Quantum Spin Noise

1
Quantum Spin Noise
QUEST 2004

• Alexander V Balatsky

JX Zhu Z. Nussinov Yishai Manassen S. Crooker,
D. Rickel, D. Smith
M. Hawley, G. Brown, H. Grube, I. Martin
2
Introduction

• Manassen experiments (1989, 2000)
• Also Cambridge Group, Durkan et al, 2001
• Spin precession STM. Hyperfine resolved.
• Noise spectroscopy examples
• Noise spectrosopy of spin. Colored noise as a
  useful information.
• Faraday rotation experiments.
• Josephson effect in presence of spin, new spin
  dynamics
• Conclusion and predictions/extensions.

3
Other techniques trying single spin detection
Impossibility of a single spin detection for a
free spin (Stern-Gerlach expt), according to N.
Bohr, W. Pauli ,late 20s
For spin force gtgtLorenz would violate
diffraction limit

4
Other techniques trying single spin detection
Impossibility of a single spin detection for a
free spin (Stern-Gerlach expt), according to N.
Bohr, W. Pauli ,late 20s
For spin force gtgtLorenz would violate
diffraction limit
NMR on a single/few spins ? Magnetic resonance
force microsope, MRFM -J. Sidles, D. Rugar (100
spins), C. Hammel Optical detection (ODMR)
5
Not a direct Spin measurement
6
Experimental setup
Energy scales of the Problem
B 100-300 Gauss Idc 1 nA 10-9A Iac 1 pA
10(-12)A
No Zeeman splitting
About 20 electrons pass Per single spin cycle

7
Experimental setup
Energy scales of the Problem
B 100-300 Gauss Idc 1 nA 10-9A Iac 1 pA
10(-12)A
This setup works as an AC generator
8
!Nodal structure of the signal!
9
esr linewidth for BDPA is few gauss At room
temperature!
10
I 1, s ½ Lines for I 0,1,-1
Bulk measurement
11
CE conduction electron Linewidth 10 Gauss 30
MHz ltlt500Mhz
12
Can single spin be detected with spin unpolarized
current?
tip
Unpolarized surface current I
13
Can single spin be detected with spin unpolarized
current?
tip
Unpolarized surface current i

14
Can single spin be detected with spin unpolarized
current?
tip
Unpolarized surface current i
The way to couple current and spin is through
spin-orbit interactions of some (more than one)
kind
N(r,t) is a scalar with respect to time reversal
S is a pseudovector and odd under time reversal
15
Onsager symmery relation
Allowed combinations
16
Theoretical set up
Dynamics of the spin is and is not addressed
here.
Hamiltonian of the system
17
Can single spin be detected with spin unpolarized
current?

• Basic point Impurity spin ( large S) produces
  Friedel oscillations of spin due to screening.
• Friedel oscillations are distorted due to net
  current in the vicinity of spin.
• SO interaction produces the net energy shift
• due to coupling between spin polarization and
• current

18
Current densities in the STM tunneling are quite
high I1 nA current per 5Angtrom by 5 Angstrom
area J 0.4 106 A/cm2 Hence electron
distribution function is nonequilibrium
ky
ky
K
kx
kx
I gt0
I0
FS displaced by K m/ne I
NB smaller the density of carriers n the larger
the displacement K
19
S
Friedel spin oscillation near magnetic impurity
S
Electron spin oscillation near magnetic impurity
follows the direction of S
Spin Orbit interaction allows time dependent (or
static) spin polarization to influence orbital
motion of the carriers and DOS. Different kinds
of SO coupling will produce the effect.
20
M. Crommie Charge Friedel oscillations on
Cu surface
21
Next, consider the perturbation due to SO term
22
Rashba spin orbit term that appears on the
surface of semiconductors and metals
Evaluating the SO term on the current carrying
state exp(iKr) we find the effective magnetic
field acting on spins
Thus we find
23
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24
Cond-mat/0112407
25
Predictions from theory

• 2 line should be present. We clearly
  see it in simulations.
• I ac scales as second power of Idc
• Direction of magnetic field does not matter
• Effect exists even in a static spin case! Our
  calculation is valid for static spin as well.
  DOS effect best seen by looking at
• asymmetric DOS map
• Nodal structure of the signal. Nodal point for
  precessing spin,
• nodal plane for static spin

Dipolar pattern In DOS
-

-
26
DOS is affected even by a static spin!

