Towards quantum metrology with N00N states enabled by ensemble-cavity interaction. Massachusetts Institute of Technology

Transcription

1 Towards quantum metrology with N00N states enabled by ensemble-cavity interaction Hao Zhang Monika Schleier-Smith Robert McConnell Jiazhong Hu Vladan Vuletic Massachusetts Institute of Technology MIT-Harvard Center for Ultracold Atoms

2 Outline N00N state (GHZ state) metrology at the Heisenberg limit; Creation of spin N00N states in mesoscopic atomic ensembles by spin squeezing around the Bloch sphere; Cavity spin squeezing in atomic ensembles: effective infinite-distance atomic spin-spin interaction mediated by light; Quantum eraser for atom-light entanglement: towards unitary cavity spin squeezing; Counting atoms in mesoscopic atomic ensembles: Experimental demonstration of readout capability for Heisenberg-limited sensing.

3 Program Themes Entangled States Creation of N00N states (GHZ states) by unitary spin squeezing Mul$- qubit Control Efficient Readout Measurement of mesoscopic atomic ensembles with single-atom resolution

4 Single-atom clock: Ramsey sequence Initialize atom in state Ψ i = 1 Prepare atom in superposition state by π/2 pulse. 1 Ψ (0) = ( ) 2 Let state evolve for time T 1 Ψ ( T) = 1 + exp ( iet / h) 2 2 ( ) Measure atomic state 1 or 2. Determine φ and hence ν=φ/t from probabilities sin 2 φ, cos 2 φ to observe atom in states 1, Apply second π/2 pulse to transform phase φ = ET/ into amplitude Ψ = + f cosφ 1 sinφ 2 At point of optimum sensitivity φ=π/4 the measurement corresponds to tossing a coin with possible outcomes 1, 2 : quantum noise. E

5 Measurement precision Repeated measurement with N independent atoms: Binomial distribution Gaussian distribution 1 N 0 1 Fraction of measurements N Signal quantum projection noise Measurement precision scales as 1/ N N Standard Quantum Limit

6 Clock operation in Bloch representation initialize π/2 about x Free evolution π/2 about y measure S z Fuzzy area represents quasiprobability distribution for quantum projection noise (tossing N atomic coins) or angular-momentum uncertainty relations.

7 Interferometry with N00N states N00N: (GHZ) state: Superposition of two opposite coherent spin states Let state evolve for time T 1 Ψ ( T) = exp / h ( ( inet ) ) Phase evolution N times faster for t N00N state; How to generate N00N state? How to measure phase?

8 How to measure phase of N00N state rotate + ( iφ ) exp( φ) exp x + i x ( φ) 2 2 m /2 S /2 exp ( 1) m e m + i e m S m For e iφ = ±1 only even/odd parity S z states populated: phase of N00N state maps onto parity of S z

9 Cavity squeezing: effective spin-spin interaction between distant atoms mediated by light

10 Spin squeezing by one-axis twisting H = S z generates rotations about the z axis by angle θ e i θ S z H = S z 2 generates rotations about the z axis by S z -dependent angle θ(s z ) e i θ ( Sz) S z one-axis twisting M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993);

11 Creation of N00N states by spin squeezing around the Bloch sphere

12 Setup for spin squeezing with light Probe laser Trap laser Intracavity probe power is proportional to S z. n 1 >1 1 n 2 <1 e 2

13 light Spin squeezing by cavity feedback n 1, n 2 S z n 1 >1 n 2 <1 atoms e S y 1 2 Clock levels S z quantum noise is mapped onto light intensity that acts back on S y variable: quantum correlation = spin squeezing

14 THE HAMILTONIAN Intracavity photon number Cavity shift by atoms or light shift by photons Atomic energy Ω Differential light shift between atomic states per photon Or Differential cavity shift per atom (cavity shift when one atom changes state) ( h a h ) ω + ΩccS ( h ) ω c + hωsz c c z

15 Dynamic squeezing: theory Initial state Full evolution No photon shot noise M. Schleier-Smith, I. Leroux, and V. Vuletic, PRA 81, (R) (2010).

16 Unitary spin squeezing: quantum eraser for light

17 Atom-light entanglement leads to decoherence light atoms Transmitted power depends on atomic state S z R<1 R=1 One-sided cavity: all power reflected, but S z information in reflected phase n R<1 R=1 For photon number state input, there is no phase information in light

18 Spin echo erasure of photon shot noise Photon shot noise in incident light broadens spin distribution when traced over incident light R<1 π R=1 One-sided cavity Use spin echo to cancel photon shot noise while keeping squeezing term: interact twice with same pulse of light with atomic π pulse in between I. D. Leroux, M. H. Schleier-Smith, and V. Vuletic, submitted; Similar idea in free space: Trail, Jessen, and Deutsch, Phys. Rev. Lett. 105, (2010).

19 Measurement of atom number in mesoscopic Magnetic microchip for trapping 87 Rb atoms ensembles Light in To detector Optical resonator Atoms in optical trap 200 µm from the chip surface. Temperature ~50 µk.

20 Atom number variance vs. integration time 108 atoms 29 atoms empty cavity Measurement of atom-induced cavity shift via Pound-Drever-Hall signal

21 State-independent and state-dependent counting of atoms vs. ensemble atom number e e Δ Δ 5S 1/2 F=2 F=1 F=2 F=1 Δ = 250 MHz Δ = 375 MHz Δ = 250 MHz Δ = 375 MHz

22 Summary We have demonstrated atom counting with singleatom resolution for mesoscopic ensembles up to ~ 100 atoms; Experimentally demonstrated spin squeezing in ensembles of distant atoms induced by light; Theoretically developed an experimentally viable setup for disentangling the light and the atoms, allowing for unitary spin squeezing; Unitary spin squeezing should allow the creation of N00N states in mesoscopic ensembles; Single-atom detection enables Heisenberg-limited interferometry.

23 Classical and quantum erasure of photon shot noise Photon shot noise in incident light broadens spin distribution when traced over incident light High QE detector R<1 R=1 One-sided cavity High QE detector: effective creation of photon Fock state by feedback or postselection