Wave packet dynamics of entangled two-mode states

C. Sudheesh; S. Lakshmibala; V. Balakrishnan

REVIEW

Wave packet dynamics of entangled two-mode states

C. Sudheesh, S. Lakshmibala, V. Balakrishnan

Quantum Physics · Submitted 2 Mar 2006 · 16 references

Pith's verdict: A numerically credible but under-supported extension of revival diagnostics to a non-symmetric two-mode model; the fractional-revival claim needs direct evidence.

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arXiv IDquant-ph/0603020 (version 1)

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CitationC. Sudheesh, S. Lakshmibala, V. Balakrishnan. "Wave packet dynamics of entangled two-mode states." arXiv:quant-ph/0603020 (2006). https://pith.science/paper/P3W3LKQM

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Editor's note

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Dear Colleague,

This is a numerical study of entanglement dynamics in a two-mode model where a field mode is coupled to an anharmonic oscillator, with weak nonlinearity. The genuinely new bit is the extension to a non-symmetric, non-diagonalizable Hamiltonian and to photon-added coherent states as initial states. The paper's own earlier single-mode results are extended to show that the subsystem von Neumann and linear entropies dip at near-revivals, and that increasing m or nu degrades revivals. That qualitative point is plausible and consistent with what you'd expect.

What the paper does well: the block-diagonalization by total excitation number is standard, the reduced density matrix expressions are carefully derived, and the numerical checks (Sa=Sb, purity of the total state) are the right sanity checks. The overlap fidelity is a reasonable way to quantify revival quality, and the quadrature variance plots show that squeezing appears near half-revival for coherent but not for PACS.

Where it wobbles: the fractional-revival claim. Section III asserts 'marked oscillatory behaviour' near 1/2, 1/3, 2/3, and 1/4 of Trev, but the paper shows no plot, no zoom, no quantitative criterion for a dip. The fidelity C(Trev) is evaluated only near the full revival and cannot corroborate fractional-time features. A local entropy dip is not by itself a fractional revival; you'd want to see the global state split into components or some equivalent measure. That said, the revival times themselves are imported from earlier work on the same model, so that part is not a serious circularity. The lack of documentation of truncation ranks and convergence thresholds for the infinite-dimensional density matrices is also a real gap, though the falling factorials in the PACS coefficients suggest convergence is probably fine.

Net: the central qualitative finding—entropy tracks collapse-revival and degrades with the initial state's departure from coherence—probably holds. The paper would benefit from a revision that shows the fractional-revival evidence directly and adds convergence details. It's an incremental but legitimate numerical contribution, worth refereeing rather than desk rejecting.

Best.

Referee report

Significance

If the claims are substantiated, the paper provides a useful diagnostic: for a non-symmetric, non-diagonalizable two-mode Hamiltonian, subsystem entropies and the overlap fidelity could serve as entanglement-based markers of near-revivals and fractional revivals. The explicit reduced-density-matrix expressions in the Appendix and the reported internal consistency checks (Sa = Sb, purity) are strengths, and the paper clearly identifies the model and the parameter regime. However, the central quantitative claim about fractional-revival counterparts is currently supported mainly by qualitative statements and band-like plots, and the numerical truncation procedure for the infinite-dimensional density matrices is not fully documented. These gaps prevent the paper from being accepted in its present form.

Major objections (5)

