Single-spin detection and quantum state readout by magnetic resonance force microscopy

Single-spin detection and quantum state readout by magnetic resonance force

microscopy

Goan, Hsi-Sheng

管希聖 Department of Physics

National Taiwan University

http://www.ntu.edu.tw/english/main.html

http://www.ntu.edu.tw/english/main.html

Silicon-based quantum computingTwo 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

• 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.

Superconducting Cooper-pair box qubit measured by a single electron transistor

Quantum electromechanical systems• Single Fullerene (C60)

molecular transistor

• Shuttle systems

• 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.

(Park et al., 2000)

(Shekhter et al., 2003)

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.

• 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).

(Curtsey of Roukes's group at Caltech)

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).

• 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).

Quantum feedback control in circuit QED

Nature 431, 162 (2004); A. Blais et al., Physical Review A 69, 062320 (2004)

Single spin detection by magnetic resonance force microscopy

• 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).

D. Rugar et al., Nature 430, 329 (2004):

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, spin-

dependent 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 conceptStep 1:Convert spin to charge

Step 2:Measure charge

• Spin magnetic moment: B =

9.310-24 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

EZ

Use Zeeman splitting EZ=gBB

time

charge

0

-e

SPIN UP

N = 1

N = 1 N = 1N = 0

SPIN DOWN

time

charge

0

-e

-1

Single spin readout

spin to charge conversion + fast charge detection

EZ = gBB

….single spin measurement ?

+

=

IQPC

DRAIN

SOURCE

200 nm M P R

Q

T

B//




• 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).

D. Rugar et al., Nature 430, 329 (2004):

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

x

y

zB

1B

z

xzB

g

1B

effB

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 non-destructive magnetic resonance imaging,

• atomic-level resolution atomic force microscopy,

• mobile scanning probe microscopy allowing in-situ 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 real-time 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.

D. Rugar et al., Nature 430, 329 (2004): demonstrated to achieve a detection sensitivity of a single electron spin.

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.

1 1

1 1

cos[ ( )],

sin[ ( )].x

y

B B t t

B B t t

The frequency modulation is a periodic function in time with the resonant frequency of the cantilever.

( )t

Spin-cantilever Hamiltonian

2 2

1 1

Lamor frequency

Rabi fre

ˆ ˆ ˆ/(2 )

quency.

/ 2,

/ ,

/ ,

z m

L z

H P m m Z

g B

g B

ˆ ˆ ˆˆ ˆ ˆ ( ) ( ) 2 ,o r ,F sz z xL z zH t H f t S S ZS

In the reference frame rotating with the RF field,

0

1

( ) [ ( )] / ,

( / 2)

where

( / ) ,

.Z

f t d t dt

g B Z

1ˆ ˆ ˆˆ ˆ ˆ( ) [ ( )] ,z

sz z L z x z

BdH t H t S S g ZS

dt Z

Principle of single-spin measurement I.

• 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

2 2 ( ) | ( ) / ( ) | spin Hamiltonian

changes

, then the

with tim adiabaticae lly.

f t f t f t

• We define an operator for the component of spin along this axis.

denoted as respectively.

• If

eff ( ) ( ,0, ( )),t f t B

( ) ,v t

ˆzS

Principle of single-spin measurement II.

• Starting at a general initial spin state in the basis

ˆzS• In the basis of the instantaneous eigen states of

where

initial angle between and z-axis direction

ˆzS

(0) a b

(0) (0) (0) ,eff effa v b v

0 0

0 0

cos( / 2) sin( / 2),

sin( / 2) cos( / 2),

eff

eff

a a b

a a b

0 (0) eff (0)B

eff

eff

( )tan[ ( )]

( ) ( )x

z

B tt

B t f t

z

x

effB

Principle of single-spin measurement III.

( )t

eff 0

eff 0

( ) ( ) exp( ( ') ')

+ ( ) exp( ( ') '),

t

t

t a v t i t dt

b v t i t dt

2

effa

• Following from the adiabatic theorem:

where are instantaneous eigenvalues.

• Probabilities and remain the same at all times.

• This provides us with an opportunity to measure the initial spin state probabilities at later times.

2

effb

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:

• 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 approaches

ˆˆ2 cos[ ( )]zZS t

.ˆzS

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

( ) ( ) ˆˆ ˆ( 2 ) ( ) .( )z z

d t f ti H ZS t

dt t

'(If th

) | ( ) |, en,

( )

f tf t

f t

2 2( ) ( )t f 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

• The term independent of frequency is the contribution from the shot noise of the photons.

• The blue curve is the ``back-action'' 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.

-1/2SNR( ) 220 sm

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

x

y

zB

1B

z

xzB

g

1B

effB

Mass-loaded cantilever

• 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.

Chui et al. (03)

SmCo

Cantilever thermal noise spectrum

Effect of spin relaxation and dephasing

1 1T s

ms2 9.8T

Interrupted OSCAR protocol

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 fr

equency 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 and determine f

2 ˆc Bz

peak

f Gf S

k x

OScillating Cantilever-driven Adiabatic Reversal (OSCAR) protocol.

Single-spin detection by MRFM

• But the required averaging time is still too long to achieve the real-time 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.

D. Rugar et al., Nature 430, 329 (2004): demonstrated to achieve a detection sensitivity of a single electron spin.

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

0 e

10 1 rf0 tip e

xt

1 24 1 0B

1

xt rf

for 106mT, 30mT, 0.3 resonance slice: 250nm below the tip

ˆ( , , ) ( , , ) / ; 2.8 10 HzT , 2.96GHz

5.5KHz, k=0.11mNm ,

,

16nm, 9.3 10 JT ,

T

2 2m ,

c peak

Bf

B x y z x y z

B

x G

B B

B

B z

5 1

1

int 1 sig int/ 64 86Hz, is turned off for

2 ˆ , 3.7

/ 2,

1.7mHz;

:

relative phase of the spin and cantilever is re

( , ,0)

i

ve

/ 2,

rs

O

2 10

SCA

R

ed aus

Tm

c

c Bz c

c

eff

c

peak

f f B T f f

f Gf S f

k x

x

Gx B

B

s

2

ig

where ( ) : 1 (random telegraph fuunction for extra sp

ing frequency shift to reverse polarity

4( ) ( )sin

in flips), ( ) 0, [ ( )

freque

(

n

2

cy noise: 25mHz in 1-Hz bandw

i

] 1,

) ,

dth

cf t f A t

A t A t A t

f t

1

m

2

2 2 2

4 [ ( )]Lorentzian spectral density: ( ) ,

spectral width at half-maximun 0.21Hz

signal averagi

, correlation time (rotating fram relaxation time) 760m

ng to detect the spin signa

l.

1

s

4

m

m

f tS f

f

Probabilitydistributionp(z) at a range of times.


• 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).



Documents

Single-spin detection and quantum state readout by magnetic resonance force microscopy