A real-time synchronization feedback for single-atom frequency standards 1

A real-time synchronization feedback for single-atom frequency standards

Pierre Rouchon

Mines ParisTech, Centre Automatique et Syst `emes [email protected]

in collaboration with Mazyar Mirrahimi(INRIA)

Quantum/Classical Control in Quantum Information:

Theory and experiments.

September 13-20, 2008 Otranto, Italy

1Partially supported by ”ANR Blanc” project CQUID

Outline

Chip-scale Atomic clock The NIST MicroClock

The principle: Coherent Population Trapping The system and its synchronization scheme Master equations and quantum trajectories

The slow/fast master equation The slow master equation Quantum trajectories Convergence analysis Concluding remarks

The NIST MicroClock

I Quartz crystal clocks: 1 second over few days.

I NIST chip-scale atomic clock: 1 second over 300 years

I High-Perf. atomic clocks: 1 second over 100 million years.

2NIST: National Institute of Standards and Technology, web-site:

http://tf.nist.gov/timefreq/index.html.

The principle: Coherent Population Trapping

3From the web-site:http://tf.nist.gov/timefreq/index.html.

The synchronization via extremum seeking

s=_dt^d, constant>0 parameters(h,k,a,ω).

Here u = ω_diode and y = f(ω_diode) where f admits a sharp maximum at the unknown value u¯=ωatom.

Extremum seeking via feedback: u(t) =v(t) +asin(ωt)where v(t)≈ωatomis adjusted via adynamic time-varying output feedback(withω,a,h,√

kωatom):

d

dtv(t) =−ksin(ωt)(y(t)−ξ(t)) with d

dtξ=h(y(t)−ξ(t)) Our contribution⁴: a real-time synchronization schemewhen the atomic cloud is replaced by a single atom.

4Mirrahimi-R, 2008, arxiv:0806.1392v1

The system and its synchronization scheme

Input: Ω˜1,Ω˜2∈C and u = _dt^d∆.

Output: photo-detector click times corresponding to stochas- tic jumps from|eito|g₁ior|g₂i.

Synchronization goal: stabilize the unknown detuning∆to 0.

Two time-scales:

|Ω˜₁|,|Ω˜₂|,|∆_e|,|∆| Γ₁,Γ₂

Modulationof Rabi complex amplitudesΩ˜₁andΩ˜₂: Ω˜₁(t) = Ω₁−ıεΩ₂cos(ωt), Ω˜₂(t) =ıεΩ₁cos(ωt) + Ω₂, withΩ1,Ω2>0 constant,ωΓ1,Γ2and 0<ε1.

Detuning update ∆_N+1= ∆_N−K^2Ω1Ω2

Ω²₁+Ω²₂cos(ωt_N) at each detected jump-timet_N. The gainK >0 fixes the standard deviationσ_K: 4σ_K² =εK^Ω_Γ^21+Ω22

1+Γ2.

Closed-loop quantum trajectory (

)

0 1 2 3 4 5

x 10⁴ 0

500 1000

ω t / 2 π

Click number

0 1 2 3 4 5

x 10⁴ 0

20 40

ω t / 2 π

Normalized de−tuning Δ/σ_K

Λ-system parameters:Γ₁= Γ₂=10,∆_e=2.0

Modulation parameters:Ω1= Ω2=1.0,ω=2.8,ε=0.14 Feedback gainK=0.0023 leading to a standard deviation σK =0.0057

Robustness

0 1 2 3 4 5

x 10⁴ 0

500 1000

ω t / 2 π

Click number

0 1 2 3 4 5

x 10⁴ 0

20 40

ω t / 2 π

Normalized de−tuning Δ/σ_K

Detector efficiencyof 50%,wrong jump detectionof 50%, feedback-loop delayofτ withω τ=π/4.

