Lecture 3 EXPERIMENT: Measuring sub-Planck state displacements in - PowerPoint PPT Presentation

Lecture 3

EXPERIMENT: Measuring sub-Planck state displacements in phase space β ≈ 1 orthogonality | α | ⇒ β ⇒ p x

Looking for a classical-like distribution in phase space We look for a distribution in phase space with the following property: Pure state: Property should be valid with rotated axes:

RADON TRANSFORM (1917) P(q θ ) determines uniquely W(q,p)! � inverse Radon transform → tomography Cormack and Quantum mechanics: P(q θ ) Hounsfield: Nobel Prize in Medicine ⇒ Wigner distribution (1979) (Bertrand and Bertrand, 1987)

Wigner distribution Wigner, 1932: Quantum corrections to classical statistical mechanics Moyal, 1949: Average of operators in symmetric form Density matrix from W:

Examples of Wigner distributions for harmonic oscillator Ground state Fock state with n=3 Mixed state (| α〉〈α |+| −α〉〈−α |)/2 Superposition ∝ | α〉 +| −α〉

Experimental procedure Damping time 65 ms Coherent state ω c /2 π = 51.1 GHz with 12.7 photons w = 5.96 mm {| g i , | e i } ! n = 50 , 51 | e i Ω 0 / 2 π = 46 kHz v=250 m/s Ω ( t ) = Ω 0 exp[ − v 2 t 2 / w 2 ] Temporal variation of the atom-cavity coupling T max ! 42 µ s -t 1 0 t 2 − T 1 T 2 Switch off Switch on resonant resonant interaction interaction Field to be measured is ∑ | α ⟩ = e − α 2 /2 β ( α n / n !) | n ⟩ injected into the cavity at t=0 n

Experimental procedure Damping time 65 ms Coherent state ω c /2 π = 51.1 GHz with 12.7 photons w = 5.96 mm {| g i , | e i } ! n = 50 , 51 | e i Ω 0 / 2 π = 46 kHz T max ! 42 µ s v=250 m/s Temporal variation of the atom-cavity coupling Modulation of atomic frequency —> induces π phase shift between and | g ⟩ | e ⟩ —> time inversion! T 2 − T 1 Field to be measured is β injected into the cavity at t=0

Measurement protocol t = − t 1 t = 0 | α − ⟩ D = 2 α sin Φ 1 p | − ⟩ x | + ⟩ x | e i = ( | + i x + | �i x ) / 2 | Ψ + ⟩ p | Ψ − ⟩ |± i x = ( | e i ± | g i ) / 2 | α + ⟩ 1 2 [ e − i Φ 1 α 2 | α + ⟩ | Ψ + ⟩ − e i Φ 1 α 2 | α − ⟩ | Ψ − ⟩ ] | Ψ⟩ ≈ | α ± ⟩ = | α e ∓ i Φ 1 ⟩ 1 | Ψ ± ⟩ = [ e ∓ i Φ 1 | e ⟩ ± | g ⟩ ] 2 Φ 1 = Ω 0 T 1 /4 α ( large) α

Measurement protocol D = 2 α sin Φ D ( ) T 1 = T 2 2 ⎛ ⎞ ⎡ ( ) ⎤ d ln P j β Geometric phase ⎣ ⎦ Δ β ≥ 1/ ν F ( β ), ⎜ ⎟ ( ) ≡ ( ) F β P j ⇒ ∑ β d β ⎜ ⎟ g , e ⎝ ⎠

Measurement protocol D = 2 α sin Φ D P g = 1 2 (1 + C cos γ ) γ = Ω 0 T 2 β + Ω 0 α ( T 2 − T 1 ) C = exp [ −Ω 2 0 ( T 1 − T 2 ) 2 /8 ] 2 ⎛ ⎞ ⎡ ( ) ⎤ d ln P j β ⎣ ⎦ Δ β ≥ 1/ ν F ( β ), ⎜ ⎟ ( ) ≡ ( ) F β P j ⇒ ∑ β Better to have large T 2 d β ⎜ ⎟ g , e ⎝ ⎠ but Ω 2 0 ( T 2 − T 1 ) 2 /8 ≪ 1

