On the statistics of coherent quantum phase-locked states

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On the statistics of coherent quantum
phase-locked states
Quantum Optics II, Cozumel, Dec 2004

• Michel Planat
• Institut FEMTO-ST, Dept. LPMO
• 32 Av. de lObservatoire, 25044 Besançon Cedex
• planat_at_lpmo.edu

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• Classical phase-locking
• 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

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THE OPEN LOOP (frequency locking)
(p,q)1 fB(t)pf0-qf(t)
THE PHASE LOCKED LOOP
beat signal
1/f noise in the IF
PLL
beat frequency fB
time
open loop

closed loop
frequency shift f-f0
2K
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Adlers equation of phase locking
input frequency shift
dynamical phase shift
1/f noise
t
output frequency shift
equation for the 1/f noise variance
experiments -- theory
M. Planat and E. Henry The arithmetic of 1/f
noise in a phase-locked loop Appl. Phys. Lett.
80 (13), 2002
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A phenomenological model of classical
phase-locking Is the Arnold map
1/1
2/3
1/2
1/3
with a desynchronization from the Mangoldt
function
M. Planat and E. Henry, Appl. Phys. Lett. 80
(13), 2002 The arithmetic of 1/f noise in a
phase-locked loop
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• How to account for Mangoldt function and 1/f
  noise
• quantum mechanically ? ? 2.
• How to avoid prime number fluctuations ? ? 3
• 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

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2. Quantum phase-locking from rational numbers
Pegg and Barnett phase operator
the states are eigenstates of the Hermitian phase
operator
the Hilbert space is of finite dimension q the ?p
are orthonormal to each other and form a
complete set
Given a state Fgt the phase probability
distribution is lt?pFgt2
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The quantum phase - locking operator (Planat
and Rosu)
One adds the coprimality condition (p,q)1
with the Ramanujan sums
M. Planat and H. Rosu Cyclotomy and Ramanujan
sums in quantum phase-locking Phys. Lett. A315,
1-5 (2003)
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Phase properties of a general state (Pegg
Barnett)
pure phase state
and for a partial phase state
phase probability distribution
phase expectation value
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Oscillations in the expectation value of quantum
locked phase ß1 (dotted line)
ß0 (plain line) ßp?(q) / ln q
(brokenhearted line) with ?(q) the Mangoldt
function
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Phase variance of a pure phase state




(p/q)2
with
peaks at pa, p a prime number
Classical variance p2/3
Plain ß0 Dotted ßp
squeezed phase noise
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Bost and Connes quantum statistical model
A dynamical system is defined from the
Hamiltonian operator
The partition function is
Given an observable Hermitian operator M, one has
the Hamiltonian evolution st(M) versus time
t and the Gibbs state is the expectation value
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In Bost and Connes approach the observables
belong to an algebra of operators
shift operator elementary phase operator
Gibbs state -gt Kubo-Martin-Schwinger state
ß0 KMS 1 high
temperature ß1 critical point ß1e
squeezing zone KMS -?(q)e/q
with ?(q) the Mangoldt function ßgtgt1 KMS
µ(q)/f(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
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Phase expectation value In Bost and Connes
model at low temperature ß3 (plain
line) µ(q)/f(q) (dotted line)

• Phase expectation value
• In Bost and Connes model
• close to critical
• ß1e (plain line)
• -?(q)e/q (dotted line)
• with e0.1

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Cyclotomic quantum algebra of time
perception (Bost et Connes 94)
ß température inverse q dimension
de lespace de Hilbert KMS état
thermique
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• How to account for Mangoldt function and 1/f
  noise
• quantum mechanically ? ? 2.
• How to avoid prime number fluctuations ? ? 3
• 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
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3. The Galois phase locking operator a.
Odd characteristic p qudits (Wootters 89,
Klappenecker 03)
PeggBarnett operator iff a0 and qp prime
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mutual unbiasedness of phase-states
Mutually unbiased bases are such that two vectors
in one base are orthogonal and two vectors in
different bases have constant inner product equal
to 1/vq.
if p is odd i.e. for characteristic ?2
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Evaluation of the Galois phase-locking
operator
Matrix elements
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Evaluation of the phase-number commutator
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Phase fluctuations and Gauss sums 1

• Let ? a multiplicative and ? an additive
  character of the Galois field Fq,
• the Gauss sums are defined as
• with the properties
• where ?0 and ?0 are the trivial characters
• 
  and
• For pure phase states
• we will use more general Gauss sums
• with indeed the property

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Phase fluctuations and Gauss sums 2
Phase probability distribution Phase expectation
value Phase variance
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3. The Galois phase locking operator b.
Characteristic 2 qubits (Klappenecker 03)
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Particular case qubits D2 GR(4)Z4Z4x/(x1
) T2(0,1)
B00gt(1,0),1gt(0,1) B1(1/v2)
(1,1),(1,-1) B2(1/v2)(1,i),(1,-i)
Particular case quartits d4
GR(42)Z4x/(x2x1) T2(0,1,x,33x)
B00gt(1,0,0,0),1gt(0,1,0,0),2gt(0,0,1,0),3gt
(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)
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MUBs and maximally entangled states
More generally they are maximally entangled two
particle sets of 2m dits obtained from the
generalization of the MUB formula for qubits
a. Special case of qubits m1
Two bases on one column are mutually
unbiased, But vectors in two bases on the same
line are orthogonal.
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b. Special case of maximally entangled bases of
2-qubits
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Conclusion

• 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
• 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)