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Classical and quantum synchronization
Christoph Bruder - University of Basel
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synchronization: synchronized events
different “agents” act synchronously, at the same time:
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synchronization: synchronized events
different “agents” act synchronously, at the same time:
• audience leaves for the coffee break
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synchronization: synchronized events
different “agents” act synchronously, at the same time:
• audience leaves for the coffee break
• rowing (e.g. coxed eight)
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synchronization: synchronized events
different “agents” act synchronously, at the same time:
• audience leaves for the coffee break
• rowing (e.g. coxed eight)
control by external “clock”
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spontaneous synchronization
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spontaneous synchronization
Huygens’ observation (1665): two pendulum clocks fastened to the same beam willsynchronize (anti-phase)
A. Pikovsky, M. Rosenblum, and J. KurthsSynchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, New York, 2001)
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spontaneous synchronization
Huygens’ observation (1665): two pendulum clocks fastened to the same beam willsynchronize (anti-phase)
A. Pikovsky, M. Rosenblum, and J. KurthsSynchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, New York, 2001)
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spontaneous synchronization
nice introduction to classical synchronization
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spontaneous synchronization
• rhythmic applause in a large audience
• heart beat (due to synchronization of 1000’s of cells)
• synchronous flashing of fireflies
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global outline
• Lecture I: classical synchronization
• Lecture II: quantum synchronization
• Lecture III: topics in quantum synchronization
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lecture I: classical synchronization
• synchronization of a self-oscillator by external forcing
• two coupled oscillators
• ensembles of oscillators: Kuramoto model
• realization in a one-dimensional Josephson array
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definition of self-oscillator
self-oscillator or self-sustained (limit-cycle) oscillator
• driven into oscillation by some energy source• maintains stable oscillatory motion when unperturbedor weakly perturbed
• intrinsic natural frequency !0
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definition of self-oscillator
self-oscillator or self-sustained (limit-cycle) oscillator
• driven into oscillation by some energy source• maintains stable oscillatory motion when unperturbedor weakly perturbed
• intrinsic natural frequency !0
x+ (��1 + �2x2)x+ !
20x = 0
examples: (i) pendulum clock(ii) van der Pol oscillator
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synchronization problem
given two (or more) self-oscillators with (slightly) different frequencies:will they agree on ONE frequency if coupled?
!0,!00, ...
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synchronization problem
given two (or more) self-oscillators with (slightly) different frequencies:will they agree on ONE frequency if coupled?
let’s start with an easier problem:
will one self-oscillator frequency-lock to an external harmonic drive of frequency !d 6= !0
!0,!00, ...
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synchronization problem
given two (or more) self-oscillators with (slightly) different frequencies:will they agree on ONE frequency if coupled?
let’s start with an easier problem:
will one self-oscillator frequency-lock to an external harmonic drive of frequency !d 6= !0
!0,!00, ...
= synchronization by external forcing
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linear oscillators
NOTE: driven linear damped harmonic oscillator
solution: damped eigenmodes +
x+ �x+ !
20x = ⌦ cos(!dt)
A cos(!dt)
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linear oscillators
NOTE: driven linear damped harmonic oscillator
solution: damped eigenmodes +
x+ �x+ !
20x = ⌦ cos(!dt)
A cos(!dt)
will always adjust to an external drive frequency(after a transient) - this is NOT synchronization
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linear oscillators
NOTE: driven linear damped harmonic oscillator
non-linearity is crucial for synchronization
solution: damped eigenmodes +
x+ �x+ !
