Lecture 3 Quantum non-demolition photon counting and quantum jumps of light

Transcription

1 Lecture 3 Quantum non-demolition photon counting and quantum jumps of light A stream of atoms extracts information continuously and non-destructively from a trapped quantum field Fundamental test of measurement theory and applications for quantum information.

2 Photon detection : a chronicle of a foretold death «clic» «clic» «clic» 1!!" 0 A clic projects the field onto the vacuum: clic the photon dies upon delivering its message This is not what textbooks of Quantum Physics tell us about ideal projective measurements! A QND measurement should realize instead:! 1!!" 1!!" 1!!" """! clic clic clic clic!" 1? We need a non-demolition detector at single photon level and a very good box to keep the photons alive long enough

3 It has been a very long quest!

4 Bloch sphere representation of the two-level Rydberg atom Equatorial plane of Bloch sphere is the dial and the spin is the hand of an atomic clock e> (n=51) π/2 microwave pulse φ=2πνt g> (n=50) e> + g> Free evolution e> + e iφ g> Phase shift per photon adjusted by changing atomcavity detuning Atoms are off-resonant and cannot absorb light, but spins are delayed by light-shift effect. One photon can make the «spin hand» miss half a turn while atom crosses cavity (π phase shift per photon).

5 Outline 3A. A super-high-q cavity as a photon trap 3B. Repetitive QND measurement of a single light quantum: Witnessing the birth, life and death of a photon 3C. QND measurement of arbitrary photon numbers: witnessing the progressive collapse of a field state 3D. Conclusion of 3 rd lecture

6 3A. A super high-q cavity as a photon trap Mirrors reflecting the face of Christine Guerlin

7 Niobium coated copper mirrors Sputter 12 µm of Nb Niobium coated copper Particles mirrors accelerator technique Sputtering at CEA, Saclay [E. Jacques, B. Visentin, P. Bosland] Copper mirrors Diamond machined ~1 µm ptv form accuracy ~10 nm roughness Toroidal single mode

8 The new cavity (half-mounted) The new cavity

9 A very good photon box S.Kuhr et al, Applied Physics Letters, 90, (2007) T c = s Atoms Q = ω T c = A photon bounces on average 1.3 billion times before decaying! largest finesse for an open FP resonator at any frequency: f = Q/9 = The best mirrors ever! Light travels between mirrors over distance equal to Earth circumference during 1/e damping. and some (lucky) photons travel over half the Moon to Earth distance!

10 An artist s view of the set-up Classical pulses (Ramsey interferometer) Rydberg atoms High Q cavity An atomic clock with photons trapped inside

11 and the real thing Atoms Cavity Cavity R 1 C 1 R 2 C 2 R 3 Cold region (at bottom of helium cryostat): 40 cm side box 40 kg copper and Niobium 0.8 K base temperature 24 hours cooling time below 2K for 18 months!

12 3B. Repetitive QND measurement of a single light quantum: Witnessing the birth, life and death of a photon S.Gleyzes et al, Nature, 446, 297 (2007)

13 Non-resonant atoms phase-shifted by light e, n g, n w! 2 (n + 1) 4"! "2 n 4# No absorption even if δ Ω (adiabatic atomic evolution) Vacuum Rabi frequency! 2" = 50kHz g e δ Light shift!" = #2 (n + 1 / 2) 2$ Lamb shift t %!" = & per photon if #2 t $ = 2&! 2 t " =!2 " 70 khz # 2 w v = 2# 250 m/s

14 Each atom s pseudo spin is a clock whose rate is affected by light 1. Reset the clock (1 st Ramsey pulse). 2. precession ot the spin through the cavity: clock ticks. e> g> π 2 n π 2 z z 5 6 y n = 0 y x 1 2 x 2 1! 0 The clock s shift is proportional to n: non-demolition photon counting by measuring spin direction (using 2 nd Ramsey pulse) phase shift per photon

15 Detecting 0 or 1 photon Strong dispersive coupling: " 0 =! e> g> π 2 0> or 1> z z y!, n = 1 y +, n = 0 y y 1 x 2 x One atom = one bit of information (+ or - spin along y) perfectly correlated with the photon number.

16 Detecting 0 or 1 photon Strong dispersive coupling: " 0 =! e> g> π 2 π 2 Detection e or g 3 z z e, n = 1 y y 1 x 2 Atom detected in e field projected on g, 1> n = 0 g field projected on 0> e field projected on 1> x

17 Repeated measurement of a small thermal field (cavity at 0.8K) Thermal field at 0.8 K fluctuates between 0 and 1 photon (n t =0.05) R 1 R 2 e or g?

