Quantum Measurements and Back Action (Spooky and Otherwise)

Transcription

1 Quantum Measurements and Back Action (Spooky and Otherwise) SM Girvin Yale University Thanks to Michel, Rob, Michael, Vijay, Aash, Simon, Dong, Claudia for discussions and comments on Les Houches notes. 1

2 Quantum Back Action is a Weird Thing control qubit CNOT gate target qubit c t c t truth table Target is affected but control is not Are you sure???

3 Quantum Back Action is a Weird Thing control qubit 0 1 CNOT gate target qubit The control qubit is flipped from to!! CNOT 1+ Z 1 Z = X + I 1 1 3

4 It s all about the measurement: Stern Gerlach Experiment Silver atom is a qubit. Silver atom has magnetic moment due to the electron spin V r F r r = µ g B r = V Magnetic moment (spin) can point in any direction and can be measured by passing 4 the atom through a magnetic field gradient.

5 Stern Gerlach Experiment: Quantum Measurement Silver atom has magnetic moment which can point in any direction, and yet. (Measured) 5

6 DISPERSIVE READOUT (MHD) qubit + readout pulses Qubit + resonator AMP dispersive shi8 χ Readout amplitude 1 f r 0 f 90 Readout phase (deg) width κ θ f r 0 f

7 MEASUREMENT HISTOGRAM (data from Devoret lab) (unlike Stern-Gerlach, the qubit is in nearly a pure state) Counts - 0

8 Stern & Gerlach did not measure spin! They entangled spin with position and measured position. (Measured) 8

9 More precisely, they used a spin-dependent force to entangle spin with momentum and waited for momentum to turn into position. (Measured) 9

10 This is a measurement of EM modes that were entangled with the qubit spin Counts - 0

11 1D toy model with spin-dependent impulsive force X B z p z H( p, x, t) = hk0 xσ δ( t) m z F( x, t) =+ hk σ δ( t) 0 z [ H, σ ] = 0 QND

12 Quantum Non-Demolition (QND) Measurements are Repeatable Z Z First result is random, rest are repeats. Z Z Z measurement measurement measurement 1

13 1D toy model with spin-dependent impulsive force X B z We will measure momentum just after impulse rather than waiting for it to turn into position. p z H( p, x, t) = hk0 xσ δ( t) m z F( x, t) =+ hk σ δ( t) 0

14 p z H( p, x, t) = hk0 xσ δ( t) m ( ) ψ ( xt, = 0 ) = a + b Φ( x) ψ Input product state Output entangled entangled state ( ik x ik x ) ( x, t 0 ) ae ( x) be ( x) = = Φ + Φ Output state in momentum basis ( ) ψ [ kt, = 0 ] = aφ[ k k] + bφ [ k+ k] 14

15 Pk ( ) Pk ( ) 0.6 Strong measurement k (gaussian input packet) Pk ( ) Pk ( ) 0.4 Weak measurement 0. k 15

16 Pk ( ) Pk ( ) k Pk ( ) is easy to understand but what we need is: P( k) 16

17 Practice on two continuous variables Pxy (, ) = ψ ( xy, ) Pxy (, ) = Px ( ypy ) ( ) Y y x PxY ( ) ψ ( xy, ) PxY (, ) = = dxʹ ψ ( xʹ, Y ) PY ( ) 17

18 Density Matrix Equivalent Reduced density matrix for spin conditioned on measurement of momentum ρ k k ρ k k ρ k = = Tr k ρ k P( k) k ρ k ρ k Full density matrix projected onto observed momentum state Reduced density matrix for spin given observed value of momentum 18

19 Density Matrix Equivalent ψ Full state a Φ = a Φ + b Φ = b Φ Full density matrix aa* Φ Φ ab* Φ Φ ρ = ψ ψ = a* b bb* Φ Φ Φ Φ 19

20 Reduced density matrix for spin conditioned on measurement of momentum ρ k k ρ k k ρ k = = Tr k ρ k P( k) Full density matrix aa* Φ Φ ab* Φ Φ ρ = a* b bb* Φ Φ Φ Φ Reduced density matrix for spin conditioned on measurement of momentum 1 aa* k Φ Φ k ab* k Φ Φ k ρk = Pk ( ) a* b k k bb* k k Φ Φ Φ Φ 0

21 Reduced density matrix for spin conditioned on measurement of momentum 1 aa* k Φ Φ k ab* k Φ Φ k ρk = Pk ( ) a* b k k bb* k k Φ Φ Φ Φ Easy to verify conditional state is pure: Det ρ = 0 k Tr ρ = 1 k 1 0 exists a basis s.t. ρk = 0 0 If we fully measure the state of the bath then the conditional state remains pure! 1