20x = ⌦ cos(!dt)
A cos(!dt)
will always adjust to an external drive frequency(after a transient) - this is NOT synchronization
the same applies to eigenmodes of coupled linearharmonic oscillators
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synchronization by external forcing
phase parametrizes motion along one cycle of the oscillatoramplitude is assumed to be constant
undisturbed dynamics:
�(t)
d�(t)
dt= �(t) = !0
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synchronization by external forcing
phase parametrizes motion along one cycle of the oscillatoramplitude is assumed to be constant
undisturbed dynamics:
�(t)
d�(t)
dt= �(t) = !0
always possible by re-parametrization: non-uniform ⇒
uniform
�(t)
�(t) = !0
Z �
0d�
d�
dt
!�1
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synchronization by external forcing
drive by a periodic force of frequency and amplitude :
where is -periodic in both arguments
✏
�(t) = !0 + ✏Q(�,!dt)
!d ⇡ !0
2⇡Q
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synchronization by external forcing
drive by a periodic force of frequency and amplitude :
where is -periodic in both arguments
define deviation �� = �� !dt
✏
�(t) = !0 + ✏Q(�,!dt)
!d ⇡ !0
2⇡Q
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synchronization by external forcing
drive by a periodic force of frequency and amplitude :
where is -periodic in both arguments
define deviation �� = �� !dt
✏
�(t) = !0 + ✏Q(�,!dt)
!d ⇡ !0
2⇡Q
Fourier-expansion of and averaging (→ vanishing of rapidly oscillating terms) leads to
Adler 1946d��(t)
dt= !0 � !d + ✏q(��)
Q
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synchronization by external forcing
is -periodic
simplest choice: q(��) = sin(��)
d��(t)
dt= !0 � !d + ✏q(��)
d��(t)
dt= !0 � !d + ✏ sin(��)
q 2⇡
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synchronization by external forcing
is -periodic
simplest choice: q(��) = sin(��)
d��(t)
dt= !0 � !d + ✏q(��)
d��(t)
dt= !0 � !d + ✏ sin(��)
q 2⇡
synchronization, i.e.,
possible for 0
0.2 0.4 0.6 0.8
1
-1 -0.5 0 0.5 1
synchronized
¡
t0<td
| ✏
!0 � !d| > 1
d��(t)
dt= 0
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Adler plot of observed frequency
(time average); differs in general from both and
!0
!d
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
(tobs- td)/¡
(t0<td)/¡
synchronization
entrainment
!obs
= !d
!obs
6= !0,!d
!obs
= hd�dt
i
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side remark
Adler equation
appears in many areas of physics: e.g.
superconductivity (Shapiro steps)
quantum optics (ring-laser gyros)
d��(t)
dt= !0 � !d + ✏ sin(��)
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current-biased Josephson junction
Ic =2e
~ EJ
d�
dt=
2e
~ VI = Ic sin�Josephson relations
EJ
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current-biased Josephson junction
I = Ic sin�+~
2eR
d�
dt+
C~2e
d2�
dt2
RSJ model
J
I I
C
R
E
Ic =2e
~ EJ
d�
dt=
2e
~ VI = Ic sin�Josephson relations
EJ
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current-biased Josephson junction
I = Ic sin�+~
2eR
d�
dt+
C~2e
d2�
dt2
C ! 0classical junction:
d�
dt=
2e
~ RI � 2e
~ RIc sin�
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current-biased Josephson junction
I = Ic sin�+~
2eR
d�
dt+
C~2e
d2�
dt2
Adler equation!
C ! 0classical junction:
d�
dt=
2e
~ RI � 2e
~ RIc sin�
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current-biased Josephson junction
I = Ic sin�+~
2eR
d�
dt+
C~2e
d2�
dt2
Adler equation!
“synchronized” state with
for “detuning” I/Ic < 1 0
0.5 1
1.5 2
2.5 3
0 0.5 1 1.5 2 2.5 3
<dq/dt>
I/Ic
hd�dt
i = 2e
~ hV i = 0
C ! 0classical junction:
d�
dt=
2e
~ RI � 2e
~ RIc sin�
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current-biased Josephson junction
• strictly speaking, J junction is a rotator (and not a self-sustained oscillator)
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current-biased Josephson junction
• strictly speaking, J junction is a rotator (and not a self-sustained oscillator)
• can be synchronized by a periodic external force (Shapiro steps) or to another J junction
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Shapiro steps
current-biased Josephson junction + harmonic current bias:
I0 + I1 sin!dt =~
2eR
d�
dt+ Ic sin�
additional synchronization plateaus
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 3 3.5
0 0.5
1 1.5
2 2.5
3 3.5
I0
I1<dq/dt> = <V>
I0
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2 oscillators, no external forcing
undistorted frequencies
weak interaction affects only the phases �1,�2
�1(t) = !1 + ✏Q1(�1,�2)
�2(t) = !2 + ✏Q2(�2,�1)
Fourier expansion, averaging to get rid of rapidly oscillating terms leads to
d��(t)
dt= !1 � !2 + ✏q(��)
Adler equation!
