Phase Measurements at Phase Measurements at the Theoretical Limit - PowerPoint PPT Presentation

Phase Measurements at Phase Measurements at the Theoretical Limit the Theoretical Limit Dominic Berry Dominic Berry Institute for Quantum Computing Institute for Quantum Computing Brendon Higgins, Howard Wiseman, Steve Bartlett, Brendon Higgins, Howard Wiseman, Steve Bartlett, Morgan Mitchell, Geoff Pryde Pryde Morgan Mitchell, Geoff

Applications of phase measurement Applications of phase measurement Frequency and time Distance measurement measurement Communication

The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle position & momentum spin Δ Δ ≥ � 1 Δ Δ ≥ p x J J J y z x 2 2 z quadratures Δ Δ ≥ X Y 1 Y Δ y Y Δ x X X

The Heisenberg limit vs vs The Heisenberg limit the standard quantum limit the standard quantum limit The Standard The Heisenberg Limit The Standard The Heisenberg Limit Quantum Limit Quantum Limit � If one uncertainty is If one uncertainty is � reduced as much as reduced as much as � If the two uncertainties If the two uncertainties � possible. possible. are equal. are equal. � Uncertainty scaling Uncertainty scaling � Uncertainty scaling Uncertainty scaling � � Δ ∝ φ Δ ∝ φ 1 N 1 N

Types of phase measurement Types of phase measurement Interferometric measurement Interferometric measurement Single- -mode phase mode phase Single atomic optical atomic optical input state z input state x z y y rotation x φ y φ z z rotation x y z y rotation x φ est y φ est J z measurement

Types of phase measurement Types of phase measurement Interferometric measurement Interferometric measurement Single- -mode phase mode phase Single atomic optical atomic optical input state z input state x z y y rotation x φ y φ z z rotation x y z y rotation x φ est y φ est J z measurement

Single- -mode measurements mode measurements Single φ est Signal is beam is mixed with Signal is beam is mixed with � � strong “ strong “local oscillator local oscillator” ”. . input φ state Heterodyne – – linear variation linear variation Heterodyne � � θ . of θ . of θ ( t ) – θ θ close to close to φ φ . Homodyne – . Homodyne � � local oscillator − Use an estimate of the Use an estimate of the input φ � � state phase to approximate a phase to approximate a homodyne measurement. homodyne measurement. θ ( t ) processor local oscillator

Adaptive phase measurement Adaptive phase measurement Task: measure an arbitrary phase Task: measure an arbitrary phase − input state φ Total phase uncertainty Δ φ Total phase uncertainty θ θ processor Δ φ 2 = (intrinsic uncertainty) 2 + (uncertainty due to measurement) 2 local oscillator final phase final phase estimate estimate H. M. Wiseman, Phys. Rev. A 56 , 944 (1997). H. M. Wiseman, Phys. Rev. A 57 , 2169 (1998).

Adaptive phase measurement Adaptive phase measurement Task: measure an arbitrary phase Task: measure an arbitrary phase − input state φ Total phase uncertainty Δ φ Total phase uncertainty θ θ processor Δ φ 2 = (intrinsic uncertainty) 2 + (uncertainty due to measurement) 2 local oscillator final phase final phase estimate estimate 1 ∝ heterodyne measurements N H. M. Wiseman, Phys. Rev. A 56 , 944 (1997). H. M. Wiseman, Phys. Rev. A 57 , 2169 (1998).

Adaptive phase measurement Adaptive phase measurement Task: measure an arbitrary phase Task: measure an arbitrary phase − input state φ Total phase uncertainty Δ φ Total phase uncertainty θ θ processor Δ φ 2 = (intrinsic uncertainty) 2 + (uncertainty due to measurement) 2 local oscillator final phase final phase estimate The standard estimate 1 quantum limit ∝ heterodyne measurements N H. M. Wiseman, Phys. Rev. A 56 , 944 (1997). H. M. Wiseman, Phys. Rev. A 57 , 2169 (1998).

Wiseman Mark I Wiseman Mark I Task: measure an arbitrary phase Task: measure an arbitrary phase ☺ ≡ ≡ best phase estimate ☺ best phase estimate − � ≡ ≡ poor phase estimate � poor phase estimate input state φ Mark I feedback phase estimate: � � processor � final phase estimate: � ⇒ ideal phase measurement for N = 1 local oscillator � � ⇒ for N >> 1 1 Worse than standard Δ ∝ φ quantum limit! 1/4 N H. M. Wiseman, Phys. Rev. A 56 , 944 (1997). H. M. Wiseman, Phys. Rev. A 57 , 2169 (1998).