• Nowhere in calculation we had assumed that spin
  has to be dynamic. It was a specific feature of
  experiment.
• Our calculation goes through even for static
  spin. For spin out of plane there will be a
  dipolar pattern for corrections to DOS. Absolute
  value

Dipolar pattern In DOS
-

-
27
Buttiker formalism
28
Spin back action and weak coupling
20 electrons per single precession cycle -gt weak
coupling between Spin and electrons. Small
parameter g_so J/E2_F ltlt1 Weak coupling allows
spin precess long compared with tuneling
rate electrons
G
tip
S
S
G
surf
Within factor of 10 from observed experimentally
29
Noise spectroscopy of spin
White noise sent through the system with peaked
susceptibility is filtered and results in colored
noise. Characteristic frequency is seen.
Hot environment small splitting random dynamics
of spin S
30
Stochastic spin noise the fluctuation-dissipatio
n theorem
Consider an ensemble of N uncorrelated
paramagnetic spins
But, fluctuations exist
In principle, then, all properties of the system
available from spin noise alone.
31
Fluctuation dissipation theorem
Noise spectrum
Dissipation spectrum
Are directly related
32
Noise spectroscopy
Penzias Wilson 1965
33
Cantielever noise in MRFM set up.
Thermally induced noise can be used to measure
Temperature, Q and eigenfrequency of resonator
without ever driving it, just monitoring
the noise.
J. Markert, UT Austin, unpublished.
34
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36
Time dependent magnetic field As picked up by
SQUID Another example of noise spectroscopy
M(t)
37
Another example of noise spectroscopy
Photocurrent modulation due to precessing spin
in
out
Transparency is modulated at Larmor frequency
38
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39
Listening to magnetization fluctuations (spin
noise)
an example using paramagnetic alkali vapors a
classical ensemble of N uncorrelated spins
S. Crooker et al, Nature, 2004 Cond-mat 0408107
dqF(t)
I45
rubidium or potassium N109 mm-3
dV(t)
spectrum analyzer
cw TiS laser
I-45
B (lt10 gauss)

• Alkali vapor (Rb, K) in thermal equilibrium
  (T350K). ltMz(t)gt0.

• Laser tuned near -- but not on resonance no
  absorption.

• Random magnetization fluctuations dMz(t)
  generate noise in Faraday rotation dqF(t).

• Measure spin correlation function, S(t)ltMz(0)
  Mz(t)gt, without perturbing system
• Spin ensemble always remains in thermal
  equilibrium
• (in contrast with conventional magnetic
  resonance)

40
Spectral density of spin noise
Peaks in spectrum of dqF(t) due to stochastic
fluctuations of ground-state spin Fluctuations
must precess in B spontaneous coherence
between Zeeman sublevels
B1.85 G
5 2P1/2
85Rb
30
Rb D1 transition 794.6 nm
Faraday rotation (nanorad/Hz1/2)
Spin noise (nV/Hz1/2)
87Rb
Atomic ground state
2
1
FI1/2
25
0
-1
5 2S1/2
-2
Dhyperfine
-1
0
FI-1/2
1
mF
Noise floor photon shot noise amplifier noise
Not very big! 1 nanoradian angle subtended by a
human hair (100mm) at 100 kilometers
41
Spin noise spectroscopy
remember, spin ensemble remains unperturbed in
thermal equilibrium
Integrated spin noise sqrtN gives isotope
abundance ratio
Spin fluctuations alone directly reveal
ground-state g-factor nuclear spin I
gFgJ /(2I1)
85Rb (467kHz/G) gF1/3, I5/2 87Rb (700kHz/G)
gF1/2, I3/2
42
Off-resonant Faraday rotation probes ltM(0) M(t)gt
Linteraction length, nlaser frequency, N0spin
density, Dn-n0 laser detuning, f oscillator
strength, b polarizability, ltJegt(N-N-)/2N
average electron spin polarization, and Abeam
area
Total measured spin noise as a function of
detuning from absorption resonance qF
couples to magnetization fluctuations
through dispersive indices of refraction
(no absorption at large detuning).
43
Spin noise scales with square root of particle
number
Fluctuations from N uncorrelated, precessing
spins sqrtN
integrated spin noise scales with sqrtN
Temperature-tune the spin density
Integrated spin noise (mV)
85Rb density (mm-3)
44
Spin noise increases when probe area shrinks
Vary beam area, keeping laser power constant
Potential for really small systems, with small N
Beam area
A
Beam area (mm2)
45
Spin fluctuations reveal ground-state hyperfine
coupling

