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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 4 — Aug. 17, 1998
  • pp: 141–146
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Sampling quantum phase space with squeezed states

Konrad Banaszek and Krzysztof Wódkiewicz  »View Author Affiliations


Optics Express, Vol. 3, Issue 4, pp. 141-146 (1998)
http://dx.doi.org/10.1364/OE.3.000141


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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.

© Optical Society of America

1. Introduction

Phase space quasidistribution functions are a convenient way of characterizing the quantum state of optical radiation [1

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

]. Over past several years, they have gained experimental significance due to the reconstruction of the Wigner function of a single light mode performed using tomographic algorithms [2

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]

]. Recently, an alternative method for measuring quasidistribution functions of a light mode has been proposed [3

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

,4

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

]. The method is based on photon counting of the signal field superposed on a probe field in a coherent state. The advantage of this method is that there is no complicated numerical processing of the experimental data. A simple arithmetic operation performed on the photocount statistics yields directly the value of a quasidistribution function at a point defined by the amplitude and the phase of the coherent field.

The purpose of this communication is to study the application of squeezed states in the proposed photon counting scheme. The most important feature of squeezed states is that quantum fluctuations in some observables are reduced below the coherent state level [5

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

]. In the context of optical homodyne tomography, the squeezing transformation has been shown to be capable of compensating for the deleterious effect of low detection efficiency [6

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]

]. Therefore, it is interesting to discuss the information on the quantum state of light which can be retrieved in a photon counting experiment using squeezed states. The advantages of using the squeezing transformation in balanced homodyne detection have been discussed in Ref. [7

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

].

2. Experimental scheme

We start with a brief description of the proposed setup, depicted in Fig. 1. The field incident on a photodetector is a combination, performed using a beam splitter with a power transmission T, of a transmitted signal mode and a reflected probe mode. The statistics of the detector counts {pn } is used to calculate an alternating series Σn=0(-1)n pn . In terms of the outgoing mode, this series is given by the expectation value of the parity operator:

̂=(1)âoutâout,
(1)

where the annihilation operator of the outgoing mode âout is a linear combination of the signal and the probe field operators:

âout=TâS1Tâp.
(2)

The expectation value of the measured observable involves statistical properties of both the signal and the probe modes. The operator ∏̂ can be written in the following normally ordered form:

̂=:exp[2(TâS1Tâp)(TâS1Tâp)]:,
(3)

which has a clear and intuitive interpretation within the Wigner function formalism: the measured quantity is proportional to the phase space integral of the product of the signal and the probe Wigner functions with relatively rescaled parameterizations [4

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

]. Hence the proposed scheme is a realization of direct sampling of the quantum phase space.

An important class of probe fields are coherent states âp |αp = α|αp. The quantum expectation value over the probe mode can be easily evaluated in this case using the normally ordered form given in Eq. (3). Thus the measured observable is given by the following operator acting in the Hilbert space of the signal mode:

α̂αP=:exp[2(TâS1Tα*)(TâS1Tα)]:.
(4)

Û(β,s)=2π(1s):exp[21s(âSβ*)(âSβ)]:.
(5)

After a simple rearrangement of parameters we finally arrive at the formula:

α̂αP=π2TÛ(1TTα;1TT).
(6)

Thus, the alternating series computed from the photocount statistics yields the value of a quasidistribution function at a point (1T)/Tα defined by the amplitude and the phase of the probe coherent field. The complete quasidistribution function can be scanned point-by-point by changing the probe field parameters.

The ordering of the measured quasidistribution function depends on the beam splitter transmission. This is a consequence of the fact that a fraction of the signal field escapes through the second unused output port of the beam splitter. These losses of the field lower the ordering of the detected observable. This effect is analogous to the one appearing in balanced homodyne detection with imperfect detectors [8

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

,9

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

]. In the limit T → 1, when the complete signal field is detected, we measure directly the Wigner function, corresponding to the symmetric ordering.

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.

3. Sampling with squeezed states

We will now consider the case when a squeezed coherent state Sp (r,φ)|αp enters through the probe port of the beam splitter. We use the following definition of the squeezing operator for an ith mode:

Si(r,φ)=exp[r(eâi2e(âi)2)/2].
(7)

The detected quantity is now given by the expectation value of the following operator acting in the Hilbert space of the signal mode:

̂P=αŜp(r,φ)̂Ŝp(r,φ)αP.
(8)

In order to find an interpretation for this observable, we will derive a formula for the squeezing transformations of the parity operator ∏̂. We start from a simple unitary transformation:

(1)âoutâoutâout2(1)âoutâout=eiπâoutâoutâout2eiπâoutâout=e2πiâout2=âout2.
(9)

This equation implies the commutator:

[(1)âoutâout,e(âout)2eâout2]=0,
(10)

which states that generation or annihilation of pairs of photons conserves parity. Therefore, the parity operator is invariant under the squeezing transformation:

