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On the WEAK MEASUREMENT of the ELECTRICAL THz CURRENT: a NEW SOURCE - - PowerPoint PPT Presentation

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2015, July 17 th UPoN 2015 On the WEAK MEASUREMENT of the ELECTRICAL THz CURRENT: a NEW SOURCE of NOISE? Damiano Marian 1 , Nino Zangh` 2 , Xavier Oriols 1 1 Autonomous University of Barcelona 2 University of Genoa Barcelona - 2015, July 17

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SLIDE 1

UPoN 2015 2015, July 17th

On the WEAK MEASUREMENT

  • f the ELECTRICAL THz CURRENT:

a NEW SOURCE of NOISE?

Damiano Marian1, Nino Zangh` ı2, Xavier Oriols1

1 Autonomous University of Barcelona 2 University of Genoa Damiano Marian Barcelona - 2015, July 17th 1 / 23

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Index

1 Open problem Damiano Marian Barcelona - 2015, July 17th 2 / 23

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Index

1 Open problem 2 Novel approach to model the measurement of THz current Damiano Marian Barcelona - 2015, July 17th 2 / 23

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SLIDE 4

Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise Damiano Marian Barcelona - 2015, July 17th 2 / 23

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SLIDE 5

Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities Damiano Marian Barcelona - 2015, July 17th 2 / 23

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Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 2 / 23

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Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 3 / 23

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1 - Open problem

How do we model the measurement of the high frequency current?

Damiano Marian Barcelona - 2015, July 17th 4 / 23

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1 - Open problem

How do we model the measurement of the high frequency current? ⇒ Year 2015 2020 2025 Cutoff Frequency (GHz) 620 1137 2062

International Technology Roadmap for Semiconductors (2011)

Damiano Marian Barcelona - 2015, July 17th 4 / 23

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1 - Open problem

How do we model the measurement of the high frequency current? ⇒ Year 2015 2020 2025 Cutoff Frequency (GHz) 620 1137 2062

International Technology Roadmap for Semiconductors (2011)

⇒ Total Current = Particle + Displacement Current Id(t) =

  • Si ǫ(r) dE(r,t)

dt

· ds

Damiano Marian Barcelona - 2015, July 17th 4 / 23

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1 - Open problem

How do we model the measurement of the high frequency current? ⇒ Year 2015 2020 2025 Cutoff Frequency (GHz) 620 1137 2062

International Technology Roadmap for Semiconductors (2011)

⇒ Total Current = Particle + Displacement Current Id(t) =

  • Si ǫ(r) dE(r,t)

dt

· ds ⇒ Continuous (or very High Frequency) Measurement ⇓ Inclusion of the Back-Action (Disturbance of quantum system)

Damiano Marian Barcelona - 2015, July 17th 4 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics?

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Initjal Wave Functjon

  • J. Von Neumann: PUP (1955)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Initjal Wave Functjon Stsong Operatpr

  • J. Von Neumann: PUP (1955)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Wave Functjon aftfr Stsong Measurement

  • J. Von Neumann: PUP (1955)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Wave Functjon aftfr Stsong Measurement

  • J. Von Neumann: PUP (1955)

Weak

Initjal Wave Functjon

  • Y. Aharonov, D. Z. Albert, and L. Vaidman: PRL (1988)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Wave Functjon aftfr Stsong Measurement

  • J. Von Neumann: PUP (1955)

Weak

Initjal Wave Functjon Weak Operatpr

  • Y. Aharonov, D. Z. Albert, and L. Vaidman: PRL (1988)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Strong

Wave Functjon aftfr Stsong Measurement

  • J. Von Neumann: PUP (1955)

Weak

Wave Functjon aftfr Weak Measurement

  • Y. Aharonov, D. Z. Albert, and L. Vaidman: PRL (1988)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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1 - Open problem

How is Measurement modelled in Quantum Mechanics? Main features of the Weak Measurement Istrong = Iweak Wave Function of the system is slightly perturbed after the interaction Weak

Wave Functjon aftfr Weak Measurement

  • Y. Aharonov, D. Z. Albert, and L. Vaidman: PRL (1988)

Damiano Marian Barcelona - 2015, July 17th 5 / 23

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SLIDE 20

1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful!

