On the statistics of coherent quantum phase-locked states Michel Planat Institut FEMTO-ST, Dept. LPMO 32 Av. de l’Observatoire, Besançon Cedex 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 frequency f B frequency shift f-f 0 2K open loop closed loop time beat signal 1/f noise in the IF PLL * THE OPEN LOOP (frequency locking) * THE PHASE LOCKED LOOP (p,q)=1 f B (t)=|pf 0 -qf(t)|

Adler’s equation of phase locking equation for the 1/f noise variance: dynamical phase shift input frequency shift output frequency shift t + experiments -- theory M. Planat and E. Henry « The arithmetic of 1/f noise in a phase-locked loop » Appl. Phys. Lett. 80 (13), /f noise

A phenomenological model of classical phase-locking Is the Arnold map with a desynchronization from the Mangoldt function 1/2 1/3 2/3 1/1 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 the states are eigenstates of the Hermitian phase operator the Hilbert space is of finite dimension q the θ p are orthonormal to each other and form a complete set Given a state |F> the phase probability distribution is | | 2

* The quantum phase - locking operator (Planat and Rosu) M. Planat and H. Rosu Cyclotomy and Ramanujan sums in quantum phase-locking Phys. Lett. A315, 1-5 (2003) with the Ramanujan sums: One adds the coprimality condition (p,q)=1

Phase properties of a general state (Pegg & Barnett) pure phase state: and for a partial phase state: phase probability distribution: phase expectation value:

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

Phase variance of a pure phase state ~ ~ (p/q) 2 ~ ~ with peaks at p α, p a prime number Plain: ß=0 Dotted: ß=π Classical variance π 2 /3 squeezed phase noise

* 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 σ t (M) versus time t and the Gibbs state is the expectation value

In Bost and Connes approach the observables belong to an algebra of operators shift operator elementary phase operator Gibbs state -> Kubo-Martin-Schwinger state ß=0 KMS = 1 high temperature ß=1 critical point ß=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/

Phase expectation value In Bost and Connes model at low temperature * ß=3 (plain line) * μ(q)/φ(q) (dotted line) Phase expectation value In Bost and Connes model close 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) ß: température inverse q: dimension de l’espace de Hilbert KMS: 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/

3. The Galois « phase – locking » operator a. Odd characteristic p: qudits (Wootters 89, Klappenecker 03) Pegg&Barnett operator iff a=0 and q=p prime

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/√q. if p is odd i.e. for characteristic ≠2

Evaluation of the Galois « phase-locking » operator Matrix elements:

Evaluation of the phase-number commutator

Phase fluctuations and Gauss sums 1 * Let Ψ a multiplicative and κ an additive character of the Galois field F q, the Gauss sums are defined as with the properties where Ψ 0 and χ 0 are the trivial characters and * For « pure » phase states we will use more general Gauss sums with indeed the property

Phase fluctuations and Gauss sums 2 Phase probability distribution Phase expectation value Phase variance

3. The Galois « phase – locking » operator b. Characteristic 2: qubits (Klappenecker 03)

* Particular case: quartits d=4; GR(4 2 )=Z 4 [x]/(x 2 +x+1); T 2 =(0,1,x,3+3x) B 0 ={|0>=(1,0,0,0),|1>=(0,1,0,0),|2>=(0,0,1,0),|3>=(0,0,0,1)} B 1 =(1/2){(1,1,1,1),(1,1,-1,-1),(1,-1,-1,1),(1,-1,1,-1)} B 2 =(1/2){(1,-1,-i,-i),(1,-1,i,i),(1,1,i,-i),(1,1,-i,i)} B 3 =(1/2){(1,-i,-i,-1),(1,-i,i,1),(1,i,i,-1),(1,i,-i,1)} B 4 =(1/2){(1,-i,-1,-i),(1,-i,1,i),(1,i,1,-i),(1,i,-1,i)} * Particular case: qubits D=2; GR(4)=Z 4 =Z 4 [x]/(x+1); T 2 =(0,1) B 0 ={|0>=(1,0),|1>=(0,1)} B 1 =(1/√2) {(1,1),(1,-1)} B 2 =(1/√2){(1,i),(1,-i)}

MUBs and maximally entangled states More generally they are maximally entangled two particle sets of 2 m dits obtained from the generalization of the MUB formula for qubits a. Special case of qubits: m=1 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:

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)