Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics

C. J. Villas-Boas; F. O. Prado; M. H. Y. Moussa; N. G. de Almeida

REVIEW

Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics

F. O. Prado, N. G. de Almeida, M. H. Y. Moussa, C. J. Villas-Boas

Quantum Physics · Submitted 20 Feb 2006 · 20 references

Pith's reading: A two-level atom with classical drives effectively produces beam-splitter, down-conversion, and squeezing Hamiltonians in a two-mode cavity, with numerical support for the squeezing case.

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arXiv IDquant-ph/0602165 (version 1)

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CitationC. J. Villas-Boas, F. O. Prado, M. H. Y. Moussa, N. G. de Almeida. "Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics." arXiv:quant-ph/0602165 (2006). https://pith.science/paper/GHJYS2LQ

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Claim paper

The paper

The paper claims

A two-level atom with classical drives effectively produces beam-splitter, down-conversion, and squeezing Hamiltonians in a two-mode cavity, with numerical support for the squeezing case.

The reading

Cavities can trap light in small spaces, and atoms inside them can exchange energy with the light. This paper describes a way to make two different light modes inside a cavity interact with each other in specific useful ways, using just one atom that has two energy levels. Previous methods needed atoms with three levels, which are harder to control and have shorter lifetimes. The authors show mathematically that with a carefully tuned two-level atom and two classical laser fields, the effective interaction between the cavity modes can become a beam-splitter type exchange (ab† + h.c.), a two-photon down-conversion (ab + h.c.), or a single-mode squeezing term (a^2 + h.c.). These interactions are useful for creating entangled and squeezed states of light, which are resources for quantum information.

The derivation uses a standard technique where fast oscillating terms are averaged out, leaving a slow effective Hamiltonian. The authors check the squeezing case numerically, solving the full time-dependent Schrödinger equation and the master equation with dissipation. They find that the squeezing produced by the full model matches the effective model to within about one percent, and that including decay of the cavity and the atom still gives 80.5 percent squeezing.

The paper does not provide numerical checks for the two bilinear interactions, and the analytic derivation skips some steps. Still, the squeezing test gives confidence that the method works. If the scheme is practical, it could simplify experiments that need to generate non-classical light states in cavities.

Extended reading notes

Core claim

The central claim is that a single driven two-level atom interacting with two cavity modes can generate the bilinear Hamiltonians (ab + h.c.) and (ab† + h.c.), and the quadratic squeezing Hamiltonian χ(a^2 + h.c.), with coupling strengths comparable to prior three-level schemes. The paper states: 'A two-level atom is able to generate both interactions, (6) and (8), with coupling strengths comparable to those obtained in [9]' and gives Eq. (11) H(↑↓) = ∓χ(e^{-2iφ1} a^2 + h.c.).

Load-bearing premise

The results depend on the second-order effective Hamiltonian approximation H = -i V(t) ∫ V(t') dt' (Eq. 4), which requires that rapidly oscillating terms vanish and that the atom remains in the chosen dressed state. The rapid-oscillation conditions are |2Ω1 ± (δα - δ1)| ≫ |~λα| (Sec. II, after Eq. 3) and |δa| ≪ |Ω2| (Sec. III, for the squeezing derivation). If these conditions are not met, the engineered interactions do not emerge. The numerical check only covers the squeezing case for r ≤ 1, leaving the bilinear Hamiltonians without direct numerical validation.

Original abstract

In this work we show how to engineer bilinear and quadratic Hamiltonians in cavity quantum electrodynamics (QED) through the interaction of a single driven two-level atom with cavity modes. The validity of the engineered Hamiltonians is numerically analyzed even considering the effects of both dissipative mechanisms, the cavity field and the atom. The present scheme can be used, in both optical and microwave regimes, for quantum state preparation, the implementation of quantum logical operations, and fundamental tests of quantum theory.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,982 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.37 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this work we show how to engineer bilinear and quadratic Hamiltonians in cavity quantum electrodynamics (QED) through the interaction of a single driven two-level atom with cavity modes. The validity of the engineered Hamiltonians is numerically analyzed even considering the effects of both dissipative mechanisms, the cavity field and the atom. The present scheme can be used, in both optical and microwave regimes, for quantum state preparation, the implementation of quantum logical operations, and fundamental tests of quantum theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work we show how to engineer bilinear and quadratic Hamiltonians in cavity quantum electrodynamics (QED) through the interaction of a single driven two-level atom with cavity modes.	contribution 0.99	0.92	0.63	asserted 1.00
s1	The validity of the engineered Hamiltonians is numerically analyzed even considering the effects of both dissipative mechanisms, the cavity field and the atom.	method 0.60	0.92	0.08	asserted 0.99
s2	The present scheme can be used, in both optical and microwave regimes, for quantum state preparation, the implementation of quantum logical operations, and fundamental tests of quantum theory.	interpretation 0.82	0.32	0.31	hedged possibility 0.68

What kind of paper the classifier reports this is: method or tool 0.86 · evidence from simulation or computation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Energy diagrams underlying the bilinear Hamiltonians (a)

Figures reproduced from arXiv:quant-ph/0602165 by the authors.