The slow/fast master equation

Master equation of theΛ-system d

dtρ=−ı

h¯[ ˜H,ρ] +1 2

2

∑

j=1

(2Q_jρQ_j^†−Q_j^†Q_jρ−ρQ^†_jQ_j),

with jump operatorsQ_j=p Γ_j

g_j

he|and Hamiltonian H˜

h¯ =∆

2(|g₂i hg₂| − |g₁i hg₁|)−

∆_e+∆ 2

|ei he|

+Ωe₁|g₁i he|+Ωe^∗₁|ei hg₁|+Ωe₂|g₂i he|+Ωe^∗₂|ei hg₂|.

Since|Ω˜₁|,|Ω˜₂|,|∆_e|,|∆| Γ₁,Γ₂we havetwo time-scales: a fast exponential decay for ”|ei”and aslow evolution for

”(|g₁i,|g₂i)”.

The slow master equation

Geometric reduction via center manifold techniques⁵leads to a reduced master equationthat is still of Lindblad type with a slow HamiltonianH andslow jump operatorsL_j:

d

dtρ=−ı

¯

h[H,ρ] +1 2

2

∑

j=1

(2L_jρL^†_j −L^†_jL_jρ−ρL^†_jL_j),

withH=^∆₂σz= ^∆(|g2ihg2₂^{|−|g1ihg1|)} andL_j=p γ˜_j

g_j b

Ωe

and whereγ˜_j=4^|eΩ_(Γ^1|2+|eΩ2|2

1+Γ₂)² Γ_j and b_Ω˜

is thebright state:

b

Ωe

= Ωe₁ q

|eΩ1|²+|eΩ2|²

|g₁i+ Ωe₂ q

|eΩ1|²+|eΩ2|²

|g₂i

For∆ =0,ρ converges towards thedark state

d_Ω˜ d_Ω˜ :

d

Ωe

=− Ωe^∗₂ q

|eΩ₁|²+|eΩ₂|²

|g₁i+ Ωe^∗₁ q

|eΩ₁|²+|eΩ₂|²

|g₂i.

5Mirrahimi-R 2008, arXiv:0801.1602v1, accepted in IEEE-AC.

Quantum trajectories for the slow approximation

In the absence of the quantum jumps,ρ evolves on theBloch sphereaccording to (˜γ=4^|Ωe1_Γ^|2+|Ωe2|2

1+Γ2 ) 1

γ˜ d

dtρ=−ı∆

2˜γ[σz,ρ]− b

Ωe b

Ωe

ρ+ρ

bΩe b

Ωe

2 +

bΩe

ρ

b

Ωe

ρ.

At each time stepdt,ρ may jump towards the state|g₁i hg₁|or

|g₂i hg₂|with ajump probabilitygiven by:

dt γ˜ bΩe

ρ

bΩe

SinceΩ˜₁(t) = Ω₁−ıεΩ₂cos(ωt)andΩ˜₂(t) =ıεΩ₁cos(ωt) + Ω₂, γ˜

bΩe b

Ωe

=γ(|bi+ıεcos(ωt)|di) (hb| −ıεcos(ωt)hd|) withγ=4^|Ω1_Γ^|2+|Ω2|2

1+Γ₂ ,|bi=^Ω1√^{|g1i+Ω2|g2i}

Ω²₁+Ω²₂ and|di= ^−Ω2√^{|g1i+Ω1|g2i}

Ω²₁+Ω²₂

Quantum trajectories in Bloch-sphere coordinates

Withβ=2 arg(Ω₁+ıΩ2)and

ρ= ^1+X(|bihd|+|dihb|)+Y(ı|bihd|−ı|dihb|)+Z(|dihd|−|bihb|)

2 :

d

dtX=−∆cosβY−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

X d

dtY= ∆cosβX−∆sinβZ+γ εcos(ωt)

−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

Y d

dtZ= ∆sinβY+γ

1−ε²cos²(ωt) 2

−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

Z Thejump probability per unit of timeis

P_jump= γ

2(1−Z−2εcos(ωt)Y+ε²cos²(ωt)(1+Z)).

Just after a jump(X,Y,Z)is reset to±(sinβ,0,cosβ).