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Which is the operator that defines the (non-unitary) evolution of the wave function when measuring the total current?

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Which is the operator that defines the (non-unitary) evolution of the wave function when measuring the total current? Is it “continuous” or “instantaneous”? Does it provide a “weak” or a “strong” perturbation on the wave function?

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Which is the operator that defines the (non-unitary) evolution of the wave function when measuring the total current? Is it “continuous” or “instantaneous”? Does it provide a “weak” or a “strong” perturbation on the wave function?

III Looking at the problem from a different point of view

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Which is the operator that defines the (non-unitary) evolution of the wave function when measuring the total current? Is it “continuous” or “instantaneous”? Does it provide a “weak” or a “strong” perturbation on the wave function?

III Looking at the problem from a different point of view

Include the apparatus and see what happens!

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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1 - Open problem

Measuring the current at high frequency Three ways: I Taking information from the system without worrying about the apparatus → Be careful! II Looking for an Operator...but...

Which is the operator that defines the (non-unitary) evolution of the wave function when measuring the total current? Is it “continuous” or “instantaneous”? Does it provide a “weak” or a “strong” perturbation on the wave function?

III Looking at the problem from a different point of view

Include the apparatus and see what happens!

V Si SA Ammeter

x y z

(t) Pointer Device Active Region Electron X(t)

𝚶

Cable

Damiano Marian Barcelona - 2015, July 17th 6 / 23

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SLIDE 27

Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 7 / 23

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2 - Approach to model the measurement of THz current

Many-Body Problem i∂Ψ(x1, x2, ..., xN, t) ∂t = [H0 + VCoul] Ψ(x1, x2, ..., xN, t)

Damiano Marian Barcelona - 2015, July 17th 8 / 23

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2 - Approach to model the measurement of THz current

Many-Body Problem i∂Ψ(x1, x2, ..., xN, t) ∂t = [H0 + VCoul] Ψ(x1, x2, ..., xN, t) Coulomb Interaction ↓ VCoul =

1 4πǫ(r) 1 2 N

  • i=1,j=i

qiqj

√

(xi−xj)2

Damiano Marian Barcelona - 2015, July 17th 8 / 23

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2 - Approach to model the measurement of THz current

Many-Body Problem i∂Ψ(x1, x2, ..., xN, t) ∂t = [H0 + VCoul] Ψ(x1, x2, ..., xN, t) Coulomb Interaction ↓ VCoul =

1 4πǫ(r) 1 2 N

  • i=1,j=i

qiqj

√

(xi−xj)2

Many Particle ↓ Numerically inaccessible

Damiano Marian Barcelona - 2015, July 17th 8 / 23

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SLIDE 31

2 - Approach to model the measurement of THz current

Conditional Wave Function i∂ψ(x1, t) ∂t = [H0 + VCond] ψ(x1, t) dX1(t) dt = mIm ∇ψ ψ

  • D. D¨

urr, S. Goldstein, and N. Zangh` ı: JSP (1992;2005) Damiano Marian Barcelona - 2015, July 17th 9 / 23

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2 - Approach to model the measurement of THz current

Conditional Wave Function i∂ψ(x1, t) ∂t = [H0 + VCond] ψ(x1, t) dX1(t) dt = mIm ∇ψ ψ

  • D. D¨

urr, S. Goldstein, and N. Zangh` ı: JSP (1992;2005)

ψ(x1, t) = Ψ(x1, X2(t), ..., XN(t))

Damiano Marian Barcelona - 2015, July 17th 9 / 23

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2 - Approach to model the measurement of THz current

Conditional Wave Function i∂ψ(x1, t) ∂t = [H0 + VCond] ψ(x1, t) dX1(t) dt = mIm ∇ψ ψ

  • D. D¨

urr, S. Goldstein, and N. Zangh` ı: JSP (1992;2005)

ψ(x1, t) = Ψ(x1, X2(t), ..., XN(t)) Conditional Coulomb Interaction VCond =

1 4πǫ N

  • i=2

q1qi

√

(x1−Xi(t))2

  • X. Oriols: PRL (2007)

Damiano Marian Barcelona - 2015, July 17th 9 / 23

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2 - Approach to model the measurement of THz current

