Coherent Control of Rydberg States in Silicon,
P. T. Greenland1, S. A. Lynch1, A. F. G. van der Meer2, , B. N. Murdin3, C. R.
Pidgeon4 B. Redlich2 N. Q. Vinh2,† , G. Aeppli1
1London Centre for Nanotechnology and Department of Physics and Astronomy,
University College London, WC1H 0AH, England
2FOM Institute for Plasma Physics “Rijnhuizen”, P.O. Box 1207, NL-3430 BE
Nieuwegein, The Netherlands
3Advanced Technology Institute, University of Surrey, Guildford GU2 7XH, England
4Heriot-Watt University, Department of Physics, Riccarton, Edinburgh, EH14 4AS,
Laser cooling and electromagnetic traps have led to a revolution in atomic physics,
yielding dramatic discoveries from Bose-Einstein condensation to quantum control
of single atoms1. Of particular interest because they can be exploited for quantum
control of one atom by another are excited Rydberg states2,3,4, where
wavefunctions are expanded from their ground state extents of below 0.1 nm to
several nm and even beyond; this allows atoms far enough apart to be non-
interacting in their ground states to strongly interact in their excited states. For
eventual application5, a solid state implementation is very desirable, and we
demonstrate here the coherent control of impurity wavefunctions in the most
ubiquitous donor in a semiconductor, namely phosphorous-doped silicon. Our
experiments take advantage of a free electron laser to stimulate and observe
photon echoes6,7, the orbital analog of the Hahn spin echo8, and Rabi oscillations
† Present address ITST, Department of Physics, University of California, Santa Barbara CA 93106-4170
familiar from magnetic resonance spectroscopy. As well as extending atomic
physicists’ explorations1,2,3,9 of quantum phenomena to the solid sate, our work
adds coherent THz radiation as a particularly precise regulator of orbitals in solids
to the list of controls, such as pressure and chemical composition, already familiar
to materials scientists10.
There are several approaches to atom trap physics in solids. One possibility is to
use quantum dots hosted by compound semiconductors11,12, which have the advantage
of addressability using conventional lasers. However, this relies on expensive
fabrication and experiments have been mostly restricted to the frequency domain. We
follow here another avenue, which requires more exotic laser technology but lends itself
to straightforward time-domain measurements and has much simpler sample
requirements. We use the ubiquitous semiconductor-donor combination Si:P , which
thus opens up the possibility of exploiting sophisticated semiconductor device
processing technologies for combined electrical and coherent optical control.
Our experiments exploit the remarkable fact that impurities with one more valence
electron than the host semiconductor display ground and excited states corresponding to
the free hydrogen atom Rydberg series, the touchstone for the Bohr formulation of
quantum mechanics. In the semiconductor13, the conduction bands, with curvature
characterized by the electron effective mass m*, play the role of the free electron
continuum, while the Coulomb interaction is reduced in proportion to its dielectric
constant εr. Thus, the binding energy scales with m*/εr2 while the orbital radii scale
with εr/m*. For typical donors in Si, the lowest energy Lyman series line is therefore in
the THz regime – in the case of phosphorus (P) the 1sA→2p0 transition is at 34.2meV,
equivalent to 36.2μm and 8.29THz (Fig 1a). There are also smaller level splittings
associated with the broken rotational symmetry in the solid. The orbitals have an order
of magnitude larger spatial extent than those for hydrogen in vacuum: the 2p0 level, for
example has an extent of ~ 10 nm, enclosing about 104 Si atoms, and is thus comparable
in size to transistors already in commercial use. Previous frequency14 and time-domain
studies15 have established the astonishing longevity of the excited states, with a
population lifetime T1 of 200 ps and corresponding oscillator quality factors of 2000 or
A two-level atom resonantly illuminated by the high intensity coherent light from
a laser undergoes Rabi oscillations at a frequency given by
where F0 is
the electric field envelope of the light beam and 12 is the transition dipole matrix
element. For a pulse of finite duration, the excited state polarization that remains in the
system after the pulse has passed varies sinusoidally with the pulse area,