22 Reduced density matrix for spin conditioned on measurement of momentum 1 aa* k Φ Φ k ab* k Φ Φ k ρk = Pk ( ) a* b k k bb* k k Φ Φ Φ Φ Easy to verify conditional state is pure: Det ρ = 0 k Sanity check: aa* ab* Φ Φ ρ = dkp( k) ρk = a* b bb* Φ Φ Averaging over measurement results yields measurement-induced dephasing

23 If the state is pure, there is a corresponding wave function for the spin alone 1 k = a k Φ + b k Φ Pk ( ) { } ψ Spooky back action: z [ H, σ ] = 0 and yet: 1 P( k) = aa* k Φ Φ k aa * Pk ( ) 1 P( k) = bb* k Φ Φ k bb* Pk ( ) 3

24 Pk ( ) Pk ( ) Gaussian packet k a P( k) = e Z b P( k) = e Z kk 0 + ( Δ k ) kk 0 ( Δ k ) 1 Φ ( x) = πσ 0 1/4 σ 0 Φ [ ± 0] = k k e π ( ) Δ k = 1 4σ 0 1/4 ( k± k ) 0 σ 0 kk kk + ( Δ ) ( ) 0 0 k 4 Δk Z a e + b e x 4 e σ 0

25 Pk ( ) Pk ( ) k S = Tr ρlog ρ (log base e) ( ) a P( k) = e Z b P( k) = e Z kk 0 + ( Δ k ) kk 0 ( Δ k ) Average Shannon entropy reduction (information gain) for a weak measurement I = k 0 ( Δk) 5

26 Summary so far: Spin-dependent force entangles spin with momentum Measurement of momentum improves knowledge of spin z z σ changes even though [ H, σ ] = 0 Spooky back action drives qubit up and down in latitude What happens if we measure x instead of k? The back action changes! Qubit moved in longitude instead. 6

27 1D toy model with spin-dependent impulsive force X B z If we measure position just after the impulse, we gain NO information about the momentum change. We DO however learn the value of the magnetic field that acted on the qubit. p z H( p, x, t) = hk0 xσ δ( t) m z F( x, t) =+ hk σ δ( t) 0

28 ψ ( ik x ik x ) ( x, t 0 ) ae ( x) be ( x) = = Φ + Φ ρ iϕ 1 aa* e ab* aa * e ab* P ( x) e a * b bb* e a * b bb* iϕ x = Φ ( x) = + iϕ + iϕ X Measuring position gives no information about momentum or spin but produces rotation of qubit: p z H( p, x, t) = hk0 xσ δ( t) m Non-spooky back action! ϕ k0x Qubit in pure state. 8

29 ρ iϕ 1 aa* e ab* aa * e ab* P ( x) e a * b bb* e a * b bb* iϕ x = Φ ( x) = + iϕ + iϕ X Measuring position gives no information about momentum or spin but produces rotation of qubit at constant latitude: Non-spooky back action! Qubit in pure state. 9 Same dephasing as before if average over x measurements.

30 DISPERSIVE READOUT is Exactly Analogous qubit + readout pulses Qubit + resonator AMP dispersive shi8 χ Readout amplitude 1 f r 0 f 90 Readout phase (deg) width κ θ f r 0 f

31 Y X z V = a a ax χ σ χ σ a a + δ a ( a real) X δa + δa z Homodyne measurement of Y is analogous to Stern-Gerlach measurement of momentum. Spooky back action. Homodyne measurement of X is analogous to Stern-Gerlach measurement of position. Non-spooky back action due to photon shot noise.

32 Y X z V = a a ax χ σ χ σ X δa + δa z ψ in { a b } = + α + iθ iθ ψout ae α be α = + out { + inθ inθ } n ψ = ae + be n α Each photon passing through the cavity rotates the qubit by θ (but only if we measure n or X!!)

33 Pseudo-heterodyne measurement vacuum port qubit + readout pulses Qubit + resonator X Y Back action has both spooky and non-spooky components. Qubit is now entangled with two independent oscillators (field modes). But still in a PURE state if we measure one quadrature of each.

34 Pseudo-heterodyne measurement vacuum port qubit + readout pulses Qubit + resonator X Back action has both spooky and non-spooky components. Measurement efficiency is 50% but added noise does not dephase if both modes are fully measured. Y

35 Thanks Many further details soon in my Les Houches notes.

36 Gerlach s Postcard to Bohr 8 February 19 Attached [is] the experimental proof of directional quantization. We congratulate [you] on the confirmation of your theory. (Historical note: they did not realize this was the discovery of electron spin.) AIP Emilio Segrè Visual Archives. 36

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