!1 ,!2
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2 oscillators, no external forcing
the two oscillators will lock in on a common frequency between and
→ spontaneous synchronization
!1 !2
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ensembles of coupled oscillators
N coupled phase oscillators ,random frequencies described by probability density
�i = !i +NX
j=1
Kij sin(�j � �i), i = 1, . . . , N
!i g(!)�i(t)
Kuramoto model (1975), solvable for infinite-range coupling Kij = ✏/N > 0
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ensembles of coupled oscillators
N coupled phase oscillators ,random frequencies described by probability density
�i = !i +NX
j=1
Kij sin(�j � �i), i = 1, . . . , N
!i g(!)�i(t)
Kuramoto model (1975), solvable for infinite-range coupling Kij = ✏/N > 0
non-equilibrium phase transition to a synchronized state as a function of ✏
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ensembles of coupled oscillators
degree of synchronicity described by order parameter
rei =1
N
NX
j=1
ei�j
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ensembles of coupled oscillators
degree of synchronicity described by order parameter
rei =1
N
NX
j=1
ei�j
0 r(t) 1 measures the coherence of the ensemble
is the average phase (t)
transition to → synchronizationr 6= 0
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ensembles of coupled oscillators
degree of synchronicity described by order parameter
�i = !i + ✏r sin( � �i), i = 1, . . . , N
substituted back in the Kuramoto equation gives
→each oscillator couples to the common average phase (t)
rei =1
N
NX
j=1
ei�j
0 r(t) 1 measures the coherence of the ensemble
is the average phase (t)
transition to → synchronizationr 6= 0
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partial coherence
�i = !i + ✏r sin( � �i), i = 1, . . . , N
interpretation of :
typical oscillator running with velocitywill become stably locked at an angle such that
! � ✏r sin(�� )
✏r sin(�� ) = ! �⇡/2 �� ⇡/2
oscillators with frequencies cannot be locked|!| > ✏r
0 < r < 1
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partial coherence
�i = !i + ✏r sin( � �i), i = 1, . . . , N
interpretation of :
typical oscillator running with velocitywill become stably locked at an angle such that
! � ✏r sin(�� )
✏r sin(�� ) = ! �⇡/2 �� ⇡/2
oscillators with frequencies cannot be locked|!| > ✏r
0 < r < 1
→ three groups: (i) synchronized(ii) unsynchronized, velocity(iii) unsynchronized, velocity
> <
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Kuramoto’s results
for , transition from incoherent state to partially coherent state occurs at
J.A. Acebron et al., Rev. Mod. Phys. 77, 317 (2005)
(r > 0)(r = 0)
✏c =2
⇡g(! = 0)
N ! 1
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Kuramoto’s results
for , transition from incoherent state to partially coherent state occurs at
J.A. Acebron et al., Rev. Mod. Phys. 77, 317 (2005)
(r > 0)(r = 0)
✏c =2
⇡g(! = 0)
N ! 1
r ⇠
s�16(✏� ✏c)
⇡✏4cg00(0)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
r
✏/✏c
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Kuramoto’s results
for , transition from incoherent state to partially coherent state occurs at
J.A. Acebron et al., Rev. Mod. Phys. 77, 317 (2005)
g(!) =1
⇡
�
�2 + !2 r =
r1� ✏c
✏for ✏ > ✏c = 2�
(r > 0)(r = 0)
✏c =2
⇡g(! = 0)
N ! 1
r ⇠
s�16(✏� ✏c)
⇡✏4cg00(0)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
r
✏/✏c
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Kuramoto’s results
for , transition from incoherent state to partially coherent state occurs at
exact solution - pretty amazing!
