Transcript Single-spin detection and quantum state readout by magnetic

Single-spin detection and
quantum state readout by
magnetic resonance force
microscopy
Goan, Hsi-Sheng
管
希
聖
Department of Physics
National Taiwan University
Silicon-based quantum computing
Two interactions: hyperfine
and exchange interactions .
Determining the strength of
these two interactions as
function of donor depth, donor
separation and surface gate
configuration and voltage.
•
•
•
L.M. Kettle, H.-S. Goan, S.C. Smith, C.J. Wellard, L.C.L. Hollenberg and C.I. Pakes, “A numerical
study of hydrogenic effective mass theory for an impurity P donor in Si in the presence of an electric
field and interfaces'', Physical Review B 68, 075317 (2003).
C.J. Wellard, L.C.L. Hollenberg, F. Parisoli, L.M. Kettle, H.-S. Goan, J.A.L. McIntosh and D.N.
Jamieson, “Electron exchange coupling for single donor solid-state spin qubits”, Physical Review B
68, 195209 (2003).
L.M. Kettle, H.-S. Goan, S.C. Smith, L.C.L. Hollenberg and C.J. Wellard, ”Effect of J-gate potential
and interfaces on donor exchange coupling in the Kane quantum computer architecture'', Journal of
Physics: Condensed Matter 16, 1011 (2004).
• C.J. Wellard, L.C.L. Hollenberg, L.M. Kettle and H.-S. Goan, “Voltage control of exchange
coupling in phosphorus doped silicon”, Journal of Physics: Condensed Matter 16, 5697 (2004).
• L. M. Kettle, H.-S. Goan, and S. C. Smith, “Molecular orbital calculations of two-electron
states for P donor solid-state spin qubits”, cond-mat/0512200.
Quantum gate operation, and quantum
algorithm modelling
CNOT
•
•
•
•
•
C. D. Hill and H.-S. Goan, “Fast non-adiabatic two-qubit gates for the Kane
quantum computer”, Physical Review A 68, 012321 (2003).
C.D. Hill and H.-S. Goan, “Comment on Grover search with pairs of
trapped ions“, Physical Review A 69, 056301 (2004).
C.D. Hill and H.-S. Goan, “Gates for the Kane quantum computer in the
presence of dephasing”, Physical Review A 70, 022310 (2004).
C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree,
and H.-S. Goan, “Global control and fast solid-state donor electron spin
quantum computing”, Physical Review B 72, 045350 (2005).
C.D. Hill and H.-S. Goan, “Fast non-adiabatic gates and quantum
algorithms on the Kane quantum computer in the presence of dephasing”,
AIP Conference Proceedings Vol. 734, pp167-170 (2004).
Mesoscopic qubit and
quantum measurement theory
Coupled quantum
dots measured by a
point contact detector
•
•
•
•
H.-S. Goan, “Quantum measurement process of a coupled quantum dot system”, invited
book chapter in “Trends in Quantum Dots Research”, Ed. By P.A. Ling (Nova Science, New
York, 2005) pp.189-227
H.-S. Goan, “An analysis of reading out the state of a charge quantum bit”, Quantum
Information and Computation 3, 121-138 (2003).
H.-S. Goan and G. J. Milburn, “Dynamics of a mesoscopic charge quantum bit under
continuous quantum measurement”, Physical Review B 64, 235307 (2001).
H.-S. Goan, G. J. Milburn, H. M. Wiseman, and H. B. Sun, “Continuous quantum
measurement of two coupled quantum dots using a point contact: A quantum trajectory
approach”, Physical Review B 63, 125326 (2001).
Monte Carlo method for a quantum
measurement process
Superconducting
Cooper-pair box qubit
measured by a single
electron transistor
•
•
•
H.-S. Goan, “Monte Carlo method for a quantum measurement process by a
single-electron transistor”, Physical Review B 70, 075305 (2004).
T.M. Stace, S.D. Barrett, H.-S. Goan and G.J. Milburn, “Parity measurement of
one- and two-electron double well system”, Physical Review B 70, 205342
(2004).
H.-S.Goan, “Monte Carlo method for a superconducting cooper-pair-box charge
qubit measured by a single-electron transistor”, in “Quantum Computation: solid
state systems” ed. by P. Delsing, C. Granata, Y. Pashkin, B. Ruggiero and P.
Silvestrini, (Springer, New York, 2005) pp.171-179.
Quantum electromechanical systems
• Single Fullerene (C60)
molecular transistor
• Shuttle systems
(Park et al., 2000)
(Shekhter et al., 2003)