Human review

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References

Works cited (20)

[1]

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974)

View on Pith 1974
[2]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995)

View on Pith 1995
[3]

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977)

View on Pith 1977
[4]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978); D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994)

View on Pith 1978
[5]

J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979); C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980)

View on Pith 1979
[6]

E. Knill, R. Laﬂamme, G. J. Milburn, Nature 409, 46 (2001); T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. Lett. 88, 257902 (2002)

View on Pith 2001
[7]

C. J. Villas-Boas, N. G. de Almeida, R. M. Serra, and M. H. Y . Moussa, Phys. Rev. A 68, 061801(R) (2003)

View on Pith 2003
[8]

R. M. Serra, C. J. Villas-Boas, N. G. de Almeida, and M. H. Y . Moussa, Phys. Rev. A 71, 045802 (2005)

View on Pith 2005
[9]

C. J. Villas-Boas and M. H. Y.Moussa, European Physical J ournal D 32,147 (2005)

View on Pith 2005
[10]

R. Guzm´ an, J. C. Retamal, E. Solano,and N. Zagury, Phys . Rev. Lett. 96, 010502 (2006); quant-ph/0407218

View on Pith Pith review arXiv 2006
[11]

E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett . 90, 027903 (2003). See also S. B. Zheng, Phys. Rev. A 66, 060303(R) (2002)

View on Pith 2003
[12]

D. F. V. James, Fortschr. Phys. 48, 823 (2000)

View on Pith 2000
[13]

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)

View on Pith 1969
[14]

C. J. Villas-Boas, F. R. de Paula, R. M. Serra, and M. H. Y. Moussa, Phys. Rev. A 68, 053808 (2003); J. Opt. B: Quantum Semiclass. Opt. 5, 391 (2003)

View on Pith 2003
[15]

C. J. Villas-Boas, N. G. de Almeida, R. M. Serra, and M. H. Y. Moussa, Phys. Rev. A 68, 061801(R) (2003)

View on Pith 2003
[16]

N. G. de Almeida, R. M. Serra, C. J. Villas-Boas, and M. H. Y. Moussa, Phys. Rev. A 69, 035802 (2004). 10

View on Pith 2004
[17]

P. Tombesi and D. Vitali, Phys. Rev. A 50, 4253 (1994)

View on Pith 1994
[18]

F. O. Prado, M. H. Y. Moussa, and C. J. Villas-Bˆ oas, to ap pear elsewhere

View on Pith
[19]

M. F. Santos, E. Solano, and R. L. de Matos Filho, Phys. Re v. Lett. 87, 093601 (2001)

View on Pith 2001
[20]

A. Rauschenbeutel, P. Bertet, S. Osnaghi, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 64, 050301(R) (2001); B. T. H. Varcoe, S. Brattke, M. Weidinger , and H. Walther, Nature 403, 743 (2000). Fig. 1. Energy diagrams underlying the bilinear Hamiltonians (a) ab + h .c. and (b) ab† + h.c. Fig. 2. Energy diagram of the scheme used to obta...

View on Pith 2001

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974)

View on Pith 1974

[2] [2]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995)

View on Pith 1995

[3] [3]

H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977)

View on Pith 1977

[4] [4]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978); D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994)

View on Pith 1978

[5] [5]

J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979); C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980)

View on Pith 1979

[6] [6]

E. Knill, R. Laﬂamme, G. J. Milburn, Nature 409, 46 (2001); T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. Lett. 88, 257902 (2002)

View on Pith 2001

[7] [7]

C. J. Villas-Boas, N. G. de Almeida, R. M. Serra, and M. H. Y . Moussa, Phys. Rev. A 68, 061801(R) (2003)

View on Pith 2003

[8] [8]

R. M. Serra, C. J. Villas-Boas, N. G. de Almeida, and M. H. Y . Moussa, Phys. Rev. A 71, 045802 (2005)

View on Pith 2005

[9] [9]

C. J. Villas-Boas and M. H. Y.Moussa, European Physical J ournal D 32,147 (2005)

View on Pith 2005

[10] [10]

R. Guzm´ an, J. C. Retamal, E. Solano,and N. Zagury, Phys . Rev. Lett. 96, 010502 (2006); quant-ph/0407218

View on Pith Pith review arXiv 2006

[11] [11]

E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett . 90, 027903 (2003). See also S. B. Zheng, Phys. Rev. A 66, 060303(R) (2002)

View on Pith 2003

[12] [12]

D. F. V. James, Fortschr. Phys. 48, 823 (2000)

View on Pith 2000

[13] [13]

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)

View on Pith 1969

[14] [14]

C. J. Villas-Boas, F. R. de Paula, R. M. Serra, and M. H. Y. Moussa, Phys. Rev. A 68, 053808 (2003); J. Opt. B: Quantum Semiclass. Opt. 5, 391 (2003)

View on Pith 2003

[15] [15]

C. J. Villas-Boas, N. G. de Almeida, R. M. Serra, and M. H. Y. Moussa, Phys. Rev. A 68, 061801(R) (2003)

View on Pith 2003

[16] [16]

N. G. de Almeida, R. M. Serra, C. J. Villas-Boas, and M. H. Y. Moussa, Phys. Rev. A 69, 035802 (2004). 10

View on Pith 2004

[17] [17]

P. Tombesi and D. Vitali, Phys. Rev. A 50, 4253 (1994)

View on Pith 1994

[18] [18]

F. O. Prado, M. H. Y. Moussa, and C. J. Villas-Bˆ oas, to ap pear elsewhere

View on Pith

[19] [19]

M. F. Santos, E. Solano, and R. L. de Matos Filho, Phys. Re v. Lett. 87, 093601 (2001)

View on Pith 2001

[20] [20]

A. Rauschenbeutel, P. Bertet, S. Osnaghi, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A 64, 050301(R) (2001); B. T. H. Varcoe, S. Brattke, M. Weidinger , and H. Walther, Nature 403, 743 (2000). Fig. 1. Energy diagrams underlying the bilinear Hamiltonians (a) ab + h .c. and (b) ab† + h.c. Fig. 2. Energy diagram of the scheme used to obta...

View on Pith 2001