Indirect measurements of a harmonic oscillator Gian Michele Graf - - PowerPoint PPT Presentation

Indirect measurements of a harmonic oscillator

Gian Michele Graf ETH Zurich Quantissima in the Serenissima III Palazzo Pesaro Papafava, Venezia August 19-23, 2019

Indirect measurements of a harmonic oscillator

Gian Michele Graf ETH Zurich Quantissima in the Serenissima III Palazzo Pesaro Papafava, Venezia August 19-23, 2019

based on joint work with M. Fraas, L. H¨ anggli discussions with J. Fr¨

• hlich, K. Hepp

Outline

Setting the stage The model defined at last Results

Setting the stage The model defined at last Results

Born rule (BR)

If an ideal measurement of the observable A = A∗ =

• λ∈spec(A)

λPλ (spectral decomposition) is done in the state ψ, then the outcome λ is found with probability ψ|Pλ|ψ.

Born rule (BR)

If an ideal measurement of the observable A = A∗ =

• λ∈spec(A)

λPλ (spectral decomposition) is done in the state ψ, then the outcome λ is found with probability ψ|Pλ|ψ. Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation (A = −id/dx) is premised (e.g. correspondence principle).

Born rule (BR)

If an ideal measurement of the observable A = A∗ =

• λ∈spec(A)

λPλ (spectral decomposition) is done in the state ψ, then the outcome λ is found with probability ψ|Pλ|ψ. Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation (A = −id/dx) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning

• f A is not even addressed. The apparatus remains exophysical;

the observable dry.

Born rule (BR)

If an ideal measurement of the observable A = A∗ =

• λ∈spec(A)

λPλ (spectral decomposition) is done in the state ψ, then the outcome λ is found with probability ψ|Pλ|ψ. Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation (A = −id/dx) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning

• f A is not even addressed. The apparatus remains exophysical;

the observable dry.

• Cf. Heisenberg 1927: If one wants to be clear about what is meant by

”position of an object,” for example of an electron (relatively to a given reference frame), then one has to specify definite experiments by which the ”position of an electron” can be measured; otherwise this term has no meaning at all.

Born rule (BR)

If an ideal measurement of the observable A = A∗ =

• λ∈spec(A)

λPλ (spectral decomposition) is done in the state ψ, then the outcome λ is found with probability ψ|Pλ|ψ. Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation (A = −id/dx) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning

• f A is not even addressed. The apparatus remains exophysical;

the observable dry. ◮ State ψ changes with time. Hence the ideal measurement is

• instantaneous. (But any real measurement takes time.)

Remarks on duration of measurement

◮ If the measurement of the intended observable A takes a time T, then the observable actually measured is (at best) the time average AT := 1 T T eiHtAe−iHtdt

Remarks on duration of measurement

◮ If the measurement of the intended observable A takes a time T, then the observable actually measured is (at best) the time average AT := 1 T T eiHtAe−iHtdt ◮ General protocol: Measure AT before it deviates a lot from A (von Neumann: Strong coupling to apparatus)

Remarks on duration of measurement

◮ If the measurement of the intended observable A takes a time T, then the observable actually measured is (at best) the time average AT := 1 T T eiHtAe−iHtdt ◮ General protocol: Measure AT before it deviates a lot from A (von Neumann: Strong coupling to apparatus) ◮ Special protocol: If A is a constant of motion, then T does not matter: AT = A. (To the contrary: the larger T, the better, because the initialization of the apparatus matters ever less; e.g. quantum non-demolition experiments (QND) and their theory, T → ∞.)

Stability of QND

A intended observable AT := 1 T T eiHtAe−iHtdt actual observable

Stability of QND

A intended observable AT := 1 T T eiHtAe−iHtdt actual observable ◮ If A = f(H), then AT = A (as seen).

Stability of QND

A intended observable AT := 1 T T eiHtAe−iHtdt actual observable ◮ If A = f(H), then AT = A (as seen). ◮ If A = f(H0) with H − H0 small (Hamiltonian error, unavoidable), then AT − A ≤ 2f(H) − f(H0)

Stability of QND

A intended observable AT := 1 T T eiHtAe−iHtdt actual observable ◮ If A = f(H), then AT = A (as seen). ◮ If A = f(H0) with H − H0 small (Hamiltonian error, unavoidable), then AT − A ≤ 2f(H) − f(H0) ∴ stability: AT − A small uniformly in T > 0 (no apparatus, though).

