Real Time Imaging of Quantum and Thermal Fluctuations (A pinch of - - PowerPoint PPT Presentation

Real Time Imaging of Quantum and Thermal Fluctuations

D.B. with M. Bauer, and (in part) T. Benoist & A. Tilloy. Kyoto, July 2013

(A pinch of quantum mechanics, a drop of probability, ...)

arXiv:1106.4953, arXiv:1210.0425, arXiv:1303.6658, to appear (hopefully soon...)

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1 e g e g Time (s) 0.90 0.0 0.5 1.0 1.5 2.0 2.5

b

0.95 1.00 1.05 1.10 1.15 1.20

a

State n

Quantum jumps of light recording the birth and death

• f a photon in a cavity

Se ´bastien Gleyzes1, Stefan Kuhr1{, Christine Guerlin1, Julien Bernu1, Samuel Dele ´glise1, Ulrich Busk Hoff1, Michel Brune1, Jean-Michel Raimond1 & Serge Haroche1,2

Non-demolition measurements and Q-jumps

• - How to measure photons without destroying them ?
• - How to record the cavity states ?
• - How to observe quantum jumps? Are they detector dependent ?
• - What determines the Q-jump dynamics ?

Figure 2 | Birth, life and death of a photon. a, QND detection of a single

• photon. Red and blue bars show the raw signal, a sequence of atoms detected

in e or g, respectively (upper trace). The inset zooms into the region where the statistics of the detection events suddenly change, revealing the quantum jump from |0æ to |1æ. The photon number inferred by a majority vote over

Courtesy of LKB-ENS.

Creation-Annihilation of a thermal photon. The photon system is probed indirectly via another quantum system.

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Cavity QED experiments

Photons in a cavity Probe measurement apparatus Preparation

• f the probes

Courtesy of LKB-ENS.

• - Indirect measurements:

Direct (Von Neumann) measurements on an auxiliary system (the probes).

• - Testing light/photon (the quantum system) with matter (the quantum probes).

System (S)= photons in a cavity. Probes (P)= Rydberg atoms (two state systems) No direct observation of the cavity (the system).

• - Probe like gyroscope (spin half system).
• - Interaction like rotation of the gyroscope

(with an angle depending on the number of photons)

U = exp[iθ σz Nphoton]

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Number of indirect measurements P.d.f. of the photon numbers

Courtesy of LKB-ENS.

Progressive field-state collapse and quantum non-demolition photon counting

Christine Guerlin1, Julien Bernu1, Samuel Dele ´glise1, Cle ´ment Sayrin1, Se ´bastien Gleyzes1, Stefan Kuhr1{, Michel Brune1, Jean-Michel Raimond1 & Serge Haroche1,2

• -- Why does the p.d.f. change after each indirect measurement ?
• -- How does it evolve? why does it become peaked (collapsed) ?
• -- How does it represent the cavity state ?
• -- What does continuous-in-time quantum measurement mean ?

(Here: a `discrete version’ of time continuous measurement)

Figure 2 | Progressive collapse of field into photon number state.

(integral of PN(n) normalized to unity). c, Photon number probabilities plotted versus photon and atom numbers n and N. The histograms evolve, as N increases from 0 to 110, from a flat distribution into n 5 5 and n 5 7 peaks.

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Outline:

• - A view on (quantum) noise.

(repeated interactions)

• - Repeated (quantum) measurements, Bayes’ law and collapses.

(via the martingale convergence theorem).

• - Pointer states, exchangeability and de Finetti’s theorem.

(objective or subjective probabilities) Classical probability theory with a bit of quantum mechanics, (or the reverse).

Part I: Part II:

• - Time continuous measurement and Q-jumps.

(Bi-stability (multi-stability) and Q-jumps)

• - Real time imaging of thermal and quantum fluctuations.

(observing quantum fluctuations continuously in time.) Making real «virtual» fluctuations

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Quantum non-demolition measurement and random bayesian updating.

up-dating of the trial/estimated p.d.f. at each step, using an a priori model for the output conditional probabilities

• - Repeated cycles of interaction plus probe measurement:

Partial gain of information at each iteration, because of system-probe entanglement.

• - Bayesian approach (encoded into Q-mechanics) :

Pestimated(Nphoton

• probe

state) = Pa priori(probe state

• Nphoton) Ptrial(Nphoton)

Pnormalisation(probe

state)

And the updating is random (because of Q-mechanics)

Photons in a cavity Probe measurement apparatus Preparation

• f the probes

Courtesy of LKB-ENS.