Conditional Wave Function i∂ψ(x1, t) ∂t = [H0 + VCond] ψ(x1, t) dX1(t) dt = mIm ∇ψ ψ

  • D. D¨

urr, S. Goldstein, and N. Zangh` ı: JSP (1992;2005)

ψ(x1, t) = Ψ(x1, X2(t), ..., XN(t)) Conditional Coulomb Interaction VCond =

1 4πǫ N

  • i=2

q1qi

√

(x1−Xi(t))2

  • X. Oriols: PRL (2007)

Id(t) =

  • SL ǫ(r) dE(r;X1(t),...,XN(t),t)

dt

· ds

Damiano Marian Barcelona - 2015, July 17th 9 / 23

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2 - Approach to model the measurement of THz current

1 2 3 4 5 50 100 150 200 250 300 y (nm) x (nm)

Conditional Wave Function Electron in the device E l e c t r

  • n

s i n t h e m e t a l

Damiano Marian Barcelona - 2015, July 17th 10 / 23

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2 - Approach to model the measurement of THz current

Evolution of the system under the interaction with the apparatus “

Damiano Marian Barcelona - 2015, July 17th 10 / 23

1 2 3 4 5 50 100 150 200 250 300 y (nm) x (nm)

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2 - Approach to model the measurement of THz current

1 2 3 4 5 50 100 150 200 250 300 y (nm) x (nm)

Electron in the device Conditional Wave Function Electrons in the metal

Damiano Marian Barcelona - 2015, July 17th 10 / 23

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2 - Approach to model the measurement of THz current

1 2 3 4 5 50 100 150 200 250 300 y (nm) x (nm)

Electron in the device Conditional Wave Function Electrons in the metal

We have a numerical method to tackle the many-body problem! ⇓ We include the Back-Action of the measuring apparatus!!!

Damiano Marian Barcelona - 2015, July 17th 10 / 23

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Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 11 / 23

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3 - New source of noise

The wave function of the system is only “slightly” perturbed

d

Different distances from the metal surface → Parameter d

Damiano Marian Barcelona - 2015, July 17th 12 / 23

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3 - New source of noise

The wave function of the system is only “slightly” perturbed

d

Different distances from the metal surface → Parameter d

10

  • 9

0.5

1

1.5 2 2.5 3 3.5 4

Distance (𝜈m) Error wave

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

Error Wave Function ErrorWave =

  • |ψint − ψfree|2dx

The error in the Wave Function decreases with the distance!

Damiano Marian Barcelona - 2015, July 17th 12 / 23

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3 - New source of noise

Total Current Measured in the Surface SL

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Current (µA) Time (ps)

Instantaneous Current (without Ammeter)

Damiano Marian Barcelona - 2015, July 17th 13 / 23

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3 - New source of noise

Total Current Measured in the Surface SL

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Current (µA) Time (ps)

Instantaneous Current (without Ammeter) Instantaneous Current (with Ammeter)

Damiano Marian Barcelona - 2015, July 17th 13 / 23

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3 - New source of noise

Total Current Measured in the Surface SL

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Current (µA) Time (ps)

Instantaneous Current (without Ammeter) Instantaneous Current (with Ammeter)

Repeating many times the experiment we reach the mean value!!!

Damiano Marian Barcelona - 2015, July 17th 13 / 23

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SLIDE 45

3 - New source of noise

Probability distribution of the total current in a large metallic surface

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Total current probability distribution ( A) 𝜈 Total Current

Output current very noisy! ⇓ Additional source of noise!!!

Damiano Marian Barcelona - 2015, July 17th 14 / 23

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SLIDE 46

3 - New source of noise

Probability distribution of the total current in a large metallic surface

Mean Value

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Total current probability distribution ( A) 𝜈 Total Current

Istrong = Iweak

Damiano Marian Barcelona - 2015, July 17th 14 / 23

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SLIDE 47

3 - New source of noise

Probability distribution of the total current in a large metallic surface

Mean Value

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Total current probability distribution ( A) 𝜈 Total Current

⇒ Output current very noisy! ⇒ The error in the Wave Function decreases with the distance! ⇒ Istrong = Iweak

Damiano Marian Barcelona - 2015, July 17th 14 / 23

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SLIDE 48

3 - New source of noise

Probability distribution of the total current in a large metallic surface

Mean Value

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Total current probability distribution ( A) 𝜈 Total Current

⇒ Output current very noisy! ⇒ The error in the Wave Function decreases with the distance! ⇒ Istrong = Iweak Weak Measurement!!!

