Quantum feedback for preparation and protection of quantum states - - PowerPoint PPT Presentation

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Quantum feedback for preparation and protection of quantum states - - PowerPoint PPT Presentation

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Quantum feedback for preparation and protection of quantum states of light I gor Dotsenko LABORATOI RE KASTLER BROSSEL de lcole Normale Suprieure, Paris Workshop on Quantum Control I HP, Paris December 8-11, 2010 1 The Cavity QED

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Quantum feedback for preparation and protection

  • f quantum states of light

I gor Dotsenko

LABORATOI RE KASTLER BROSSEL de l‘École Normale Supérieure, Paris

Workshop on Quantum Control I HP, Paris December 8-11, 2010

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

The Cavity QED team

Clément Sayrin Xingxing Zhou Bruno Peaudecerf Théo Rybarczyk Michel Brune Jean-Michel Raimond Serge Haroche Hadis Amini Alain Sarlette Mazyar Mirrahimi Pierre Rouchon Igor Dotsenko Sébastien Gleyzes ENS team Ecole des Mines team

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Cavity QED quantum feedback scheme

state injection

Goal:

  • Steering the trapped microwave

field (harmonic oscillator) to a desired quantum state

  • Preserving this state from

decoherence

Elements of feedback loop

  • Quantum measurement:

performed with (spin ½) atoms followed by cavity state estimation

  • Quantum filter:

estimation of what is best to do for becoming closer to the target

  • Actuator – Classical:

microwave injection with a classical source Quantum: resonant interaction with a single two-level atom

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Quantum feedback in cavity QED

Outline

  • Cavity QED setup
  • Quantum non-demolition measurement
  • Quantum feedback proposal:

Workshop on Quantum Control, 2010 4

generation of photon-number states

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Microwave superconducting cavity: Storage box for photons

  • Resonance @ νcav = 51 GHz
  • Lifetime of photons Tcav= 130 ms
  • Q factor = ωcav Tcav = 4.2 ⋅ 1010

5 cm

  • best Fabry-Pérot resonator so far
  • 1.4 billion bounces on the mirrors
  • a light travel distance of 39 000 km

(one full turn around the Earth) 2.8 cm

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

  • Rydberg atoms: large

principle quantum number n

  • Circular states: l=|m|=n-1
  • Mesoscopic orbit size
  • Large dipole moment

Advantages:

  • Almost ideal two-level system
  • Long lifetime (30 ms)
  • Tunable via the Stark effect
  • Large coupling to radiation (orbit diameter of 0.25 µm)
  • Efficient state sensitive detection by ionization

Circular Rydberg atoms: Field microprobes

n = 51 (level e) n = 50 (level g) 51 GHz = cavity resonance

85 Rubidium atom

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Meeting atoms and photons

Field ionization detector Microwave source Lasers Circular atoms preparation Low Q cavities: classical field pulses

(Ramsey interferometer to manipulate spin ½ atoms )

High Q cavity (harmonic oscillator) Atom source cooled to 0.8 K; thermal, acoustic, magnetic & electric isolation

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Quantum feedback in cavity QED

Outline

  • Cavity QED setup
  • Quantum non-demolition measurement
  • Quantum feedback proposal:

Workshop on Quantum Control, 2010 8

generation of photon-number states

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Dispersive interaction

cavity atom

Phase shift of atomic coherence (light shift) Energy conservation + adiabatic coupling ⇒ the field (i.e. photon number) is preserved

phase shift per photon

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number

  • f photons
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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

QND measurement of photon number

  • 1. Trigger of the atom clock:

resonant π/2 pulse

π 2

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Bloch vector representation for spin ½ particle