=/F(t)dt. If the laser is at resonance with the 1s to 2p0 transition it will produce
a linear superposition of 1s and 2p0 wavefunctions – a very simple wavepacket which
oscillates in time as the superposition precesses around the Bloch sphere (see fig 1b),
representing the quantum mechanical state space for two-level systems. For an
ensemble, all the wavepackets initially radiate in phase, and therefore strongly, to
produce coherent radiation. This coherence is lost, owing to small offsets in the
resonant frequencies resulting from differences in the local environment, and the
radiation weakens as the dipoles dephase on a timescale given by the inverse of the 1s-
2p0 inhomogeneous linewidth, measureable in the frequency domain using conventional
continuous wave infrared spectroscopy. However, their relative phases can be restored
by a subsequent laser pulse leading to a second burst of coherent radiation – the photon
echo – which appears later by a time equal to the time difference between the initial and
rephasing pulses, in precise analogy to the well-known Hahn spin echo7 (fig 1c). In
general, the amplitude of the echo will decrease as function of time delay of the
rephasing pulse with a characteristic time T2, due to population decay and stochastic
phase jumps of the oscillators. The emission appears not only at a well-defined time,
but also at a well-defined angle: its direction is given by the vector equation
kE = 2k2 − k1 where k1 is the wave-vector of the pump, k2 is that of the rephasing
beam6. This means that a genuine echo, as opposed to more conventional four-wave
mixing effects where different coherent beams are present in the sample simultaneously,
has signatures in space and time. For our measurements to discover the orbital echo in
Si:P, we accordingly set out to establish both the direction of the echo beam relative to
the pump and rephasing beams (Fig 2a ANGLE) and also the arrival time of the echo
relative to the pump and rephasing pulses by interfering each of them with a reference
pulse (Fig 2a TIMING).
We performed most of our experiments on a Czochralski-grown 110 natural Si
wafer, of thickness 200μ and doped with 1.5 X 10 15 P donors/cm3, but also verified the
key results on several other samples, as described in the Supplementary Material. The
THz source was the FELIX free electron laser at the FOM institute in Nieuwegein16,
which produces trains of radiation pulses that for the present experiment were tuned so
that their frequency matched the 1sA2p0 transition of Si:P and had durations of ~10
ps, as also verified below.
The first experiment was simply to measure the angular distribution of intensity
emitted by the sample. We did this by recording the intensity while moving a small
mirror across the collecting parabolic mirror. Fig. 2b shows the resulting profile, which
does indeed show the expected echo peak at echo. There are also sharp maxima at
angles corresponding to the directions of pump and rephasing pulses.
Having shown the appearance of an echo signal with the correct wavevector, we
turn to verification of its arrival time, which we determine – taking advantage of the
coherence of the radiation from the free electron laser – using a reference pulse split
from the rephasing pulse and a delay line. The transmitted pump, rephasing and emitted
echo pulses, as well as the reference pulse are all focussed onto the detector through a
pinhole to produce a characteristic interference pattern in time. We exploit the angular
dispersion of the pump, rephasing and echo pulses and block all but one of them,
thereby obtaining the interference patterns of the reference beam with the pump,
rephasing and echo beams separately. By subtracting the mean intensity and squaring
the result, the arrival times and shapes of the pump, rephasing and echo pulses can then
be determined as a function of time (Fig. 2c). All three pulses take the form of well-
defined peaks, with the maxima occurring at the times (Fig. 2d) anticipated for echos.
We can now measure the dephasing time, T2, by observing the dependence of the
echo intensity on the time difference between the pump and rephasing pulses17. As
shown in Fig. 2e, the intensity of the echo decays exponentially as exp -12/Texp and
T2=4Texp, where the factor of 4 arises because the time between the emission of the echo
pulse and the pump pulse that caused it is twice the pump-rephasing pulse delay, and the
intensity of the echo decays twice as fast as the polarization amplitude.