• Increasing magnetic field spin noise broadens
  and breaks up into separate peaks
• -due to quadratic Zeeman effect Zeeman
  sublevels become unequally spaced
• Fluctuations generate coherence between every
  allowed DF0, DmF1 transition

39K
2
B0.81 gauss (W00.57MHz)
1
FIJ2
0
mF
-1
4 2S1/2
-2
2.88G (2.01 MHz)
Dhf1.90mV (461.7MHz)
Spin noise (nV/Hz1/2)
39K, I3/2
-1
5.27G (3.69 MHz)
0
dW
FI-J1
1
8.85 gauss (6.20 MHz)
-200 0 200
kHz from W0
46
Spin fluctuations also reveal nuclear magnetism
(mI, gI)
Coupling of electron to nuclear magnetic moment
Zeeman transitions within F2, F1
hyperfine levels are no longer exactly degenerate
2
1
F2
mF
0
-1
5 2S1/2
-2
Dhf6835 MHz
87Rb, I3/2
-1
0
F1
1
Additional splitting reveals nuclear moment
47
Spin fluctuations generate high-frequency
inter-hyperfine coherences
Spontaneous spin coherences also exist between
hyperfine levels
2
1
F2
0
mF
-1
4 2S1/2
-2
Dhf
-1
0
F1
1
39K (I3/2)
Dhf461.7 MHz
48
I 1, s ½ Lines for I 0,1,-1
Bulk measurement
49
Noise Spectroscopy
N
i1
When N1 signal is large

50
We believe that similar noise spectroscopy
is what is measured in STM on single/few
spins experiments
51
Fluctuation dissipation theorem
Noise spectrum
Dissipation spectrum
Are directly related
52
Relevant time scales
Imagine we have a situation of one electron per
cycle. Even if there is no net spin polarization,
there is a fluctuating temporal polarization on a
time scale relevant for spectrum at Larmor
frequency! Truly a noise spectrocsopy!
53
Temporal spin polarization of tunneling electrons
B z
Tip
Tunneling current electrons
S(t)
surface

54
Modulation of tunneling barrier by JS/U
55
Modulation of tunneling barrier by JS/U
Localized spin S(t)
56
Nonstationary white noise
S
t
S
57
Tunneling out of the tip withparamagnetic cluster
Basic idea is that if one has a paramagnetic
center (say easy axis) on the tip then tunneling
electrons will acquire a time dependent spin
polarization. We do not have a steady spin
polarization of tunneling Spins. Polarization
long enough on the scale of precessing S spin is
sufficient.
1/f spin noise, Manassen , AVB Cond-mat, to be
published
Cond electron peak
S peak
58
Cond-mat/ 0301032. PRB, B 68 (2003),
Main idea dc spin polarization of tunneling
electrons is all that is required to measure the
dynamics of a single spin
tip
polarized tunneling current I
Current dynamics reflects the spin Dynamics S/N
1
59
Spin back action and weak coupling
20 electrons per single precession cycle -gt weak
coupling between Spin and electrons. Small
parameter g_so J/U ltlt1 Weak coupling allows spin
precess long compared with tuneling rate electrons
G
tip
S
S
G
surf
Within factor of 10 from observed
experimentally S/N 1
60
1/f spin noise and noise spectroscopy
Basic idea is that we do not have a steady spin
polarization of tunneling Spins. Polaization
long enough on the scale of precessing S spin is
sufficient. Assume spin current out of the tip
has 1/f spectrum. if one has a paramagnetic
center (say easy axis) on the tip then tunneling
electrons will acquire a time dependent spin
polarization.
AVB and Manassen Israel Jopurnal of Chemistry,
2004
Cond electron peak
S peak
61
1/f noise in unpolarized STM current
62
Phil Mag B82, p. 1291, (2002) PRB, submitted.
63
spectrum
T gtgtTK
T TK
T ltltTK
frequency
Line shifts because of spin renormalization Due
to Kondo effect?? A possibility. ESR STM allows
to address the Kondo dynamics directly.
64
Can Transport really Read out a Single Precessing
Spin?--1
Direct exchange interaction
65
Quantum wire or dot with spins

