On the statistics of coherent quantum phase-locked states Michel Planat Institut FEMTO-ST, Dept....
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Transcript of On the statistics of coherent quantum phase-locked states Michel Planat Institut FEMTO-ST, Dept....
On the statistics of coherent quantum phase-locked states
Michel PlanatInstitut FEMTO-ST, Dept. LPMO
32 Av. de l’Observatoire, 25044 Besançon [email protected]
Quantum Optics II, Cozumel, Dec 2004
1. Classical phase-locking
2. Quantum phase-locking from rational numbers cyclotomic field over Q, Ramanujan sums and prime number theory
3. Quantum phase-locking from Galois fields mutual unbiasedness and Gauss sums
beat frequencyfB
frequency shift f-f02K
open loop
closed loop
time
beat signal
1/f noise in the IF
PLL
* THE OPEN LOOP (frequency locking)
* THE PHASELOCKED LOOP
(p,q)=1fB(t)=|pf0-qf(t)|
BtKt )(sin)(.
Adler’s equation of phase locking
2/122 )(~ KBB
BB
BBB
K
K
~/
)~/1(~ 2/122
equation for the 1/f noise variance:
dynamicalphase shift
inputfrequency shift
output frequency shift
B~
t
B~
B~
+ experiments-- theory
M. Planat and E. Henry « The arithmetic of 1/f noise in a phase-locked loop »Appl. Phys. Lett. 80 (13), 2002
1/f noise
A phenomenological model of classical phase-lockingIs the Arnold map
n
f
Kc
f
f
c
nn
nnn
/)(lim
;
sin2
0
00
1
with a desynchronization from the Mangoldt function
1/2
1/3
2/3
1/1
otherwiseandprimebbnif
bnnwith
qpnqpncc
k
iiii
0,
ln)()1,1,(
mod,),;(*
M. Planat and E. Henry, Appl. Phys. Lett. 80 (13), 2002« The arithmetic of 1/f noise in a phase-locked loop »
* How to account for Mangoldt function and 1/f noise quantum mechanically ? → 2.* How to avoid prime number fluctuations ? → 3
2. Quantum phase-locking from rational numbers, cyclotomic field over Q, Ramanujan sums and prime number theory
3. Quantum phase-locking from Galois fields, mutual unbiasedness and Gauss sums
2. Quantum phase-locking from rational numbers
* Pegg and Barnett phase operator
nnq
pi
q
q
np
1
0
)2exp(1
the states are eigenstates of the Hermitian phase operator
qpwith pp
q
ppp /20
1
0
the Hilbert space is of finite dimension qthe θp are orthonormal to each other
and form a complete set
qp
q
pp 1
1
0
Given a state |F> the phase probability distribution is |<θp|F>|2
lnlncP
with
lnqqmeasmeas
qp
pp
q
qpp
ppmeas
)(
2
,
1
1),(0
* The quantum phase - locking operator (Planat and Rosu)
M. Planat and H. RosuCyclotomy and Ramanujan sums in quantum phase-lockingPhys. Lett. A315, 1-5 (2003)
)),/((
)()),/((
)2exp()(1
1),(0
mqq
qmqq
mimcq
qpp q
pq
with the Ramanujan sums:
One adds the coprimality condition (p,q)=1
21
12,
4111
1411
1141
1114
65 cc
Phase properties of a general state (Pegg & Barnett)
pure phase state:
1
0
q
nn nuf
and for a partial phase state: )exp(1 Inuqn
phase probability distribution:
phase expectation value:2
fpp
pmeas
2fp
))(exp()()(
0,2
q
lnqqmeas lninlc
Oscillations in the expectation value of quantum locked phase* ß=1 (dotted line) * ß=0 (plain line)* ß=πΛ(q) / ln q (brokenhearted line) with Λ(q) the Mangoldt function
))(exp()()(
0,2
q
lnqqmeas lninlc
Phase variance of a pure phase state
)2(42
2
measmeas
measmeas
~
))(exp()()(
0,2
q
lnqqmeas lninlc
~
1
1),(0 )2exp()(
q
qpp q
pq mimc (p/q)2
~
~
with
peaks at pα,
p a prime number
Plain: ß=0Dotted: ß=π
Classical variance π2/3
squeezed phase noise
* Bost and Connes quantum statistical model
A dynamical system is defined from the Hamiltonian operator
nnnH ln0
The partition function is
0
0 )())(exp(n
nHTrace
Given an observable Hermitian operator M, one has the Hamiltonianevolution σt(M) versus time t
and the Gibbs state is the expectation value
00)( itHitHt MeeM
)(/)()( 00 HH etraceMeTraceMGibbs
In Bost and Connes approach the observables belong to an algebra of operators
nnq
pine
qann
pq
a
)2exp(
)(mod
)(
shift operator
elementary phase operator
Gibbs state -> Kubo-Martin-Schwinger state
primep
qdividesp
pq p
pqeKMS
1
1)(
1
1)(
ß=0 KMS = 1 high temperatureß=1 critical pointß=1+ε squeezing zone KMS ≈ -Λ(q)ε/q with Λ(q) the Mangoldt function ß>>1 KMS = μ(q)/φ(q) low temperature zone
Invitation to the « spooky » quantum phase-locking effect and its link to 1/f fluctuations M. Planat ArXiv quant-ph/0310082
Phase expectation value In Bost and Connes modelat low temperature* ß=3 (plain line)* μ(q)/φ(q) (dotted line)
Phase expectation value In Bost and Connes modelclose to critical* ß=1+ε (plain line)•-Λ(q)ε/q (dotted line) with ε=0.1
Cyclotomic quantum algebra of time perception(Bost et Connes 94)
primep
k
kprimep
k
p
p
p
pq
p
ppKMS
0
1
1
1
1
ß: température inverseq: dimension de l’espace de HilbertKMS: état thermique
* How to account for Mangoldt function and 1/f noise quantum mechanically ? → 2.* How to avoid prime number fluctuations ? → 3
2. Quantum phase-locking from rational numbers, cyclotomic field over Q, Ramanujan sums and prime number theory
3. Quantum phase-locking from Galois fields, mutual unbiasedness and Gauss sums
Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements, M. Planat et al,ArXiv quant-ph/0409081
3. The Galois « phase – locking » operator a. Odd characteristic p: qudits(Wootters 89, Klappenecker 03)
characteradditivean)(and
charactertivemultiplicaa)(with
)2
exp(prime,p,)(
,...)(with
)(1
oddpifand
,2with
)(
p
)(
1
2
xtrp
k
qm
ppp
Fn
bnantrpk
ab
qqb
bab
Fb
a
bbGal
x
n
p
iFpqGFx
Fxxxxtr
nnq
Fa
m
q
q
Pegg&Barnettoperatoriff a=0and q=p prime
mutual unbiasedness of phase-states
Mutually unbiased bases are such that two vectors in one baseare orthogonal and two vectors in different bases have constantinner product equal to 1/√q.