Wiseman Mark II Wiseman Mark II Task: measure an arbitrary phase Task: measure an arbitrary phase ☺ ≡ ≡ best phase estimate ☺ best phase estimate − � ≡ ≡ poor phase estimate � poor phase estimate input state φ Mark II feedback phase estimate: � � processor � final phase estimate: ☺ ⇒ ideal phase measurement for N = 1 local oscillator ☺ ☺ ⇒ for N >> 1 1 Beats the standard Δ ∝ φ quantum limit 3/4 N H. M. Wiseman, Phys. Rev. A 56 , 944 (1997). H. M. Wiseman, Phys. Rev. A 57 , 2169 (1998).

Optimal adaptive Optimal adaptive Task: measure an arbitrary phase Task: measure an arbitrary phase ☺ ≡ ≡ best phase estimate ☺ best phase estimate � ≡ ≡ poor phase estimate � poor phase estimate − ☯ ≡ ≡ intermediate intermediate ☯ input state φ Mark II feedback phase estimate: ☯ ☯ processor ☯ final phase estimate: ☺ ⇒ ideal phase measurement for N = 1 local oscillator ☺ ☺ ⇒ for N >> 1 Almost the ln N Δ ∝ φ Heisenberg limit N D. W. Berry and H. M. Wiseman, Phys. Rev. A 63 , 013813 (2001).

Types of phase measurement Types of phase measurement Interferometric measurement Interferometric measurement Single- -mode phase mode phase Single atomic optical atomic optical input state z input state x z y y rotation x φ y φ z z rotation x y z y rotation x φ est y φ est J z measurement

Optical interferometry Optical interferometry � Theoretical limit Theoretical limit � � Squeezed states Squeezed states 1 1 � � NOON states NOON states 2 2 � � Theoretical Theoretical- -limit adaptive measurements limit adaptive measurements 3 3 � � Theoretical Theoretical- -limit limit non non adaptive adaptive measurements measurements 4 4 � � Hybrid measurements Hybrid measurements 4 4 � 3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, 1 C. M. Caves, Phys. Rev. D 23 , 1693 (1981). and G. J. Pryde, Nature 450, 393 (2007). 2 B. C. Sanders, Phys. Rev. A 40 , 2417 (1989). 4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

Optical interferometry Optical interferometry � Theoretical limit Theoretical limit � � Squeezed states Squeezed states 1 1 � � NOON states NOON states 2 2 � � Theoretical Theoretical- -limit adaptive measurements limit adaptive measurements 3 3 � � Theoretical Theoretical- -limit limit non non adaptive adaptive measurements measurements 4 4 � � Hybrid measurements Hybrid measurements 4 4 � 3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, 1 C. M. Caves, Phys. Rev. D 23 , 1693 (1981). and G. J. Pryde, Nature 450, 393 (2007). 2 B. C. Sanders, Phys. Rev. A 40 , 2417 (1989). 4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

Optimal measurements Optimal measurements φ φ est N photons optimal two-mode optimal two-mode entangled input state 1,2 joint measurement 3 Δ φ ≈ π N The theoretical limit 1 A. Luis and J. Pe ř ina, Phys. Rev. A 54, 4564 (1996). 3 B. C. Sanders and G. J. Milburn, PRL 75 , 2944 (1995). 2 D. W. Berry and H. M. Wiseman, PRL 85 , 5098 (2000).

How to perform the measurement? How to perform the measurement? � θ θ ( ( t ) is adjusted to minimise the expected variance after is adjusted to minimise the expected variance after t ) � the next detection. the next detection. Δ φ � Gives uncertainty Gives uncertainty ∼ 1 N � φ θ ( t ) N photons processor D. W. Berry and Wiseman, PRL 85 , 5098 (2000).

How to create the input state? How to create the input state? Two problems: Two problems: The state needs to be a special coherent superposition The state needs to be a special coherent superposition 1. 1. of the form of the form N ∑ ψ − n N n n = n 0 There is no known way of producing such a state. There is no known way of producing such a state. θ The input mode needs to be very long so that θ ( t ) can can The input mode needs to be very long so that ( t ) 2. 2. be adjusted between detections. be adjusted between detections.

Optical interferometry Optical interferometry � Theoretical limit Theoretical limit � � Squeezed states Squeezed states 1 1 � � NOON states NOON states 2 2 � � Theoretical Theoretical- -limit adaptive measurements limit adaptive measurements 3 3 � � Theoretical Theoretical- -limit limit non non adaptive adaptive measurements measurements 4 4 � � Hybrid measurements Hybrid measurements 4 4 � 3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, 1 C. M. Caves, Phys. Rev. D 23 , 1693 (1981). and G. J. Pryde, Nature 450, 393 (2007). 2 B. C. Sanders, Phys. Rev. A 40 , 2417 (1989). 4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).