18 Birth and death of a photon e 0,90 0,95 1,00 1,05 1,10 1,15 1,20 g Quantum jump e g 1 Hundreds of atoms see same photon 0 0,0 0,5 1,0 1,5 2,0 2,5 time (s)

19 Other thermal photons n th = 0.05 (Planck law at 0.8K) This is an absolute radiation thermometer!

20 QND measurement, quantum gate and mesoscopic entanglement Prepare with 1 st resonant atom a superposition of 0 and 1 photon (π Rabi pulse ): ( g + e )! 0 " #" g! ( ) Probe field with sequence of non-resonant atoms (QND). The photon (0/1) is a control qubit for atoms. Before detection, massive entanglement is generated (atomic Schrödinger cat). Information carried by field is shared in a quantum way by all atoms. + 0;g,g,g,gLg + 1;e,e,e,eLe Requires a change in set-up (move detector downstream)

21 QND detection of 0 or 1 photon is a measurement of photon number parity n = 1,3,5 n = 0,2,4 Interferometer set with π phase-shift per photon: atom in e! "! n odd atom in g! "! n even For <n> << 1, probability for n > 1 is small and measuring parity is equivalent to counting n (one bit of information) Extending the method to larger photon numbers requires more atoms per measuring sequence (at least as many as number of bits to write n in binary form!)

22 3. QND measurement of arbitrary photon numbers: progressive collapse of field state P(n) Δn n A coherent field (Glauber state) has uncertain photon number: ΔnΔφ 1/2 Heisenberg relation A small coherent state with Poissonian uncertainty and 0 n 7 is initially injected in the cavity and its photon number is progressively pinned-down by QND atoms Experiment illustrates on light quanta the three postulates of measurement: state collapse, statistics of results, repeatability. C.Guerlin et al, Nature, 448, 889 (2007)

23 Counting larger photon numbers: 1 st atom effect on inferred photon distribution Chose Φ 0 =π/4 P(n) z e> g> π 2 z n π 2 ϕ 2 nd Ramsey pulse maps a direction in equatorial plane back into Oz before detection If «spin» found in state -+ (j=1) (j=0) (along n=6) n=2) n P(n) x probability multiplied y by a cosine function of n n Random decimation of photon number projection postulate (or Bayes law) j = 1 6 j = 0 x 2 Detection direction 1 7 n = 0 y! 0 phase shift per photon

24 A step-by-step acquisition of information! n c n n n = 5 n = 6 n = 4 n = 3 n = 2 a n = 7 n = 1 d b n = 0 c To pin down photon number, send a sequence of atoms one by one. and change direction of spin detection to decimate different numbers P (N ) (n) = P(0) (n) N 2Z k=1! 1+ cos n" 0 # $(k) # j(k)% &' ( )( ) / 2 a/b/c/d 0/1 Spin reading Direction "! abdcadb cbadcaa bcbacd b" P (N ) (n)! "! #(n $ n 0 ) Progressive collapse!

25 Convergence of coherent state towards Fock state: wave function collapse in real time! Spin " reading Direction! abdcadb cbadcaa bcbacd b" P(n) distribution obtained from one experimental sequence, as the number of detected atom increases. The initial distribution is flat (no a priori knowledge is assumed, besides n<8). Result is random since it depends upon the unpredictable outcomes of individual spin measurements.

26 A progressive collapse: which number wins the race? n = n =

27 Statistical analysis of 2000 sequences: histogram of the Fock states obtained after collapse Coherent field with n=3.43 Illustrates quantum measurement postulate about statistics

28 Evolution of the photon number probability distribution in a long measuring sequence Field state collapse Repeated measurement Quantum jumps (field decay) { Number of detected atoms 1500 Single realization of field trajectory: real Monte Carlo

29 Evolution of mean photon number in a long measuring sequence n =! np mesure (n) Trajectory corresponding to the n Repeated measurements confirm n=5 counting of 5 photons Projection of coherent state on n=5 Quantum jumps towards vacuum due to field decay in cavity

30 Other photon number trajectories Similar QND trajectories observed between oscillatorlike cyclotron states of an electron (Peil and Gabrielse, PRL 83, 1287 (1999). Two trajectories following collapses into n=5 and 7 Four trajectories following collapse into n=4 An inherently random process (durations of steps widely fluctuate and only their statistics can be predicted)

31 An exotic non-classical state Photon loss increases average energy!!! n = 8 n = 0 QND detection modulo 8 collapses field into a coherent superposition of vacuum and 8 photons! State decays according to: c c 8 8! "! 7 Interferometer counts n modulo 8: does not distinguish 0 and 8 c c 8 8 c c 8 2

32 3D. Conclusion of 3 rd lecture

33 It is not photon s fate to die upon delivering information! the development of super high Q Fabry-Perot microwave cavities opens the way to a new way to look at light: Single photons can be continuously observed over macroscopic times without being destroyed. The field becomes an object of investigation as ions in traps. Individual Monte Carlo field trajectories are observable. This amounts to the repeated action of a CNOT gate in which the photon (or the atom which has deposited it) is the control and the successive QND atoms are the targets. Hundreds of gate operations realized in succession. Several photons can be counted in a QND way. The progressive collapse of the wave function in a continuous QND measurement observed for the first time. Experiment generates, also for first time, Fock states with N>2 and other non-classical states. In QND measurement of the photon number, the phase of the field (conjugate variable) is subjected to back action. It leads to the generation of Schrödinger phase cat states with two or more components. The decoherence of these states can be studied directly (via Wigner function measurements). See last lecture Experiments soon extended to two cavities (study of non-locality in mesoscopic field systems)