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

QND measurement of photon number

  • 1. Trigger of the atom clock:

resonant π/2 pulse

π 2

  • 2. Dephasing of the clock:

interaction with the cavity field

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π 2

n=2 n=0 n=1 n=3 n=4 n=5 n=6 n=7

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phase

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

QND measurement of photon number

  • 1. Trigger of the atom clock:

resonant π/2 pulse

π 2 π 2

  • 2. Dephasing of the clock:

interaction with the cavity field

  • 3. Measurement of the clock:

second π/2 pulse & atom's state detection phase

n=2 n=0 n=1 n=3 n=4 n=5 n=6 n=7

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Single atom detection

initial knowledge

atom in |e〉 atom in |g〉

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

photon number probability number of photons photon number probability

1 2 3 4 5 6 7

photon number probability number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

detection direction

atom detection changes photon-number distribution

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

From weak to projective measurement

  • Initial coherent field with 3.7 photon
  • Progressive collapse of the field

state vector during information acquisition

Many repeated weak measurements result in the ideal projective measurement of the photon number

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Another sequence

  • Final photon number fluctuates

randomly from sequence to sequence

  • Statistics of final photon number

should reveal the statistics of the initial quantum field

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Many repeated weak measurements result in the ideal projective measurement of the photon number

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Photon number statistics

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|n〉=3 state preparation by post-selection

Coherent state of a harmonic oscillator in a photon-number basis

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Quantum feedback in cavity QED

Outline

  • Cavity QED setup
  • Quantum non-demolition measurement
  • Quantum feedback proposal:

generation of photon-number states on demand

Workshop on Quantum Control, 2010 17

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

photon number probability number of photons

Single atom measurement

photon number probability photon number probability number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

atom detection changes photon-number distribution initial field

1 2 3 4 5 6 7 atom in |e〉 atom in |g〉

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Back-action of weak measurement

initial state projected state detection direction phase shift photon number

  • perator

photon number probability photon number probability number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

Two POVMs correspond to two possible experimental

  • utcomes

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atom detection changes photon-number distribution

atom in |e〉 atom in |g〉

⇒ good ⇒ bad

  • n average, good/bad outcomes

are equally probable

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Back-action of weak measurement

initial state projected state detection direction phase shift photon number

  • perator

photon number probability photon number probability number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

number of photons

1 2 3 4 5 6 7 8 0,00 0,05 0,10 0,15 0,20 0,25

Two POVMs correspond to two possible experimental

  • utcomes

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atom in |e〉 atom in |g〉

Deterministically prepare state |n〉= 3 ?

⇒ good ⇒ bad I dea: Let us alter the distribution, i.e. increase P(n= 3), depending on measurement outcome before the next measurement

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Field displacement as feedback control

We modify the photon-number distribution by displacing the field's state:

displacement operator : injection of a coherent pulse into the cavity displacement amplitude: complex amplitude of the injection pulse

Displacement amplitude is chosen to maximize the fidelity to the desired photon number (i.e. population of this state):

desired photon number state

Efficient feedback law (using Lyapunov function approach) reads:

  • ptimal gain

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Proposal: Quantum feedback loop

state injection

Standard closed-loop components:

  • Sensor (quantum):

atoms and QND measurement

  • Controller (classical):

classical computer

  • Actuator (classical):

microwave injection Feedback protocol:

  • Inject initial coherent field into the cavity
  • Send one-by-one atoms in a Ramsey configuration
  • Detection of each atom projects cavity field ρ into a new state ρproj
  • Calculate displacement α, which maximizes overlap F between ρtarget and ρdisp
  • Close feedback loop by injecting a control coherent field |α〉
  • Repeat feedback cycles until success when F ≈1

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Quantum feedback in cavity QED

Feedback performance: ideal case

n > ntarget n = ntarget n < ntarget

Workshop on Quantum Control, 2010 23

Monte-Carlo simulation with ntarget = 3 photons average over 104 quantum trajectories

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Quantum feedback in cavity QED

Feedback performance: realistic case

  • target Fock state: |3〉
  • Hilbert space size: 10
  • cavity decay time: 130 ms
  • separation of atomic pulses: 100 µs
  • delay in atom detection of 4 atoms
  • black-body thermal field: 0.05 photons
  • average number of atoms per pulse: 0.4
  • detection efficiency: 80%
  • false state detection: 10%

Workshop on Quantum Control, 2010 24

Simulation parameters:

n > ntarget n = ntarget n < ntarget

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Population

Feedback performance

initial state final state photon loss convergence state recovery Simulation results for a target Fock state ρtarget = |3〉

about 3 atoms in 1 ms

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Average over many trajectories

What is fidelity of the state production at arbitrary time? about 63% fidelity result of cavity decay

average over 104 quantum trajectories

Population

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Convergence rate

How fast the target state is prepared ?