[Section III, paragraph after Fig. 2] The statement that 'the SVNE and SLE display marked oscillatory behaviour near 1/2 Trev, 1/3 Trev and 1/4 Trev' and that this constitutes counterparts of fractional revivals is not supported by quantitative evidence. No zoomed time-resolved plots, no numerical dip criterion, and no comparison with the behavior away from those times are provided. Since a local minimum of an entanglement measure can arise from interference in the reduced density matrix without any genuine wave-packet splitting, the abstract's headline claim that fractional-revival counterparts are 'clearly identifiable' is not established. Please add a quantitative characterization of the dips (for example, dip depth relative to baseline, and a threshold for identifying a dip) and, if possible, a global-state diagnostic such as the state overlap or a phase-space distribution at the fractional times.
[Appendix, after Eqs. (A30) and (A31)] The numerical procedure for the infinite-dimensional reduced density matrices is incompletely specified. The manuscript gives no truncation rank, no number of retained N-blocks, and no convergence threshold; the statement that 'rapid convergence in numerical computation is provided by the factorials in the denominators' and the use of 'double precision arithmetic with an accuracy of 1 part in 10^6' do not constitute a convergence criterion. For PACS initial states with ν = 5 and m = 5 the matrix elements decay only polynomially in N for fixed Fock indices before the factorial suppression, so the truncation is material. Please report the cutoffs and a convergence check, e.g. stability of the reported entropies to a specified number of digits as the truncation rank increases.
[Section III, Eq. (11) and Figs. 3(a)-(b)] The overlap fidelity C(T_rev) is defined as the maximum of |⟨ψ(0)|ψ(t)⟩|² in the 'vicinity' of the first near-revival time, but the size of this vicinity is never specified, which makes the quantitative values in Figs. 3(a)-(b) dependent on an undocumented window. More importantly, this fidelity is evaluated only near the first full revival, so it cannot corroborate the claimed entropy dips at 1/3, 2/3, or 1/4 of T_rev. The introduction's statement that the overlap fidelity corroborates the entropy behavior at fractional revivals is therefore unsupported. Please define the window explicitly and, if fractional-time corroboration is intended, provide fidelity values at the fractional times as well.
[Section II, text after Eq. (5), and Section III] The revival times T_rev ≃ 2π/γ (Fock initial state) and T_rev ≃ 4π/γ (coherent initial state) are imported from Refs. [9,13] and then used to locate and label the entropy dips. If these revival times carry an uncertainty or depend on parameter regimes, the identifications at 1/2, 1/3, 2/3, and 1/4 of T_rev inherit that uncertainty. Please state explicitly that T_rev is taken from the mean-photon-number revival studies of Refs. [9,13] and, ideally, verify for the specific parameters used here that the full-state overlap |⟨ψ(0)|ψ(t)⟩|² has maxima at those times.
[Section III, paragraph on higher moments of ξ and η] The manuscript claims that the higher moments of ξ and η carry signatures that selectively pin-point the analogues of the different fractional revivals, but no plot, numerical table, or quantitative example is provided. Since this is one of the stated objectives of the paper, please present at least one representative result supporting this claim, or explicitly mark it as a qualitative observation requiring further evidence.

Minor objections (5)

[Section III, first paragraph] The text refers to 'Section 1' but should refer to 'Section I'.
[Fig. 3] The curves for different photon-added indices m are difficult to distinguish, and in Fig. 3(b) the axis labels containing 'm=' appear garbled. Please use distinct line styles or a legend and ensure the labels render correctly.
[Appendix, Eq. (A31)] In Eq. (A31) the phase factor uses λ_{Ns} - λ_{N′s′}, but the summation indices are written in terms of n+l and n+l′; for clarity define N = n+l and N′ = n+l′ explicitly in the formula.
[Section III, paragraph after Fig. 4] The sentence 'The expectation value of η also displays these signatures' is stated without a corresponding figure or quantitative description; please either add the plot or state that it is not shown.
[Section III, paragraph on subsystem quadratures] The statement that the subsystem variables xa, xb, pa, pb 'do exhibit near-revivals in their expectation values' needs qualification, since for Fock initial states the odd moments of ξ and η vanish identically; please clarify which initial states and which moments are being referred to.

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The paper

The paper claims

Entanglement entropies carry clear signatures of near-revivals and fractional revivals in a non-symmetric two-mode system, and the signatures fade as the initial field state departs from coherence.

The reading

This paper asks whether wave-packet revivals, the instants when a quantum state returns close to its initial form, can be detected through entanglement rather than through the usual overlaps of wave functions. The authors study a two-mode Hamiltonian, a field mode coupled to an anharmonic 'atom' oscillator, that is not symmetric in the two modes and is not diagonalisable in general, so exact revivals are not expected and only near-revivals can occur. They claim that, under weak nonlinearity, the subsystem von Neumann entropy and the linear entropy of the field mode develop sharp dips at the near-revival time $T_{\rm rev}$ and at the fractions $1/2$, $1/3$, $2/3$ and $1/4$ of $T_{\rm rev}$, and that these dips can be regarded as entanglement-based counterparts of fractional revivals. A second indicator, the overlap fidelity $C(T_{\rm rev})$, rises toward unity only when the field-medium coupling $g$ is large relative to the nonlinearity $\gamma$, and the required coupling grows as the initial field state departs from coherence, that is, for photon-added coherent states with larger $m$ and larger mean photon number $\nu$. The result offers a practical quantitative indicator of near-revivals that works where direct overlap computations are cumbersome.

