Abstract

We propose an interferometric setup that permits to tune the quantity of radiation absorbed by an object illuminated by a fixed light source. The method can be used to selectively irradiate portions of an object based on their transmissivities or to accurately estimate the transmissivities from rough absorption measurements.

© 2006 Optical Society of America

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References

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  1. A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
    [CrossRef]
  2. P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
    [CrossRef]
  3. A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
    [CrossRef]
  4. P. Kwiat, "Experimental and theoretical progress in interaction-free measurements," Physica Scripta T76, 115- 121 (1998).
    [CrossRef]
  5. P. Facchi, Z. Hradil, G. Krenn, S. Pascazio, and J . Rěhaćěk, "Quantum Zeno tomography," Phys. Rev. A 66, 12110 (2002).
    [CrossRef]
  6. S. Inoue and G. Björk, "Experimental demonstration of exposure-free imaging and contrast amplification," J. Opt. B: Quantum Semiclass. Opt. 2, 338-345 (2000).
    [CrossRef]
  7. J.-S. Jang, "Optical interaction-free measurement of semitransparent objects," Phys. Rev. A 59, 2322-2329 (1999).
    [CrossRef]
  8. Rigorously speaking, the transformation βn+1 ->√ηβn+1 does not correspond to the linear mapping |βn+1i - > |√ηβ n+1i but to its density matrix counterpart: |βn+1ihβn+1| ->|√ηβn+1ih√ηβn+1|. The latter accounts for decoherence effect, whereas the former does not. Since in our analysis we are always dealing with factorized states, the two transformations coincide for us.
  9. <other> This effect can be explained intuitively as follows. If the object is transparent, at the first round trip a small amount of radiation leaks into the R modes, at the second round trip a higher amount leaks there and so on constantly increasing through constructive interference until all the radiation moves into such modes after N =π/φ round trips. If, instead, the object is opaque, the little radiation that has leaked into the R modes at the first round trip is absorbed and does not contribute to the constructive interference that would draw more radiation into these modes at the second round trip. As a result very little radiation transfers and most of it remains in the L modes.</other>

1999 (1)

J.-S. Jang, "Optical interaction-free measurement of semitransparent objects," Phys. Rev. A 59, 2322-2329 (1999).
[CrossRef]

1998 (2)

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

P. Kwiat, "Experimental and theoretical progress in interaction-free measurements," Physica Scripta T76, 115- 121 (1998).
[CrossRef]

1996 (1)

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

1993 (1)

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

Elizur, A.

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

Herzog, T.

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

Jang, J.-S.

J.-S. Jang, "Optical interaction-free measurement of semitransparent objects," Phys. Rev. A 59, 2322-2329 (1999).
[CrossRef]

Kasevich, M. A.

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

Kwiat, P.

P. Kwiat, "Experimental and theoretical progress in interaction-free measurements," Physica Scripta T76, 115- 121 (1998).
[CrossRef]

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

Kwiat, P. G.

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

Mitchell, J. R.

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

Nairz, O.

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

Vaidman, L.

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

Weinfurter, H.

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

White, A. G.

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

Zeilinger, A.

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

Found. Phys. (1)

A. Elizur and L. Vaidman, "Quantum mechanical interaction-free measurements," Found. Phys. 23, 987-997 (1993); L. Vaidman, "On the realization of interaction-free measurements," Quantum Opt. 6, 119 (1994).
[CrossRef]

Phys. Rev. A (2)

J.-S. Jang, "Optical interaction-free measurement of semitransparent objects," Phys. Rev. A 59, 2322-2329 (1999).
[CrossRef]

A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat, "Interaction-free imaging," Phys. Rev. A 58, 605-613 (1998).
[CrossRef]

Physica Scripta (1)

P. Kwiat, "Experimental and theoretical progress in interaction-free measurements," Physica Scripta T76, 115- 121 (1998).
[CrossRef]

Sci. Am. (1)

P. Kwiat, H. Weinfurter, A. Zeilinger, "Quantum Seeing in the Dark," Sci. Am. 275, 72-77 (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, "Interaction-free measurement," Phys. Rev. Lett. 74, 4763-4767 (1995).
[CrossRef]

Other (4)

Rigorously speaking, the transformation βn+1 ->√ηβn+1 does not correspond to the linear mapping |βn+1i - > |√ηβ n+1i but to its density matrix counterpart: |βn+1ihβn+1| ->|√ηβn+1ih√ηβn+1|. The latter accounts for decoherence effect, whereas the former does not. Since in our analysis we are always dealing with factorized states, the two transformations coincide for us.

