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www.sciencemag.org/cgi/content/full/1181193/DC1
Supporting Online Material for
Gigahertz Dynamics of a Strongly Driven Single Quantum Spin G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, D. D. Awschalom*
*To whom correspondence should be addressed. E-mail: [email protected]
Published 19 November 2009 on Science Express DOI: 10.1126/science.1181193
This PDF file includes:
Materials and Methods Figs. S1 to S5 References
Page 1
Supporting Online Material
Materials and methods
The sample is a type Ib single‐crystal diamond grown by high‐pressure, high‐temperature
methods by Sumitomo Electric. The short‐terminated coplanar waveguide (CPW) structures are
fabricated with optical lithography and lift‐off of Ti 10 /Pt 50 /Au 120 (in nm). We fabricated several of
these CPWs on similar Ib diamonds and found uniform and reproducible microwave characteristics. The
CPW creates oscillating, microwave‐frequency magnetic fields with local amplitudes of ~350 G peak‐to‐
peak along the [001] crystallographic direction during the pulses (Fig. 1D). The projection perpendicular
to the [111] NV symmetry axis manipulates the spin with a Rabi frequency of up to 440 MHz. Room
temperature operation in combination with the high thermal conductivity of diamond allows us to apply
on‐pulse powers to the sample in the range of a Watt without damaging the CPW or substantially
heating the sample. We also studied a number of different NV centers in different samples, and always
observed qualitatively similar behavior.
Pulsed microwave frequency radiation is generated using the interleave output channel of a
Tektronix AWG7102 running at 20 GS/s. The microwave signal is amplified an Amplifier Research 1000B
amplifier with an operational bandwidth of 0‐1 GHz. To extinguish reflections between the sample and
the amplifier, an additional 10 dB attenuator was inserted before the sample. The amplifier is specified
with 25 dBc harmonic distortion. To reduce the second harmonic we also inserted a low‐pass filter with
a cut‐off of 0.8 GHz (minicircuits VLF‐800+). The microwave pulses were continuously monitored in a
300 MHz oscilloscope using a directional coupler and an RF diode. Spectral properties of the pulses and
the amplifiers were characterized using an Agilent E4448A spectrum analyzer and the individual pulses
were measured using a Tektronix DPO 72004 digital capture oscilloscope that operates at 50 GS/s with a
Page 2
20 GHz bandwidth. Illumination is provided by a 532 nm continuous wave laser that was gated using an
acousto‐optic modulator (Isomet 1250c). Photoluminescence intensity (IPL) is measured using a home‐
built confocal microscope (S1). Individual measurements of IPL determine the final spin state after the
microwave pulses, and are repeated many (~105) times. The frequencies of the spin transitions at B=850
G are determined using continuous‐wave optically detected spin resonance (Fig. S1).
Isolation of a single NV center
To insure we are studying a single NV center, photon anti‐bunching was performed in the
Hanbury‐Brown‐Twiss geometry. In addition to the dark counts of our detector (Perkin‐Elmer AQR‐
SPCM‐13), there is background luminescence in the diamond as well as reflections of the laser off the
CPW edges near to the NV center under study (Fig. S2A). Since the NV center is close to the surface of
the sample, this effect is at a maximum. The uncorrected photon correlation data g(2)(τ) dips below 0.5,
which indicates we are studying a single quantum emitter (S2, S3) (Fig. S2B). When we correct g(2)(τ) for
the uncorrelated background light, g(2)(τ) goes to zero at τ=0 (Fig. S2C). We estimated the background
level from the photoluminescence intensity (IPL) near the NV center, both at deeper focus and at the
same focus but translated, and found that it comprises roughly 40% if the detected photons. As an
additional test that we are not collecting any spurious spin‐dependent luminescence, we rotated the
laser polarization and found that there is another NV center with a different dipole axis at roughly the
same X‐Y coordinates as the one we are studying, but at a focus ~2.2 μm deeper into the sample. Since
the focal resolution of our microscope is about 1.5 μm (the X‐Y resolution is about 400 nm), we reject
most of the light from the lower NV center due to geometry and the polarization of the laser. Also, the
coupling (both dipolar and exchange) between the spins of the two centers is negligible at the timescale
of our experiments. Nevertheless, we were concerned that it may contribute a small spin‐dependent
signal that would be a systematic error in our data. Therefore, in order to estimate the upper bound of
Page 3
this error, we performed a continuous wave spin resonance measurement at B=100 G (Fig. S2D). If the
lower NV center, which has a symmetry axis that is misaligned with the magnetic field, was contributing
spurious IPL to our measurements we would expect to see resonances near the zero‐field splitting of 2.87
GHz in addition to the two spin transitions for the NV center we are studying. These extra resonances
are not present within the signal‐to‐noise ratio, so we estimate that the contamination of the signal by
the lower NV center is below 3%. Moreover, since in our experiment we work at a much higher static
magnetic field, the spin‐dependent signal from the lower NV center is even further suppressed. We
finally note a systematic error due to mechanical drift of the sample during the measurement that we
estimate is less than 5%.
