Abstract

An absorber placed into a laser beam is expected to cast a shadow. This expectation is confirmed not only by research, but also by our everyday experience. However, the shadow will not be produced if a weak absorber is known with certainty to have removed a photon from an optical state. We show this effect by distributing a state of light, via a beam splitter, between two parties. When one of the parties applies the photon annihilation operator to its portion of the state, the photon gets removed from the entire initial state, leaving the spatial and temporal structure of its mode undisturbed. In this way, the local application of the photon annihilation operator has a nonlocal effect, occurring without local state collapse by either party.

© 2015 Optical Society of America

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References

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  1. Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
    [Crossref]
  2. B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
    [Crossref]
  3. This can be seen easily from Eq. (1): if a^ is replaced with a^†, the term λa^⊥†|ψ〉a^ no longer vanishes.
  4. J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
    [Crossref]
  5. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
    [Crossref]
  6. A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
    [Crossref]
  7. H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
    [Crossref]
  8. A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
    [Crossref]
  9. R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
    [Crossref]
  10. Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
    [Crossref]
  11. A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
    [Crossref]
  12. E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
    [Crossref]
  13. R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
    [Crossref]
  14. A. I. Lvovsky, M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
    [Crossref]
  15. A. I. Lvovsky, J. Opt. B 6, S556 (2004).
    [Crossref]
  16. J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
    [Crossref]
  17. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
    [Crossref]
  18. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, Opt. Lett. 34, 2739 (2009).
    [Crossref]
  19. Mode a^ is split equally between the horizontal and vertical polarizations; so the 6% transmissivity of the PPBS for the horizontal polarization implies a 3% mean loss for mode a^.
  20. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
    [Crossref]

2014 (1)

Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
[Crossref]

2013 (2)

Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
[Crossref]

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

2012 (2)

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

2010 (2)

E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
[Crossref]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

2009 (3)

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, Opt. Lett. 34, 2739 (2009).
[Crossref]

A. I. Lvovsky, M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

2008 (1)

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

2007 (2)

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

2006 (2)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
[Crossref]

2004 (2)

J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

A. I. Lvovsky, J. Opt. B 6, S556 (2004).
[Crossref]

2001 (1)

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Aichele, T.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Babichev, S. A.

Baragiola, B. Q.

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

Barrios, E.

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Bimbard, E.

E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
[Crossref]

Branczyk, A. M.

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

Cairns, E.

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

Caves, C. M.

Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
[Crossref]

Combes, J.

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

Cook, R. L.

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

Dantan, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

Ferreyrol, F.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

Furusawa, A.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Grangier, P.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
[Crossref]

J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

Hansen, H.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Hayasaka, K.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Hradil, Z.

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

Huisman, S. R.

Huntington, E. H.

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

Jain, N.

Jiang, Z.

Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
[Crossref]

Knill, E.

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

Kumar, R.

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

Kupchak, C.

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

Kurochkin, Y.

Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
[Crossref]

Lang, M. D.

Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
[Crossref]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

Lvovsky, A. I.

Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
[Crossref]

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
[Crossref]

A. I. Lvovsky, M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, A. I. Lvovsky, Opt. Lett. 34, 2739 (2009).
[Crossref]

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

A. I. Lvovsky, J. Opt. B 6, S556 (2004).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

MacRae, A.

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
[Crossref]

Mlynek, J.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Neergaard-Nielsen, J. S.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

Ourjoumtsev, A.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
[Crossref]

Prasad, A. S.

Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
[Crossref]

Raymer, M. G.

A. I. Lvovsky, M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

Rehàcek, J.

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

Sasaki, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Schiller, S.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Suzuki, S.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Takahashi, H.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Takeoka, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Takeuchi, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

Tualle-Brouri, R.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
[Crossref]

J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

Vewinger, F.

Wakui, K.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Wenger, J.

J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

Youn, S. H.

Zhang, A. N.