What carries the argument

The argument runs on three objects. First, the model Hamiltonian, which conserves the total number operator $N_{\rm tot} = a^\dagger a + b^\dagger b$; in the Fock basis $|N-n;n\rangle$ it decomposes into finite tridiagonal blocks, one for each $N$, which are diagonalised numerically to obtain the eigenvalues and eigenstates that drive the time evolution in Eq. (5). Second, the entanglement indicators: the subsystem von Neumann entropy $S_k(t) = -\mathrm{Tr}_k\{\rho_k\ln\rho_k\}$ and the subsystem linear entropy $\delta_k(t) = 1 - \mathrm{Tr}_k\{\rho_k^2\}$ of the field mode, computed from numerically diagonalised reduced density matrices; because the total state is pure, the field and atom entropies coincide at every instant, and their vanishing marks a return of the two-mode state to a product form. Third, the overlap fidelity $C(T_{\rm rev})$, defined as the maximum of $|\langle\psi(0)|\psi(t)\rangle|^2$ in the vicinity of the first near-revival time, which ties the entropy dips to an actual closeness of the total state to its initial form. The entropies act as the revival clock, and the fidelity as the confirmation that the clock is telling the right time.

Extended reading notes

Core claim

For the non-symmetric Hamiltonian $H = \omega a^\dagger a + \omega_0 b^\dagger b + \gamma (b^\dagger)^2 b^2 + g(a^\dagger b + b^\dagger a)$ with weak nonlinearity $\gamma/g = 10^{-2}$, the paper shows that the dynamics of entanglement carries the full fingerprint of wave-packet revivals. Starting from a product state in which the atom is in its ground state and the field is in a Fock state, a coherent state, or an $m$-photon-added coherent state, the subsystem von Neumann entropy $S_k(t) = -\mathrm{Tr}_k\{\rho_k(t)\ln\rho_k(t)\}$ and the linear entropy $\delta_k(t) = 1 - \mathrm{Tr}_k\{\rho_k^2(t)\}$ of the field mode both return to values close to zero at $t \simeq 2\pi/\gamma$ for a Fock initial state and at $t \simeq 4\pi/\gamma$ for a coherent initial state, signalling near-revivals; at $1/2$, $1/3$, $2/3$ and $1/4$ of these times the entropies show smaller marked dips, the entanglement counterparts of fractional revivals. The overlap fidelity $C(T_{\rm rev}) = \max_{t\sim T_{\rm rev}} |\langle\psi(0)|\psi(t)\rangle|^2$ corroborates this behaviour, approaching unity only for sufficiently large $g/\gamma$. When the initial field state is a photon-added coherent state, the revival dips are suppressed and the entropies saturate at larger values, the suppression growing with both $m$ and $\nu$. The combined quadrature $\xi = (x_a + x_b)/2$ shows squeezing, $\Delta\xi < 1/2$, near $T_{\rm rev}/2$ for a coherent initial state but not for photon-added states, and amplitude-squared squeezing is absent at the fractional-revival time in the cases considered, distinguishing this two-mode system from single-mode dynamics.

Load-bearing premise

The load-bearing premise is that the near-revival times $T_{\rm rev} \simeq 2\pi/\gamma$ for Fock states and $T_{\rm rev} \simeq 4\pi/\gamma$ for coherent states, established in earlier single-mode and coupled studies, remain valid for this non-symmetric two-mode Hamiltonian; the paper imports these times to label the entropy dips, and if the actual near-return times differ, the claimed revival counterparts are misidentified.