Measurement protocol D = 2 α sin Φ D Measured Fisher information approaches the quantum Fisher ( ) ⇒ ˆ a † − ˆ β ˆ information limit for large a ˆ a † − ˆ h = − i ˆ ( ) ( ) = e D β a enough values of D (the difference is below 1.8% for D>2) ℱ Q = 4 ⟨ ( Δ ̂ h ) 2 ⟩ = 4(1 + D 2 ) Coherent state: D=0 —> —> Standard quantum limit: Δ β SQL = 1/ F ( β ) = 0.5 ℱ Q = 4 ℱ Q = 4(1 + 4 α 2 ) ≈ 6 α 2 ⇒ Maximum value: D=2 —> Heisenberg scaling α

Experimental results Theoretical Fisher information Δ β SQL = 0.5 Δ β Q = 1/ F Q ( ) ≈ 2.4 dB 10log 10 F exp / F SQL Best result: F exp = 3SQL

QUANTUM METROLOGY IN LOSSY SYSTEMS

RECALLING: QUANTUM FISHER INFORMATION In the first lecture, we defined, for a given measurement corresponding { ˆ to the POVM , the Fisher information, E ( ξ ) } � 2 � 2  ∂ ln p ( ξ | X ) 1  ∂ p ( ξ | X ) Z Z F [ X ; { ˆ E ( ξ ) } ] = d ξ p ( ξ | X ) = d ξ p ( ξ | X ) ∂ X ∂ X and we have also defined the “Quantum Fisher information,” which is obtained by maximizing the above expression with respect to all quantum measurements: E ( ξ ) } F [ X ; { ˆ F Q ( X ) = max { ˆ E ( ξ ) } ] The lower bound for the precision in the measurement of the parameter X is then , where N is the number of p p ⇥ ( ∆ X est ) 2 ⇤ � 1 / N F Q ( X ) repetitions of the experiment. The quantum Fisher information for pure states that evolve according to , where X is the parameter to be estimated and | ψ ( X ) � = ˆ ˆ U ( X ) | ψ (0) � U ( X ) is a unitary operator, is i 2 h F Q ( X ) = 4 ⇤ ( ∆ ˆ ⇤ ( ∆ ˆ H ( X ) � ⇤ ˆ ˆ H ) 2 ⌅ 0 , H ) 2 ⌅ 0 ⇥ ⇤ ψ (0) | H ( X ) ⌅ 0 | ψ (0) ⌅ H ( X ) ≡ i d ˆ U † ( X ) d ˆ U † ( X ) where U ( X ) ˆ U ( X ) = − i ˆ ˆ dX dX

Parameter estimation with losses η η ʹ Loss of a single photon transforms NOON state into a separable state! | ψ ( N ) ⇤ = | N, 0 ⇤ + | 0 , N ⇤ ⌅ ⇥ | N � 1 , 0 ⇤ or | 0 , N � 1 ⇤ 2 No simple analytical expression for Fisher information! For small N, more robust states can be numerically calculated Experimental test with more robust states (for N=2):

Parameter estimation with losses - experiments States leading to minimum uncertainty in the presence of noise: x 2 20 + x 1 11 − x 0 02 ψ = Coefficients are determined numerically for each value of . η Losses simulated by a beam splitter in the upper arm. These states are prepared by two beam splitters. η = 1 → no losses η = 0 → complete loss NOON What happens when N increases? SQL ψ