2))(deg(,))((0)(
)(1
)(
)(1
2/1
))()((2 2
xpqxpandx
sumsWeilofpropertythetodue
ssunbiasedneacifq
ityorthogonalacif
nq
q
FxFx
bd
nbdnactrp
Fnk
cd
ab
if p is odd i.e. for characteristic ≠2
Evaluation of the Galois « phase-locking » operator
q
q
Fb
mnbtrp
FmnkGal
bmnS
mnmnSmnq
)(
,2
),(with
),()(2
Matrix elements:
mnmn
qmn
q
qnn
mntrp
kGal
Gal
,)(2
)1(
)(
Evaluation of the phase-number commutator
1
)()(2,
0,
)(
mntrp
kGal
Gal
mnmn
qmn
nn
N
N
operator)(phase
),()(2
operator)(number
,2
mnmnSmnq
lll
q
q
FmnkGal
FlN
Phase fluctuations and Gauss sums 1
* Let Ψ a multiplicative and κ an additive character of the Galois field Fq,
the Gauss sums are defined as
with the properties
where Ψ0 and χ0 are the trivial characters and
* For « pure » phase states
we will use more general Gauss sums
with indeed the property
qFc
ccG )()(),(
0),(;1),(;1),( 0000 GGqG
2/1),( qG
)exp(1
with Inq
unuf nFn
n
q
qFc
RcccIG ),()()exp(),(
2/1),( qG
Phase fluctuations and Gauss sums 2
Phase probability distribution
Phase expectation value
Phase variance
),(12
2 Gq
fb
),(/
),(0)),(1(
2
)1(2
2with,
0
3
2
forq
forG
q
q
bf
Gal
bbFb
bGalq
),(2
),(0
)( 2
0222
forq
for
fbFb
GalbGal
q
3. The Galois « phase – locking » operator b. Characteristic 2: qubits (Klappenecker 03)
characteradditivean)(and
charactertivemultiplicaa)(with
,2)2(,)4(
,)(...)()()(with
)(1
2pifand
,2with
)(
22
412
))2(
xtr
k
qm
m
Rn
nbatrk
ab
qqb
bab
Rb
a
bbGal
ix
n
babaRqGRx
Zxxxxxtr
ninq
Ra
q
q
* Particular case: quartitsd=4; GR(42)=Z4[x]/(x2+x+1);T2=(0,1,x,3+3x)
B0={|0>=(1,0,0,0),|1>=(0,1,0,0),|2>=(0,0,1,0),|3>=(0,0,0,1)}B1=(1/2){(1,1,1,1),(1,1,-1,-1),(1,-1,-1,1),(1,-1,1,-1)}B2=(1/2){(1,-1,-i,-i),(1,-1,i,i),(1,1,i,-i),(1,1,-i,i)}B3=(1/2){(1,-i,-i,-1),(1,-i,i,1),(1,i,i,-1),(1,i,-i,1)}B4=(1/2){(1,-i,-1,-i),(1,-i,1,i),(1,i,1,-i),(1,i,-1,i)}
* Particular case: qubitsD=2; GR(4)=Z4=Z4[x]/(x+1); T2=(0,1)
B0={|0>=(1,0),|1>=(0,1)}B1=(1/√2) {(1,1),(1,-1)}B2=(1/√2){(1,i),(1,-i)}
MUBs and maximally entangled states
More generally they are maximally entangled two particle sets of 2m ditsobtained from the generalization of the MUB formula for qubits
hnniBm
n
nbatr
m
abh
,2
1 12
0
])2[(,
a. Special case of qubits: m=1
)1001,1001()1100,1100(
)1001,1001()1100,1100(
2
1
iiii
Two bases on one column are mutually unbiased,But vectors in two bases on the same line are orthogonal.
b. Special case of maximally entangled bases of 2-qubits:
...
);32211003,32211003
,32211003,32211003(
...
);31201302,31201302
,31201302,31201302(
...
);30231201,30231201
,30231201,30231201(
...
);33221100,33221100
,33221100,33221100(
);33221100,33221100
),33221100,33221100(
2
1
iiii
iiii
Conclusion
1. Classical phase_locking and its associated 1/f noise is related to standard functions of prime number theory
2. There is a corresponding quantum phase-locking effect over the rational field Q with similar phase fluctuations, which are possibly squeezed
3. The quantum phase states over a Galois field (resp. a Galois ring) are fascinating, being related to
* maximal sets of mutually unbiased bases* minimal phase uncertainty* maximally entangled states* finite geometries (projective planes and ovals)