• D.W. Utami, H.-S. Goan, and G.J. Milburn, “Transport properties of a quantum
electromechanical system”, Physical Review B 70, 075303 (2004).
• D.W. Utami, H.-S. Goan, C.A. Holmes and G.J. Milburn, “Quantum noise in the
electromechanical shuttle”, cond-mat/0509748.
• J. Twamley, D.W. Utami, H.-S. Goan, and G. J. Milburn, “Spin detection in a
quantum electromechanical system”, cond-mat/0601448.
Quantum jump transitions in
mesoscopic mechanical systems
Investigate the possibility of
experimental observation of
quantized energy state
(discrete Fock state)
transitions (i.e., quantum
jumps) in a nano-mechanical
oscillator.
(Curtsey of Roukes's group at Caltech)
• D.H. Santamore, H.-S. Goan, and G.J. Milburn and M.L. Roukes,
“ Phonon number measurement of a quantum electromechanical system”,
Physical Review A 70, 052105 (2004).
Resonator and cold ion system
W.K. Hensinger, D.W. Utami, H.-S. Goan, K. Schwab, C. Monroe, and G.J.
Milburn, “Ion trap transducers for quantum electromechanical oscillators”, Physical
Review A (Rapid Communications) 72, 041405 (2005).
Quantum feedback control in circuit QED
Nature 431, 162 (2004); A. Blais et al., Physical Review A 69, 062320 (2004)
• M. Sarovar, H.-S. Goan,
T. P. Spiller, and G. J.
Milburn, “High fidelity
measurement and
quantum feedback
control in circuit QED” ,
Physical Review A 72,
062327(2005).
Single spin detection by magnetic
resonance force microscopy
D. Rugar
et al.,
Nature
430, 329
(2004):
• T.A. Brun and H.-S. Goan, “Realistic simulations of single-spin nondemolition measurement
by magnetic resonance force microscopy”, Physical Review A 68, 032301 (2003).
• G.P. Berman, F. Borgonovi, H.-S. Goan, S.A. Gurvitz, and V.I. Tsifrinovich, “Single-spin
measurement and decoherence in magnetic resonance force microscopy”, Physical Review
B 67, 094425 (2003).
• H.-S. Goan, and T.A. Brun, “Single spin measurement by magnetic resonance force
microscopy: Effect of measurement device, thermal noise and spin relaxation”, Proceedings
of SPIE, 5276, 250-261 (2004).
• T. A. Brun and H.-S. Goan, “Realistic simulations of single-spin measurement via magnetic
resonance force microscopy”, International Journal of Quantum Information 3, 1-9 Suppl.
(2005).
Research
• Solid-state quantum computing (impurity donors in semiconductors;
superconductor Josephson junctions; quantum dots)
• Theoretical mesoscopic physics
• Theoretical condensed matter physics
• Nano (Quantum) electro-mechanical systems
• Quantum measurement theory and quantum feedback control
• Single-spin and single-charge detection
• Spintronics
• …
People
• 3 Ph.D. students and 2 Master students
• Getting 1 postdoctoral fellow
Collaborators
•
•
•
•
Prof. Gerard J. Milburn (U. of Queensland, Australia)
Prof. Todd A. Brun (U. of Southern California, USA)
Prof. Jason Twamley (U. of Macquarie, Australia)
Former UQ Ph.D. students: Louise Kettle, Charles Hill, Dian Utami
Single-spin detection
• Single-spin measurement is an extremely important
challenge, and necessary for the future successful
development of several recent spin-based proposals for
quantum information processing.
• There are both direct and indirect single-spin measurement
proposals:
– Direct proposals: SQUID, MRFM
– Indirect proposal: Spin-dependent charge transport, spindependent optical transition.
• The idea behind some indirect proposals is to transform the
problem of detecting a single spin into the task of measuring
charge transport since the ability to detect a single charge is
now available.
• Magnetic resonance force microscopy (MRFM) has been
suggested as a promising technique for single-spin
detection [Sidles (’92), Berman et.al.(’02)].
• To date, MRFM technique has demonstrated with
D. Rugar’s group (’04)
Readout concept
Step 1:
Convert spin to charge
Step 2:
Measure charge
•
•
•
Spin magnetic moment: mB =
9.310-24 J T-1 is very small!
Use spin to charge conversion
with fast charge read-out
Apply magnetic field to split the
spin up and down by the
Zeeman energy with
appropriate dot potential.
Spin-to-charge conversion
B=0
B>0

