On the statistics of coherent quantum phase-locked states

Michel PlanatInstitut FEMTO-ST, Dept. LPMO

32 Av. de l’Observatoire, 25044 Besançon Cedexplanat@lpmo.edu

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

qq

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

qq

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)