Abstract

In the presence of a longitudinal magnetic field B, a beam of linearly polarized light incident from a Faraday medium of Verdet constant V refracts at its interface with a medium of negligible Verdet constant and emerges as two opposite circularly polarized beams that are separated by a small divergence angle δ that is proportional to the product BV. Judicious postselection of the polarization state of the emergent light can be used to amplify the measured value of δ by several orders of magnitude. This technique makes it possible to optically measure either very small V values when B is known or small magnetic fields when V is known.

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  1. A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
    [CrossRef]
  2. Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
    [CrossRef]
  3. Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
    [CrossRef]
  4. Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).
    [CrossRef]
  5. N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
    [CrossRef]
  6. A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
    [CrossRef]
  7. K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
    [CrossRef]
  8. Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
    [CrossRef]
  9. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [CrossRef]
  10. I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
    [CrossRef]
  11. N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurement or standard interferometry,” Phys. Rev. Lett. 105, 010405 (2010).
    [CrossRef]
  12. A. Parks and J. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A 84, 012116 (2011).
    [CrossRef]
  13. B. Carnahan, H. Luther, and J. Wilkes, Applied Numerical Methods (Wiley, 1969), p. 171–175.
  14. M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19, 16508–16517 (2011).
    [CrossRef]
  15. S. Wu and Y. Li, “Weak measurements beyond the Aharonov–Albert–Vaidman formalism,” Phys. Rev. A 83, 052106 (2011).
    [CrossRef]

2011

A. Parks and J. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A 84, 012116 (2011).
[CrossRef]

M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19, 16508–16517 (2011).
[CrossRef]

S. Wu and Y. Li, “Weak measurements beyond the Aharonov–Albert–Vaidman formalism,” Phys. Rev. A 83, 052106 (2011).
[CrossRef]

2010

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurement or standard interferometry,” Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

2008

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

2007

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

2006

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

2004

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[CrossRef]

1998

A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
[CrossRef]

1991

N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[CrossRef]

1990

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).
[CrossRef]

1989

I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

1988

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef]

1986

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Aharonov, Y.

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).
[CrossRef]

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef]

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Albert, D.

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef]

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Brunner, N.

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurement or standard interferometry,” Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

Carnahan, B.

B. Carnahan, H. Luther, and J. Wilkes, Applied Numerical Methods (Wiley, 1969), p. 171–175.

Casher, A.

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Cullin, D.

A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
[CrossRef]

Duck, I.

I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Fischer, P.

M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19, 16508–16517 (2011).
[CrossRef]

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

Ghosh, A.

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

Gray, J.

A. Parks and J. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A 84, 012116 (2011).
[CrossRef]

Guo, G.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Hill, W.

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Huang, Y.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Hulet, R.

N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[CrossRef]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Li, J.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Li, Y.

S. Wu and Y. Li, “Weak measurements beyond the Aharonov–Albert–Vaidman formalism,” Phys. Rev. A 83, 052106 (2011).
[CrossRef]

Lundeen, J.

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[CrossRef]

Luther, H.

B. Carnahan, H. Luther, and J. Wilkes, Applied Numerical Methods (Wiley, 1969), p. 171–175.

Parks, A.

A. Parks and J. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A 84, 012116 (2011).
[CrossRef]

A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
[CrossRef]

Pfeifer, M.

Resch, K.

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[CrossRef]

Ritchie, N.

N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[CrossRef]

Simon, C.

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurement or standard interferometry,” Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

Steinberg, A.

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[CrossRef]

Stevenson, P.

I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Storey, J.

N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[CrossRef]

Stoudt, D.

A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
[CrossRef]

Sudarshan, E.

I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Sun, F.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Vaidman, L.

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).
[CrossRef]

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef]

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Wang, Q.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Wilkes, J.

B. Carnahan, H. Luther, and J. Wilkes, Applied Numerical Methods (Wiley, 1969), p. 171–175.

Wu, S.

S. Wu and Y. Li, “Weak measurements beyond the Aharonov–Albert–Vaidman formalism,” Phys. Rev. A 83, 052106 (2011).
[CrossRef]

Zhang, Y.

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Ann. N.Y. Acad. Sci.

Y. Aharonov, D. Albert, A. Casher, and L. Vaidman, “Novel properties of preselected and postselected ensembles,” Ann. N.Y. Acad. Sci. 480, 417–421 (1986).
[CrossRef]

Opt. Express

Phys. Lett. A

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[CrossRef]

Phys. Rev. A

Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, and G. Guo, “Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon,” Phys. Rev. A 73, 023814 (2006).
[CrossRef]

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).
[CrossRef]

S. Wu and Y. Li, “Weak measurements beyond the Aharonov–Albert–Vaidman formalism,” Phys. Rev. A 83, 052106 (2011).
[CrossRef]

A. Parks and J. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A 84, 012116 (2011).
[CrossRef]

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

Phys. Rev. D

I. Duck, P. Stevenson, and E. Sudarshan, “The sense in which a ‘weak measurement’ of a spin 1/2 particle’s spin component yields a value of 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Phys. Rev. Lett.

