crystal field varies for the differentions, this transition is inhomogeneously broadened to a width of a few GHz. The lifetime ofthe excited state 1D2 is T1 = 164 µs [26]. Relaxation of the excited ions occurs mainly via

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S349

Figure 2. Level scheme of Pr:YSO. Ground and excited states are split into three Kramer’sdoublets by the crystal field. Before the preparation the population is distributed almost equallybetween the ground state doublets. A preparation pulse sequence drives population to the groundstate |1〉 = 3H4(± 5

2 ). The two-level system of state |1〉 = 3H4(± 52 ) and |2〉 = 1D2(± 5

2 ) servesthen as an appropriate quantum system to study coherent excitation processes.

short-lived intermediate levels (not shown in figure 2) by emitting fluorescence in a wavelengthrange from 610 to 640 nm [26].

The interaction of the nuclear spin (I = 5/2) with the crystal field splits the ground andthe excited state into three Kramer’s doublets identified by the quantum number mI of thenuclear spin orientation. Due to the low site symmetry the nuclear wave functions mix andthe �m = 0 selection rule for electric dipole transitions breaks down [27]. Therefore, all ninedifferent transitions between the ground state and excited state sublevels are possible. As theenergy splitting of the sublevels in the order of a few MHz is considerably smaller than theinhomogeneous width, the resonance frequencies of nine different ensembles of ions coincide,but in each case they correspond to a different transition, i.e. a different combination of groundand excited state sublevels. However, although all ground state sublevels can be coupled toall excited state sublevels, the mixing of the nuclear wave functions is much stronger for thesublevels with mI = ±1/2 and mI = ±3/2, while the degree of mixing of the sublevelswith mI = ±5/2 with the other sublevels is quite low (cf relative oscillator strengths in [28]).Therefore, the six-level system shown in figure 2 falls apart into a four- and a two-level systemwith weak coupling between them.

The dipole moment of the transition of the two-level system, defined by the strongest ofall nine transitions, is µ = 2.6 × 10−32 Cm [29]. Due to the anisotropy of the crystal fieldthe dipole transition moments are oriented and the ions mainly absorb radiation, which ispolarized along one of the crystal axes.

In thermal equilibrium all ground state sublevels are equally populated. Therefore, a singlelaser field with a frequency within the inhomogeneous linewidth of the transition 3H4 →1D2

excites nine different ensembles of ions at the same time. Hence, Pr:YSO occurs as a complexspectroscopic system. However, the medium can be prepared by optical pumping to facilitateselective excitation and simplify the interpretation of spectroscopic measurements.

3.2. Pulse sequence of preparation, coherent interaction and probing

In the experiment, an appropriate sequence of preparation, pump and probe laser pulsesinteracts with the medium (see figure 3). The preparation is based on the technique introducedby Nilsson et al [28].

S350 J Klein et al

Figure 3. Pulse sequence. Frequency and intensity modulation is shown. The combination of bothwaveforms yields the two preparation pulses prep 1 (creating pit) and prep 2 (placing anti-hole inpit), the (chirped) pump pulse as well as the probe pulse. Note, that the time scale is stretched fort > 88.7 ms to increase the visibility of the short pump and probe pulses.

For the preparation an intense laser pulse (prep 1, rectangular temporal shape, duration84 ms) with frequency centred respective to the inhomogeneous bandwidth is repetitivelyswept over a range 0 MHz < �ν < 18 MHz (see figure 3). This process creates a spectralpit, i.e. a spectral range within the inhomogeneous width, in which no ions can absorb lightdue to optical pumping. All ensembles of ions, which initially exhibit absorption in thatspectral range, are pumped to different ground state sublevels, from which no transitions withresonance frequencies within the frequency range of the pit exist.

The preparation of the spectral pit is followed by a second preparation pulse (prep 2,rectangular temporal shape, duration 200 µs) at a fixed relative frequency �ν = −18 MHzinteracting with nine individual ensembles of ions. A part of them was already opticallypumped by the first preparation pulse. Consequently, the second preparation pulse pumpsthese ensembles back to the ground state sublevels emptied by the first preparation pulse. Dueto differences in coupling strength for the different sublevels this creates only one considerableabsorption peak (anti-hole) within the spectral range of the pit. The absorption due to thatanti-hole at the relative frequency �ν = +9.5 MHz is exclusively from ions, originally inthe 3H4(±5/2) state, now excited to the 1D2(±5/2) state. Therefore, a spectrally isolatedtwo-level system is prepared. For this relevant ensemble of ions the laser pulses of thesequence couple the transitions indicated in figure 2. The preparation pulse sequence yieldsan absorption spectrum, as depicted in figure 4.