Calibration of spin rotation with adiabatic passage
In the case of strong‐driving or other potentially complex spin dynamics, it is critical to calibrate
IPL for each spin eigenstate to accurately determine the ms=0 population P during Rabi driving. We
employ an adiabatic passage (15) to produce a precise spin‐flip independent of the Rabi oscillations. An
adiabatic passage is performed by applying a moderate driving field with a frequency strongly de‐tuned
from the spin resonance. The frequency is then swept adiabatically through the resonance causing the
spin to flip. This procedure can be mapped to a Landau‐Zener transition (S4, S5) with the probability of
remaining in the ms=0 state given by the Landau‐Zener formula, ]/2[ 21
•
− fHExp π . Here H1 is the
amplitude of the microwave driving field, expressed in units of the Rabi frequency it produces on
resonance, and •
f is the Landau velocity, or sweep rate of the frequency. To verify our adiabatic
passage pulses reliably flip the spin, we measure IPL as we sweep across the ms=0 to ‐1 transition. For
this measurement, we hold H1 and the sweep interval constant at 29 MHz and 300 MHz respectively,
Page 4
and step the sweep duration to vary the Landau velocity (Fig. S3A). We fit the data to a simple
exponential decay (red line) and find a decay constant of 0.020±0.001 ns‐1, which agrees with the value
of 0.018 ns‐1 calculated from the Landau‐Zener formula. For t<150 ns (•
f =2 MHz/ns) the transition is
non‐adiabatic and is accompanied by oscillations in the transition probability that are expected when
using a finite sweep range. At t>150 ns the transition probability saturates to an adiabatic spin‐flip.
Next, we fix the sweep duration at t=600 ns (•
f =0.5 MHz/ns), even further in the adiabatic regime, and
perform a series of partial adiabatic passages where we stop at increasingly longer times to project the
spin and see the progress of the spin‐flip (Fig. S3B). As expected, the spin flips as the driving frequency
passes through the resonance. In all measurements we calibrate the measured IPL level with the value
measured after initialization (P=1) and IPL for a spin‐flip via adiabatic passage (P=0).
Microwave pulses
Figure S3 and S4 show the measured pulses of nominal width 0 through 10 ns in 0.5 ns
increments, used for square and Gaussian pulses, respectively. We note that the sampling rate used to
sequence the pulses is larger than the frequency of the driving field, so the actual duration can be
slightly longer than the nominal pulse width since the voltage must relax to zero. In addition, here we
employ Gaussian pulses to mitigate pulse edge effects, while they are typically used to avoid exciting
higher levels (S6, S7).