DEZ
Use Zeeman splitting DEZ=gmBB

charge
SPIN UP
0
-e
time
N=1
charge
SPIN DOWN
0
-e
N=1
N=0
N=1
~G-1
time
B//
Single spin
readout
spin to charge conversion +
fast charge detection
EZ =
gmBB


….single spin
measurement ?
=
T
DRAIN
IQPC
+
Q
G
200 nm
MP R
SOURCE
Single spin detection by magnetic
resonance force microscopy
D. Rugar
et al.,
Nature
430, 329
(2004):
• T.A. Brun and H.-S. Goan, “Realistic simulations of single-spin nondemolition measurement
by magnetic resonance force microscopy”, Physical Review A 68, 032301 (2003).
• G.P. Berman, F. Borgonovi, H.-S. Goan, S.A. Gurvitz, and V.I. Tsifrinovich, “Single-spin
measurement and decoherence in magnetic resonance force microscopy”, Physical Review
B 67, 094425 (2003).
• H.-S. Goan, and T.A. Brun, “Single spin measurement by magnetic resonance force
microscopy: Effect of measurement device, thermal noise and spin relaxation”, Proceedings
of SPIE, 5276, 250-261 (2004).
• T. A. Brun and H.-S. Goan, “Realistic simulations of single-spin measurement via magnetic
resonance force microscopy”, International Journal of Quantum Information 3, 1-9 Suppl.
(2005).
Magnetic resonance imaging
• Magnetic Resonance Imaging (MRI) principle: if the
precessing frequency of magnetic moments in a uniform
magnetic field is driven on resonance by an external ac
magnetic field, the resulting signal reveals something
about the spin state of the magnetic moments and the
external magnetic environment in which they are placed.
• At least approximate amount of 1012 nuclear spins or 107
electron spins is required to generate a measurable MRI
signal (via conventional inductive detection techniques).
• Compared to MRI, MRFM technique provides
considerable improvements in sensitivity (minimum force
detectable) and spatial resolution.
Laboratory frame and rotating
reference frame
Laboratory frame
Reference frame
z
z
Bz
y
Bz 
B1
x