Damiano Marian Barcelona - 2015, July 17th 14 / 23

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SLIDE 49

Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 15 / 23

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4 - Measurement of the local (Bohmian) velocities

Total Current (conduction plus displacement components) I(t) =

  • Si

Jc(r, t) · ds +

  • Si

ǫdE(r, t) dt · ds

Damiano Marian Barcelona - 2015, July 17th 16 / 23

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SLIDE 51

4 - Measurement of the local (Bohmian) velocities

Total Current (conduction plus displacement components) I(t) =

  • Si

Jc(r, t) · ds +

  • Si

ǫdE(r, t) dt · ds Or from the Ramo-Schockley-Pellegrini theorem I(t) = −

  • Ω

F(r)·Jc(r, t)·dv +

  • S

ǫ·F(r)· dV (r, t) dt ·ds

Damiano Marian Barcelona - 2015, July 17th 16 / 23

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SLIDE 52

4 - Measurement of the local (Bohmian) velocities

Total Current (conduction plus displacement components) I(t) =

  • Si

Jc(r, t) · ds +

  • Si

ǫdE(r, t) dt · ds Or from the Ramo-Schockley-Pellegrini theorem I(t) = −

  • Ω

F(r)·Jc(r, t)·dv +

  • S

ǫ·F(r)·

✟✟✟✟✟ ❍❍❍❍❍

dV (r, t) dt ·ds

Si Wave Function X (t)

1 d

Ammeter Ω SA X (t)

3

X (t)

2

V Lx

In the case of two metallic surfaces Si at distance Si ≫ L2

x:

I(t) = q mLx p(t)Ω

Damiano Marian Barcelona - 2015, July 17th 16 / 23

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SLIDE 53

4 - Measurement of the local (Bohmian) velocities

Si Wave Function X (t)

1 d

Ammeter Ω SA X (t)

3

X (t)

2

V Lx

In the case of two metallic surfaces Si at distance Si ≫ L2

x:

I(t) = q mLx p(t)Ω Total Current in a large surface ⇓ Weak Measurement of the Momentum!!!

Damiano Marian Barcelona - 2015, July 17th 16 / 23

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SLIDE 54

4 - Measurement of the local (Bohmian) velocities

Weak measurement: a new way for measuring incompatible observables

Damiano Marian Barcelona - 2015, July 17th 17 / 23

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SLIDE 55

4 - Measurement of the local (Bohmian) velocities

Weak measurement: a new way for measuring incompatible observables ⇓ Measurement of position and momentum (for an ensemble of experiments)

Damiano Marian Barcelona - 2015, July 17th 17 / 23

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SLIDE 56

4 - Measurement of the local (Bohmian) velocities

Weak measurement: a new way for measuring incompatible observables ⇓ Measurement of position and momentum (for an ensemble of experiments) ⇓ Experimental measurement of the local velocity

From: S. Kocis, et al., Science, 332 (2011).

Experimental Bohmian trajectories: Photons in a double slit set-up

Damiano Marian Barcelona - 2015, July 17th 17 / 23

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SLIDE 57

4 - Measurement of the local (Bohmian) velocities

Weak measurement: a new way for measuring incompatible observables ⇓ Measurement of position and momentum (for an ensemble of experiments) ⇓ Experimental measurement of the local velocity

From: S. Kocis, et al., Science, 332 (2011).

Experimental Bohmian trajectories: Photons in a double slit set-up

From: C. Philippidis, et al., Il nuovo cimento B (1979)

Theoretical Bohmian trajectories

Damiano Marian Barcelona - 2015, July 17th 17 / 23

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SLIDE 58

4 - Measurement of the local (Bohmian) velocities

Weak measurement: a new way for measuring incompatible observables ⇓ Measurement of position and momentum (for an ensemble of experiments) ⇓ Experimental measurement of the local velocity

From: S. Kocis, et al., Science, 332 (2011).