A pump pulse of gaussian temporal shape with a duration of 10 µs (FWHM of intensity)and a delay of 500 µs with respect to the preparation pulses is used to drive a coherent excitationprocess in the two-level system. This pump pulse can be chosen at a fixed frequency or witha chirp of approximately dν/dt = 0.2 MHz/µs. A probe pulse of rectangular temporal shapewith a duration of 1 µs and variable delay is used to measure the absorption, i.e. the populationdifference n1 − n2, where nk is the population in state |k〉 (k = 1, 2).

3.3. Experimental setup

A Pr:YSO crystal (Scientific Materials) with a dopant concentration of 0.05 at% is held at3.9 K in a closed-cycle cryostat (Janis Model SHI-4-1-331S). The crystal is 3 mm thick

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S351

Figure 4. Absorption spectrum of the medium after the preparation process. A spectral pit witha well-defined anti-hole is prepared. The width of the absorption peak (∼2.8 MHz (FWHM)) ismainly due to the jitter of the laser frequency.

Cryostat@4K

Pr:YSO Photodiode

Iris diaphragm

λ /4-Waveplate

R=90%

R=90%

Polarizing BeamSplitter cube

Telescope

606 nmCoherent CR 699-21 dye laser

Preparation, Pump

Probe

Monochromator

Bandpass filter( =620 nm)λ c

PMT

Attenuator

Figure 5. Experimental setup.

along the direction of light propagation. A dye laser (Coherent 699) provides radiationat a wavelength of λ = 605.977 nm to drive the relevant transition in the Pr3+ ions (seefigure 2). The spectral resolution is limited by frequency jitter of the dye laser. The jitter rateis in the order of a few MHz/millisecond. The radiation from the dye laser is split up in twobeam lines—an intense beam for the preparation and pump pulses and a weak probe beam(see figure 5). Both beams are intensity modulated and frequency shifted by acousto-opticalmodulators (AOM, Brimrose BRI-TEF-80-50-.606) in double-pass configuration. Due to thenonlinear response of the AOM radio frequency drivers to the modulation waveforms suppliedby waveform generators (Agilent 33220A) the temporal shape of the pump pulse is not exactlygaussian but slightly distorted. However, the temporal variation of the pump intensity issmooth and therefore suitable to drive adiabatic processes.

The two beams are counter-propagating with parallel linear polarization and overlap inthe Pr:YSO sample (see figure 5). The polarization direction is chosen in the direction of

S352 J Klein et al

−2 −1 0 1 2 30.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5no chirpwith chirp

LIF

Sig

na

l/a.u

.

Pump laser detuning/MHz

Figure 6. Transfer efficiency versus pump laser detuning for the case of resonant excitation(open circles) and chirped excitation (solid triangles), measured by the LIF yield. In the case ofchirped excitation the detuning is defined as the deviation of the pump laser centre frequency fromresonance. The peak intensity of the pump pulse is IP = 18.8 W cm−2. This corresponds toa peak Rabi frequency �P = 2π × 470 kHz (using the electric dipole moment of the relevanttransition in Pr3+ ions µ = 2.6 × 10−32 Cm [28]) The data clearly show the enhancement in thetransfer efficiency in the case of RAP, i.e. chirped excitation.

maximum absorption in the Pr:YSO crystal. The diameter (FWHM of intensity) of the beamfor preparation and coherent excitation at the position of the sample is 565 µm, while thediameter of the probe beam is only 430 µm. For the intense beam, laser powers of up to90 mW are available, while the probe beam power is typically 8 µW, after attenuation.

After passing the sample, a small fraction of the probe laser intensity is directed onto asilicon photodiode for absorption measurements. Additionally, laser-induced fluorescence canbe observed in the setup. Fluorescence emitted under a slight angle with respect to the laserpropagation axis is deflected by a prism, imaged onto the entrace slit of a monochromator, setto a wavelength of approx. 620 nm, and detected by a photomultiplier (Hamamatsu R7400U-02). To enhance the suppression of scattered light, induced by the excitation laser, a bandpassfilter (centre wavelength 620 nm, bandwidth ±5 nm) is used in front of the entrance slit.