B eff

gm
B1
x
MRFM setup
• A uniform magnetic field in
the z-direction.
• A ferromagnetic particle
(small magnetic material)
mounted on the cantilever tip
producing a magnetic field
gradient on the single spin.
• As a result, a reactive force
(interaction) acts back on the
magnetic cantilever tip in the
z-direction from the single
spin.
Schematic illustration of MRFM
(John Sidles’s group at UW, Seattle, USA)
MRFM animation
http://www.almaden.ibm.com/vis/models/
models.html#mrfm
Mrfm.mpg
What is the use of MRFM?
MRFM combines four
different technologies to
serve as a sensing and
imaging device:
• 3-dimensional nondestructive magnetic
resonance imaging,
• atomic-level resolution
atomic force microscopy,
• mobile scanning probe
microscopy allowing insitu and direct observation,
• continuous observation or
readout technique.
• the direct observation of individual
molecules (or other nanoscale devices
or materials),
• in situ, in their native forms and
native environments,
• with three-dimensional atomic-scale
resolution,
• by a nondestructive observation
process.
Single-spin detection by MRFM
• But the required averaging time is
still too long to achieve the realtime readout of the single electron
spin quantum state.
• The ability to accomplish the
single spin magnetic resonance
detection at a spatially resolved
location would fulfil an important
requirement for many quantum
computation schemes.
•
D. Rugar et al., Nature 430, 329
(2004): demonstrated to
achieve a detection sensitivity
of a single electron spin.
Moreover, the ability to detect a single
nuclear spin would have tremendous
impacts on the fields of quantum
information processing, quantum
computation, data storage,
nanometre-scale electronics,
materials sciences, biology,
biomedicine, and etc.
MRFM CAI technique
• The interaction between the single spin and the
cantilever is rather weak.
• In the MRFM cyclic adiabatic inversion (CAI) , the
cantilever is driven at its resonance frequency to
amplify the otherwise very small vibrational amplitude.
• This is achieved by a modulation scheme using the
frequency modulation of a rotating radio-frequency
(RF) magnetic field in the x-y plane.
B1x  B1 cos[t  D (t )],
B1 y   B1 sin[t  D (t )].
The frequency modulation D  ( t ) is a periodic function
in time with the resonant frequency of the cantilever.
Spin-cantilever Hamiltonian
In the reference frame rotating with the RF field,
d
 B z  ˆ ˆ
ˆ
ˆ
ˆ
ˆ
H sz ( t )  H z  [ L   
D  ( t )] S z   1 S x  g m 
 ZS z ,
dt
 Z 
2
2
Hˆ z  Pˆ /(2 m )  m  m Zˆ / 2,
 L  g m B z / , Lam or frequency
 1  g m B1 / ,
R abi fre quency.
F o r    L , Hˆ sz ( t )  Hˆ z  f ( t ) Sˆ z   Sˆ x  2 Zˆ Sˆ z ,
w here f ( t ) 
d [ D  ( t )] / dt ,
  ( g m / 2) (  B Z /  Z ) 0 ,
  1.
Principle of single-spin
measurement I.
• If
2
2
  f ( t )  | f ( t ) / f ( t ) | , then the spin H am iltonian
changes w ith tim e adiabatica lly.
• In the case when the adiabatic approximation is exact, the
instantaneous eigenstates of the spin Hamiltonian in the
rotating reference frame of the RF field are the spin states
parallel or antiparallel to the direction of the effective
magnetic field
eff
B ( t )  ( , 0,  f ( t )),
denoted as
v (t ) ,
respectively.
• We define an operator Sˆ z for the component of spin along this axis.
Principle of single-spin
measurement II.
• Starting at a general initial spin state in the Sˆ z basis
 (0)  a   b 
• In the basis of the instantaneous Sˆ z eigen states of
 (0)  a eff v  (0)  b eff v  (0) ,
where a eff  a cos( 0 / 2)  b sin( 0 / 2),
z
a eff   a sin( 0 / 2)  b cos( 0 / 2),
 0   (0) initial angle between B eff (0)
and z-axis direction
B
eff
eff
tan[  ( t )] 
x
B x (t )
B
eff
z
(t )