Experimental Bohmian trajectories: Photons in a double slit set-up

From: C. Philippidis, et al., Il nuovo cimento B (1979)

Theoretical Bohmian trajectories

⇓ Is it possible to envisage an analogous experiment for electrons?

Damiano Marian Barcelona - 2015, July 17th 17 / 23

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SLIDE 59

4 - Measurement of the local (Bohmian) velocities

Experimental proposal for measuring the local (Bohmian) velocity v(x) = 1 mRex|ˆ p|ψ x|ψ

H M Wiseman 2007 New J. Phys. 9 165

E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= v(xs, τ)

Damiano Marian Barcelona - 2015, July 17th 18 / 23

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SLIDE 60

4 - Measurement of the local (Bohmian) velocities

Experimental proposal for measuring the local (Bohmian) velocity v(x) = 1 mRex|ˆ p|ψ x|ψ

H M Wiseman 2007 New J. Phys. 9 165

E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= v(xs, τ)

s-ammeter g-ammeter w-ammeter

x y z

s-cable g-cable w-cable V Ss Sg Sw Lx Ly Lz L

z

L

y

electron Damiano Marian Barcelona - 2015, July 17th 18 / 23

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SLIDE 61

4 - Measurement of the local (Bohmian) velocities

Experimental proposal for measuring the local (Bohmian) velocity v(x) = 1 mRex|ˆ p|ψ x|ψ

H M Wiseman 2007 New J. Phys. 9 165

E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= v(xs, τ)

s-ammeter g-ammeter w-ammeter

x y z

s-cable g-cable w-cable V Ss Sg Sw Lx Ly Lz L

z

L

y

electron

Sw where Ly, Lz ≫ Lx → Iw ∝ px WM of total current = WM of momentum

Damiano Marian Barcelona - 2015, July 17th 18 / 23

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SLIDE 62

4 - Measurement of the local (Bohmian) velocities

Experimental proposal for measuring the local (Bohmian) velocity v(x) = 1 mRex|ˆ p|ψ x|ψ

H M Wiseman 2007 New J. Phys. 9 165

E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= v(xs, τ)

s-ammeter g-ammeter w-ammeter

x y z

s-cable g-cable w-cable V Ss Sg Sw Lx Ly Lz L

z

L

y

electron

Sw where Ly, Lz ≫ Lx → Iw ∝ px WM of total current = WM of momentum Ss where L′

y, L′ z ≪ Lx

→ Is ∝ |rs|ψ|2 Post-selection with position measurement

Damiano Marian Barcelona - 2015, July 17th 18 / 23

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SLIDE 63

4 - Measurement of the local (Bohmian) velocities

Numerical Experiments

Current (nA) Position (nm)

With ammeter Without ammeter

Reconstruction of the Bohmian velocity from an ensemble of 55000 numerical experiments E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= =

J(xs,τ) |ψ(xs,τ)|2 ≡ v(xs, τ)

Weak Measurement Iw ∝ px Strong Measurement Is ∝ |rs|ψ|2

Damiano Marian Barcelona - 2015, July 17th 19 / 23

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SLIDE 64

4 - Measurement of the local (Bohmian) velocities

Numerical Experiments

Electron Position (nm) Time (fs) Wave function

Wave function E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= =

J(xs,τ) |ψ(xs,τ)|2 ≡ v(xs, τ)

Weak Measurement Iw ∝ px Strong Measurement Is ∝ |rs|ψ|2

Damiano Marian Barcelona - 2015, July 17th 19 / 23

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SLIDE 65

4 - Measurement of the local (Bohmian) velocities

Numerical Experiments

Electron Position (nm) Time (fs) Bohmian trajectories

Bohmian trajectories E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= =

J(xs,τ) |ψ(xs,τ)|2 ≡ v(xs, τ)

Weak Measurement Iw ∝ px Strong Measurement Is ∝ |rs|ψ|2

Damiano Marian Barcelona - 2015, July 17th 19 / 23

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SLIDE 66

4 - Measurement of the local (Bohmian) velocities

Numerical Experiments

0.1 0.2 0.3 0.4 0.5 50 100 150 200

Position (nm) T i m e ( p s )

Modulus Squared Wave Funcion Trajectories

Wave function and Bohmian trajectories E[pw|xs] =

  • dpwpwP(pw∩xs)

P(xs)

= =

J(xs,τ) |ψ(xs,τ)|2 ≡ v(xs, τ)

Weak Measurement Iw ∝ px Strong Measurement Is ∝ |rs|ψ|2

  • D. M., X. Oriols and N. Zangh`

ı: Weak Values from Displacement Currents in Multiterminal Electron Devices, in preparation.

