Abstract

We present the generation of approximated coherent state superpositions—referred to as Schrödinger cat states—by the process of subtracting single photons from picosecond pulsed squeezed states of light. The squeezed vacuum states are produced by spontaneous parametric down-conversion (SPDC) in a periodically poled KTiOPO4 crystal while the single photons are probabilistically subtracted using a beamsplitter and a single photon detector. The resulting states are fully characterized with time-resolved homodyne quantum state tomography. Varying the pump power of the SPDC, we generated different states which exhibit non-Gaussian behavior.

© 2014 Optical Society of America

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2013

M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5309–5317 (2013).
[CrossRef]

J. B. Brask, R. Chaves, and N. Brunner, “Testing nonlocality of a single photon without a shared reference frame,” Phys. Rev. A 88, 012111 (2013).
[CrossRef]

A. Laghaout, J. S. Neergaard-Nielsen, I. Rigas, C. Kragh, A. Tipsmark, and U. L. Andersen, “Amplification of realistic Schrödinger-cat-state-like states by homodyne heralding,” Phys. Rev. A 87, 043826 (2013).
[CrossRef]

2012

R. Blandino, F. Ferreyrol, M. Barbieri, P. Grangier, and R. Tualle-Brouri, “Characterization of a π-phase shift quantum gate for coherent-state qubits,” New J. Phys. 14, 013017 (2012).
[CrossRef]

C. Weedbrook, S. Pirandola, R. Garca-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[CrossRef]

A. Mandilara and N. J. Cerf, “Quantum bit commitment under Gaussian constraints,” Phys. Rev. A 85, 062310 (2012).
[CrossRef]

M. Ježek, A. Tipsmark, R. Dong, J. Fiurášek, L. Mišta, R. Filip, and U. L. Andersen, “Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,” Phys. Rev. A 86, 043813 (2012).
[CrossRef]

2011

T. C. Ralph, “Quantum error correction of continuous-variable states against Gaussian noise,” Phys. Rev. A 84, 022309 (2011).
[CrossRef]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[CrossRef]

A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Ježek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A 84, 050301 (2011).
[CrossRef]

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–333 (2011).
[CrossRef]

2010

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent state superpositions by number-resolved photon subtraction from squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[CrossRef]

N. Namekata, Y. Takahashi, G. Fujii, D. Fukuda, S. Kurimura, and S. Inoue, “Non-Gaussian operation based on photon subtraction using a photon-number-resolving detector at a telecommunications wavelength,” Nat. Photonics 4, 655–660 (2010).
[CrossRef]

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Optical continuous-variable qubit,” Phys. Rev. Lett. 105, 053602 (2010).
[CrossRef]

P. Marek and J. Fiurášek, “Elementary gates for quantum information with superposed coherent states,” Phys. Rev. A 82, 014304 (2010).
[CrossRef]

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[CrossRef]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

M. Lassen, M. Sabuncu, A. Huck, J. Niset, G. Leuchs, N. J. Cerf, and U. L. Andersen, “Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code,” Nat. Photonics 4, 700–705 (2010).
[CrossRef]

M. Ohliger, K. Kieling, and J. Eisert, “Limitations of quantum computing with Gaussian cluster states,” Phys. Rev. A 82, 042336 (2010).
[CrossRef]

L. Magnin, F. Magniez, A. Leverrier, and N. J. Cerf, “Strong no-go theorem for Gaussian quantum bit commitment,” Phys. Rev. A 81, 010302 (2010).
[CrossRef]

2009

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef]

A. I. Lvovsky, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

R. Tualle-Brouri, A. Ourjoumtsev, A. Dantan, P. Grangier, M. Wubs, and A. Sørensen, “Multimode model for projective photon-counting measurements,” Phys. Rev. A 80, 013806 (2009).
[CrossRef]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