f (t )
Principle of single-spin
measurement III.
• Following from the adiabatic theorem:
t
 ( t )  a eff v  ( t ) exp(  i    ( t ') dt ')
0
t
+ beff v  ( t ) exp(  i    ( t ') dt '),
0
where   ( t ) are instantaneous eigenvalues.
• Probabilities a eff
2
and
beff
2
remain the same at all times.
• This provides us with an opportunity to measure the initial
spin state probabilities at later times.
How do we measure these spin
state probabilities?
• The idea is to transfer the information of the spin state to
the state of the driven cantilever.
• In the interaction picture in which the state is rotating with
the instantaneous eigenstates of the spin Hamiltonian, the
spin-cantilever interaction can be written as:
2 Zˆ Sˆ  co s[ ( t )]
z
• The phase of the driven cantilever vibrations depends on
the orientation of the spin states.
• Numerical simulations (with reasonable parameters for
the CAI approximations) indicate that as the amplitude of
the cantilever vibrations increases with time, the phase
difference in the oscillations for the two different initial
spin eigenstates of Sˆ  approaches  .
z
Phases in <Z> for spin up and
down states
Parameters
• We chose our parameters based on those used by G. Berman
et. al. , J. Phys. A: Math. Gen. 36, 4417 (2003).
• These values are
(in arbitrary units):
• The frequency modulation (driving force):
Effective CAI Hamiltonian
 (t ) 
If | f ( t ) |, 
f '( t )
f (t )
f (t )  
2
d  (t )
, th en
dt
2
f (t ) ˆ ˆ
ˆ
  i ( H z  2
Z S z )  ( t ) .
 (t )
Measurement scheme and device
• The cleaved end of the fiber and the vibrating cantilever
form a cavity. As the cantilever moves, the resonant
frequency of the cavity changes.
• Because the time scale of the cantilever's motion is very
long compared to the optical time scale, we can treat the
effects of this in the adiabatic limit.
• The cavity mode is also subject to driving by an external
laser, and has a very high loss rate.
• In the bad cavity limit, the dynamics of field quadrature (x)
adiabatically follows that of cantilever position.
• Phase-sensitive homodyne measurement on the field
quadrature of the cavity mode by a fiber-optic
interferometer:
phases of the cantilever vibrations.
state of the single spin.
MRFM setup
• A uniform magnetic field in
the z-direction.
• A ferromagnetic particle
(small magnetic material)
mounted on the cantilever tip
producing a magnetic field
gradient on the single spin.
• As a result, a reactive force
(interaction) acts back on the
magnetic cantilever tip in the
z-direction from the single
spin.
Stochastic master equation
approach
• Consider various relevant sources of noise:
– cantilever in a thermal bath and interacting with the cavity mode
– cavity mode subject to driving by an external laser and interacting
with continuum of electromagnetic modes outside the cavity
• Develop a continuous measurement model: a stochastic
master equation represents the evolution conditioned on the
photocurrent measurement record.
• For numerical purpose, it is often easier to unravel the
master equation to a stochastic Schrodinger equation.
• Additional stochastic process represents a fictitious additional
measurement, whose outcome we average over to recover
the state which is conditioned on the actual measurement.
• Present some simulation results for the single-spin
measurement process.
Phases in <Z> for spin up and
down states
Photocurrent output
Trajectories for the
superposition state
The spins
quickly localize
onto either up or
down state, but
the cantilever
takes longer
time to register
this visibly.
Output Noise Spectrum
S N R ( m )  2 2 0 s
-1 /2
• The term independent of
frequency is the contribution
from the shot noise of the
photons.
• The blue curve is the ``backaction'' noise on the position
of the cantilever by the
radiation, due to the random
way in which photons bounce
off the cantilever.
• The red curve is the thermal
noise, due to the thermal
Brownian-motion fluctuation
of the cantilever.
High-frequency vibrational noise
• High frequency vibrational noise of the cantilever tip may
cause fast spin relaxation [Mozyrsky et al. (03)]
Laboratory frame
Reference frame
z
z
Bz
y
Bz 
B1
x