Editorial extensions

If this is right

Under weak nonlinearity, the entropy dips at $t \simeq 2\pi/\gamma$ (Fock) and $t \simeq 4\pi/\gamma$ (coherent) give a direct way to read off the revival time from entanglement dynamics alone, without computing wave-packet overlaps.
The dips at fractions $1/2$, $1/3$, $2/3$ and $1/4$ of $T_{\rm rev}$ give an entanglement-based diagnostic of fractional revivals that does not require visualising the wave packet.
The overlap fidelity $C(T_{\rm rev})$ as a function of the coupling $g$ defines a quantitative threshold in $g/\gamma$ above which near-revivals occur, and this threshold grows with the departure of the initial field state from coherence as measured by the photon-added-state parameters $m$ and $\nu$.
The standard deviation $\Delta\xi$ of the combined quadrature $\xi = (x_a + x_b)/2$ bursts into rapid oscillations at $T_{\rm rev}/2$ and drops below $1/2$ there for a coherent initial state, so squeezing serves as an independent observable signature of the fractional revival; amplitude-squared squeezing is predicted to be absent in the cases examined.
Expectation values of the subsystem quadratures $x_a$, $x_b$, $p_a$, $p_b$ do not carry reliable fractional-revival signatures when the modes are entangled, so the combined quadratures $\xi$ and $\eta$ are the appropriate observables for this system.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The entropy-dip method should continue to work for initially entangled two-mode states, not just the product states studied here; if the dips persist, the revival clock is a property of the Hamiltonian and of the initial state's coherence rather than of the initial product structure.
Since low-order photon-added coherent states have been produced in the laboratory, the predicted suppression of the revival dip with increasing $m$ suggests a direct experiment: prepare a field state of unknown coherence, let it interact with the nonlinear medium, and infer the departure from coherence from the depth of the entropy dip at $T_{\rm rev}$.
The same block-diagonalisation and entropy-dip analysis could be extended to the regimes $\gamma/g \sim 1$ and $\gamma/g \gg 1$, where revivals become irregular, to test whether residual entropy dips still mark partial recurrences and whether their positions track any generalised revival time.
One could check whether the dip at $T_{\rm rev}/2$ corresponds to an actual splitting of the two-mode wave packet into two sub-packets, as in the single-mode case; direct visualisation of the sub-packets would connect the entanglement-dip language to the standard revival picture.

Original abstract

We consider a model Hamiltonian describing the interaction of a single-mode radiation field with the atoms of a nonlinear medium, and study the dynamics of entanglement for specific non-entangled initial states of interest: namely, those in which the field mode is initially in a Fock state, a coherent state, or a photon-added coherent state. The counterparts of near-revivals and fractional revivals are shown to be clearly identifiable in the entropy of entanglement. The ``overlap fidelity'' of the system is another such indicator, and its behaviour corroborates that of the entropy of entanglement in the vicinity of near-revivals. The expectation values and higher moments of suitable quadrature variables are also examined, with reference to possible squeezing and higher-order squeezing.

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We consider a model Hamiltonian describing the interaction of a single-mode radiation field with the atoms of a nonlinear medium, and study the dynamics of entanglement for specific non-entangled initial states of interest: namely, those in which the field mode is initially in a Fock state, a coherent state, or a photon-added coherent state. The counterparts of near-revivals and fractional revivals are shown to be clearly identifiable in the entropy of entanglement. The ``overlap fidelity'' of the system is another such indicator, and its behaviour corroborates that of the entropy of entanglement in the vicinity of near-revivals. The expectation values and higher moments of suitable quadrature variables are also examined, with reference to possible squeezing and higher-order squeezing.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider a model Hamiltonian describing the interaction of a single-mode radiation field with the atoms of a nonlinear medium, and study the dynamics of entanglement for specific non-entangled initial states of interest: namely, those in which the field mode is initially in a Fock state, a coherent state, or a photon-added coherent state.	contribution 0.74	0.85	0.06	asserted 0.78
s1	The counterparts of near-revivals and fractional revivals are shown to be clearly identifiable in the entropy of entanglement.	result 1.00	0.90	0.38	asserted 1.00
s2	The ``overlap fidelity'' of the system is another such indicator, and its behaviour corroborates that of the entropy of entanglement in the vicinity of near-revivals.	result 0.96	0.89	0.11	asserted 0.99
s3	The expectation values and higher moments of suitable quadrature variables are also examined, with reference to possible squeezing and higher-order squeezing.	contribution 0.54	0.88	0.05	asserted 0.54

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Figures

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References

Works cited (16)

[1]

See, for instance, R. W. Robinett, Phys. Rep. 392, 1 (2004)

View on Pith 2004
[2]

K. Tara, G. S. Agarwal and S. Chaturvedi, Phys. Rev. A 47, 5024 (1993)

View on Pith 1993
[3]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, Phys. L ett. A 329, 14 (2004)

View on Pith 2004
[4]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, Europh ys. Lett. 71, 744 (2005)

View on Pith 2005
[5]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, J. Opt. B: Quant. Semiclass. Opt. 7, S728 (2005)

View on Pith 2005
[6]

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991)

View on Pith 1991
[7]