J.A. Acebron et al., Rev. Mod. Phys. 77, 317 (2005)
g(!) =1
⇡
�
�2 + !2 r =
r1� ✏c
✏for ✏ > ✏c = 2�
(r > 0)(r = 0)
✏c =2
⇡g(! = 0)
N ! 1
r ⇠
s�16(✏� ✏c)
⇡✏4cg00(0)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
r
✏/✏c
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simulation with N=900 oscillators
example:
synchronization transition at
for ✏ = 4
� = 1 g(!) =1
⇡
1
1 + !2
✏c = 2
r =
r1� ✏c
✏⇡ 0.71
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simulation with N=900 oscillators
example:
synchronization transition at
for ✏ = 4
� = 1 g(!) =1
⇡
1
1 + !2
✏c = 2
r =
r1� ✏c
✏⇡ 0.71
nice simulation program: Synched by Per Sebastian Skardal
https://sites.google.com/site/persebastianskardal/software/synched
“K” corresponds to ✏
gives amplitude and phase of the order parameter
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realization in a one-dim Josephson array
uncoupled Josephson junctions = rotators with
K. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
!i =2e
~ RS
qI2 � I2Ci
ICR
IL
...R
C1
S
I
R
C2
S
I
RS
I Cn
global coupling by RCL branch ⇒ natural realization of the Kuramoto model
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realization in a one-dim Josephson arrayK. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
LQ+RQ+Q
C=
~2e
NX
k
�k
ICR
IL
...R
C1
S
I
R
C2
S
I
RS
I Cn
for each junction I � Q = ICk sin�k +~
2eRS�k
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realization in a one-dim Josephson arrayK. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
LQ+RQ+Q
C=
~2e
NX
k
�k
ICR
IL
...R
C1
S
I
R
C2
S
I
RS
I Cn
for each junction
uniformly rotating phases in the uncoupled case Q = 0✓kd✓k!k
= dt =d�k
(2eRS/~)(I � ICk sin�k)
I � Q = ICk sin�k +~
2eRS�k
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realization in a one-dim Josephson arrayK. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
combined with
˙✓k = !k � !k˙Q
I2 � I2Ck
(I � ICk cos ✓k)
yieldsI � Q = ICk sin�k +~
2eRS�k
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realization in a one-dim Josephson arrayK. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
first-order averaging ⇒
✓k = !k � K
N
NX
j
sin(✓k � ✓j + ↵) Kuramoto!
combined with
˙✓k = !k � !k˙Q
I2 � I2Ck
(I � ICk cos ✓k)
yieldsI � Q = ICk sin�k +~
2eRS�k
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realization in a one-dim Josephson arrayK. Wiesenfeld, P. Colet, and S.H. Strogatz, PRL 76, 404 (1996)
first-order averaging ⇒
✓k = !k � K
N
NX
j
sin(✓k � ✓j + ↵) Kuramoto!
cos↵ =
L!2 � 1/C
[(L!2 � 1/C)
2+ !2
(R+NRS)2]
1/2
K =NRS!(2eRSI/~� !)
[(L!2 � 1/C)2 + !2(R+NRS)2]1/2
combined with
˙✓k = !k � !k˙Q
I2 � I2Ck
(I � ICk cos ✓k)
yieldsI � Q = ICk sin�k +~
2eRS�k
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quantum synchronization
• experimental situation?
• does it exist at all?
• how to quantify and measure it?
• relation to other measures of `quantumness’ (entanglement, mutual information, ...)
so far only classical non-linear systems
synchronization in quantum systems:
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conclusion
• classical synchronization is well-studied,
• simplest model: one self-oscillator + external forcing→ Adler equation
• frequency locking if detuning < drive strength
• two oscillators lock if detuning < coupling
• synchronization (phase) transition in ensembles ofmutually coupled self-oscillators: Kuramoto model
d��(t)
dt= !0 � !d + ✏ sin(��)
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appendix