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

Quantum feedback for preparation and protection of 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 1

The Cavity QED team ENS team Clément Sayrin Xingxing Zhou Bruno Peaudecerf Théo Rybarczyk Igor Dotsenko Sébastien Gleyzes Michel Brune Jean-Michel Raimond Serge Haroche Ecole des Mines team Hadis Amini Alain Sarlette Mazyar Mirrahimi Pierre Rouchon Quantum feedback in cavity QED Workshop on Quantum Control, 2010 2

Cavity QED quantum feedback scheme Goal:  Steering the trapped microwave field (harmonic oscillator) to a desired quantum state  Preserving this state from decoherence injection state 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 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 3

Outline • Cavity QED setup • Quantum non-demolition measurement • Quantum feedback proposal: generation of photon-number states Quantum feedback in cavity QED Workshop on Quantum Control, 2010 4

Microwave superconducting cavity: Storage box for photons - Resonance @ ν cav = 51 GHz - Lifetime of photons T cav = 130 ms - Q factor = ω cav T cav = 4.2 ⋅ 10 10 2.8 cm - best Fabry-Pérot resonator so far - 1.4 billion bounces on the mirrors - a light travel distance of 39 000 km 5 cm (one full turn around the Earth) Quantum feedback in cavity QED Workshop on Quantum Control, 2010 5

Circular Rydberg atoms: Field microprobes 85 Rubidium atom - Rydberg atoms: large n = 51 (level e) principle quantum number n - Circular states: l=|m|=n-1 51 GHz - Mesoscopic orbit size = cavity resonance - Large dipole moment n = 50 (level g) Advantages: - Almost ideal two-level system - Long lifetime (30 ms) - Tunable via the Stark effect - Large coupling to radiation (orbit diameter of 0.25 µ m ) - Efficient state sensitive detection by ionization Quantum feedback in cavity QED Workshop on Quantum Control, 2010 6

Meeting atoms and photons Microwave source High Q cavity (harmonic oscillator) Atom source Field Lasers ionization detector Circular atoms preparation cooled to 0.8 K; thermal, acoustic, Low Q cavities: classical field pulses magnetic & electric (Ramsey interferometer to manipulate spin ½ atoms ) isolation Quantum feedback in cavity QED Workshop on Quantum Control, 2010 7

Outline • Cavity QED setup • Quantum non-demolition measurement • Quantum feedback proposal: generation of photon-number states Quantum feedback in cavity QED Workshop on Quantum Control, 2010 8

Dispersive interaction atom cavity number of photons Phase shift of atomic coherence (light shift) phase shift per photon Energy conservation + adiabatic coupling ⇒ the field (i.e. photon number) is preserved 9 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 9

QND measurement of photon number 1. Trigger of the atom clock: π resonant π /2 pulse 2 Bloch vector representation for spin ½ particle 10 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 10

QND measurement of photon number 1. Trigger of the atom clock: π π resonant π /2 pulse 2 2 2. Dephasing of the clock: phase interaction with the cavity field n=6 n=5 n=7 n=4 n=0 n=3 n=1 n=2 11 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 11

QND measurement of photon number 1. Trigger of the atom clock: π π resonant π /2 pulse 2 2 2. Dephasing of the clock: phase interaction with the cavity field 3. Measurement of the clock: second π /2 pulse & atom's state detection n=1 n=0 n=2 n=7 n=3 n=6 n=5 n=4 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 12

Single atom detection atom detection changes photon-number distribution 2 3 1 4 photon number probability 0,25 detection 0 direction 0,20 5 0,15 atom in | e 〉 7 6 0,10 0,05 0,00 0 1 2 3 4 5 6 7 8 number of photons 0,25 photon number probability initial knowledge 0,20 0,25 photon number probability 0,15 atom in | g 〉 0,20 0,10 0,15 0,05 0,00 0,10 0 1 2 3 4 5 6 7 8 number of photons 0,05 0,00 0 1 2 3 4 5 6 7 8 number of photons Quantum feedback in cavity QED Workshop on Quantum Control, 2010 13

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 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 14

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 Many repeated weak measurements result in the ideal projective measurement of the photon number Quantum feedback in cavity QED Workshop on Quantum Control, 2010 15

Photon number statistics Coherent state of a harmonic oscillator in a photon-number basis | n 〉 =3 state preparation by post-selection Quantum feedback in cavity QED Workshop on Quantum Control, 2010 16

Outline • Cavity QED setup • Quantum non-demolition measurement • Quantum feedback proposal: generation of photon-number states on demand Quantum feedback in cavity QED Workshop on Quantum Control, 2010 17

Single atom measurement atom detection changes photon-number distribution 2 3 1 4 photon number probability 0,25 0 0,20 5 0,15 atom in | e 〉 7 6 0,10 0,05 0,00 0 1 2 3 4 5 6 7 8 number of photons initial field 0,25 photon number probability 0,25 photon number probability 0,20 0,20 0,15 atom in | g 〉 0,15 0,10 0,05 0,10 0,00 0 1 2 3 4 5 6 7 8 0,05 number of photons 0,00 0 1 2 3 4 5 6 7 8 number of photons Quantum feedback in cavity QED Workshop on Quantum Control, 2010 18

Back-action of weak measurement initial state atom detection changes photon-number distribution detection photon number probability 0,25 direction projected state 0,20 0,15 atom in | e 〉 0,10 ⇒ bad 0,05 photon number Two POVMs 0,00 operator 0 1 2 3 4 5 6 7 8 correspond to number of photons two possible 0,25 photon number probability experimental phase shift 0,20 outcomes 0,15 atom in | g 〉 0,10 ⇒ good 0,05 0,00 0 1 2 3 4 5 6 7 8 number of photons on average, good/bad outcomes are equally probable Quantum feedback in cavity QED Workshop on Quantum Control, 2010 19

Back-action of weak measurement initial state Deterministically prepare state |n 〉 = 3 ? detection photon number probability 0,25 direction projected state 0,20 0,15 atom in | e 〉 0,10 ⇒ bad 0,05 photon number Two POVMs 0,00 operator 0 1 2 3 4 5 6 7 8 correspond to number of photons two possible 0,25 photon number probability experimental phase shift 0,20 outcomes 0,15 atom in | g 〉 0,10 ⇒ good 0,05 0,00 0 1 2 3 4 5 6 7 8 number of photons I dea: Let us alter the distribution, i.e. increase P(n= 3), depending on measurement outcome before the next measurement Quantum feedback in cavity QED Workshop on Quantum Control, 2010 20

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: optimal gain Quantum feedback in cavity QED Workshop on Quantum Control, 2010 21

Proposal: Quantum feedback loop Standard closed-loop components:  Sensor (quantum): atoms and QND measurement  Controller (classical): classical computer  Actuator (classical): injection microwave injection state 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 ρ proj  Calculate displacement α , which maximizes overlap F between ρ target and ρ disp  Close feedback loop by injecting a control coherent field | α〉  Repeat feedback cycles until success when F ≈ 1 Quantum feedback in cavity QED Workshop on Quantum Control, 2010 22

Feedback performance: ideal case Monte-Carlo simulation with n target = 3 photons n > n target n = n target n < n target average over 10 4 quantum trajectories Quantum feedback in cavity QED Workshop on Quantum Control, 2010 23