Abstract

We study the application of squeezed states in a quantum optical scheme for direct sampling of the phase space by photon counting. We prove that the detection setup with a squeezed coherent probe field is equivalent to the probing of the squeezed signal field with a coherent state. An example of the SchrÖdinger cat state measurement shows that the use of squeezed states allows one to detect clearly the interference between distinct phase space components despite losses through the unused output port of the setup.

© 1998 Optical Society of America

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References

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  1. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
    [CrossRef]
  2. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef] [PubMed]
  3. S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
    [CrossRef] [PubMed]
  4. K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
    [CrossRef] [PubMed]
  5. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
    [CrossRef]
  6. U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
    [CrossRef] [PubMed]
  7. M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A 53, 3694–3697 (1996).
    [CrossRef] [PubMed]
  8. U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A 48, 4598–4604 (1993).
    [CrossRef] [PubMed]
  9. K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
    [CrossRef]
  10. W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
    [CrossRef] [PubMed]
  11. V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, ed. by E. Wolf (north-Holland, Amsterdam, 1995), 1–158.
    [CrossRef]

1997 (1)

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[CrossRef]

1996 (3)

M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A 53, 3694–3697 (1996).
[CrossRef] [PubMed]

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
[CrossRef] [PubMed]

1994 (1)

U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
[CrossRef] [PubMed]

1993 (2)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A 48, 4598–4604 (1993).
[CrossRef] [PubMed]

1991 (1)

W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
[CrossRef] [PubMed]

1987 (1)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

1969 (1)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Banaszek, K.

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[CrossRef]

K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
[CrossRef] [PubMed]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Bužek, V.

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, ed. by E. Wolf (north-Holland, Amsterdam, 1995), 1–158.
[CrossRef]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Glauber, R. J.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Kim, M. S.

M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A 53, 3694–3697 (1996).
[CrossRef] [PubMed]

Knight, P. L.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, ed. by E. Wolf (north-Holland, Amsterdam, 1995), 1–158.
[CrossRef]

LeKien, F.

W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A 48, 4598–4604 (1993).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

Paul, H.

U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A 48, 4598–4604 (1993).
[CrossRef] [PubMed]

Pernigo, M.

W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
[CrossRef] [PubMed]

Raymer, M. G.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Sanders, B. C.

M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A 53, 3694–3697 (1996).
[CrossRef] [PubMed]

Schleich, W.

W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
[CrossRef] [PubMed]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Vogel, W.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef] [PubMed]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef] [PubMed]

Wódkiewicz, K.

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[CrossRef]

K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

Phys. Rev. (1)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Phys. Rev. A (5)

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[CrossRef] [PubMed]

M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A 53, 3694–3697 (1996).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A 48, 4598–4604 (1993).
[CrossRef] [PubMed]

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[CrossRef]

W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172–2187 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
[CrossRef] [PubMed]

Other (1)

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics XXXIV, ed. by E. Wolf (north-Holland, Amsterdam, 1995), 1–158.
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

The setup for direct probing of the quantum phase space. The detector measures the photocount statistics {pn } of a signal âs combined with a probe field âp using a beam splitter with a power transmission T.

Fig. 2.
Fig. 2.

Sampling the SchrÖdigner cat state |ψ;〉 ∝ |3i〉 + | - 3i〉 with: (a) coherent states |α p and (b) squeezed states Ŝp (r = l,0)|α p . The plots show the expectation value of the parity operator 〈∏̂〉 as a function of the rescaled complex probe field amplitude β = ( 1 T ) / . The beam splitter transmission is T = 80%.

Equations (19)

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

̂ = ( 1 ) a ̂ out a ̂ out ,
a ̂ out = T a ̂ S 1 T a ̂ p .
̂ = : exp [ 2 ( T a ̂ S 1 T a ̂ p ) ( T a ̂ S 1 T a ̂ p ) ] : ,
α ̂ α P = : exp [ 2 ( T a ̂ S 1 T α * ) ( T a ̂ S 1 T α ) ] : .
U ̂ ( β , s ) = 2 π ( 1 s ) : exp [ 2 1 s ( a ̂ S β * ) ( a ̂ S β ) ] : .
α ̂ α P = π 2 T U ̂ ( 1 T T α ; 1 T T ) .
S i ( r , φ ) = exp [ r ( e a ̂ i 2 e ( a ̂ i ) 2 ) / 2 ] .
̂ P = α S ̂ p ( r , φ ) ̂ S ̂ p ( r , φ ) α P .
( 1 ) a ̂ out a ̂ out a ̂ out 2 ( 1 ) a ̂ out a ̂ out = e i π a ̂ out a ̂ out a ̂ out 2 e i π a ̂ out a ̂ out = e 2 πi a ̂ out 2 = a ̂ out 2 .
[ ( 1 ) a ̂ out a ̂ out , e ( a ̂ out ) 2 e a ̂ out 2 ] = 0 ,
S ̂ out ( r , φ ) ̂ S ̂ out ( r , φ ) = . ̂
S ̂ S ( r , φ ) S ̂ P ( r , φ ) ̂ S ̂ P ( r , φ ) S ̂ S ( r , φ ) = ̂
S ̂ P ( r , φ ) ̂ S ̂ P ( r , φ ) = S ̂ S ( r , φ ) ̂ S ̂ S ( r , φ )
̂ P = S ̂ S ( r , φ ) α ̂ α P S ̂ S ( r , φ )
= π 2 T S ̂ S ( r , φ ) U ̂ ( 1 T T α ; 1 T T ) S ̂ S ( r , φ ) .
ψ = + 2 + 2 exp ( 2 κ 2 ) ,
ψ S ̂ S ( r , 0 ) U ̂ ( q + ip ; s ) S ̂ S ( r , 0 ) ψ
= exp ( 2 q 2 e 2 r s ) π [ 1 + exp ( 2 κ 2 ) ] 1 2 s cosh 2 r + s 2 { exp [ 2 ( p e r κ ) 2 e 2 r s ]
+ exp [ 2 ( p + e r κ ) 2 e 2 r s ] + 2 exp ( 2 s κ 2 e 2 r s 2 p 2 e 2 r s ) cos ( 4 e r κq e 2 r s ) } .

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