B eff

gm
B1
x
Mass-loaded cantilever
Chui et al. (03)
• Mass-loaded design to reduce the high
frequency vibrational noise
• Superconducting RF resonator
• Cantilever perpendicular to the sample
• Interrupted OScillating Cantilever-driven
Adiabatic Reversal (iOSCAR) protocol.
SmCo
Cantilever thermal noise spectrum
Effect of spin relaxation and
dephasing
T1   1 s
T 2   9 .8 ms
Interrupted OSCAR protocol
Df 
2 fcG m B
k  x peak
Sˆ z
Near the surface, the cantilever
frequency is affected
• not only by the presence of
the spins
• but also by the more
dominant electrostatic and
van der Waals forces.
To make the spin signal distinctive:
• periodically reverse the sign of the
frequency shift by interrupting the
microwave power for 1/2 cycle of the
cantilever vibration every Nint cycles.
• the spin signal fsig = fc/2Nint.
• Use a lock-in amplifier referenced to
fsig to demodulate the frequency shift
OScillating Cantilever-driven Adiabatic
and determine Df
Reversal (OSCAR) protocol.
Single-spin detection by MRFM
D. Rugar et al., Nature 430, 329
(2004): demonstrated to
achieve a detection sensitivity
of a single electron spin.
• But the required averaging time is
still too long to achieve the realtime readout of the single electron
spin quantum state.
• The ability to accomplish the single
spin magnetic resonance detection
at a spatially resolved location
would fulfil an important
requirement for many quantum
computation schemes.
• Moreover, the ability to detect a
single nuclear spin would have
tremendous impacts on the fields
of quantum information processing,
quantum computation, data
storage, nanometre-scale
electronics, materials sciences,
biology, biomedicine, and etc.
Recent experiments on MRFM
Improvements in detection
signal-to-noise ratio should
allow real-time quantum
state detection and
feedback control of
individual electron spins
Conclusion
•
•
•
Simulation results indicate that the single-spin
readout by MRFM is possible.
The parameters we assumed for the simulations
were somewhat optimistic (field gradient: 107 T/m vs.
105 T/m; T: 0.1K vs. 0.2K); Steady improvement in
these techniques, however, should make single-spin
measurement more efficient and effective.
To be a good quantum measurement, the effects
which can flip the spin must remain small.
Future work
•
Take into account
– effect of higher-order cantilever vibrational modes
– domain motion, thermal magnetic noise in the tip
– other modulation schemes, multiple spins
– quantum feedback control
Animation of single spin detection using
interrupted OScillating Cantilever-driven
Adiabatic Reversal protocol
Parameters and iOSCAR protocol
B 0 ( x , y , z )  B tip ( x , y , z )  B e xt zˆ   rf /  ;

2
1
 2.8  10 H zT ,
10
 rf
2
 2.96G H z ,
for B 0  106 m T , B e xt  30 m T , B1  0.3 m T , resonance slice: 250nm below the tip
1
f c  5.5K H z, k= 0.11m N m , x peak  16 nm , m B  9.3  10
Df 
2 fcG m B
k  x peak
Sˆ z ,
 f c  3.7  1.7 m H z;
 24
1
JT , G 
B0
x
 2  10 T m
5
1
B eff  ( G x , B1 , 0)
i O S C A R : f int  f c / 64  86 H z, B1 is turned off for T c / 2, f sig  f int / 2,
 relative phase of the spin and cantileve r is re ve rs ed  c aus ing frequency shift to reverse polarity
D f (t ) 
4

 f c A ( t ) sin ( 2  f s ig t )    ,
w here A ( t ) :  1 (random telegraph fuunction for extra sp in flips),
A ( t )  0,
[ A (t )]
2
 1,
freque n cy noise: 25m H z in 1-H z bandw i dth  signal averagi ng to detect the spin signa l.
L orentzian spectral density: S ( f ) 
4 m [ D f 1 ( t )]
1  4 
2
2
m
f
2
2
,
spectral w idth at half-m axim un  0.21H z , correlation tim e (rotating fram relaxa tion tim e)  m  760 m s
Probability
distribution
p(z) at a
range of
times.
Single spin detection by magnetic
resonance force microscopy
• H.-S. Goan, and T.A. Brun, “Single spin measurement by magnetic resonance force
microscopy: Effect of measurement device, thermal noise and spin relaxation”,
Proceedings of SPIE: Device and Process Technologies for MEMS, Microelectronics,
and Photonics III, 5276, 250-261 (2004).
• T.A. Brun and H.-S. Goan, “Realistic simulations of single-spin nondemolition
measurement by magnetic resonance force microscopy”, Physical Review A 68,
032301 (2003).
• G.P. Berman, F. Borgonovi, H.-S. Goan, S.A. Gurvitz, and V.I. Tsifrinovich, “Singlespin measurement and decoherence in magnetic resonance force microscopy”,
Physical Review B 67, 094425 (2003).