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.

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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. Wodkiewicz, "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. Wodkiewicz, "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. Buzek 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]

Other

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

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]

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

K. Banaszek and K. Wodkiewicz, "Direct sampling of quantum phase space by photon counting," Phys. Rev. Lett. 76, 4344-4347 (1996).
[CrossRef] [PubMed]

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

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]

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. Wodkiewicz, "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]

V. Buzek 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)

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̂ = ( 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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