4. Results

4.1. Transfer efficiency, monitored by laser-induced fluorescence (LIF)

The fluorescence, emitted after decay of the excited state, is proportional to the population ofthe excited state. Thus, the LIF signal is a direct measure of the population transfer efficiency.No separate probe pulse to monitor the transfer efficiency is necessary. Figure 6 shows theLIF signal recorded versus the detuning of the (centre) frequency of the pump pulse fromresonance, for the case of resonant excitation (no chirp), or for a chirped pump laser pulse(RAP). The data show an enhancement of the fluorescence signal by a factor of 2.5 for thecase of RAP with regard to the fluorescence yield for resonant excitation.

In the case of strong resonant excitation the transfer efficiency varies between 0%and 100%, depending upon the exact value of the experimental parameters (see above).Fluctuations in the peak laser intensity, variations of the laser intensity across the spatial profileand fluctuations of the laser frequency lead to an averaged maximum transfer efficiency of50%. This is also the value expected for strong, incoherent excitation. Thus the RAP processshould lead to an enhancement of a factor of 2 with respect to the case of resonant excitation.

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S353

However, the enhancement, observed in our experiment, is even a bit larger. This implies thatthe averaged transfer efficiency is less than 50% for resonant excitation in our experiment.

In contrast to the pulse sequence, discussed above (see 3.2), a pump pulse with rectangulartemporal shape and a duration of 20 µs was used in this particular experiment to yield strongersignal for the case of resonant excitation. Due to jitter in the laser frequency, the excitationprocess, driven by the pump laser is not always resonant. Therefore also off-resonant excitationmay occur, i.e. CPR takes place at the end of the pump pulse, removing population from theexcited state. As data acquisition implies averaging over several excitation cycles, the transferefficiency at resonant excitation will average to a lower value. However, for a rectangularpulse shape the sudden change in pump laser intensity induces diabatic coupling and thereforeperturbs the CPR process. This gives rise to a stronger fluorescence signal. Moreover,simulations show that for the given experimental parameters the population transfer efficiencyof RAP is barely affected.

Due to the limited temporal response of the AOM the evolution of the pump laser intensityis not discontinuous as assumed for pump pulses with rectangular temporal shape. Therefore,the CPR effect is not completely suppressed and population is partially returned to the groundstate. This results in a reduced signal for excitation with pump laser pulses at fixed frequency.Consequently, the enhancement of population transfer efficiency for RAP is larger than theexpected factor of 2. Moreover, the resonant excitation is not saturated in the weaker regionsof the spatial pump laser profile. In contrast, RAP suffers less from these variations in thelaser intensity, although strong excitation is also needed. This also results in a higher ratio ofLIF yield for the two excitation methods.

Though a precise, absolute calibration of the transfer efficiency is difficult, the significantenhancement of the LIF signal for chirped excitation already exhibits striking evidence forefficient coherent population transfer by RAP.

4.2. Transfer efficiency, monitored by absorption spectroscopy

As discussed above, an absolute calibration of the transfer efficiency in the case of LIF detectionsuffers from both CPR and excitation in the weaker regions of the pump laser. This problemcan be overcome in a setup, involving absorption measurements of an additional probe laser. Ifthe spatial profile of this probe laser is smaller than the profile of the pump laser, it is possible tomonitor the population transfer process in the intense centre of the spatial pump laser profilewithout contributions from the weaker wings. Moreover, the absolute transmission of theprobe laser, detuned from the resonance, can be used for absolute calibration of the transfersignal. The transfer in the case of resonant excitation is not needed as reference.

In the experiment the transmission signal of a weak probe pulse, well delayed withrespect to the pump pulse, is recorded versus the detuning of the probe laser frequency fromresonance with the anti-hole. Thus the probe laser excites the same transition as the pumplaser. In contrast to the pump laser, the probe laser is weak, i.e. well below the saturationintensity.

No absorption by the Pr3+ ions occurs, if the probe laser is sufficiently far detuned fromresonance. Using this signal level as reference, i.e. transmission equals unity, Beer’s lawpermits to calculate the absorption coefficient. The absorption on resonance is proportionalto the difference of population in the ground and excited state, i.e. n1 − n2. Hence, we relatethe observed absorption with respect to absolute transfer efficiencies. From the absorption therelative excited state population P2 can be calculated by

P2 = 1

2

(1 − α′

α0

), (6)

S354 J Klein et al

(a)

(b)

(c)

Figure 7. Absorption coefficient versus probe laser detuning. The probe laser is delayed by17 µs with respect to the pump pulse. (a) The pump laser pulse is switched off. (b) The pump laserpulse is switched on and set on resonance. (c) The pump laser pulse is chirped through resonance(RAP). In (b,c) the peak intensity of the pump pulse is IP = 24.9 W cm−2, which corresponds toa Rabi frequency �P = 2π × 530 kHz. In the case of RAP (c) the absorption coefficient becomesnegative, i.e. the probe laser is amplified. This is clear evidence for population inversion, drivenby RAP.

where α′ and α0 are the absorption coefficients with and without the pump pulse,respectively.