  • We chose the feedback to converge if F > 95%
  • Convergence probability of about 50% after 20 ms
  • Inevitably, all trajectories converge

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Quantum feedback in cavity QED

Resonant atoms as field injectors

Workshop on Quantum Control, 2010 28 5 10 15 20 0,0 0,2 0,4 0,6 0,8 1,0

Emission probability P(t) Time t [µs]

Probability to inject a photon for an atom initially in |e〉:

Rabi oscillation in a photon number state

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Quantum feedback in cavity QED

Resonant atoms as field injectors

Rabi oscillation in a photon number state

At t =10 µs, atoms perform 2π pulse in 3 photons: “trapping state” situation

Workshop on Quantum Control, 2010 29 5 10 15 20 0,0 0,2 0,4 0,6 0,8 1,0

Emission probability P(t) Time t [µs]

n=0 n=1 n=2 n=3 n=4 n=5

Probability to inject a photon for an atom initially in |e〉: “Photon pumping” of the cavity: 1. Start from empty cavity (vacuum state with n=0) 2. Send atoms in |e〉 and set interaction time to 2π for |3〉 state 3. After several atoms, the cavity will be “pumped” and “trapped” in |3〉

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Quantum feedback in cavity QED

BUT: As soon as n>3 (e.g. due to thermal field excitation), field will continue to uncontrollably increase and run away from the desired state ! SOLUTION: Probe the field with QND atoms and start to send resonant atoms in state |g〉 if n>3 in order to absorb the excess field !

Resonant atoms as field injectors

Rabi oscillation in a photon number state

At t =10 µs, atoms perform 2π pulse in 3 photons: “trapping state” situation

Workshop on Quantum Control, 2010 30 5 10 15 20 0,0 0,2 0,4 0,6 0,8 1,0

Emission probability P(t) Time t [µs]

n=0 n=1 n=2 n=3 n=4 n=5

Probability to inject a photon for an atom initially in |e〉:

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Quantum feedback in cavity QED

Atomic feedback convergence

initial state: coherent with 〈n〉=3 initial state: vacuum

Very fast convergence toward the target

Workshop on Quantum Control, 2010 31

Monte-Carlo simulations

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Quantum feedback in cavity QED

Stabilization of decoherence

Cavity decay: |3〉 → |2〉 quantum jump ⇓ Intrinsic passive stability Black-body field emission: |3〉 →|4〉 quantum jump ⇓ Active feedback action

Workshop on Quantum Control, 2010 32

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Quantum feedback in cavity QED

Atomic feedback convergence

Convergence to |n〉=3 with 90% fidelity in 15 ms

Average over many trajectory

Time (ms) Photon number probability

|n〉=3 |n〉=2

Workshop on Quantum Control, 2010 33

initial state: coherent with 〈n〉=3

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Quantum feedback in cavity QED

Trapping state instability

Photon number evolution in trapping state condition without feedback: instability due to blackbody radiation

Workshop on Quantum Control, 2010 34

initial state: coherent with 〈n〉=3

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Quantum feedback in cavity QED

  • Other work in progress:

non-local state preparation in two cavities

EPR pair of Schrödinger cats

Conclusion / Perspectives

  • Quantum Non-Demolition photon counting for quantum

state preparation :

weak and projective measurement

  • Quantum feedback proposals – coherent and atomic:

deterministic preparation of number states with high fidelity

protection of these states with respect to decoherence

Realization in progress

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Quantum feedback in cavity QED Workshop on Quantum Control, 2010

Thank you

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Clément Sayrin Xingxing Zhou Bruno Peaudecerf Théo Rybarczyk Michel Brune Jean-Michel Raimond Serge Haroche Hadis Amini Alain Sarlette Mazyar Mirrahimi Pierre Rouchon Igor Dotsenko Sébastien Gleyzes ENS team Ecole des Mines team