Ŝout(r,φ)̂Ŝout(r,φ)=.̂
(11)

This identity has nontrivial consequences when written in terms of the signal and the probe modes. It is equivalent to the equation:

ŜS(r,φ)ŜP(r,φ)̂ŜP(r,φ)ŜS(r,φ)=̂
(12)

which, after moving the signal squeezing operators to the right hand side, yields the following result:

ŜP(r,φ)̂ŜP(r,φ)=ŜS(r,φ)̂ŜS(r,φ)
(13)

This formula shows that squeezing of the probe mode is equivalent to squeezing of the signal mode with the opposite sign of the parameter r. This change of the sign swaps the field quadratures that get squeezed or antisqueezed under the squeezing transformation.

Finally we obtain the following explicit expression for the detected signal field observable:

̂P=ŜS(r,φ)α̂αPŜS(r,φ)
=π2TŜS(r,φ)Û(1TTα;1TT)ŜS(r,φ).
(14)

Thus, the setup delivers again an s = -(1 - T)/T-ordered quasidistribution function at a phase space point (1T)/T, but corresponding to a squeezed signal field.

Let us note that it was possible to carry the squeezing transformation from the probe to the signal degree of freedom only due to a specific form of the measured observable. We have explicitly used the conservation of the parity operator during generation or annihilation of pairs of photons. For a general observable defined for the outgoing mode aout, there is no formula analogous to Eq. (13).

4. Detection of SchrÖdinger cat state

As an illustration, we will consider a photon counting experiment for a SchrÖdinger cat state, which is a quantum superposition of two coherent states [10

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

]:

ψ=+2+2exp(2κ2),
(15)

where κ is a real parameter. The Wigner function of such a state contains, in addition to two positive peaks corresponding to the coherent states, an oscillating term originating from quantum interference between the classical-like components. This nonclassical feature is extremely fragile, and disappears very quickly in the presence of dissipation [11

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]

].

ψŜS(r,0)Û(q+ip;s)ŜS(r,0)ψ
=exp(2q2e2rs)π[1+exp(2κ2)]12scosh2r+s2{exp[2(perκ)2e2rs]
+exp[2(p+erκ)2e2rs]+2exp(2sκ2e2rs2p2e2rs)cos(4erκqe2rs)}.
(16)

In Fig. 2 we depict the expectation value ofthe parity operator 〈∏̂〉 as a function of the rescaled complex probe field amplitude β=(1T)/. For comparison, we show two cases: when the SchrÖdigner cat state is probed with coherent states |αp and squeezed coherent states ŜP (r = 1,0)|αp . The beam splitter transmission is T = 80%. When coherent states are used, only a faint trace of the oscillatory pattern can be noticed due to losses of the signal field. In contrast, probing of the SchrÖdinger cat state with suitably chosen squeezed states yields a clear picture of quantum coherence between distinct phase space components. This effect is particularly surprising if we realize that 20% of the signal field power is lost through the unused output port of the beam splitter.

The visibility of the oscillatory pattern depends substantially on the sign of the squeezing parameter r. This can be most easily understood using the Wigner phase space description of the discussed scheme [4

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

]. In order to detect the interference, fluctuations in the probe squeezed states have to be reduced in the direction corresponding to the rapid oscillations of the Wigner function corresponding to the Schrodinger cat state. The width of the rescaled probe Wigner function along the squeezed direction must be smaller than the spacing between the interference fringes.

5. Conclusions

We have studied the quantum optical scheme for direct sampling of the quantum phase space using squeezed coherent states. We have shown that squeezing transformations performed on the signal and the probe input ports of the setup are equivalent. The application of squeezed states with the appropriately chosen squeezing direction allows one to detect quantum interference despite losses through the unused output port of the setup.

Fig. 2. Sampling the SchrÖdigner cat state |ψ;〉 ∝ |3i〉 + | - 3i〉 with: (a) coherent states |αp and (b) squeezed states Ŝp (r = l,0)|αp. The plots show the expectation value of the parity operator 〈∏̂〉 as a function of the rescaled complex probe field amplitude β=(1T)/. The beam splitter transmission is T = 80%.

Acknowledgements

This work has been partially supported by the Polish KBN grants 2P03B 006 11 and 2P03B 002 14. K.B. would like to acknowledge fruitful discussions with E. Czuchry.

References

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]

OCIS Codes
(270.5570) Quantum optics : Quantum detectors
(270.6570) Quantum optics : Squeezed states

ToC Category:
Focus Issue: Quantum noise reduction in optical systems

History
Original Manuscript: March 27, 1998
Published: August 17, 1998

Citation
Konrad Banaszek and Krzysztof Wodkiewicz, "Sampling quantum phase space with squeezed states," Opt. Express 3, 141-146 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-4-141


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References

  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]

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