• - Probe measurements give values + or - ;

Recursion for the photon number p.d.f. from data of sequences

ω = (+, +, −, +, · · · ) = (ǫ1, ǫ2, ǫ2, · · · )

• - A two step analysis :
• - What happens during one cycle ?
• - What happens for an iteration of cycles ?

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Quantum mechanics implies «classical» Bayes’ rules

Or what happens during one interaction + measurement cycle?

• - Interaction: A delicate point : we suppose that there is a basis of system states

*preserved* by the probe-system interaction, i.e.: U |α ⊗ |φ = |α ⊗ Uα|φ

for U the evolution operator of the probe-system interaction

i.e. Bayes’ rules |φ

• - probe:
• - system:
• - Preparation:
• - After interaction:
• - probe + system :
• α C(α) |α ⊗ Uα|φ

|ψ =

α C(α)|α

• - After probe measurement: If output probe measurement is |i
• - probe + system :∝

α C(α)i|Uα|φ |α

• ⊗ |i
• - New state distribution :

... and new system state.

with p(i|α) = |i|Uα|α|2 , Q0(α) = |C(α)|2 This will be related to the existence of *pointer states*, to exchangeability,... Qnew(α) = p(i|α) Q0(α) Z(i)

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Evolution of the probability distributions:

Random Bayesian up-dating...

Q0(α),

• α

Q0(α) = 1,

Pick a basis a of states of the Q-system. Start with a probability distribution (initial system state): Let i be the output measurements on the probes. Data (probe-system interaction) are probabilities to measure i conditioned on the Q-system to be in state a. p(i|α),

• i

p(i|α) = 1 The output of the n-th probe measurement is i_n with probability : Let Q_{n-1}(a) be the probability distribution of the Q-system after (n-1) cycles, Then, Qn(α) = 1 Zn p(in|α) Qn−1(α), with Zn =

• α

p(in|α) Qn−1(α)

πn(i)

P := Probes

Iterations...

Out-going probes after interaction with the Q-system S := Quantum System

• utputs....

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Collapse of the p.d.f. Qn(α)

as n → ∞

* Peaked distributions are stable (stability of the pointer states):

Qn(α) = δα;γ are solutions. Then, outputs i_n are i.i.d. with probability: p(in|γ)

* The convergence is exponential :

Qn(α) ≃ exp[−n S(γω|α)] (α = γω)

with a relative entropy. S(γ|α) = −

• i

p(i|γ) log p(i|α) p(i|γ)

* Probability distributions converge a.s. (and in L1) towards peaked distributions

(collapse of the wave function):

lim

n→∞ Qn(α) = δα,γω

with a realisation dependent target γω

Prob[γω = β] = Q0(β)

(Von Neumann rules for quantum measurements)

(A classical statement...)

Claim:

Qn(α) = p(in|α) Qn−1(α) Zn , with proba πn := Zn =

• α

p(in|α) Qn−1(α)

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Mesoscopic collapses :

What about if we don’t record the probe outputs?

• - statistical ensemble of peaked distribution = diagonal density matrix

Progressive decoherence: α|ρn|α = Q0(α) = const. α|ρn|β =

• U †

αUβ

n α|ρ0|β → 0 Quantum to classical transition. "Collapse is nothing morethan the updating of that calculational device

• n the basis of additional experience".
• D. Mermin, arXiv:1301.6551 (posterior to these works)

... A Bayesian point of view. ... and some robustness or universality in quantum measurements Mesoscopic measurements.

Elements of a proof :

• - Qn(α) are bounded martingales, i.e. E[Qn(α)|Fn−1] = Qn−1(α)

as such they converge a.s. and in L1

• - Asymptotically, the outputs are i.i.d. with asymptotic frequencies : Nn(i) ≃n→∞ n p(i|γω)
• - The limit is independent of the initial trial distribution. (Important for the experiment).

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S := Quantum System P := Probes

Iterations...

Out-going probes after interaction with the Q-system

• - Gain of information : by testing output observable on the n-th first probes,

but a probabilistic gain because of Q.M.

• - Hilbert space:
• - Algebras of observable:

H = Hs ⊗ H1 ⊗ · · · ⊗ Hn ⊗ · · ·

• - Filtration:

As ⊂ Bn ⊂ Bm, for n < m

Bn := As ⊗ A1 ⊗ · · · ⊗ An ⊗ I

Quantum noise and repeated quantum interaction.

• - Measure some observable on the (n-th) output probes (not on the Q-system):

The quantum filtration is reduced to a classical filtration. (Quantum Trajectories) (classical random process, the events are the out-put measurements)

with measurements

• r not?

