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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+ :Y2 SiO5 crystal
Jens Klein, Fabian Beil and Thomas Halfmann
Fachbereich Physik, Universität Kaiserslautern, Erwin-Schrödinger-Strasse,
D-67663 Kaiserslautern, Germany
E-mail: [email protected]
Received 27 October 2006
Published 16 May 2007
Online at stacks.iop.org/JPhysB/40/S345
Abstract
We report on rapid adiabatic passage (RAP) in a Pr3+ :Y2 SiO5 crystal, cooled
to cryogenic temperatures. The medium is prepared by optical pumping and
spectral hole burning, creating a spectrally isolated two-level system within
the inhomogeneous bandwidth of the 3 H4 →1 D2 transition of the Pr3+ ions.
A chirped laser pulse drives a RAP process in the medium, i.e. inverts the
initial population distribution. We study the properties and dynamics of RAP
by means of fluorescence detection, absorption spectroscopy and amplified
spontaneous emission. Time-resolved absorption measurements serve to
monitor the adiabatic population dynamics during the excitation process.
In addition, we compare the results with coherent excitation at fixed laser
frequency 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 in
coherently driven quantum systems is a major topic in quantum optics at present. Adiabatic,
coherent processes, e.g. electromagnetically induced transparency (EIT) (see [1] and references
therein), 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. In
contrast, solid state media are very attractive for applications. Due to their high density and
scalability, solid media offer significant advantages for, e.g. optical data storage or quantum
information processing. Usually ultra-fast decoherence processes in solids are an obstacle,
as they prohibit successful implementation of coherent excitations. However, these problems
can be overcome in some special solid state media, e.g. quantum dots and rare earth ion doped
inorganic crystals. Such media permit the combination of the advantages of solids and the
coherence properties of atoms. Therefore, a growing number of studies on coherent excitations
in 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
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J Klein et al
In rare earth ion doped crystals (RE materials) the coherent properties of the dopant ions
are preserved since the optically active electrons (4f) are shielded from the environment by
outer electrons. Homogeneous linewidths in the order of a few kHz are typical at cryogenic
temperatures. However, as the crystal field varies for the different dopant sites, the crystalline
host causes a inhomogeneous broadening of the optical transition. This broadening is typically
in the order of a few GHz. Therefore, a large number of different ensembles of ions can
be addressed individually using different laser frequencies with narrow bandwidth, e.g. by
spectral hole burning. The latter offers attractive possibilities for data storage and quantum
computing. In fact, RE materials were proposed as quantum computer hardware [3–5] and the
implementation of qubit distillation and qubit operations in these materials is subject of current
research [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 demonstrated
by Ham et al [9–12]. Also, the storage of light pulses in these solid state materials has been
achieved using EIT [13]. Storage times of more than one second were obtained applying
dynamic decoherence control [14].
Adiabatic population transfer using RAP was studied by de Sèze et al in a Tm3+ :YAG
crystal [15]. These authors observed population transfer by absorption measurements
subsequent to the excitation process. First results showed a population transfer efficiency
only slightly above the limit of incoherent excitation. However, in an experiment involving a
modified repetitive RAP process to prepare a narrow absorption line within the inhomogeneous
bandwidth of the crystal, the coherent nature of the process was proven. In subsequent
publications [16, 17] improved population transfer efficiency close to 100% was reported. The
transfer was driven by a highly stabilized diode laser and laser pulses of complex hyperbolic
secant (in the following we refer to them as CHS pulses) rather than a gaussian temporal
profile.
In [17] it was demonstrated, that the insensitivity of population transfer by CHS pulses
with regard to fluctuations in the pulse amplitude (above a critical threshold) is due to the
adiabaticity of the process. CHS pulses, known from nuclear magnetic resonance (NMR)
[18], were proposed for qubit manipulation in quantum computing with RE materials by Roos
and Mølmer [19]. This was experimentally demonstrated by Rippe et al [8] in a Pr3+ :Y2 SiO5
crystal. These authors present convincing data, showing population inversion due to excitation
with CHS pulses, observed by absorption spectroscopy. In the experiment a dye laser, highly
stabilized in frequency (jitter ∼30 kHz) and intensity, was used. The population transfer
efficiency was found to be better than 90%.
