Two-Mode Squeezed LG Beams

• The paper demonstrates that two-mode squeezing redistributes phase-space weight, linking quadrature correlations to the formation of vortex structures in LG beams.
• It employs methodologies including SPDC vortex generation, PSR-induced multimode squeezing, and su(1,1) beam mapping to precisely engineer the LG modal structure.
• The work highlights how squeezing-driven vortex formation and radial mode redistribution enable quantum metrology and communications while contending with diffraction and detection limits.

Searching arXiv for the cited papers and closely related work on two-mode squeezed Laguerre-Gaussian beams. Two-mode squeezed Laguerre–Gaussian beams are structured optical fields in which continuous-variable squeezing is carried by transverse modes with Laguerre–Gaussian (LG) spatial structure, so that quadrature correlations, orbital-angular-momentum content, and radial-mode structure become coupled within a single paraxial framework. In the literature represented here, the concept appears in three closely connected forms: as vortex-bearing output states obtained by rotating two-mode photon-number squeezed inputs in quadrature space (Puentes et al., 2021); as an experimentally observed and theoretically modeled multimode LG decomposition of atom-generated squeezed light that can be projected into effective two-mode LG pairs (Zhang et al., 2015); and as a group-theoretic beam-engineering construction in which continuous-variable two-mode squeezed states are mapped directly onto scalar paraxial beams expanded in an LG basis (Rodríguez-Lara et al., 15 Sep 2025). Across these formulations, the central theme is that squeezing redistributes phase-space weight and induces correlations among transverse degrees of freedom while preserving the paraxial modal structure characteristic of LG beams.

1. Conceptual definition and modal setting

In paraxial optics, LG modes form a complete orthonormal basis for cylindrically symmetric beams. A standard expression is

$$\mathrm{LG}_{p}^{l}(r,\phi,z) = \frac{C_{pl}}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|} L_{p}^{|l|}\!\left(\frac{2r^2}{w^2(z)}\right)\, \exp\!\left(-\frac{r^2}{w^2(z)}\right)\, \exp\!\left(i l\,\phi\right)\, \exp\!\left[i k\frac{r^2}{2R(z)} - i\,(2p+|l|+1)\,\zeta(z)\right],$$

where $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$, $w(z)=w_0\sqrt{1+(z/z_R)^2}$, $R(z)=z\,[1+(z_R/z)^2]$, $\zeta(z)=\arctan(z/z_R)$, and $z_R=\pi w_0^2/\lambda$ (Zhang et al., 2015). In quantum-optical notation, the positive-frequency field operator may be decomposed as

$$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$

with $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$ and bosonic mode operators $\hat a_{p,l}$ (Zhang et al., 2015).

The “two-mode” descriptor is used in two technically distinct but related senses. In spontaneous parametric down-conversion, it refers to two correlated bosonic modes $a$ and $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$0, whose squeezing is generated by

$C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$1

producing the two-mode squeezed vacuum

$C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$2

(Puentes et al., 2021). In the spatial beam-engineering framework, the two modes are the two transverse degrees of freedom $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$3 and $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$4, represented by canonical quadratures $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$5 and $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$6, with two-mode squeezing implemented through an $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$7 algebraic action (Rodríguez-Lara et al., 15 Sep 2025).

This dual usage is not contradictory. It indicates that two-mode squeezing may be represented either in Hilbert-space mode labels or in coupled transverse canonical variables. A plausible implication is that the structured-light interpretation and the nonclassical-state interpretation are different projections of the same underlying covariance structure.

2. Two-mode squeezing formalisms and quadrature correlations

For the ideal two-mode squeezed vacuum, the covariance matrix in canonical ordering $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$8 is

$C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$9

which yields the EPR-like correlations

$w(z)=w_0\sqrt{1+(z/z_R)^2}$0

(Puentes et al., 2021). The associated Wigner function is Gaussian,

$w(z)=w_0\sqrt{1+(z/z_R)^2}$1

(Puentes et al., 2021).

In the beam-theoretic formulation, the $w(z)=w_0\sqrt{1+(z/z_R)^2}$2 generators for the two transverse degrees of freedom are

$w(z)=w_0\sqrt{1+(z/z_R)^2}$3

with commutators

$w(z)=w_0\sqrt{1+(z/z_R)^2}$4

(Rodríguez-Lara et al., 15 Sep 2025). Under the squeezing action generated by $w(z)=w_0\sqrt{1+(z/z_R)^2}$5,

$w(z)=w_0\sqrt{1+(z/z_R)^2}$6

$w(z)=w_0\sqrt{1+(z/z_R)^2}$7

$w(z)=w_0\sqrt{1+(z/z_R)^2}$8

$w(z)=w_0\sqrt{1+(z/z_R)^2}$9

(Rodríguez-Lara et al., 15 Sep 2025).