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurement or standard interferometry,” Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

N. Ritchie, J. Storey, and R. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[CrossRef]

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef]

Proc. R. Soc. A

A. Parks, D. Cullin, and D. Stoudt, “Observation and measurement of an optical Aharonov–Albert–Vaidman effect,” Proc. R. Soc. A 454, 2997–3008 (1998).
[CrossRef]

Science

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Other

B. Carnahan, H. Luther, and J. Wilkes, Applied Numerical Methods (Wiley, 1969), p. 171–175.

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Figures (3)

Fig. 1.
Fig. 1.

Geometry of the Faraday differential refraction effect.

Fig. 2.
Fig. 2.

Schematic of an apparatus for amplifying and measuring the divergence angle.

Fig. 3.
Fig. 3.

Theoretical relationships between the measurement pointer’s effective rotation angle and amplification factor for various divergence angles. The dashed line represents the smallest divergence angle measured by Ghosh et al. [1] using only preselection.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

δ λ sin θ 0 π n 0 cos θ B V .
H ^ = γ ( t ) A ^ p ^ ,
| Φ = e i H ^ d t | ψ i | ϕ = e i γ A ^ p ^ | ψ i | ϕ .
| Ψ = ψ f | Φ = ψ f | e i γ A ^ p ^ | ψ i | ϕ .
| Ψ ψ f | ( 1 ^ i γ A ^ p ^ ) | ψ i | ϕ ψ f | ψ i S ^ ( γ A w ) | ϕ ,
A w ψ f | A ^ | ψ i ψ f | ψ i , ψ f | ψ i 0
Δ q γ | A w | 1 and Δ q min ( n = 2 , 3 , ) γ | A w ( A ^ n ) w | 1 n 1 .
| q | Ψ | 2 | ψ f | ψ i | 2 | ϕ ( q γ Re A w ) | 2
Ψ | q ^ | Ψ = ϕ | q ^ | ϕ + γ Re A w ;
σ ^ | ± = ± | ±
± | ± = 1 and ± | = 0 .
A ^ ( θ + θ 0 ) | + + | + ( θ θ 0 ) | | ,
H ^ = A ^ J ^ z δ ( t t 0 )
| Ψ = ψ f | e i A ^ J ^ z | ψ i | ϕ .
A ^ n = ( θ + θ 0 ) n | + + | + ( θ θ 0 ) n | | , n = 0 , 1 , 2 , ,
e i A ^ J ^ z = n = 0 [ i ( θ + θ 0 ) J ^ z ] n n ! | + + | + n = 0 [ i ( θ θ 0 ) J ^ z ] n n ! | | .
| Ψ ψ f | { [ | + + | + | | ] i [ ( θ + θ 0 ) | + + | + ( θ θ 0 ) | | ] J ^ z } | ψ i | ϕ ψ f | ( 1 ^ i A ^ J ^ z ) | ψ i | ϕ ψ f | ψ i e i A w J ^ z | ϕ
| Ψ ψ f | ψ i R ^ z ( Re A w ) | ϕ ,
R ^ z ( Re A w ) e i Re A w J ^ z .
| θ | Ψ | 2 | ψ f | ψ i | 2 | θ | R ^ z ( Re A w ) | ϕ | 2 .
| ψ i = 1 2 [ ( sin α + cos α ) | H + i ( sin α cos α ) | V ] ,
| ψ f = cos β | + sin β | .
A w = ( θ + θ 0 ) sin α cos β ( θ θ 0 ) cos α sin β sin ( α β ) = Re A w .
β = α ϵ , where ϵ is the amplification factor , and α = π 4 .
A w = ( θ + θ 0 ) ( cos ϵ + sin ϵ ) ( θ θ 0 ) ( cos ϵ sin ϵ ) 2 sin ϵ
A w = δ 2 cot ϵ + θ θ 0 .
A w δ 2 ϵ .
(a) Δ θ | A w | and (b) Δ θ { min ( n = 2 , 3 , ) | A w ( A ^ n ) w | 1 n 1 } 1 .
(a)  Δ θ | δ | 2 ϵ and (b) Δ θ { min ( n = 2 , 3 , ) | ( θ + θ 0 ) ( θ θ 0 ) ( θ + θ 0 ) n ( θ θ 0 ) n | 1 n 1 } 1 .
( θ + θ 0 ) ( θ θ 0 ) ( θ + θ 0 ) n ( θ θ 0 ) n = ( θ ± θ 0 ) ( θ ± θ 0 ) n · 1 ( θ θ 0 θ ± θ 0 ) 1 ( θ θ 0 θ ± θ 0 ) n X · Y .
Δ θ | δ | 2 ϵ when 0 < ϵ ϵ B 1 ( this ensures that | δ | 2 ϵ 2 | θ θ 0 | )
Δ θ 2 | θ θ 0 | when ϵ B < ϵ 1 ( this ensures that 2 | θ θ 0 | > | δ | 2 ϵ ) .
ϵ B 1 4 | δ θ θ 0 |
log ( A w θ + θ 0 ) = log ( cot ϵ 2 ) + log δ
Δ θ | δ est | 2 ϵ when 0 < ϵ ϵ B est 1 ,
Δ θ 2 | θ est θ 0 | when ϵ B est < ϵ 1 ,
ϵ B est 1 4 | δ est θ est θ 0 |
B est ( V est ) [ π n 0 cos θ est λ V ( B ) sin θ 0 ] δ est .
B est ( V est ) δ est δ ref B ref ( V ref ) ,

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