Moreover, the technique was applied to demonstrate qubit distillation, i.e. the preparation
of the medium to facilitate selective interaction with different mutually interacting qubits. The
excitation of an ion influences the resonance frequencies of neighbouring ions by dipole–dipole
interaction. This allows qubit–qubit interaction, which is necessary to implement quantum
gates. Therefore, efficient and robust transfer to an excited state, e.g. by RAP, is interesting
for 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+ :Y2 SiO5
crystal with linearly chirped laser pulses derived from a less complex laser system, without
any additional frequency stabilization involved (frequency jitter ∼2 MHz). We observe and
study adiabatic population transfer by different and independent techniques, i.e. laser-induced
fluorescence, absorption spectroscopy and amplified spontaneous emission. Moreover,
the population dynamics during the excitation process were investigated by time-resolved
absorption measurements. For excitation with laser pulses at fixed frequency, i.e. without
chirp, 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+ :Y2 SiO5 crystal
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2. Dynamics of a coherently driven two-level system
Consider a two-level quantum system, driven by a radiation field at a detuning from
resonance. The coupling strength is defined by the Rabi frequency (t) = µE(t)/h̄, with the
electric field E of the laser and the transition dipole moment µ. The population dynamics in
this system is described by the time-dependent Schrödinger equation
d
= H (t)(t),
(1)
ih̄ (t)
dt
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 Hamiltonian
H is given by
h̄ − 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 system
ends 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 of
the 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]. The
transfer efficiency does not depend upon variations in the experimental parameters, provided
some 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 are
the instantaneous eigenvectors of (2) and can be expressed as coherent superpositions of the
bare 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
(t)
(t)2
+ 1+
tan θ (t) =
.
(t)
(t)2
(5)
From equations (3)–(5) we see, that for < 0 and → 0 the mixing angle θ is equal to
0 and therefore the adiabatic state |+ is identical to state |1. However, the same adiabatic
state is identical to state |2, if > 0 and → 0. We assume conditions for adiabatic
evolution, i.e. the system stays in the initial adiabatic state during the interaction. Then
complete population transfer is achieved, if the laser frequency is chirped across resonance
during the interaction. The process is known as rapid adiabatic passage (RAP). It does not
depend upon the sign of the chirp.
Figure 1(a) shows the population dynamics for RAP obtained from the analytical
expressions (3)–(5) by calculating |± (t)|k|2 (k = 1, 2). The calculation shows, that
all 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 drive
a transition and an additional off-resonant laser pulse to induce dynamic Stark shifts, also
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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 the
analytical expressions (3)–(5) for a pump pulse of gaussian temporal shape (duration 10 µs (FWHM
of 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 adiabatic
passage (SCRAP). In the SCRAP process, the transition frequency is varied rather than the
laser frequency [20–22].
2.2. Coherent population return
RAP provides a way to drive complete persistent adiabatic population transfer between a
ground and an excited state. In addition, the dynamics, governed by equations (3)–(5) also
permit the observation of a transient adiabatic population transfer process.
For = const = 0 one of the adiabatic states is identical to the ground state, before and
after the interaction. Therefore, no net population transfer takes place, although the excited
state |2 is populated during the interaction. However, this population completely returns to
the ground state towards the end of the interaction. This effect is known as coherent population
return (CPR). To avoid diabatic couplings and loss of adiabaticity, the detuning must be
larger than the bandwidth of the Fourier-limited laser pulse ( > 1/τPulse ). CPR can be
exploited 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 uses
parameters 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 the
ground to the excited state and back again.
3. Experiment
3.1. Spectroscopic properties of Pr3+ :Y2 SiO5
For our studies on rapid adiabatic passage a Pr3+ :Y2 SiO5 crystal (in the following abbreviated
as Pr:YSO) serves as the medium. The manipulation of the Pr3+ ions by coherent radiation at
a wavelength of λ = 605.977 nm involves the substructure of the transition from the ground
state 3 H4 to the excited state 1 D2 (see figure 2). As the crystal field varies for the different
ions, this transition is inhomogeneously broadened to a width of a few GHz. The lifetime of
the excited state 1 D2 is T1 = 164 µs [26]. Relaxation of the excited ions occurs mainly via
Rapid adiabatic passage in a Pr3+ :Y2 SiO5 crystal
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Figure 2. Level scheme of Pr:YSO. Ground and excited states are split into three Kramer’s
doublets by the crystal field. Before the preparation the population is distributed almost equally
between the ground state doublets. A preparation pulse sequence drives population to the ground
state |1 = 3 H4 (± 52 ). The two-level system of state |1 = 3 H4 (± 52 ) and |2 = 1 D2 (± 52 ) serves
then as an appropriate quantum system to study coherent excitation processes.
short-lived intermediate levels (not shown in figure 2) by emitting fluorescence in a wavelength
range from 610 to 640 nm [26].