These exact operator relations show that two-mode squeezing does not merely rescale a single quadrature pair. It mixes the transverse degrees of freedom hyperbolically, thereby creating joint spatial and spectral correlations. In the LG basis, those correlations are represented as redistribution among radial orders at fixed azimuthal index $R(z)=z\,[1+(z_R/z)^2]$0 (Rodríguez-Lara et al., 15 Sep 2025).

3. Generation routes

Two main physical routes are described in the cited work.

The first route begins from two-mode photon-number squeezed states produced by spontaneous parametric down-conversion. The working finite-$R(z)=z\,[1+(z_R/z)^2]$1 input used in the vortex-generation scheme is

$R(z)=z\,[1+(z_R/z)^2]$2

with normalization factor $R(z)=z\,[1+(z_R/z)^2]$3 and squeezing parameter $R(z)=z\,[1+(z_R/z)^2]$4 (Puentes et al., 2021). This state exhibits sub-Poissonian number statistics and a Gaussian quadrature-space profile without vortices (Puentes et al., 2021). The quadrature-space wavefunction is built from Hermite–Gaussian functions,

$R(z)=z\,[1+(z_R/z)^2]$5

so that

$R(z)=z\,[1+(z_R/z)^2]$6

(Puentes et al., 2021).

The second route is polarization self-rotation in dense rubidium vapor. In that experiment, a squeezed vacuum field is generated via polarization self-rotation interaction of an ensemble of Rb atoms with a strong near-resonant linearly polarized laser field. The orthogonal polarization after the cell contains a squeezed vacuum state measured by balanced homodyne detection using the pump as the local oscillator, with typical results of $R(z)=z\,[1+(z_R/z)^2]$7 dB of squeezing and $R(z)=z\,[1+(z_R/z)^2]$8 dB of anti-squeezing at $R(z)=z\,[1+(z_R/z)^2]$9 MHz (Zhang et al., 2015). The experiment found that the spatial structure is multimode rather than single-mode and can be decomposed into LG modes (Zhang et al., 2015).

The PSR theory derives an inhomogeneous paraxial propagation equation,

$\zeta(z)=\arctan(z/z_R)$0

and, in a weak-scattering approximation, obtains a superposition of radial LG modes,

$\zeta(z)=\arctan(z/z_R)$1

(Zhang et al., 2015). Selection rules favor $\zeta(z)=\arctan(z/z_R)$2 because the pump is cylindrically symmetric and the source term carries no $\zeta(z)=\arctan(z/z_R)$3 dependence (Zhang et al., 2015).

A more abstract but exact construction is given by the scalar-beam mapping of continuous-variable states. There, two-mode squeezed LG beams are defined directly by the expansion

$\zeta(z)=\arctan(z/z_R)$4

with coefficients

$\zeta(z)=\arctan(z/z_R)$5

(Rodríguez-Lara et al., 15 Sep 2025).

Taken together, these routes show three experimentally or formally relevant regimes: direct quantum-state generation in SPDC, atom-generated multimode squeezing in PSR, and mode-engineered optical analogs based on $\zeta(z)=\arctan(z/z_R)$6 beam synthesis.

4. Hermite–Gaussian to Laguerre–Gaussian conversion and vortex formation

A defining mechanism in the SPDC-based construction is a quadrature-space “$\zeta(z)=\arctan(z/z_R)$7 converter,” implemented by

$\zeta(z)=\arctan(z/z_R)$8

(Puentes et al., 2021). In the Heisenberg picture,

$\zeta(z)=\arctan(z/z_R)$9

so that at $z_R=\pi w_0^2/\lambda$0,

$z_R=\pi w_0^2/\lambda$1

(Puentes et al., 2021). In quadrature space this corresponds to the rotation

$z_R=\pi w_0^2/\lambda$2

(Puentes et al., 2021).

The paper invokes the standard astigmatic HG$z_R=\pi w_0^2/\lambda$3LG conversion result that Hermite–Gaussian mode products are converted into single Laguerre–Gaussian modes by imparting a $z_R=\pi w_0^2/\lambda$4 phase shift in the mode profile, and states that choosing $z_R=\pi w_0^2/\lambda$5 implements exactly that phase condition (Puentes et al., 2021). After rotation, the state becomes a coherent superposition of Fock states $z_R=\pi w_0^2/\lambda$6 with $z_R=\pi w_0^2/\lambda$7-dependent coefficients (Puentes et al., 2021). In quadrature space, the relevant Hermite–Gaussian pairs $z_R=\pi w_0^2/\lambda$8 map to LG terms $z_R=\pi w_0^2/\lambda$9 under the Gaussian envelope, yielding

$$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$0

(Puentes et al., 2021).