S. J. van Enk, Phys. Rev. Lett. 91, 017902 (2003); for a discussion on the decoherence of entanglement, see also S. J. van Enk, Phys. Rev. A 72 , 022308, (2005)

View on Pith 2003
[8]

I. Sanz, R. M. Angelo and K. Furuya, J. Phys. A: Math. Gen. 36, 9737 (2003)

View on Pith 2003
[9]

G. S. Agarwal and R. R. Puri, Phys. Rev. A39, 2969 (1989)

View on Pith 1989
[10]

A. Zavatta, S. Viciani and M. Bellini, Science 306, 660 (2004)

View on Pith 2004
[11]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Fl annery, Numerical Recipes in C , Cambridge University Press, Cambridge, 2001, pp. 475-481

View on Pith 2001
[12]

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth and F. Rossi, GNU Scientiﬁc Library , GNU Software, 2005

View on Pith 2005
[13]

That the revival time is inversely proportional to the c oeﬃcient of the term in the Hamiltonian that is quadratic in the quantum numbers concerned is a gener al feature. See S. Seshadri, S. Lakshmibala and V. Balakrishnan, J. Stat. Phys. 101, 213 (2000)

View on Pith 2000
[14]

A quantitative measure of entanglement is provided by t he logarithm of the trace norm of the reduced density matrix (see G. Vidal and R. F. Werner, Phys. R ev. A 65, 032314 (2002)). This is also the measure of entanglement used, for instance, in Ref. [7]

View on Pith 2002
[15]

S. D. Du and C.-D. Gong, Phys. Rev. A 48, 2198 (1993)

View on Pith 1993
[16]

C. Sudheesh, Non-classical eﬀects in wave packet dynamics , Thesis, Indian Institute of Tech- nology Madras (unpublished), (2005). 18

View on Pith 2005

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]

See, for instance, R. W. Robinett, Phys. Rep. 392, 1 (2004)

View on Pith 2004

[2] [2]

K. Tara, G. S. Agarwal and S. Chaturvedi, Phys. Rev. A 47, 5024 (1993)

View on Pith 1993

[3] [3]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, Phys. L ett. A 329, 14 (2004)

View on Pith 2004

[4] [4]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, Europh ys. Lett. 71, 744 (2005)

View on Pith 2005

[5] [5]

C. Sudheesh, S. Lakshmibala and V. Balakrishnan, J. Opt. B: Quant. Semiclass. Opt. 7, S728 (2005)

View on Pith 2005

[6] [6]

G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991)

View on Pith 1991

[7] [7]

S. J. van Enk, Phys. Rev. Lett. 91, 017902 (2003); for a discussion on the decoherence of entanglement, see also S. J. van Enk, Phys. Rev. A 72 , 022308, (2005)

View on Pith 2003

[8] [8]

I. Sanz, R. M. Angelo and K. Furuya, J. Phys. A: Math. Gen. 36, 9737 (2003)

View on Pith 2003

[9] [9]

G. S. Agarwal and R. R. Puri, Phys. Rev. A39, 2969 (1989)

View on Pith 1989

[10] [10]

A. Zavatta, S. Viciani and M. Bellini, Science 306, 660 (2004)

View on Pith 2004

[11] [11]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Fl annery, Numerical Recipes in C , Cambridge University Press, Cambridge, 2001, pp. 475-481

View on Pith 2001

[12] [12]

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth and F. Rossi, GNU Scientiﬁc Library , GNU Software, 2005

View on Pith 2005

[13] [13]

That the revival time is inversely proportional to the c oeﬃcient of the term in the Hamiltonian that is quadratic in the quantum numbers concerned is a gener al feature. See S. Seshadri, S. Lakshmibala and V. Balakrishnan, J. Stat. Phys. 101, 213 (2000)

View on Pith 2000

[14] [14]

A quantitative measure of entanglement is provided by t he logarithm of the trace norm of the reduced density matrix (see G. Vidal and R. F. Werner, Phys. R ev. A 65, 032314 (2002)). This is also the measure of entanglement used, for instance, in Ref. [7]

View on Pith 2002

[15] [15]

S. D. Du and C.-D. Gong, Phys. Rev. A 48, 2198 (1993)

View on Pith 1993

[16] [16]

C. Sudheesh, Non-classical eﬀects in wave packet dynamics , Thesis, Indian Institute of Tech- nology Madras (unpublished), (2005). 18

View on Pith 2005