2008

P. Marek and M. S. Kim, “Suitability of the approximate superposition of squeezed coherent states for various quantum protocols,” Phys. Rev. A 78, 022309 (2008).
[CrossRef]

H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
[CrossRef]

P. Marek, H. Jeong, and M. S. Kim, “Generating “squeezed” superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef]

R. Dong, J. Heersink, J. F. Corney, P. D. Drummond, U. L. Andersen, and G. Leuchs, “Experimental evidence for Raman-induced limits to efficient squeezing in optical fibers,” Opt. Lett. 33, 116–118 (2008).
[CrossRef]

2007

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of −9  dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express 15, 4321–4327 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, and A. Dantan, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical “Schrödinger cats” from photon number stataes,” Nature 448, 784–786 (2007).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

2006

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

N. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef]

S. Scheel, W. Munro, J. Eisert, K. Nemoto, and P. Kok, “Feed-forward and its role in conditional linear optical quantum dynamics,” Phys. Rev. A 73, 034301 (2006).
[CrossRef]

S. Suzuki, K. Tsujino, F. Kannari, and M. Sasaki, “Analysis on generation schemes of Schrödinger cat-like states under experimental imperfections,” Opt. Commun. 259, 758–764 (2006).
[CrossRef]

2005

U. L. Andersen, V. Josse, and G. Leuchs, “Unconditional quantum cloning of coherent states with linear optics,” Phys. Rev. Lett. 94, 240503 (2005).
[CrossRef]

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T. Hirano, K. Kotani, T. Ishibashi, S. Okude, and T. Kuwamoto, “3  dB squeezing by single-pass parametric amplification in a periodically poled KTiOPO4 crystal,” Opt. Lett. 30, 1722–1724 (2005).
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2004

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2003

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H. Jeong, W. Son, M. S. Kim, D. Ahn, and C. Brukner, “Quantum nonlocality test continuous-variable states with dichotomic observable,” Phys. Rev. A 67, 012106 (2003).
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2002

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2001

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2000

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1994

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1968

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T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
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S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
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J. B. Brask, R. Chaves, and N. Brunner, “Testing nonlocality of a single photon without a shared reference frame,” Phys. Rev. A 88, 012111 (2013).
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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent state superpositions by number-resolved photon subtraction from squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D. G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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M. Ježek, A. Tipsmark, R. Dong, J. Fiurášek, L. Mišta, R. Filip, and U. L. Andersen, “Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,” Phys. Rev. A 86, 043813 (2012).
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A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Ježek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A 84, 050301 (2011).
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R. Dong, J. Heersink, J. F. Corney, P. D. Drummond, U. L. Andersen, and G. Leuchs, “Experimental evidence for Raman-induced limits to efficient squeezing in optical fibers,” Opt. Lett. 33, 116–118 (2008).
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R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
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R. Blandino, F. Ferreyrol, M. Barbieri, P. Grangier, and R. Tualle-Brouri, “Characterization of a π-phase shift quantum gate for coherent-state qubits,” New J. Phys. 14, 013017 (2012).
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M. Ježek, A. Tipsmark, R. Dong, J. Fiurášek, L. Mišta, R. Filip, and U. L. Andersen, “Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,” Phys. Rev. A 86, 043813 (2012).
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Y. Miwa, J. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Unconditional conversion between a single-photon state and a coherent-state superposition via squeezing operation,” Quantum Electronics and Laser Science Conference, San Jose, California, United States, May6–11, 2012.

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M. Ježek, A. Tipsmark, R. Dong, J. Fiurášek, L. Mišta, R. Filip, and U. L. Andersen, “Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,” Phys. Rev. A 86, 043813 (2012).
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P. Marek and J. Fiurášek, “Elementary gates for quantum information with superposed coherent states,” Phys. Rev. A 82, 014304 (2010).
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J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
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R. Garca-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
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M. Ježek, J. Fiurášek, and Z. Hradil, “Quantum inference of states and processes,” Phys. Rev. A 68, 012305 (2003).
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J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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A. Furusawa, S. L. Braunstein, J. L. Sørensen, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
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N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–333 (2011).
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R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
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[CrossRef]

Y. Miwa, J. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Unconditional conversion between a single-photon state and a coherent-state superposition via squeezing operation,” Quantum Electronics and Laser Science Conference, San Jose, California, United States, May6–11, 2012.