Figure 7(a) shows the absorption coefficient for the anti-hole transition, if no pump laserpulse is present. The probe laser monitors the population, prepared in the initial state by thepreparation pulse sequence. As all population of the two-level system is in the ground state,stimulated absorption occurs. The absorption coefficient reaches a maximum at the probelaser driven resonance. If the pump laser pulse is switched on and set with fixed frequencyon resonance in the two-level system, a significant amount of population is transferred to theexcited state (see figure 7(b)). In the ideal case of strong excitation, i.e. saturation, half ofthe population should be driven to the excited state. In this case stimulated emission andstimulated absorption are equally strong. The absorption coefficient is expected to approachzero. The experiment shows, that the absorption coefficient for resonant excitation is indeedsignificantly reduced with respect to the case of the pump laser switched off. However, it doesnot reach zero, as the resonant excitation seems to be not saturated. We attribute this to theinfluence of CPR, which is stronger than for the experiment of section 4.1, as we now usesmooth pump pulses. From the data we calculate the relative population in the excited state(according to equation (6)) as 16%.

For the case of RAP, i.e. chirped excitation, the absorption coefficient becomes negative,as the population distribution is inverted now (see figure 7(c)).

From the data we deduce an absolute transfer efficiency of (90 ± 10)%, which clearlyexceeds the limit of 50% in the case of incoherent excitation. We note, that the transferefficiency corresponds to the population at the time of the probe laser pulse, which is delayedwith respect to the pump pulse. The radiative decay starts to reduce the population in the

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S355

Figure 8. Time-resolved observation of RAP. The population transfer efficiency calculated fromthe absorption of a probe pulse (duration 1 µs, intensity IP r = 3.8 mW cm−2)) is plotted versusthe time delay with respect to the pump pulse. Data are monitored for resonant excitation at fixedfrequency (open circles) and chirped excitation, i.e. RAP (solid triangles). The solid lines areexponential decay functions with the lifetimes given in the graph. The peak intensity of the pumppulse is IP = 24.9 W cm−2, corresponding to a Rabi frequency of �P = 2π × 530 kHz.

excited state after the transfer process. Thus the population in the excited state directly afterthe pump pulse is indeed even larger than the probed transfer of (90 ± 10)%.

4.3. Time-resolved observation of RAP

In the data presented above the population transferred to the excited state after the RAP processwas measured. By changing the delay between the pump pulse and the probe pulse the excitedstate population can be measured at different times. As the probe pulse is weak and short(�τ ∼ 0.04) its influence on the transfer process is negligible and the population dynamicscan be well resolved during the coherent interaction process. Figure 8 shows the transferefficiency versus the delay between probe and pump pulse. The efficiency was calculatedfrom absorption measurements for the case of resonant excitation at fixed frequency (opencircles) and chirped excitation, i.e. RAP (solid triangles). For resonant excitation, a maximumtransfer of about 35% is observed during the interaction process. The efficiency is reducedto below 20% towards the end of the pump pulse. Again, off-resonant excitation due to jitterin the laser frequency and, consequently, CPR prevents the observation of saturation. Theeffect of the fast CPR process, taking place on the time scale of the pump laser pulse duration,is visible in the slope of the data points for resonant excitation, right after the excitationprocess.

We note that also some population remains in the excited state after the interaction, whichoriginates from decay, dephasing, and near resonant diabatic couplings. The decay of the thisresidual population is well described by an exponential decay with a time constant of 164 µs.

Figure 9 shows numerical results for the population dynamics of CPR and RAP. Weapplied density matrix calculations including the decay of the excited state population anddephasing. For CPR (figure 9(a)), the simulation clearly reproduces the characteristics of theexperimentally observed population dynamics for fixed pump laser frequency (see above).For chirped excitation, i.e. RAP the transfer process starts slightly later than for resonantexcitation. This is due to the fact, that the excitation process in RAP is off-resonant in thebeginning. When chirping over the resonance, the excited state population shows a rapid

S356 J Klein et al

(a) (b)

Figure 9. Numerical results for population dynamics of (a) coherent population return and (b) rapidadiabatic passage, including decay of the excited state population (T1 = 164 µs) and dephasing(T2 = 111 µs). We performed the simulation for a pump pulse of gaussian temporal shape(duration 10 µs (FWHM of intensity)) with a Rabi frequency of � = 2π × 530 kHz and a linearchirp of dω/dt = 2π × 200 kHz/µs (a) or fixed detuning of � = 2π × 50 kHz (b).

increase to approximately 100%. The RAP process is followed by a slow decay due to thelimited lifetime of the upper state.