Photons in a cavity Probe measurement apparatus Preparation

• f the

probes Courtesy of LKB-ENS.

e.g. as in cavity QED experiments....

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Macroscopic measurement apparatus:

Measure whether the system is in state a, i.e. measure observable with eigenstates a. Data of the apparatus: the p.d.f. p(i|α) on I, for all α. S := Quantum System

For each infinite cycle, the apparatus provide the infinite sequence,

(i1, · · · , in, · · · )

Compare the empirical histogram of the output measurements, with the given distributions p(i|a)

Target state = result of the measure

Generalizations :

• - with different probes, probe measurements, randomly chosen, etc..
• - continuous in time description, continuous measurements, etc....

Partial collapse for mesoscopic measurements. But also «classical Bayesian measurement apparatus».

Applications: e.g. control and state manipulations......

Reader

i.e. the frequencies N(i|α)

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Time continuous measurement and Q-jumps:

• - As a model for discrete repeated measurement but with short time interval
• - hamiltonian evolution (T=0):
• - measurement (POVM) :

ρ → Uhamilton ρ U †

hamilton

ρ → (Fi ρ F †

i )/πi, with Fi := i|Umeas.|ψ

• - If time duration of «probe+system interaction cycles» is small:

Random time continuous measurements, dρ = i[H, ρ] dt + (dρ)meas (a condition on probe data), these are diffusive like equations (Belavkin’s eqs.) If i|ψ = 0

• (randomness due to (random) output probe measurements)

For spin 1/2 probes : discrete (+, +, −, +, −, · · · ) → Bt := brownian motion. For a Q-bit system : dQt = γ Qt(1 − Qt) dBt, for Qt = 0|ρt|0.

• - Non-demolition measurement : H and the measured observable commute.

Making real Bohr’s «virtual» quantum jumps τcollapse ≃ γ−2 (dρ)meas = Lmeas(ρ) dt + Dmeas(ρ) dBt |0, |1

(in law)

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• - Measuring an observable commuting with H : (progressive) collapse.
• - Measuring an observable not commuting with H : Q-jumps.

Time continuous measurement and Q-jumps (II):

• - What happens if the H and the measured observable do not commute?

A system (spin half) under continuous measurement.

H = ω0 σ2

With:

ρ = 1

2

• 1 + cos θσ3 + sin θσ1

Take

Sz = σ3

and measure

• - Technically: Kramers like transition for a two well potential random process.

Conclusion

Slightly deformed Rabi oscillations (measurement does take place) γ2 ≪ ω0 γ2 ≫ ω0, τcollpase ≪ τevolution Q-jumps between the two eigen-states τflip ≃ τ 2

evolution/τcollapse

dθt = −(ω0 + γ2 sin 2θt)dt − 2γ sin θt dBt

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• - Quantum/Thermal fluctuations and Q-jumps:
• - For system in contact

with a thermal reservoir and under continuous measurements.

Real Time Imaging of Quantum and Thermal Fluctuations.

Quantum System Reservoir Outputs

Energy cascade and Q-jumps

• - Evolution under thermal contact:
• - Recursive indirect measurements:

with probability ρn → ˜ ρn := Mtherm[ρn] :=

• k

Bk ρn B†

k

What are the quantum trajectories ? ˜ ρn → ρn+1 := M in

meas[˜

ρn] := Fin ˜ ρnF †

in/πin

πin := Tr(Fin ˜ ρnF †

in)

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Real Time Imaging of Quantum and Thermal Fluctuations (II).

• - Time continuous formulation:

dρ = (dρ)therm. + (dρ)mes. Deterministic thermal evolution (Lindblad) Random time continuous measurements T wo time scales :

τcollapse ≪ τtherm.

• - For two states systems

(with spin half probes): ρ = Q |00| + (1 − Q) |11| dQt = λ [p − Qt] dt + γ Qt(1 − Qt) dBt. * Waiting times : * Jump times : * Stationary measure:

and exponentially distributed.

τjump ≃ τcollapse log(τtherm/τcollapse)

controlled by the measurement process, Close to Gibbs but not quite.

Q-trajectories and Q-jumps

Generalisations with many states and arbitrary probes (OK), and Applications...... (γ2 ≫ λ)

T1 ≃ τtherm/p, T0 ≃ τtherm/(1 − p), T0/T1 = eβ,

with more thermal bath (off-equilibrium).

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Thank you.

Understanding repeated QND measurements and mesoscopic measurements : An interesting exercise in probability/Q.M. theory, with some (probable) quantum applications...

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