Parameter estimation with losses - theory C. W. Helstrom, Quantum detection and estimation theory (Academic Press, New York, 1976); A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North- Holland, Amsterdam, 1982); S. L. Braunstein and C. M. Caves, PRL 72, 3439 (1994). We have now (Asymptotically attainable when N → ∞ ) ( ) ρ , ˆ N F ⎦ , F ( ) ( ) ≡ max ˆ ˆ Q ˆ E j F ˆ δ X ≥ 1/ ρ X real E j ⎡ ⎤ ρ ⎣ Q 2 ⎛ ( ) ⎞ d ln p j X ⎡ ⎤ ⎣ ⎦ ( ) ≡ ρ , ˆ ( ) ˆ F ˆ ( ) ( ) = Tr ˆ ∑ ⎡ ⎤ E j p j X , p j X ρ X E j ⎜ ⎟ ⎣ ⎦ dx ⎝ ⎠ j General expression for the quantum Fisher information: h i where the operator (“symmetric logarithmic ρ ( X )ˆ ˆ L 2 ( X ) F Q [ˆ ρ ( X )] = Tr ˆ L ρ ( X )ˆ L ( X ) + ˆ derivative”) is defined by the equation d ˆ ρ ( X ) = ˆ L ( X )ˆ ρ ( X ) 2 dX For pure states: so that, from , one gets the previous result ρ ( X ) = ˆ ρ (0) ˆ U † ( X ) ˆ U ( X )ˆ H ( X ) ≡ i d ˆ U † ( X ) , with . F Q ( X ) = 4 � ( ∆ ˆ ˆ ˆ H ) 2 ⇥ 0 U ( X ) dX General case: difficult to evaluate - analytic expression not known. ˆ L

Parameter estimation in open systems: Extended space approach B. M. Escher, R. L. Matos Filho, and L. D., Nature Physics 7, 406 (2011); Braz. J. Phys . 41, 229 (2011) Given initial state and non-unitary evolution, define in S+E | Φ S , E ( x ) 〉 = ˆ (Purification) U S , E ( x )| ψ 〉 S | 0 〉 E E Then ( S ) ⊗ ˆ ( ) ≤ max ˆ ( ) ≡ C 1 F ˆ ( S , E ) F ˆ S F ( S , E ) Q ≡ max ˆ E j 1 E j ( S ) ⊗ ˆ Q E j E j since measurements on S+E should yield more information than measurements on S alone. Physical meaning of this bound: information obtained about Least upper bound: Minimization over all p a r a m e t e r w h e n S + E i s unitary evolutions in S+E - difficult problem monitored Bound is attainable - there is always a Then, monitoring S+E yields same information as monitoring S purification such that C Q = F Q

Minimization procedure There is always an unitary operator acting only on E ρ S that connects two different purifications of E | Φ S , E ( x ) 〉 = ˆ Given , U S , E ( x )| ψ 〉 S | 0 〉 E id | Φ S,E ( x ) i S , = ˆ H S,E ( x ) | Φ S,E ( x ) i dx then any other purification can be written as: ( ) | Φ S , E ( x ) 〉 | Ψ S , E ( x ) 〉 = u E x u † h E ( x ) = id ˆ E ( x ) ˆ Define u E ( x ) ˆ dx Minimize now over all Hermitian operators that act on E. Above h E ( x ) C Q paper proposes iterative procedure for doing this.

Quantum limits for lossy optical interferometry η = 1 → no absorption θ ʹ η = 0 → complete absorption η One uses here a similar strategy: a phase displacement on the environment so as to remove additional information on the phase . θ Minimization of the quantum Fisher information of system + environment yields an upper bound for the Fisher information of the system: n i 0 ∆ 2 ˆ 4 η h ˆ n 0 C Q (ˆ ρ 0 ) = (1 � η ) ∆ 2 ˆ n 0 + η h ˆ n i 0 Note that if then , the quantum Fisher (1 � η ) ∆ 2 ˆ C Q → ∆ 2 ˆ n 0 ⌧ η h ˆ n i 0 n 0 information for pure states. On the other hand, in the high-dissipation limit , one has , yielding a standard-limit scaling: (1 � η ) ∆ 2 ˆ η ⌧ 1 n 0 � η h ˆ n i 0 p δθ � (1 � η ) / 4 η h ˆ n i 0