The experimental data clearly reveal the smooth population dynamics of the RAP process,i.e. show very good agreement with the simulation of the RAP process (see figure 9(b)).However, the time constant of the experimentally observed decay is found to be ∼91 µs byfitting an exponential decay (figure 8) to the data. This substantial reduction compared withthe expected lifetime of 164 µs is due to amplified spontaneous emission (ASE), which occursin the system, driven to population inversion. The decay of an ion leads to stimulated decayof other ions. Thus the decay rate increases and the lifetime is reduced.

4.4. Population inversion, observed by amplified spontaneous emission

In our setup, RAP prepares population inversion in a cylindrical volume around the propagationaxis of the pump laser light. ASE is emitted in both longitudinal directions into a small solidangle defined by the pump beam diameter and the crystal length. Therefore, for delayedprobing ASE may produce a signal directly after the excitation process, although only lightcounter-propagating with respect to the pump pulse is detected and no probe pulse is present atthat time. In fact, we observed such radiation pulses, i.e. ASE, during the excitation process.

To reduce the solid angle of the ASE and enhance the detection efficiency, the RAPprocess was implemented with laser beams of reduced diameters, i.e. 350 µm for the pumpand 175 µm for the probe beam. Figure 10 shows the ASE pulses, obtained from two singleexperimental cycles, i.e. two ‘single-shot measurements’. In both cases an ASE pulse isobserved towards the end of the pump pulse, i.e. after the transfer process. The intensity ofthe two pulses is different, as ASE is initiated by a spontaneous process and therefore theefficiency fluctuates.

The explanation of the observed radiation pulses by ASE is also confirmed by thetransmission (or amplification) of the delayed probe pulse (see figure 10). ASE competeswith the probe pulse for the population inversion. Thus, when the ASE pulse is strong, theprobe pulse is less amplified, as ASE reduces the population inversion. In contrast, when theASE pulse is weak, the probe beam is better amplified.

Hence, the amplification detected by the probe pulse, from which we can infer the excitedstate population, is fluctuating. Therefore, single shot data are not reliable to determine the

Rapid adiabatic passage in a Pr3+:Y2SiO5 crystal S357

Figure 10. Observation of amplified spontaneous emission (ASE). The traces show two single-shotmeasurements of the temporal profiles of an ASE pulse and the transmitted (rp. amplified) probelaser pulse. Also the temporal evolution of the pump pulse is indicated (black dotted line). ASEoccurs at the end of the interaction with the pump pulse. The intensities of the ASE pulse andof the transmitted (rp. amplified) probe pulse are anti-correlated. The peak intensity of the pumppulse is IP = 56.2 W cm−2, corresponding to a Rabi frequency if �P = 2π × 810 kHz.

transfer efficiency of RAP. However, for averaged data, as presented in section 4.1–4.3, ASEcorresponds to an additional decay mechanism which reduces the observed lifetime as seen insection 4.3.

5. Conclusion

We implemented, observed and extensively studied rapid adiabatic passage (RAP) in a dopedsolid, i.e. Pr:YSO by laser-induced fluorescence, absorption spectroscopy and observationof amplified spontaneous emission. By these different techniques we collected strikingdata for the preparation of coherently driven complete population inversion in the dopedsolid. We observed amplification of a probe laser pulse by stimulated emission as wellas amplified spontaneous emission in the inverted medium. Time-dependent absorption(or amplification) measurements revealed the population dynamics of RAP, in very goodagreement with theoretical predictions. RAP was successfully implemented in our experimentswith laser pulses of different temporal profiles. Even pulses with rectangular temporal shapepermitted efficient adiabatic population transfer, irrespective of residual diabatic couplings.We compared RAP with resonant, coherent excitation at fixed frequency. In this case the datashow good agreement with predictions, also involving excitation processes based on coherentpopulation return (CPR).

Our investigations serve as another small, but hopefully valuable, step towards theunderstanding of robust adiabatic interactions in solid media. The combination of coherentexcitations (such as RAP) and media for high-density optical data storage (such as rare-earthsolids) will provide exciting possibilities for the implementation of quantum informationprocessing in the future.

S358 J Klein et al

Acknowledgments

We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG), technical supportby Jochen Klein, and most valuable discussions with Klaas Bergmann.

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