The value of T2 at low laser intensity was 160 ±20 ps. The laser excitation
introduces additional sources of decoherence, and so we expect that T2 should fall as the
laser intensity increases. Fig 3a demonstrates this – a particular echo decay is shown in
the inset. The extra decoherence arises because electrons produced by multiphoton
excitation of some donors can collide with the un-ionized oscillators which produce the
echo, and so increase their dephasing rate. The effects of this were calculated (see
Supplementary Materials) using a two-level reduced density matrix18, including
photoionization and photoelectron collisions, which represents a slight extension of the
more usual Bloch equations for the spin echo. The spatial profile of the laser beam
must also be considered because it will induce a corresponding spatial distribution of
coherent excited state polarizations as well as incoherent processes. The density matrix
calculations predict the behaviour of Texp shown in Fig. 3a, and reproduce the
experimental observations well.
We have demonstrated orbital echos and long decoherence times for Si:P. To
determine how well we can actually control impurity wavefunctions, i.e. the extent to
which we can dial in coherent superpositions of different Rydberg states, we tracked
Rabi oscillations, in the standard echo detection mode, by measuring the magnitude of
the echo as a function of pump pulse area. Results for several pulse durations and
rephasing pulse areas were obtained (see supplementary material). Fig 3b shows that
the experimental echo intensity (magenta squares in the figure) actually displays one
complete oscillation, and agrees well with the theoretical prediction (magenta line),
which takes into account the decoherence mechanisms also needed to account for T2, as
above, and is indicated by the dashed line in fig. 3b. Fig. 3b also shows what would be
observed if the extra decoherence, and spatial intensity variation were absent.
Using the theory to fit many results of this type, and taking into account the beam
attenuation due to transmission through the cryostat window, and the Si air interface
gives a dipole matrix element, μ12=0.28 ± 0.03 nm. This is much lower than the scaled
hydrogenic value, a consequence of the central cell correction19, but comparable to
values derived from low field absorption measurements20, which give values in the 0.33
- 0.5 nm range. Since our value relies on an absolute measurement of the FELIX pulse
energy – notoriously unreliable at THz frequencies – the agreement is satisfactory. For
the photoionization cross-section from the excited state we find σ=1.2810-20 m2, about
twice what would be expected from hydrogenic scaling, and for the collision cross-
section for free electrons with un-ionised donors we obtain σe= 810-16 m2. This is
similar to electron-donor recombination cross sections in Si21.
We have directly observed photon echoes and Rabi oscillations produced by
coherent optical excitation of phosphorus donors in silicon with intense THz pulses
from a free-electron laser. Fig 4, which compares Si:P to an isolated H atom,
summarizes the key parameters that we have deduced from our experiments. All
electromagnetic parameters scale to within factors of two to what is expected based on
the Bohr model with renormalized effective mass and dielectric constant, except the
1s2p dipole matrix element, which is affected by central cell corrections. This
includes not only the Rydberg series itself but also the photoionization cross-section of
the 2p0 state. The only substantive differences between free hydrogen and Si:P are then
the much shorter T1 and T2 times for the latter, due to phonons which are characteristic
of a solid, but not of the vacuum.
Our work shows that we can prepare coherent mixtures of different orbital states
for one of the most common impurities in the most common semiconductor. These
mixtures have dephasing times T2 in excess of 100 ps, three orders of magnitude larger
than the 100 femtoseconds corresponding to the frequency of the transition between the
orbital states. The frequency domain linewidth associated with our measured T2 of 160
ps is 8.2 μeV. This is about twice the linewidth reported for P in isotopically pure float
zone Si14, so there is reason to believe that more carefully prepared samples will have a
longer T2 than the Czochralski Si used here.