<other> This effect can be explained intuitively as follows. If the object is transparent, at the first round trip a small amount of radiation leaks into the R modes, at the second round trip a higher amount leaks there and so on constantly increasing through constructive interference until all the radiation moves into such modes after N =π/φ round trips. If, instead, the object is opaque, the little radiation that has leaked into the R modes at the first round trip is absorbed and does not contribute to the constructive interference that would draw more radiation into these modes at the second round trip. As a result very little radiation transfers and most of it remains in the L modes.</other>

P. Facchi, Z. Hradil, G. Krenn, S. Pascazio, and J . Rěhaćěk, "Quantum Zeno tomography," Phys. Rev. A 66, 12110 (2002).
[CrossRef]

S. Inoue and G. Björk, "Experimental demonstration of exposure-free imaging and contrast amplification," J. Opt. B: Quantum Semiclass. Opt. 2, 338-345 (2000).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Proposed apparatus. It consists of N Mach-Zehnder (MZ) interferometers concatenated so that the output ports of the nth MZ is fed into the input ports of the successive one (for the sake of clarity the first interferometer is graphically enhanced). All interferometers act on the radiation with the same phase shift ϕ. The object to be irradiated is placed outside the MZs and it interacts only with the R beams. Initially the radiation enters from the input a 0. After N round trips, it exits through the outputs aN and bN .

Fig. 2.
Fig. 2.

Mach-Zehnder interferometer constituting the nth element in the Mach-Zehnder sequence in the apparatus of Fig 1. The first of the two 50–50 beam splitters transforms the input annihilation operators an, bn into c =(an+bn )/√2 and d=(bn-an )/√2 respectively. The second beam splitter transforms the annihilation operators c and d into a n+1=(c+d)/√2 and b n+1=(d-c)/√2.

Fig. 3.
Fig. 3.

Plot of the (rescaled) output amplitudes of the nth MZ interferometer |αn /α 0|2 in the a-modes at L (continuous line) and |βn /α 0|2 in the b-modes at R (dashed line). Initially all the radiation is in mode a 0, but, as the evolutions progresses, more and more radiation is transferred to the b-modes, until (for n=π/ϕ) the radiation is entirely transferred. Here ϕ=π/10 so that the total transfer occurs for n=10 (vertical line). Left: The object is completely transparent (η=1), so that the total energy (dotted line) is constant; Right the object is semi-transparent (η=.9), so that the total energy decreases as the evolution progresses.

Fig. 4.
Fig. 4.

Above left: Plot of the (rescaled) apparatus output amplitudes |αN0 |2 at the output aN (continuous line) and |βN0 |2 at bN (dashed line) as a function of the transmissivity η of the object with ϕ=π/60. For η=1 (total transparency) all the output radiation is at bN , whereas for η=0 (total absorption) all the output radiation is at aN and a small amount of radiation has been absorbed. The absorbed radiation is proportional to the effective absorption constant r, defined in Eq. (6), which is depicted by the dotted lines. Above right: Plot of the absorption r only, in the case ϕ=π/200. This plot illustrates how the setup can be used to measure the absorption coefficient η by measuring r, attaining a higher precision than with a direct measure (in which case the absorption is given by the dashed line): Starting from the same uncertainty Δr in the measurement of r with our procedure we can obtain a lower uncertainty Δη in the measurement of η than the one, Δη, obtainable from a direct measurement. Below: Plot of r as a function of η and N=π/ϕ. Notice how the absorption peak shifts as a function of ϕ. The absorption peak moves to higher η for decreasing ϕ.

Fig. 5.
Fig. 5.

Left: Maximum ηmaxN (circles) and average value ηav (squares) of the absorption peak r of Fig. 4 as a function of N=π/ϕ. (The maximum and the average follow different evolutions because of the asymmetry in the absorption curves). Increasing N (i.e. decreasing ϕ), the maximum in the absorption peak moves to higher values of η. The graph also details which are the actual values of η that can be achieved through the proposed setup as a function of N. The dotted line is the function [(N-1)/N]4 that gives a good interpolation of the peak evolution. Right: Selectivity in the irradiation as a function of the transmissivity peak. The width of the peaks of the dotted line in Fig. 4 is not uniform. Here we plot the Root Mean Square (stars) and the width at half maximum (circles) of the absorption curve as a function of the absorption curve maximum. Notice that the RMS is almost constant over the whole range.

Equations (10)

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( α n + 1 β n + 1 ) = S ( α n β n ) ,
S = e i ϕ 2 ( cos ( ϕ 2 ) i sin ( ϕ 2 ) i sin ( ϕ 2 ) cos ( ϕ 2 ) ) ,
( α n + 1 β n + 1 ) = S ( η ) ( α n β n ) ,
S ( η ) = e i ϕ 2 ( cos ( ϕ 2 ) i sin ( ϕ 2 ) i η sin ( ϕ 2 ) η cos ( ϕ 2 ) ) .
( α N β N ) = S N ( η ) ( α 0 0 ) .
I ab = α 0 2 ( α N 2 + β N 2 ) r α 0 2 ,
α N = α 0 e iN ϕ 2 cos ( N ϕ 2 )
β N = i α 0 e iN ϕ 2 sin ( N ϕ 2 ) ,
α N = α 0 e iN ϕ 2 cos N ( ϕ 2 )
β N = 0 ,

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