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)?

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes.

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A, in order to analyze the system proper S (indirect measurement).

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A, in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut).

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A, in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A, in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A ◮ We’ll see: The stability of QND is lost for T → ∞, because A is feeding back on S.

The dilemma and the way out

◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A, in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A ◮ We’ll see: The stability of QND is lost for T → ∞, because A is feeding back on S. There is a time window of opportunity (t ∈ [0, T]) for measurement.

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical.

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. → Dyson: “As a general rule, knowledge about the past can only be expressed in classical terms.”

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. → Fr¨

• hlich: “ETH approach”

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. ◮ A is macroscopic: Part of the record is still a record.

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. ◮ A is macroscopic: Part of the record is still a record. ◮ Both requisites implemented by repeated measurements (e.g. Coleman-Hepp model):

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. ◮ A is macroscopic: Part of the record is still a record. ◮ Both requisites implemented by repeated measurements (e.g. Coleman-Hepp model): Units of A (commuting observables) coupled to S successively in time t: A S

Requisites for the apparatus A and their implementation

◮ A is quantum, but establishes a record that is classical. ◮ A is macroscopic: Part of the record is still a record. ◮ Both requisites implemented by repeated measurements (e.g. Coleman-Hepp model): Units of A (commuting observables) coupled to S successively in time t: A S We’ll take a continuum limit on A.

Setting the stage The model defined at last Results

The system S: A harmonic oscillator

◮ Hilbert space H spanned by |n, (n = 0, 1, 2, . . .) ◮ Harmonic oscillator H0 = ωa∗a

The system S: A harmonic oscillator

◮ Hilbert space H spanned by |n, (n = 0, 1, 2, . . .) ◮ Harmonic oscillator H0 = ωa∗a ◮ Observable: Excitation number N = a∗a

The system S: A harmonic oscillator

◮ Hilbert space H spanned by |n, (n = 0, 1, 2, . . .) ◮ Harmonic oscillator H0 = ωa∗a ◮ Observable: Excitation number N = a∗a ◮ Hamiltonian: Harmonic oscillator with center α ∈ C in phase space H = ω((a∗ − ¯ α)(a − α) − |α|2)

α Im a Re a

Classical trajectory (frequency ω) for Hamiltonian error α (QND: α = 0)

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·
• ◮ Instantaneous measurement of N:

N = n ,

• N

= 0

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·
• ◮ Instantaneous measurement of N:

N = n ,

• N

= 0 ∴ Outcome n is sharp.

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·
• ◮ Instantaneous measurement of N:

N = n ,

• N

= 0 ∴ Outcome n is sharp. ◮ Time-averaged measurement (T → ∞):

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·
• ◮ Instantaneous measurement of N:

N = n ,

• N

= 0 ∴ Outcome n is sharp. ◮ Time-averaged measurement (T → ∞): NT → n + 2|α|2 , (spacing is 1)

• NT

→ (2n + 1)|α|2 ≪ 1 for α small and for finitely many n’s.

Observables of S (dry run)

States |n (recall N|n = n|n). ◮ Notation for expectation and variance in |n: · ,

• ·
• ◮ Instantaneous measurement of N:

N = n ,

• N

= 0 ∴ Outcome n is sharp. ◮ Time-averaged measurement (T → ∞): NT → n + 2|α|2 , (spacing is 1)

• NT

→ (2n + 1)|α|2 ≪ 1 for α small and for finitely many n’s. ∴ Different states |n can be told apart forever.

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+))

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1.

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t])

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t]) ◮ Field quadratures Qt = At + A∗

t , Pt = −i(At − A∗ t )

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t]) ◮ Field quadratures Qt = At + A∗

t , Pt = −i(At − A∗ t )

◮ Pointers Qt, (t ≥ 0)

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t]) ◮ Field quadratures Qt = At + A∗

t , Pt = −i(At − A∗ t )

◮ Pointers Qt, (t ≥ 0) ◮ Initial state: Fock vacuum |Ω

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t]) ◮ Field quadratures Qt = At + A∗

t , Pt = −i(At − A∗ t )

◮ Pointers Qt, (t ≥ 0) ◮ Initial state: Fock vacuum |Ω ◮ Total Hamiltonian (informally)

• H(t) = H ⊗ 1 + γN ⊗ ∆Pt

∆t The pointer associated to interval ∆t (label of A) is coupled to S during ∆t (time).