The interaction of the nuclear spin (I = 5/2) with the crystal field splits the ground and
the excited state into three Kramer’s doublets identified by the quantum number mI of the
nuclear spin orientation. Due to the low site symmetry the nuclear wave functions mix and
the m = 0 selection rule for electric dipole transitions breaks down [27]. Therefore, all nine
different transitions between the ground state and excited state sublevels are possible. As the
energy splitting of the sublevels in the order of a few MHz is considerably smaller than the
inhomogeneous 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 ground
and excited state sublevels. However, although all ground state sublevels can be coupled to
all excited state sublevels, the mixing of the nuclear wave functions is much stronger for the
sublevels with mI = ±1/2 and mI = ±3/2, while the degree of mixing of the sublevels
with 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 system
with weak coupling between them.
The dipole moment of the transition of the two-level system, defined by the strongest of
all nine transitions, is µ = 2.6 × 10−32 Cm [29]. Due to the anisotropy of the crystal field
the dipole transition moments are oriented and the ions mainly absorb radiation, which is
polarized along one of the crystal axes.
In thermal equilibrium all ground state sublevels are equally populated. Therefore, a single
laser field with a frequency within the inhomogeneous linewidth of the transition 3 H4 →1 D2
excites nine different ensembles of ions at the same time. Hence, Pr:YSO occurs as a complex
spectroscopic system. However, the medium can be prepared by optical pumping to facilitate
selective 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 pulses
interacts with the medium (see figure 3). The preparation is based on the technique introduced
by Nilsson et al [28].
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J Klein et al
Figure 3. Pulse sequence. Frequency and intensity modulation is shown. The combination of both
waveforms yields the two preparation pulses prep 1 (creating pit) and prep 2 (placing anti-hole in
pit), the (chirped) pump pulse as well as the probe pulse. Note, that the time scale is stretched for
t > 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, duration
84 ms) with frequency centred respective to the inhomogeneous bandwidth is repetitively
swept over a range 0 MHz < ν < 18 MHz (see figure 3). This process creates a spectral
pit, i.e. a spectral range within the inhomogeneous width, in which no ions can absorb light
due to optical pumping. All ensembles of ions, which initially exhibit absorption in that
spectral range, are pumped to different ground state sublevels, from which no transitions with
resonance 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 MHz
interacting with nine individual ensembles of ions. A part of them was already optically
pumped by the first preparation pulse. Consequently, the second preparation pulse pumps
these ensembles back to the ground state sublevels emptied by the first preparation pulse. Due
to differences in coupling strength for the different sublevels this creates only one considerable
absorption peak (anti-hole) within the spectral range of the pit. The absorption due to that
anti-hole at the relative frequency ν = +9.5 MHz is exclusively from ions, originally in
the 3 H4 (±5/2) state, now excited to the 1 D2 (±5/2) state. Therefore, a spectrally isolated
two-level system is prepared. For this relevant ensemble of ions the laser pulses of the
sequence couple the transitions indicated in figure 2. The preparation pulse sequence yields
an 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 excitation
process in the two-level system. This pump pulse can be chosen at a fixed frequency or with
a chirp of approximately dν/dt = 0.2 MHz/µs. A probe pulse of rectangular temporal shape
with a duration of 1 µs and variable delay is used to measure the absorption, i.e. the population
difference 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 at
3.9 K in a closed-cycle cryostat (Janis Model SHI-4-1-331S). The crystal is 3 mm thick
Rapid adiabatic passage in a Pr3+ :Y2 SiO5 crystal
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Figure 4. Absorption spectrum of the medium after the preparation process. A spectral pit with
a well-defined anti-hole is prepared. The width of the absorption peak (∼2.8 MHz (FWHM)) is
mainly due to the jitter of the laser frequency.