PRL, v 89 286802,(2002)
Imagine single wire or single dot with spins
geometry. SO coupling will produce a modulation
in the conductivity of this wire. We also find
twice Larmor frequency signal
66
Josephson effect and single spin
Zhu et al, Cond mat 0306710
67
Spin Dynamics in a Josephson Junction Zhu et al.,
cond-mat/0306710 (accepted to PRL)
Two time scales
68
Effective Spin Action
Josephson tunneling contribution
Classical action
EOM
Josephson freq. wJ2eV
69
Spin Motion
AC JJ (a0.4)
DC JJ (a0)
SQUID sensitivity!
70
Spin Motion Josephson nutation
- AC J.J. (a0.4) - DC J.J. (a0)
71
Possible Observation
Spin cluster S100 measured distance r1 mm
SQUID sensitivity!
72
Effective action SC bath with no dissipation and
Josephson period
S(t)S(t)F(t,t)F(t,t)
Alpha term is allowed by symmetry sin t is T
odd dn/dt is T even, n is T odd
Non damping retardation due to SC bath
73
Wess-Zumino-Witten-Novikovor Berry phase term
for spin
74
Berry term reminder
Measures solid angle spanned by unit vector n(t).
In external Magnetic field
Equation of motion
75
Integrable equations of motion
76
Multiple Spins and Josephson-Spin coupling
Coupling for a set of spins
S_i
S_j
In the case of two spins
77
Solid body precession of a spin
B
Larmor precession in XY plane oscillations in
polar direction With Josephson frequency.
78
Josephson effect with single vibrational mode
Cond mat 0306107
We have two coupled harmonic Oscillators 1.
Josephson frequency 2. Vibrational mode. There
are beatings and Shapiro steps at combined
frequencies.
79
Q1 What is the underlying mechanism responsible
for the current modulation? Q2 What is the spin
dynamics in a Josephson junction?
80
Can Transport really Read out a Single Precessing
Spin?--3
Case I Both leads are normal metals
Reproduce the result from the Kubo formula in
imaginary time!
No time dependent modulation in current!
Spin flip scattering does the work!
81
Can Transport really Read out a Single Precessing
Spin?--4
Case II Right lead is replaced by a
superconductor
Spin down channel
Spin up channel
dG
82
STM-ESR Spectra (Spin Sensitivity)

• Modulation of the tunnel current at the Larmor
  frequency
• when tunneling into a molecule
• (b) No evidence of any modulation when tunneling
  into HOPG
• (c) The Larmor frequency indeed increases
  linearly with the
• applied field strength

83
Electron Transport Through a Quantum Dot Coupled
with a Precessing Spin
Dot level
Zhu and Balatsky, PRL 89, 286802 (2002)
84
Resonant Tunneling Independent of the
Instantaneous Direction of the Precessing Spin
Linewidth is beating at Larmor frequency!
85
DC voltage bias generates a AC current
Twice Larmor frequency mode for PH symmetry
86
I 1, s ½ Lines for I 0,1,-1
Bulk measurement
87
E gemBms(Bo S aimIi) - gNmNBomI
88
Potential applications of ESP-STM