The Bloch‐Siegert shift
A well known correction to the rotating wave approximation is the Bloch‐Siegert shift, which
accounts for the counter‐rotating field to leading order (S8). The resonance frequency in the presence
of an oscillating driving field shifts by a factor of:
Page 5
⎟⎟⎠
⎞⎜⎜⎝
⎛+ 2
0
21
41
HH
. (S1)
Since the resonance shifts, we considered this effect in the analysis and interpretation of our data. First,
we note that this correction is essentially a way to extend the viable range of the rotating wave
approximation and demonstrate that, under weak or even moderate driving, the Bloch‐Siegert shift is
negligible. For H1 comparable to H0 which is the case of our measurements, the approximations used to
derive Eqn. S1 are themselves invalid, which is why we don’t see much change in the visibility.
Numerical simulations
The simulations have been performed for a spin S=1, not only taking into account the two levels
ms=0 and ms= ‐1 that are resonantly coupled by the oscillating field, but also ms=+1. All major
experimental details have been included in numerical modeling. The Hamiltonian of the NV center was
taken in the form (in the laboratory coordinate frame):
( ) )(10 tHSrSrHH XXZZ ++= , (S2)
where H0 accounts for all static splittings (longitudinal anisotropy and Zeeman terms). We take into
account that the Z‐axis (the symmetry axis of the NV center) is along the [111] direction, while the
oscillating driving field H1(t) is applied along the [001] direction. The oscillating field has both
components along the Z‐axis and along the X‐axis, described by the projections 3/1=Zr and
3/2=Xr . The simulations used the driving field H1(t) which was obtained directly from experiments
(measured by the digital capture oscilloscope). We checked the possible influence of the extra noise
introduced by the oscilloscope by filtering the experimental signal (cutting off the spectrum of the
Page 6
driving field above 1 GHz), but found that the filtering leaves the results practically the same, so that the
high‐frequency noise is negligible.
In order to investigate the effect of the pulse tails, we also performed the simulations using the
experimental pulses, but with the tail of the pulse cut off (i.e., with the driving field amplitude dropping
discontinuously to zero at the time equal to the nominal pulse width). Cutting the pulse tail leads to
noticeable, but not drastic, changes. All qualitative features of the spin dynamics mentioned in the text
persist for the pulses with the cut‐off tails.
Page 7
Fig. S1. Continuous‐wave electron spin‐resonance measurement used to establish the transition
frequencies between ms=0 either ms=‐1 (above) or ms=+1 (below). The solid red line is a fit to a
Lorentzian line shape.
Page 8
Fig. S2. (A) Spatial photoluminescence map of the sample with the NV center discussed marked with an
arrow. (B) Photon anti‐bunching measurement uncorrected for the background. The dashed line
indicates the level of g(2)(τ)=0.5, below which indicates a single quantum emitter. (C) Photon anti‐
bunching with a background correction. (D) Continuous‐wave spin resonance measurement at B=100 G.
Page 9
Fig. S3. (A) Plot of photoluminescence intensity as a function of the passage duration as described in the
text. The driving field is fixed at H1=29 MHz and the sweep range is fixed at Δf=300 MHz. The solid line
is a fit to an exponential decay. (B) Plot of P from measurements of partial passages through the
resonance. The bottom scale gives the total duration of the partial passage, and the top scale gives the
instantaneous frequency of the driving field for each time during the passage.
Page 10
Fig. S4. Square microwave pulses used in the experiment.
Page 11
Fig. S5. Gaussian microwave pulses used in the experiment.
Page 12
References:
S1. R. J. Epstein, F. M. Mendoza, Y. K. Kato, D. D. Awschalom, Nat. Phys. 1, 94 (2005).
S2. Kurtsiefer, C., Mayer, S., Zarda, P. Weinfurter, Phys. Rev. Lett. 85, 290–293 (2000).
S3. A. Beveratos et al., Euro. Phys. J. D 18, 191 (2002).
S4. L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).
S5. C. Zener, Proc. R. Soc. A 137, 696 (1932).
S6. M. Steffen, J. M. Martinis, I. L. Chuang, Phys. Rev. B 68, 224518 (2003).
S7. L. M. K. Vandersypen, I. L. Chuang, Rev. Mod. Phys. 76, 1037 (2005).
S8. F. Bloch, A. Siegert, Phys. Rev. 57, 522 (1940).