PMT
Monochromator
Cryostat @4K
Iris diaphragm
R=90%
Probe
λ /4-Waveplate
Attenuator
Polarizing Beam
Splitter cube
Bandpass filter
(λ c=620 nm)
Pr:YSO
Photodiode
Preparation, Pump
Telescope
R=90%
Coherent CR 699-21 dye laser
606 nm
Figure 5. Experimental setup.
along the direction of light propagation. A dye laser (Coherent 699) provides radiation
at a wavelength of λ = 605.977 nm to drive the relevant transition in the Pr3+ ions (see
figure 2). The spectral resolution is limited by frequency jitter of the dye laser. The jitter rate
is in the order of a few MHz/millisecond. The radiation from the dye laser is split up in two
beam 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-optical
modulators (AOM, Brimrose BRI-TEF-80-50-.606) in double-pass configuration. Due to the
nonlinear response of the AOM radio frequency drivers to the modulation waveforms supplied
by waveform generators (Agilent 33220A) the temporal shape of the pump pulse is not exactly
gaussian but slightly distorted. However, the temporal variation of the pump intensity is
smooth and therefore suitable to drive adiabatic processes.
The two beams are counter-propagating with parallel linear polarization and overlap in
the Pr:YSO sample (see figure 5). The polarization direction is chosen in the direction of
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J Klein et al
3.5
no chirp
with chirp
LIF Signal/a.u.
3.0
2.5
2.0
1.5
1.0
0.5
0.0
−2
−1
0
1
2
3
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 of
chirped excitation the detuning is defined as the deviation of the pump laser centre frequency from
resonance. The peak intensity of the pump pulse is IP = 18.8 W cm−2 . This corresponds to
a peak Rabi frequency P = 2π × 470 kHz (using the electric dipole moment of the relevant
transition in Pr3+ ions µ = 2.6 × 10−32 Cm [28]) The data clearly show the enhancement in the
transfer efficiency in the case of RAP, i.e. chirped excitation.
maximum absorption in the Pr:YSO crystal. The diameter (FWHM of intensity) of the beam
for preparation and coherent excitation at the position of the sample is 565 µm, while the
diameter of the probe beam is only 430 µm. For the intense beam, laser powers of up to
90 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 a
silicon photodiode for absorption measurements. Additionally, laser-induced fluorescence can
be observed in the setup. Fluorescence emitted under a slight angle with respect to the laser
propagation axis is deflected by a prism, imaged onto the entrace slit of a monochromator, set
to a wavelength of approx. 620 nm, and detected by a photomultiplier (Hamamatsu R7400U02). To enhance the suppression of scattered light, induced by the excitation laser, a bandpass
filter (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 of
the 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 the
LIF signal recorded versus the detuning of the (centre) frequency of the pump pulse from
resonance, 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 the
case 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 profile
and fluctuations of the laser frequency lead to an averaged maximum transfer efficiency of
50%. This is also the value expected for strong, incoherent excitation. Thus the RAP process
should lead to an enhancement of a factor of 2 with respect to the case of resonant excitation.
Rapid adiabatic passage in a Pr3+ :Y2 SiO5 crystal
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However, the enhancement, observed in our experiment, is even a bit larger. This implies that
the 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 rectangular
temporal shape and a duration of 20 µs was used in this particular experiment to yield stronger
signal for the case of resonant excitation. Due to jitter in the laser frequency, the excitation
process, driven by the pump laser is not always resonant. Therefore also off-resonant excitation
may occur, i.e. CPR takes place at the end of the pump pulse, removing population from the
excited state. As data acquisition implies averaging over several excitation cycles, the transfer
efficiency at resonant excitation will average to a lower value. However, for a rectangular
pulse shape the sudden change in pump laser intensity induces diabatic coupling and therefore
perturbs the CPR process. This gives rise to a stronger fluorescence signal. Moreover,
simulations show that for the given experimental parameters the population transfer efficiency
of RAP is barely affected.
Due to the limited temporal response of the AOM the evolution of the pump laser intensity
is 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 ground
state. 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 the
expected factor of 2. Moreover, the resonant excitation is not saturated in the weaker regions
of the spatial pump laser profile. In contrast, RAP suffers less from these variations in the
laser intensity, although strong excitation is also needed. This also results in a higher ratio of
LIF yield for the two excitation methods.