The main numerical finding is that, for sufficiently small squeezing parameter $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$1, the transformed quadrature-space field develops $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$2 phase singularities for an input state truncated at total photon number $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$3 (Puentes et al., 2021). In the reported numerics, the dominant component at small $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$4 corresponds to $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$5, hence $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$6, so the observed number of vortices equals the input photon number $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$7 (Puentes et al., 2021). The article’s own interpretation is that reduced uncertainty in photon number in Fock space is mapped to reduced uncertainty in vortex number in quadrature space (Puentes et al., 2021).

This mapping should not be confused with a generic statement that any squeezed LG beam must contain $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$8 vortices. The claim is specific to the truncated input family, the $$\hat{E}^{(+)}(\mathbf{r},z) = \sum_{p,l} u_{p,l}(\mathbf{r},z)\, \hat{a}_{p,l}(z),$$9 converter, and the small-$u_{p,l}\equiv \mathrm{LG}_{p}^{l}$0 regime treated numerically in that work (Puentes et al., 2021). By contrast, the general $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$1 two-mode squeezed LG construction preserves $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$2 while redistributing radial order $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$3 (Rodríguez-Lara et al., 15 Sep 2025).

5. Phase-space structure, Wigner analysis, and inseparability

The cited literature treats two-mode squeezed LG beams as phase-space objects as much as spatial modes. In the scalar-beam mapping, the optical Wigner distribution is defined as

$u_{p,l}\equiv \mathrm{LG}_{p}^{l}$4

(Rodríguez-Lara et al., 15 Sep 2025). Squeezing deforms this distribution hyperbolically along coupled quadratures while preserving area, and for LG beams at fixed $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$5 yields tilted or elliptical contours in spatial and spectral marginals, signaling intermodal correlations analogous to continuous-variable entanglement (Rodríguez-Lara et al., 15 Sep 2025).

For non-Gaussian states, the same work states that Wigner marginals can exhibit negativity due to phase-space interference from nodal structure; in that framework, such negativity acts as a witness of classical continuous-variable inseparability rather than quantum nonclassicality (Rodríguez-Lara et al., 15 Sep 2025). This distinction is important because the paper explicitly constructs classical analogs of continuous-variable quantum states, not quantum light sources (Rodríguez-Lara et al., 15 Sep 2025).

A Duan–Simon-type criterion is also presented. Defining

$u_{p,l}\equiv \mathrm{LG}_{p}^{l}$6

the separability bound for Gaussian states is

$u_{p,l}\equiv \mathrm{LG}_{p}^{l}$7

(Rodríguez-Lara et al., 15 Sep 2025). The paper states that for the Gaussian case $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$8, any $u_{p,l}\equiv \mathrm{LG}_{p}^{l}$9 yields a violation of the bound in the optical analog (Rodríguez-Lara et al., 15 Sep 2025).

The PSR work formulates the same issue in a quantum-optical multimode setting. Its effective Hamiltonian can include cross-mode bilinear couplings,

$\hat a_{p,l}$0

leading to a two-mode squeezer

$\hat a_{p,l}$1

with joint quadrature variances

$\hat a_{p,l}$2

for appropriate combinations (Zhang et al., 2015). The same paper notes that the reported experiment demonstrates multimode squeezing and LG content, but does not directly extract two-mode entanglement (Zhang et al., 2015). This is a useful boundary condition on interpretation: multimode LG squeezing and the availability of a two-mode LG description do not, by themselves, constitute a direct entanglement measurement.

6. Beam parameters, propagation, and numerical-aperture constraints

For the squeezed Gaussian seed $\hat a_{p,l}$3, the $\hat a_{p,l}$4 mapping yields a closed-form propagated field,

$\hat a_{p,l}$5

with exact modified parameters

$\hat a_{p,l}$6

$\hat a_{p,l}$7

$\hat a_{p,l}$8

(Rodríguez-Lara et al., 15 Sep 2025). The minimum waist at focus occurs for $\hat a_{p,l}$9 (Rodríguez-Lara et al., 15 Sep 2025).

At the same time, the work stresses that finite numerical aperture limits the highest accessible radial order and therefore the achievable squeezing. The relevant cutoff relations are

$a$0

(Rodríguez-Lara et al., 15 Sep 2025). The operational bound is expressed through truncation of the LG expansion at $a$1, with $a$2 set by the requirement

$a$3

(Rodríguez-Lara et al., 15 Sep 2025). The paper states that stronger squeezing demands higher numerical aperture because squeezing pushes energy into higher-order LG modes with larger divergence and higher spatial-frequency content (Rodríguez-Lara et al., 15 Sep 2025). Numerical propagation of aperture-truncated beams confirms that diffraction remains the ultimate limit (Rodríguez-Lara et al., 15 Sep 2025).