Garca-Patrón, R.

C. Weedbrook, S. Pirandola, R. Garca-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
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R. Garca-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent state superpositions by number-resolved photon subtraction from squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Fidelity between a photon-subtracted squeezed state and an ideal coherent state superposition, |α|α, for different degrees of squeezing. It can be seen that the fidelity remains high, F>0.9, for α up to 1, provided the squeezing degree is not too large.

Fig. 2.
Fig. 2.

Schematic of the experimental setup. The cavity dumped laser emits 4.6 ps optical pulses at 830 nm with a repetition frequency of 815 kHz. ABS1, 90/10 beamsplitter; BF, blue filter; DM, dichroic mirror; RF, red filter; ABS2, asymmetric beamsplitter with R=7.7%; SMF, single-mode fiber; IF, interference filter; APD, avalanche photodiode; PBS, polarizing beamsplitter; HWP, half-wave plate.

Fig. 3.
Fig. 3.

(a) Blue diamonds correspond to the second-harmonic generation conversion efficiency for different input powers of the fundamental beam, PF. These points are fitted with η=ηtanh2(gPF), with η=0.53 and g=0.18. Red triangles are the total generated SHG power for different power levels of the fundamental beam.

Fig. 4.
Fig. 4.

Classical parametric gain versus average pump power, Pp. The squares correspond to the measured deamplification values, whereas the diamonds are the measured amplification values. The red and orange lines are fits using Eq. (1), with the intrinsic parametric gain r=0.28Pp(mW) and ε=0.77±0.01.

Fig. 5.
Fig. 5.

(a) Quadrature variance measurement of the shot noise as a function of the LO power. (b) SNR of the shot noise variance as a function of the LO power. ([mio] stands for “million.”)

Fig. 6.
Fig. 6.

Measured squeezing and antisqueezing variances as a function of the pump power. The blue diamonds are the measured maximum variances (antisqueezing), the black squares are the measured minimum variances (squeezing). The orange and red lines are the fittings according to Eq. (7), where ηtot=0.62±0.01 are given.

Fig. 7.
Fig. 7.

Plot of the Wigner functions, the projected Wigner functions, the density matrices, and the photon distributions of the reconstructed states for pump powers from 2 to 8 mW. There are no corrections for losses in these plots.

Fig. 8.
Fig. 8.

Plot of the Wigner functions, the projected Wigner functions, the density matrices, and the photon distributions of the loss-corrected reconstructed states for pump powers from 2 to 8 mW.

Tables (4)

Tables Icon

Table 1. Estimation of the Parameters Characterizing the Photon-Subtracted Squeezed State for Different Pump Power Levelsa

Tables Icon

Table 2. Estimation of the Parameters Characterizing the Photon-Subtracted Squeezed State for Different Pump Power Levelsa

Tables Icon

Table 3. Parameters Characterizing the Prepared Photon-Subtracted Squeezed States

Tables Icon

Table 4. Parameters Characterizing the Prepared Photon-Subtracted Squeezed States after Correction for Imperfect Detection

Equations (7)

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

g{min,max}=εexp({+2r,2r})+(1ε),
ηhd=ηopηmm2ηphηel,
W0(X^)=ex2p2π.
Ws(X^)=ex2Vxp2VpπVxVp,
Wout(X^1)=12πW(X^1,X^2)WΛ(X^2)dX^2,
Wout,Ξ(X^)=ΞWout(X^)+(1Ξ)Ws(X^),
V{min,max}=ηtot(gmin,gmax)+(1ηtot)Vvac,

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