Though a precise, absolute calibration of the transfer efficiency is difficult, the significant
enhancement of the LIF signal for chirped excitation already exhibits striking evidence for
efficient 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 detection
suffers from both CPR and excitation in the weaker regions of the pump laser. This problem
can be overcome in a setup, involving absorption measurements of an additional probe laser. If
the spatial profile of this probe laser is smaller than the profile of the pump laser, it is possible to
monitor the population transfer process in the intense centre of the spatial pump laser profile
without contributions from the weaker wings. Moreover, the absolute transmission of the
probe laser, detuned from the resonance, can be used for absolute calibration of the transfer
signal. 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 with
respect to the pump pulse, is recorded versus the detuning of the probe laser frequency from
resonance with the anti-hole. Thus the probe laser excites the same transition as the pump
laser. In contrast to the pump laser, the probe laser is weak, i.e. well below the saturation
intensity.
No absorption by the Pr3+ ions occurs, if the probe laser is sufficiently far detuned from
resonance. Using this signal level as reference, i.e. transmission equals unity, Beer’s law
permits to calculate the absorption coefficient. The absorption on resonance is proportional
to the difference of population in the ground and excited state, i.e. n1 − n2 . Hence, we relate
the observed absorption with respect to absolute transfer efficiencies. From the absorption the
relative excited state population P2 can be calculated by
α
1
1−
,
(6)
P2 =
2
α0
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J Klein et al
(a)
(b)
(c)
Figure 7. Absorption coefficient versus probe laser detuning. The probe laser is delayed by
17 µs with respect to the pump pulse. (a) The pump laser pulse is switched off. (b) The pump laser
pulse 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 to
a Rabi frequency P = 2π × 530 kHz. In the case of RAP (c) the absorption coefficient becomes
negative, i.e. the probe laser is amplified. This is clear evidence for population inversion, driven
by 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 laser
pulse is present. The probe laser monitors the population, prepared in the initial state by the
preparation 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 probe
laser driven resonance. If the pump laser pulse is switched on and set with fixed frequency
on resonance in the two-level system, a significant amount of population is transferred to the
excited state (see figure 7(b)). In the ideal case of strong excitation, i.e. saturation, half of
the population should be driven to the excited state. In this case stimulated emission and
stimulated absorption are equally strong. The absorption coefficient is expected to approach
zero. The experiment shows, that the absorption coefficient for resonant excitation is indeed
significantly reduced with respect to the case of the pump laser switched off. However, it does
not reach zero, as the resonant excitation seems to be not saturated. We attribute this to the
influence of CPR, which is stronger than for the experiment of section 4.1, as we now use
smooth 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 clearly
exceeds the limit of 50% in the case of incoherent excitation. We note, that the transfer
efficiency corresponds to the population at the time of the probe laser pulse, which is delayed
with respect to the pump pulse. The radiative decay starts to reduce the population in the
Rapid adiabatic passage in a Pr3+ :Y2 SiO5 crystal
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Figure 8. Time-resolved observation of RAP. The population transfer efficiency calculated from
the absorption of a probe pulse (duration 1 µs, intensity IP r = 3.8 mW cm−2 )) is plotted versus
the time delay with respect to the pump pulse. Data are monitored for resonant excitation at fixed
frequency (open circles) and chirped excitation, i.e. RAP (solid triangles). The solid lines are
exponential decay functions with the lifetimes given in the graph. The peak intensity of the pump
pulse 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 after
the 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 process
was measured. By changing the delay between the pump pulse and the probe pulse the excited
state 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 dynamics
can be well resolved during the coherent interaction process. Figure 8 shows the transfer
efficiency versus the delay between probe and pump pulse. The efficiency was calculated
from absorption measurements for the case of resonant excitation at fixed frequency (open
circles) and chirped excitation, i.e. RAP (solid triangles). For resonant excitation, a maximum
transfer of about 35% is observed during the interaction process. The efficiency is reduced
to below 20% towards the end of the pump pulse. Again, off-resonant excitation due to jitter
in the laser frequency and, consequently, CPR prevents the observation of saturation. The
effect 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 excitation
process.