Coherent control of donor orbitals in silicon opens up many possibilities currently
under examination using atom traps1,2,3, such as entanglement of pairs of impurities
whose ground state wavefunctions are too compact to interact. Modern nanotechnology,
which has recently been used for deterministic positioning of individual impurities in
silicon22, will also enable such control to be used for the regulation of magnetism5 by
opening and closing exchange pathways by the timed preparation of excited states.
Supplementary Information accompanies the paper on www.nature.com/nature
We are grateful for helpful conversations with A.J. Fisher, A.M. Stoneham, C. Kay and G. Morley, R.
Hulet for pointing out reference , and experimental assistance from K Litvinenko and G. Morley, and
the financial support of NWO and EPSRC.
NQV and CRP initiated this work, PTG, SAL, BNM, NQV, and GA designed the research programme;
NQV, PTG, SAL, LvdM and BR performed the experiments; PTG performed the theory and analysis;
PTG, BNM, SAL and GA wrote the paper.
Correspondence should be addressed to PTG (email@example.com).
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Fig 1(a) The spectrum of an isolated P donor in Si. The primary excitation and
two-photon ionization paths are shown in red; dephasing paths – photoelectron
collisions and phonon decay – are shown in blue. (b) The Bloch sphere, with
some sample wavepackets. The ground 1sA state is at the south pole, the 2p0
excited state at the north pole. Round the equator we show how the
wavepacket varies as the relative phase of a 50-50 mixture evolves in time. This
is the time dependent combination we excite to make the photon echo. (c) The
classic Hahn sequence, and the corresponding behaviour of the Bloch vector.
Ideally the first pulse has an area of π/2, and the second an area of π.
Fig 2 (a) Schematic showing the experimental geometry. The pump (k1) beam
(green) and rephasing (k2) beam (blue) intersect in the sample (at ~5○ in the
experiment) and the echo kE (orange) is emitted in the direction 2k2-k1.
Although the paths are shown correctly, their total lengths are not drawn to
scale. (b) The intensities of the spatially dispersed signals are recorded by
translating the detector across the far-field. Simple geometry enables the spatial
dispersion to be converted to an angle of emission to show that kE = 2k2 - k1 as
predicted (c) Result of cross-correlation of a reference beam (k*) (dashed) with
the pump k1 (top), the rephasing pulse k2 (centre) and the echo kE (bottom) by
interfering them on the detector. The abscissa is the arrival time of the
reference pulse k* in ps, relative to an arbitrary zero. On the left is the detector
signal showing the interference pattern. A moving average has been
subtracted, in order to remove the background and laser drift. The pump,
rephasing and echo temporal profiles can be obtained from the square of these
interference patterns. Gaussian fits of the pump and rephasing pulses had full-
widths-at-half-maximum intensity of 7.7±1.5ps, consistent with the inverse of the
spectral width of 0.28%, showing that the pulses are indeed band-width limited.
The echo duration, 27.8±7.6ps, is somewhat longer than would be expected
from the measured inhomogeneous frequency domain line-width (~ 200 μeV).
(d) The echo arrival time τ2E as a function of τ12 derived in the same way for
several different pump areas, showing that, within experimental error, the echo
arrives when expected. Finally (e) shows a set of echo profiles for different
pump-rephasing pulse delays τ12. The ordinate is a very crude estimate of the
echo power in W; the temporal width of the echo implies an inhomogeneous
linewidth of ~400 μeV, suggesting the sample is strained. The black line is an
exponential fit, with time constant Texp =28.4 ps showing that the echo intensity
falls with τ12, as expected.