The apparatus A: A bosonic field

◮ Continuous spatial label t ∈ R+ for the units of A ◮ Fock space F = F(L2(R+)) ◮ Creation and annihilation operators A∗(f), A(g) (f, g ∈ L2(R+)) satisfying CCR: [A(g), A∗(f)] = g|f1. In particular At = A(1[0,t]) , ∆At = A(1[t,t+∆t]) ◮ Field quadratures Qt = At + A∗

t , Pt = −i(At − A∗ t )

◮ Pointers Qt, (t ≥ 0) ◮ Initial state: Fock vacuum |Ω ◮ Total Hamiltonian (informally)

• H(t) = H + γN lim

∆t→0

∆Pt ∆t The pointer associated to interval ∆t (label of A) is coupled to S during ∆t (time).

Pointers at work (wet run)

◮ Total Hamiltonian

• H(t) = H + γN ∆Pt

∆t The pointer associated to interval ∆t (label of A) is coupled to S during ∆t (time). ◮ Change of pointer during ∆t (Heisenberg picture) (∆Qt)

• t+∆t

t

= 2γN∆t

Pointers at work (wet run)

◮ Total Hamiltonian

• H(t) = H + γN ∆Pt

∆t The pointer associated to interval ∆t (label of A) is coupled to S during ∆t (time). ◮ Change of pointer during ∆t (Heisenberg picture) (∆Qt)

• t+∆t

t

= 2γN∆t ◮ Cumulative change (Schr¨

• dinger picture)

U∗

TQTUT − QT = 2γ

T dt U∗

t NUt

with propagator Ut for H(t) on [0, t]

Pointers at work (wet run)

◮ Total Hamiltonian

• H(t) = H + γN ∆Pt

∆t The pointer associated to interval ∆t (label of A) is coupled to S during ∆t (time). ◮ Change of pointer during ∆t (Heisenberg picture) (∆Qt)

• t+∆t

t

= 2γN∆t ◮ Cumulative change (Schr¨

• dinger picture)

U∗

TQTUT − QT = 2γ

T dt U∗

t NUt

with propagator Ut for H(t) on [0, t] ◮ NT := U∗

TQTUT

2γT Then NT (hopefully) ought to be a pointer mirroring the (dry) time-averaged observable NT, and indirectly N.

The precise model

The propagator Ut solves the Quantum Stochastic Differential Equation (Hudson-Parthasarathy) idUt =

• (H − (γN)2/2)dt + γN(dPt)
• Ut

based on Itˆ

• calculus.

What dynamics to expect heuristically?

• H(t)∆t = H∆t + γN∆Pt

◮ H∆t induces motion on solid circle of angle ω∆t γN∆Pt induces motion on dashed circle of angle ∆ψ := γ∆Pt

What dynamics to expect heuristically?

◮ H∆t induces motion on solid circle of angle ω∆t γN∆Pt induces motion on dashed circle of angle ∆ψ := γ∆Pt ◮ Notation: · = Ω| · |Ω ∆ψ = 0 , (∆ψ)2 = γ2∆t , ˜ r 2 = r 2 + |α|2γ2∆t

What dynamics to expect heuristically?

◮ H∆t induces motion on solid circle of angle ω∆t γN∆Pt induces motion on dashed circle of angle ∆ψ := γ∆Pt ◮ Notation: · = Ω| · |Ω ∆ψ = 0 , (∆ψ)2 = γ2∆t , ˜ r 2 = r 2 + |α|2γ2∆t ◮ For large t N ≈ |α|2γ2t , NT ≈ |α|2γ2T/2

What dynamics to expect heuristically?

◮ H∆t induces motion on solid circle of angle ω∆t γN∆Pt induces motion on dashed circle of angle ∆ψ := γ∆Pt ◮ Notation: · = Ω| · |Ω ∆ψ = 0 , (∆ψ)2 = γ2∆t , ˜ r 2 = r 2 + |α|2γ2∆t ◮ For large t N ≈ |α|2γ2t , NT ≈ |α|2γ2T/2 ∴ Expect diffusion: Noise eventually dominates through feedback.