  • D. D¨

urr, S. Goldstein, and N. Zangh´ ı, On the Weak Measurement of Velocity in Bohmian Mechanics, Journal of Statistical Physics 134, 1023 (2009). Damiano Marian Barcelona - 2015, July 17th 19 / 23

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SLIDE 67

Index

1 Open problem 2 Novel approach to model the measurement of THz current 3 New source of noise 4 Measurement of the local (Bohmian) velocities 5 Concluding Remarks Damiano Marian Barcelona - 2015, July 17th 20 / 23

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SLIDE 68

Concluding Remarks

The Conditional Wave Function formalism provides a natural way to define the interaction between the quantum system and the metal (There is no need to define the operator)

Damiano Marian Barcelona - 2015, July 17th 21 / 23

slide-69
SLIDE 69

Concluding Remarks

The Conditional Wave Function formalism provides a natural way to define the interaction between the quantum system and the metal (There is no need to define the operator) A new source of noise at THz frequency

Damiano Marian Barcelona - 2015, July 17th 21 / 23

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SLIDE 70

Concluding Remarks

The Conditional Wave Function formalism provides a natural way to define the interaction between the quantum system and the metal (There is no need to define the operator) A new source of noise at THz frequency The measurement of the Total Current (in a large surface) is a Weak Measurement of the Momentum

Damiano Marian Barcelona - 2015, July 17th 21 / 23

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SLIDE 71

Concluding Remarks

The Conditional Wave Function formalism provides a natural way to define the interaction between the quantum system and the metal (There is no need to define the operator) A new source of noise at THz frequency The measurement of the Total Current (in a large surface) is a Weak Measurement of the Momentum Reconstruction of the Bohmian trajectories in a solid state device

Damiano Marian Barcelona - 2015, July 17th 21 / 23

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SLIDE 72

UPoN 2015 2015, July 17th

On the WEAK MEASUREMENT

  • f the ELECTRICAL THz CURRENT:

a NEW SOURCE of NOISE?

Damiano Marian, Nino Zangh` ı, Xavier Oriols

THANKS FOR THE ATTENTION!!!

Damiano Marian Barcelona - 2015, July 17th 22 / 23

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SLIDE 73

Question: At which frequency is this additional noise relevant?

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

  • 0.15
  • 0.1
  • 0.05

0.05 0.1

Probability Total Current Total Current (µA)

f = 5 THz

Damiano Marian Barcelona - 2015, July 17th 23 / 23

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SLIDE 74

Question: At which frequency is this additional noise relevant?

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

  • 0.15
  • 0.1
  • 0.05

0.05 0.1

Probability Total Current Total Current (µA)

f = 500 GHz

Damiano Marian Barcelona - 2015, July 17th 23 / 23

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SLIDE 75

Question: At which frequency is this additional noise relevant?

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

  • 0.15
  • 0.1
  • 0.05

0.05 0.1

Probability Total Current Total Current (µA)

f = 100 GHz

Damiano Marian Barcelona - 2015, July 17th 23 / 23

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SLIDE 76

Question: At which frequency is this additional noise relevant?

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

  • 0.15
  • 0.1
  • 0.05

0.05 0.1

Probability Total Current Total Current (µA)

f = 50 GHz

Damiano Marian Barcelona - 2015, July 17th 23 / 23

slide-77
SLIDE 77

Question: At which frequency is this additional noise relevant?

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

  • 0.15
  • 0.1
  • 0.05

0.05 0.1

Probability Total Current Total Current (µA)

f = 50 GHz

0.005 0.01 0.015 0.02 0.025 0.03 0.05 0.1 1

Dispersion (µA) Frequency (THz)

Without Ammeter With Ammeter

⇓ For the configuration considered ≈ 50 GHz

Damiano Marian Barcelona - 2015, July 17th 23 / 23