Fig. 3. (a) The echo dephasing lifetime as a function of pump pulse area AP for
pump:rephasing pulse area ratio fixed at 1:2 and pulse duration of 5.9 ps (red
points), and for rephasing pulse area fixed at 0.96 π for a pulse duration of 7.5
ps (green). The red and green lines show the corresponding results from the
density matrix calculation described in the text. The blue line illustrates the
effect on the red line of changing the radial profile of the laser beam to a top-hat
function. The inset shows a typical echo signal, detected via momentum
selection (“ANGLE” in Fig 2a) with a constant background subtracted, as a
function of the delay 12 between pump and rephasing pulses. The data show a
simple exponential decay with decay constant Texp (just as the time-resolved
echoes of Fig 2e (b). Time-integrated photon echo signal S as a function of
pump pulse area (in units of 2π) for a rephasing pulse area of 0.54 π and a
pulse length of 6.79 ps. The dotted line is the uncorrected theoretical (Rabi)
result. The black line shows the prediction including the non-uniform spatial
profile of the laser beam, and the magenta line includes the effect of both
photoionization and the profile. The experimental results for the same
conditions are shown as points, with the ordinate normalized by the factor of 1.3
(determined from the global fit of many similar experiments with different pulse
lengths and rephasing pulse areas - see supplementary material).
Fig 4. Comparison between H and Si:P, showing spectra and the principal
excitation and decoherence mechanisms. The crystal environment leads to an
asymmetric effective mass, so that the H 2p state splits into a 2p0 and 2p± state
(green dashed line). Additionally, the crystal field further splits the H-like levels
into states of cubic symmetry – the splitting of the 1s level into its A, T and E
components is shown, but the splitting in the excited states is too small to show.
The main decay channel of the 2p0 state for Si:P, in contrast to H, is phonon
decay: furthermore photoionization of this state is also relatively much stronger
than in H, and since the photoelectrons are confined to the conduction band
they can cause decoherence through phase changing collisions.
The Dutch free-electron laser (FELIX) provides short coherent pulses of THz
radiation. The laser pulses come in bursts called macropulses with duration of 5μs and
repetition rate of 5 Hz. The macropulses contained trains of micropulses of duration ~
10 ps and 25 MHz repetition rate. Figure 2a is a schematic diagram of our experiment.
The incoming FELIX beam was directed onto a beam-splitter (BS1). The transmitted
beam which we call k2 travelled along a fixed optical path. The intensity of this beam
could be controlled by a set of calibrated wire-grid attenuators (A). The reflected beam,
k1 travelled down an optical path with controllable length (R1) to provide a variable
delay. It also passed through a variable attenuator consisting of a motorised polariser
(P1) and a fixed analyser (P2). The k1 beam emerged from the analyser parallel to, but
spatially separated from k2. All of these optical elements were contained in a closed
metal box which was evacuated and purged several times with dry nitogen gas and then
sealed. The spatially separated beams were reflected from a large off-axis parabolic
mirror to a focus on the sample, which was mounted on the cold-finger of a continuous
flow liquid helium cooled cryostat. The angular separation of the beams at the sample
was ~5 degrees. The emerging pump and re-phasing beams and the echo beam were
collected on a second off-axis parabolic mirror and collimated. A small mirror oriented
at 45◦ and mounted on a translation stage was used to pick off the different beams for
measurement of the echo angular position relative to the other beams. The light was
focussed onto a helium-cooled Ge:Ga detector (crystal size 3x3 mm2), by a third off-
axis parabolic mirror.
To observe the pulse arrival times each of the beams (k1, k2, and kE) were made
to interfere with a fourth, reference, pulse k* derived from a small portion of k2. For this
experiment we used the alternative detection setup shown in figure 2a TIMING; k* was
sent through a second optical delay line (R2) and then recombined with the echo beam
line through a second pellicle beam-splitter, and focused through a 0.4 mm diameter
pinhole at the detector. The pinhole facilitated interference of the beams by restricting
the number of fringes seen by the detector to one. The length of the optical delay was
adjusted to give an appropriate interval between the excitation pulse k1 and the
rephasing pulse k2. The optical delay line in the reference beam was scanned through its
full range while pairwise combinations of the three main beams k1, k2 and the echo were
To control the intensity of the excitation pulse k1we used the motorised polariser
shown in figure 2a. We calibrated the k1 intensity (in terms of pulse area) against the
orientation angle of the polariser using a double Malus’ Law (taking account of the
polarisation orientation of FELIX with respect to the first polariser, and then the
orientation of this polariser with respect to the analyser). We then recorded the intensity
of the THz signal in the direction of the echo as a function of calibrated pulse area. We
also recorded the intensity of the THz signal along the same direction when the
temporal delay between k1 and k2 was reversed (i.e. we changed the length of the optical
delay so that k2 arrives at the sample before k1). Since the echo is directional this
provided us with a measure of the background signal from the scattered excitation pulse
k1 and rephasing pulse k2. This was important because at high excitation powers the
scattered signal began to exceed the echo signal.