The PSR study presents a related but experimentally grounded propagation constraint: different LG radial orders acquire different Gouy-phase shifts near the focus, so their squeezing angles become mode dependent (Zhang et al., 2015). This is directly relevant to two-mode LG beams because joint-quadrature measurements require those modal squeezing axes to be phase aligned. The reported data showed strong asymmetries and sharp noise changes when scanning an iris along $a$4 near the focus, behavior consistent with Gouy-phase evolution (Zhang et al., 2015).

7. Detection, applications, limitations, and comparison of regimes

In the SPDC-based vortex scheme, the resulting LG vortex states can be characterized by measuring the OAM spectrum via mode projectors or phase-flattening and coupling into single-mode fibers, by interferometric diagnostics of azimuthal phase and topological charge such as fork holograms, by direct camera imaging of intensity and phase to count vortices and verify azimuthal phase winding, and by homodyne tomography in quadrature space (Puentes et al., 2021). In the PSR setting, homodyne detection with a shaped or masked local oscillator performs mode-selective projection onto desired transverse profiles, with photocurrent variance given as a weighted sum of modal quadrature variances through overlap integrals $a$5 (Zhang et al., 2015).

The following comparison organizes the roles played by LG structure in the three cited formulations.

Regime	Principal LG feature	Key technical point
SPDC plus $a$6 converter (Puentes et al., 2021)	Vortex formation in quadrature space	Small-$a$7 numerics give $a$8 vortices for input truncated at photon number $a$9
PSR in Rb vapor (Zhang et al., 2015)	Multimode LG decomposition dominated by $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$00	Spatial masks and LO shaping reveal hidden LG squeezing and enable effective two-mode mode pairs
$C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$01 beam mapping (Rodríguez-Lara et al., 15 Sep 2025)	Exact two-mode squeezed LG expansion	Squeezing redistributes radial order at fixed $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$02, with finite NA limiting achievable squeezing

Applications highlighted across the sources include quantum metrology and sensing, quantum cryptography, OAM- and radial-mode–based communications, imaging, and interfaces to atomic memories (Puentes et al., 2021, Zhang et al., 2015, Rodríguez-Lara et al., 15 Sep 2025). The SPDC paper argues that the scheme “has the potential to improve measurement sensitivity beyond the Standard Quantum Limit (SQL $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$03),” but it does not provide an explicit Fisher-information or Cramér–Rao analysis and does not claim Heisenberg scaling (Puentes et al., 2021). The PSR paper emphasizes that spatial multimode structure is not merely a complication but can be harnessed as a resource by selective projection onto LG components (Zhang et al., 2015). The beam-mapping paper frames the corresponding correlations as a classical analog of continuous-variable entanglement and explicitly states that diffraction constraints remain in force (Rodríguez-Lara et al., 15 Sep 2025).

Several limitations recur. Optical loss and detector inefficiency wash out number correlations and reduce effective squeezing; imperfect HG$C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$04LG conversion and misalignment lead to mixed $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$05-content and degraded vortex visibility; decoherence and phase noise smear azimuthal phase and obscure singularities (Puentes et al., 2021). In PSR, near-resonant absorption and technical noise limit observable squeezing, while mode-dependent Gouy-phase rotations misalign squeezing angles and degrade Gaussian-LO measurements (Zhang et al., 2015). In beam-engineered analogs, finite numerical aperture truncates the high-order modal content required for strong squeezing and enforces the diffraction limit (Rodríguez-Lara et al., 15 Sep 2025).

A common misconception is that LG squeezing necessarily implies single-mode OAM squeezing in a pure $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$06 state. The cited work instead shows three more nuanced possibilities: squeezing may be encoded in correlated number-state inputs and revealed as vortex structure after quadrature rotation (Puentes et al., 2021); it may be distributed across multiple LG radial orders and hidden by an unshaped local oscillator (Zhang et al., 2015); or it may be represented as an exact radial-order superposition at fixed $C_{pl}=\sqrt{\frac{2\,p!}{\pi\,(p+|l|)!}}$07 in a classical analog of continuous-variable two-mode squeezing (Rodríguez-Lara et al., 15 Sep 2025). This suggests that “two-mode squeezed Laguerre–Gaussian beams” are best understood not as a single canonical object but as a family of structured squeezed states and beam constructions linked by common transverse-mode algebra, quadrature correlations, and phase-space geometry.