Convergence of the no-jump dynamics

d

dtX=−∆cosβY−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

X d

dtY= ∆cosβX−∆sinβZ+γ εcos(ωt)−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

Y d

dtZ= ∆sinβY+γ

1−ε²cos²(ωt) 2

−γ

εcos(ωt)Y+1−ε²cos²(ωt)

2 Z

Z

For|∆|<₂^γ and 0<ε1, the above time-periodic nonlinear system admits aquasi-global asymptotically stable periodic orbit(proof: Poincar ´e-Bendixon and perturbation). It reads

(X,Y,Z) =

0 , −sinβ∆

γ +γ²cos(ωt) +γ ωsin(ωt)

ω²+γ² ε , 1

up to second order terms inε and ^∆_γ. Whenωγ,P_jump≈γ

εcos(ωt) +^∆sin_2γ^β 2

if the last jump occurs more that few−logε/γ second(s) ago.⁶.

6ReplaceZ by 1−^X2+Y₂ ² in previous formula givingP_jump.

Detuning update as a discrete-time stochastic process

Our analysis neglects the transient just after a jump.

When a jump occurs att_N, we have

∆_N+1=∆_N−Ksinβcos(ωt_N) and its probability was proportional to

εcos(ωt_N) +^∆N_2γ^sinβ 2

. The phaseϖ=ωt_N can be seen as a stochastic variable in [0,2π]with the following probability densityP_∆N(ϖ)on[0,2π]:

P∆_N(ϖ) =

εcos(ϖ) +^∆N_2γ^sinβ2

2π

ε² 2 +^∆

2 Nsin²β

4γ²

The de-tuning update is thus adiscrete-time stochastic process

∆_N+1=∆_N−Ksinβcosϖ where the probability ofϖ∈[0,2π]depends on∆_N.

Convergence proof

We assume here|∆| ε γ (rememberγωΓ₁+ Γ₂):

∆_N+1=∆_N−Ksinβcosϖ

withϖ of probability densityP_∆N(ϖ)≈ _2π¹ +^∆N_{π ε γ}^sinβcosϖ . Simple computations yield to⁷

E(∆N+1|∆_N) = 1−Ksin²β ε γ

!

∆N

For 0<K ≤ ^{ε γ}

sin²β,E(∆_N)tends to zero.

Similarly, we have

E(∆²_N+1|∆_N) = 1−2Ksin²β ε γ

!

∆²_N+K²sin²β 2 For 0<K ≤ ^{ε γ}

2 sin²β,E(∆²_N)converges toσ_K² =^{ε γK}₄ .

7E(∆_N+1|∆_N)stands for the conditional expectation-value of∆_N+1 knowing∆_N.

Summary: scales and feedback-gain design

Rabi frequency modulations:

Ω˜₁(t) = Ω₁−ıεΩ₂cos(ωt) Ω˜₂(t) =ıεΩ₁cos(ωt) + Ω₂ withΩ1,Ω2Γ = Γ1+ Γ2, 0<ε1 and

Ω²₁+Ω²₂

Γ1+Γ2 =γωΓ Detuning update

∆_N+1 = ∆_N −Ksinβcos(ωt_N) withK >0,β =2 arg(Ω₁+ıΩ₂).

Adiscrete-time stochastic processwhere the gainK >0 drives

I the convergence speedwith a contraction of

1−^Ksin2β

ε γ

forE(∆_N)at each iteration

I the precisionvia the asymptotic root-mean-square σ_K =

√

ε γK 2 .

Concluding remarks

I For anonlinear convergence proofwith∆<γ/2,ε small enough and well tuned gainK, see Mirrahimi-R 2008, arxiv:0806.1392v1. Sensitivity analysis to wrong jump detection and noise remains to be done.

I Such simple feedback can be also developed for other single quantum systems such as the 3-level system illustrating theDehmelt’s electron shelving scheme⁸

I Such feedback scheme could be a preliminary guide for inventing the ”quantum regulator”, a quantum analogue of theclassical PID regulator.

8C. Cohen-Tannoudji, J. Dalibard: Single atom Laser spectroscopy:

looking for dark periods in fluorescent light. Europhys. Lett. 1 (9), pp:441-448, 1986.