Setting the stage The model defined at last Results

Expectations in the state |n ⊗ |Ω

Let (as before) NT := U∗

TQTUT

2γT

Expectations in the state |n ⊗ |Ω

Let (as before) NT := U∗

TQTUT

2γT Expectations and variances in the state |n ⊗ |Ω ◮ 0 ≤ NT − n ≤ |α|2(4 + γ2T/2)

Expectations in the state |n ⊗ |Ω

Let (as before) NT := U∗

TQTUT

2γT Expectations and variances in the state |n ⊗ |Ω ◮ 0 ≤ NT − n ≤ |α|2(4 + γ2T/2) ◮ Asymptotics (regime γ2 ≪ ω) NT − n ∼ = |α|2     

(ωT)2 3

, (T ≪ ω−1) , 2 , (ω−1 ≪ T ≪ γ−2) ,

γ2T 2 ,

(T ≫ γ−2) .

Distinguishability of states |n ⊗ |Ω

States |n can be told apart, i.e., |NT − n| ≪ 1 ,

• NT

≪ 1 , in two cases:

Distinguishability of states |n ⊗ |Ω

States |n can be told apart, i.e., |NT − n| ≪ 1 ,

• NT

≪ 1 , in two cases: ◮ Strong coupling γ (cf. von Neumann) γ−2 ≪ T ≪ (|α|ω)−1 Note: For |α| ≈ 1 (non-QND) we have ωT ≪ 1, i.e. measurement time is much less than a period.

Distinguishability of states |n ⊗ |Ω

States |n can be told apart, i.e., |NT − n| ≪ 1 ,

• NT

≪ 1 , in two cases: ◮ Strong coupling γ (cf. von Neumann) γ−2 ≪ T ≪ (|α|ω)−1 Note: For |α| ≈ 1 (non-QND) we have ωT ≪ 1, i.e. measurement time is much less than a period. ◮ Weak coupling γ γ−2 ≪ T ≪ (|α|γ)−2 Note: Requires |α| ≪ 1 (almost QND). Then T can be much larger, but not infinite.

Distinguishability of states |n ⊗ |Ω

States |n can be told apart, i.e., |NT − n| ≪ 1 ,

• NT

≪ 1 , ◮ Weak coupling γ γ−2 ≪ T ≪ (|α|γ)−2 Note: Requires |α| ≪ 1 (almost QND). Then T can be much larger, but not infinite. ◮ Moreover, 0 < T γ−2 : NT is getting in sync with N γ−2 ≪ T ≪ (|α|γ)−2 : NT closely mirrors N (window of opportunity) T (|α|γ)−2 : NT loses correlation with N

The long time limit: Classical noise from quantum fluctuations

• Theorem. The pointer variables (NT)T>0 converge to Brownian

motion as a process upon rescaling: εNε−1T − − − →

ε→0 |κ|2 1

T T |Bt|2dt where κ = ωαγ(iω − γ2/2)−1 and Bt is Brownian motion in the plane (phase space) with |dBt|2 = dt.

The long time limit: Classical noise from quantum fluctuations

• Theorem. The pointer variables (NT)T>0 converge to Brownian

motion as a process upon rescaling: εNε−1T − − − →

ε→0 |κ|2 1

T T |Bt|2dt where κ = ωαγ(iω − γ2/2)−1 and Bt is Brownian motion in the plane (phase space) with |dBt|2 = dt. Likewise, but for fixed t, does the (wet) excitation number N(t) εN(ε−1t) − − − →

ε→0 |κ|2|Bt|2

Summary

◮ The shift of the Heisenberg cut is not always innocent: The Born rule may, depending on cases, yield the same or substiantially different results before and after the shift. ◮ As an example, a strict QND experiment is such regardless of the position of the cut. ◮ A slightly perturbed QND experiment is not: If the cut excludes the apparatus the perturbation remains ineffective indefinitely in time; if it includes it, it will become effective eventually. The apparatus will then produce just noise.

Summary

◮ The shift of the Heisenberg cut is not always innocent: The Born rule may, depending on cases, yield the same or substiantially different results before and after the shift. ◮ As an example, a strict QND experiment is such regardless of the position of the cut. ◮ A slightly perturbed QND experiment is not: If the cut excludes the apparatus the perturbation remains ineffective indefinitely in time; if it includes it, it will become effective eventually. The apparatus will then produce just noise. Thank you for your attention!