J. Opt. B (1)

A. I. Lvovsky, J. Opt. B 6, S556 (2004).
[Crossref]

Nat. Photonics (2)

E. Bimbard, N. Jain, A. MacRae, A. I. Lvovsky, Nat. Photonics 4, 243 (2010).
[Crossref]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010).
[Crossref]

Nat. Phys. (1)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009).
[Crossref]

Opt. Commun. (1)

R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington, A. I. Lvovsky, Opt. Commun. 285, 5259 (2012).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (3)

J. Řehàček, Z. Hradil, E. Knill, A. I. Lvovsky, Phys. Rev. A 75, 042108 (2007).
[Crossref]

Z. Jiang, M. D. Lang, C. M. Caves, Phys. Rev. A 88, 044301 (2013).
[Crossref]

B. Q. Baragiola, R. L. Cook, A. M. Brańczyk, J. Combes, Phys. Rev. A 86, 013811 (2012).
[Crossref]

Phys. Rev. Lett. (7)

J. Wenger, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

R. Kumar, E. Barrios, C. Kupchak, A. I. Lvovsky, Phys. Rev. Lett. 110, 130403 (2013).
[Crossref]

Y. Kurochkin, A. S. Prasad, A. I. Lvovsky, Phys. Rev. Lett. 112, 070402 (2014).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Rev. Mod. Phys. (1)

A. I. Lvovsky, M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

Science (1)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, Science 312, 83 (2006).
[Crossref]

Other (2)

This can be seen easily from Eq. (1): if a^ is replaced with a^†, the term λa^⊥†|ψ〉a^ no longer vanishes.

Mode a^ is split equally between the horizontal and vertical polarizations; so the 6% transmissivity of the PPBS for the horizontal polarization implies a 3% mean loss for mode a^.

Supplementary Material (1)

» Supplement 1: PDF (1842 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

The quantum vampire effect. (a) When state | ψ in the mode defined by the annihilation operator a ^ is split between two remote parties, the application of the photon annihilation operator a ^ 1 by one of these parties affects the state | ψ globally. This can be verified by recombining the modes a ^ 1 and a ^ 2 on another beam splitter and analyzing the state in the output. (b) Implementation with a cloud of absorptive atoms. The detection of a re-emitted photon heralds a photon annihilation event and triggers the recording of the image on a CCD camera. Photon subtraction will not cast a shadow on the resulting quantum state, so its intensity distribution (solid orange line) does not change. This contrasts with regular linear absorption, which would cause a local shadow to appear in the intensity distribution (dashed blue line).

Fig. 2.
Fig. 2.

The experimental setup. Mode a ^ is prepared in the signal output of parametric downconversion in either the one- or two-photon Fock state. The wave plates form a Mach–Zehnder interferometer in the polarization basis, with the modes a ^ 1 and a ^ 2 being its horizontally and vertically polarized arms, respectively. Photon is subtracted from mode a ^ 1 on the PPBS. Its improvised realization is shown in the inset, with arrows and numbers indicating the polarizations of modes and their normalized intensities. The PPBS transmission, which is a compromise between the count rate and the fidelity of subtraction, can be tuned by tilting the mirror. BHD, balanced homodyne detection [13]; LO, local oscillator; SPCM, single-photon counting module.

Fig. 3.
Fig. 3.

Experimental results for the initial state of mode a ^ being (a) the one-photon and (b) the two-photon Fock state. The left column corresponds to the case without photon annihilation, whereas the right column shows the result of the photon annihilation operator applied to mode a ^ 1 (i.e., conditioned on SPCM 3 events). In each panel, the experimentally observed quadrature distributions and reconstructed diagonal elements of the density matrix are displayed together with those expected theoretically (see Supplement 1 for calculation details). The dashed lines in the right column show, for comparison, the results that would be observed if mode a ^ 1 were completely absorbed rather than subjected to photon annihilation. The error bars for the experimentally reconstructed density matrices are obtained via bootstrapping.

Equations (1)

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a ^ 1 | ψ a ^ = ( μ * a ^ + λ a ^ ) | ψ a ^ = μ * a ^ | ψ a ^ ,

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