For the echo lifetime experiments k* was not used. Instead the total echo signal
was measured as the pump-rephasing delay was varied, and the pump intensity
controlled with the polarizer-analyser system. For some experiments both pump and
rephasing beam were sent through the polarizer-analyser system, so that the ratio of
their intensities was controlled by the beamsplitter (at 1:4). Similarly, the Rabi
oscillation experiments were done by fixing the pump-rephasing pulse delay, and
controlling the pump intensity with the polarizer-analyser, and the rephasing beam with
the wire attenuator. The data presented here were obtained using a 200 μm thick CZ
sample at a doping density of 1.5 X 1015 cm-3, but we have also observed echos in both
other CZ as well as FZ samples of broadly similar specification.
All our results can be described using a theoretical model which augments the
standard echo theory with an extra source of decoherence due to photoionization. The
basic information is in the two-level reduced density matrix (t), which represents a
very slight extension of the more usual Bloch equations. We have
Here is the 22 density matrix for the 2-level atom whose dynamics are being
considered. The parameters have the following significance: is the rate at which the
excited state population decays to the ground state — it is the reciprocal of T1; is the
decoherence rate, i.e. 1/T2 . The instantaneous laser coupling is given by Ω/2 with
where F is the (time-dependent) electric field envelope due to the laser and is the
1sA →2p0 dipole moment so that (t) is the instantaneous Rabi frequency. Of course,
is a function of time – it reflects the echo pulse sequence – though we have not
explicitly indicated this, and is the detuning of the laser from the atomic line centre.
Finally γ' is the rate of irreversible population loss from the upper level. For γ'=0 we
have the usual Bloch equations, but we relate γ' to the photoionization rate from level 2,
where σ2p0 is the photoionization cross section from the excited state, I(t) is the laser
intensity at time t and ε is the photon energy (so I(t)/ε is the photon flux). The
photoelectrons thus produced also represent a source of decoherence which adds to the
off-diagonal decay rate Γ, so that, for a photoelectron density ne given by ne = n0 pe,
where n0 is the donor density, and pe=1-ρ11-ρ22 is the electron ionization probability
Γ=Γ0 +ne ve σe=Γ0 + n0 pe ve σe
with Γ0 the intrinsic off-diagonal rate, ve the mean photoelectron velocity, and σe the
electron-donor scattering cross-section. This leads to an intensity-dependent echo decay
rate. Finally, to compare with experiment we must average over the spatial distribution
of the FELIX beam. This model can then be used as the basis of a fit to the experimental
profiles as a function of pulse area. Figure S1 shows the global comparison between the
theoretical and experimental results.
Fig S1 Comparison between theory and experiment. The upper panel shows
the value of the experimental maximum of the echo signal intensity (such as
that in fig 3b) (ordinate) as a function of the corresponding theoretical value
(abscissa). The best fit line through the origin has a gradient of 1.3±0.06, which
gives the best calibration of the experimental echo magnitude in terms of the
theoretical value as calculated above.
The lower panel compares the experimental value of the pump pulse area (in
units of 2π) for which the peak echo signal occurs (again taken from profiles
such as that shown in fig 3b) (ordinate) with the theoretical prediction (abscissa)
The line through the origin with gradient 1 is also shown. (It is not a fit.)
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