We note that also some population remains in the excited state after the interaction, which
originates from decay, dephasing, and near resonant diabatic couplings. The decay of the this
residual 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. We
applied density matrix calculations including the decay of the excited state population and
dephasing. For CPR (figure 9(a)), the simulation clearly reproduces the characteristics of the
experimentally observed population dynamics for fixed pump laser frequency (see above).
For chirped excitation, i.e. RAP the transfer process starts slightly later than for resonant
excitation. This is due to the fact, that the excitation process in RAP is off-resonant in the
beginning. When chirping over the resonance, the excited state population shows a rapid
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J Klein et al
(a)
(b)
Figure 9. Numerical results for population dynamics of (a) coherent population return and (b) rapid
adiabatic 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 linear
chirp 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 the
limited 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 by
fitting an exponential decay (figure 8) to the data. This substantial reduction compared with
the expected lifetime of 164 µs is due to amplified spontaneous emission (ASE), which occurs
in the system, driven to population inversion. The decay of an ion leads to stimulated decay
of 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 propagation
axis of the pump laser light. ASE is emitted in both longitudinal directions into a small solid
angle defined by the pump beam diameter and the crystal length. Therefore, for delayed
probing ASE may produce a signal directly after the excitation process, although only light
counter-propagating with respect to the pump pulse is detected and no probe pulse is present at
that 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 RAP
process was implemented with laser beams of reduced diameters, i.e. 350 µm for the pump
and 175 µm for the probe beam. Figure 10 shows the ASE pulses, obtained from two single
experimental cycles, i.e. two ‘single-shot measurements’. In both cases an ASE pulse is
observed towards the end of the pump pulse, i.e. after the transfer process. The intensity of
the two pulses is different, as ASE is initiated by a spontaneous process and therefore the
efficiency fluctuates.
The explanation of the observed radiation pulses by ASE is also confirmed by the
transmission (or amplification) of the delayed probe pulse (see figure 10). ASE competes
with the probe pulse for the population inversion. Thus, when the ASE pulse is strong, the
probe pulse is less amplified, as ASE reduces the population inversion. In contrast, when the
ASE pulse is weak, the probe beam is better amplified.
Hence, the amplification detected by the probe pulse, from which we can infer the excited
state population, is fluctuating. Therefore, single shot data are not reliable to determine the
Rapid adiabatic passage in a Pr3+ :Y2 SiO5 crystal
S357
Figure 10. Observation of amplified spontaneous emission (ASE). The traces show two single-shot
measurements of the temporal profiles of an ASE pulse and the transmitted (rp. amplified) probe
laser pulse. Also the temporal evolution of the pump pulse is indicated (black dotted line). ASE
occurs at the end of the interaction with the pump pulse. The intensities of the ASE pulse and
of the transmitted (rp. amplified) probe pulse are anti-correlated. The peak intensity of the pump
pulse 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, ASE
corresponds to an additional decay mechanism which reduces the observed lifetime as seen in
section 4.3.
5. Conclusion
We implemented, observed and extensively studied rapid adiabatic passage (RAP) in a doped
solid, i.e. Pr:YSO by laser-induced fluorescence, absorption spectroscopy and observation
of amplified spontaneous emission. By these different techniques we collected striking
data for the preparation of coherently driven complete population inversion in the doped
solid. We observed amplification of a probe laser pulse by stimulated emission as well
as amplified spontaneous emission in the inverted medium. Time-dependent absorption
(or amplification) measurements revealed the population dynamics of RAP, in very good
agreement with theoretical predictions. RAP was successfully implemented in our experiments
with laser pulses of different temporal profiles. Even pulses with rectangular temporal shape
permitted efficient adiabatic population transfer, irrespective of residual diabatic couplings.
We compared RAP with resonant, coherent excitation at fixed frequency. In this case the data
show good agreement with predictions, also involving excitation processes based on coherent
population return (CPR).
Our investigations serve as another small, but hopefully valuable, step towards the
understanding of robust adiabatic interactions in solid media. The combination of coherent
excitations (such as RAP) and media for high-density optical data storage (such as rare-earth
solids) will provide exciting possibilities for the implementation of quantum information
processing in the future.
S358
J Klein et al
Acknowledgments
We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG), technical support
by Jochen Klein, and most valuable discussions with Klaas Bergmann.
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