• Emergent technique for detection of magnetic
  defects
• Fully capable of single spin detection. Quantum
  computing.
• General sample characterization. Surface
  science.
• Study the dynamics of the Kondo spin temperature
  evolution
• of the ESP line ( above and below TK)
• Potential for a new imaging technique ( similar
  to MRI)

89
Conclusion

• Noise spectroscopy is a mechanism for single spin
  detection. System is in thermal equilibrium.
• Noise measurements are well suited for quantum
  systems.
• Zeeman, hyperfine, g-factors are revealed in
  the noise.
• Other applications in solid state systems Bose
  condensation in a bilayer QH systems, as revealed
  by noise ( Joglekar et al , PRL 92, p
  086803,(2004))
• Novel dynamics of the spin in a Josephson
  junction combined Larmor and Josephson
  precession (Zhu et al , PRL 92, 107001, (2004)).

90
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91
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92
Tunneling Electron SpectroscopySTM
Single-Molecule Vibrational Spectroscopy and
Microscopy B. C. Stipe, M. A. Rezaei, W. Ho
Science 1998 June 12 280 1732-1735.
Tunneling electrons excite the local vibrational
modes of molecule once energy threshold is passed
93
Fig. 2. (A) I-V curves recorded with the STM tip
directly over the center of molecule (1) and over
the bare copper surface (2). The difference curve
(1 - 2) shows little structure. Each scan took
10 s. (B) dI/dV from the lock-in amplifier
recorded directly over the center of the molecule
(1) and over the bare surface (2). The difference
spectrum (1 - 2) shows a sharp increase at a
sample bias of 358 mV (arrow). Spectra are the
average of 25 scans of 2 min each. A sample bias
modulation of 5 mV was used. (C) d2I/dV2 recorded
at the same time as the data in (B) directly over
the molecule (1) and over the bare surface (2).
The difference spectrum shows a peak at 358 mV.
The area of the peak gives the conductance
change,     (    /    4.2).
94
Possible extensions
tip
Even function of S_x hence it will exhibit twice
Larmor Frequency signal
Local spin
sample
95
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96
S
Friedel spin oscillation near magnetic impurity
97
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98
Quantum wire with few spins
Imagine that tip touches surface near impurity
spin. Then we are in single wire geometry. SO
coupling will produce a modulation in the
conductivity of this wire.
99
Future Directions

• Higher (double frequency) and possibly lower
  harmonics to be studied
• Noise spectroscopy
• Role of substrate and atom( large spin Gd, Fe,
  P) to be addressed in a systematic way. Organic
  molecules BDPA, DPPH
• Effects of single spin can be seen both in dc and
  in ac experiment through DOS modulation.
• We are working on a theory of backaction. Weak
  measurement. Possible extensions of this
  technique.

100
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101
Topo (larger image) and ESR intensity on surface
of Fe-Si
102
JOURNAL OF MAGNETIC RESONANCE 126, 133137 (
1997) Real-Time Response and Phase-Sensitive
Detection to Demonstrate the Validity of ESR-STM
Results Yishay Manassen
103
ESR spectrum off single site
FM modulated ESR signal
frequency of the AC current scales with magnetic
field (Durkan et al)
104
Stochastic spin noise the fluctuation-dissipatio
n theorem
Consider an ensemble of N uncorrelated
paramagnetic spins
But, fluctuations exist
In principle, then, all properties of the system
available from spin noise alone.
105
Listening to magnetization fluctuations (spin
noise)
an example using paramagnetic alkali vapors a
classical ensemble of N uncorrelated spins
dqF(t)
I45
rubidium or potassium N109 mm-3
dV(t)
spectrum analyzer
cw TiS laser
I-45
B (lt10 gauss)

• Alkali vapor (Rb, K) in thermal equilibrium
  (T350K). ltMz(t)gt0.

• Laser tuned near -- but not on resonance no
  absorption.

• Random magnetization fluctuations dMz(t)
  generate noise in Faraday rotation dqF(t).

• Measure spin correlation function, S(t)ltMz(0)
  Mz(t)gt, without perturbing system
• Spin ensemble always remains in thermal
  equilibrium
• (in contrast with conventional magnetic
  resonance)

106
Spectral density of spin noise
Peaks in spectrum of dqF(t) due to stochastic
fluctuations of ground-state spin Fluctuations
must precess in B spontaneous coherence
between Zeeman sublevels
B1.85 G
5 2P1/2
85Rb
30
Rb D1 transition 794.6 nm
Faraday rotation (nanorad/Hz1/2)
Spin noise (nV/Hz1/2)
87Rb
Atomic ground state
2
1
FI1/2
25
0
-1
5 2S1/2
-2
Dhyperfine
-1
0
FI-1/2
1
mF
Noise floor photon shot noise amplifier noise
Not very big! 1 nanoradian angle subtended by a
human hair (100mm) at 100 kilometers
107
Spin noise spectroscopy
remember, spin ensemble remains unperturbed in
thermal equilibrium
Integrated spin noise sqrtN gives isotope
abundance ratio
Spin fluctuations alone directly reveal
ground-state g-factor nuclear spin I
gFgJ /(2I1)
85Rb (467kHz/G) gF1/3, I5/2 87Rb (700kHz/G)
gF1/2, I3/2
108
Off-resonant Faraday rotation probes ltM(0) M(t)gt
Linteraction length, nlaser frequency, N0spin
density, Dn-n0 laser detuning, f oscillator
strength, b polarizability, ltJegt(N-N-)/2N
average electron spin polarization, and Abeam
area
Total measured spin noise as a function of
detuning from absorption resonance qF
couples to magnetization fluctuations
through dispersive indices of refraction
(no absorption at large detuning).
109
Spin noise scales with square root of particle
number
Fluctuations from N uncorrelated, precessing
spins sqrtN
integrated spin noise scales with sqrtN
Temperature-tune the spin density
Integrated spin noise (mV)
85Rb density (mm-3)
110
Spin noise increases when probe area shrinks
Vary beam area, keeping laser power constant
Potential for really small systems, with small N
Beam area
A
Beam area (mm2)
111
Spin fluctuations reveal ground-state hyperfine
coupling

• Increasing magnetic field spin noise broadens
  and breaks up into separate peaks
• -due to quadratic Zeeman effect Zeeman
  sublevels become unequally spaced
• Fluctuations generate coherence between every
  allowed DF0, DmF1 transition

39K
2
B0.81 gauss (W00.57MHz)
1
FIJ2
0
mF
-1
4 2S1/2
-2
2.88G (2.01 MHz)
Dhf1.90mV (461.7MHz)
Spin noise (nV/Hz1/2)
39K, I3/2
-1
5.27G (3.69 MHz)
0
dW
FI-J1
1
8.85 gauss (6.20 MHz)
-200 0 200
kHz from W0
112
Spin fluctuations also reveal nuclear magnetism
(mI, gI)
Coupling of electron to nuclear magnetic moment
Zeeman transitions within F2, F1
hyperfine levels are no longer exactly degenerate
2
1
F2
mF
0
-1
5 2S1/2
-2
Dhf6835 MHz
87Rb, I3/2
-1
0
F1
1
Additional splitting reveals nuclear moment
113
Spin fluctuations generate high-frequency
inter-hyperfine coherences
Spontaneous spin coherences also exist between
hyperfine levels
2
1
F2
0
mF
-1
4 2S1/2
-2
Dhf
-1
0
F1
1
39K (I3/2)
Dhf461.7 MHz
114
Conclusions

• General result Most properties of spin ground
  state revealed in the spin noise.
• - g-factors, nuclear spin, hyperfine
  energy, isotopes, nuclear moment, coherence
  times, etc
• - in accord with fluctuation-dissipation
  theorem

• Ensemble remains unperturbed in thermal
  equilibrium
• no excitation, no pumping.
• - sourceless magnetic resonance

• Noise spectroscopy as a useful probe of small
  solid-state systems with few spins?
• - Eg, only 1000 electrons in typical 2D electron
  gas in 1mm area (cf 109 here)
• - Intrinsic sqrtN fluctuations are
  increasingly important as N-gt1