Geometrically Squeezed States

• Geometrically squeezed states are quantum states defined by explicit geometric structures such as uncertainty ellipses in phase space and hyperbolic state manifolds.
• They exhibit a deterministic evolution of quadrature uncertainties while the state’s displacement is influenced by measurement noise, enabling precise control over quantum dynamics.
• Applications span quantum optics, metrology, and field theory, with implementations ranging from structured light modulation to digital simulations and many-body spin systems.

A geometrically squeezed state is a squeezed state whose defining properties are expressed through an explicit geometry: the shape and orientation of an uncertainty ellipse in phase space, a point on a hyperbolic state manifold such as the Poincaré disk or a Bloch hyperboloid, or a deliberately engineered spatial profile in real space. In the literature, this label covers single-mode and multi-mode Gaussian states, higher-symmetry spin and qutrit constructions, twist-generated states in field theory and crystalline systems, and structured-light analogies in which squeezing becomes directly visible as spatial anisotropy or mode deformation (Dabrowska et al., 2012, Chi et al., 2024, Wang et al., 2022).

1. Canonical phase-space formulation

For the harmonic oscillator, a squeezed vacuum state is defined by

$$|\xi\rangle=\hat S(\xi)|0\rangle, \qquad \hat S(\xi)=\exp\left[\frac{1}{2}(\xi^* \hat a^2-\xi \hat a^{\dagger 2})\right],$$

with $\xi=r e^{i\phi_s}$, where $r$ is the squeezing parameter and $\phi_s$ is the squeezing phase. The geometric meaning is the standard one: the uncertainty distribution is contracted along one quadrature and expanded along the conjugate quadrature, with the orientation of the ellipse determined by $\phi_s$ (Lo et al., 2014).

Using quadrature operators

$$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$

or, more generally,

$$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$

the squeezed quadrature variance becomes $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$ when $\theta=\phi_s/2$. In this representation, a geometrically squeezed state is one for which the non-isotropic uncertainty profile is itself the primary descriptor of the state’s structure (Dabrowska et al., 2012, Lo et al., 2014).

The same geometry governs displaced-squeezed states,

$$|\alpha,\xi\rangle=\hat D(\alpha)\hat S(\xi)|0\rangle,$$

for which squeezing fixes the local shape of the wavepacket and displacement fixes its center. This separation between center and shape recurs in measurement theory, entanglement generation, control, and state engineering across the later literature.

2. Deterministic geometric evolution under continuous observation

A particularly explicit use of the term appears in continuous nondemolition observation of a squeezed coherent state. If the initial state is

$\xi=r e^{i\phi_s}$0

then under single or double heterodyne detection the conditioned posterior state remains of the form

$\xi=r e^{i\phi_s}$1

The state therefore remains a squeezed coherent state throughout the stochastic evolution (Dabrowska et al., 2012).

The geometric content of this result is that the center of the state and the shape of the state evolve differently. The quadrature means are stochastic and depend on the measurement noise, while the quadrature uncertainties are deterministic. For double heterodyne detection, the squeeze phase and modulus evolve as

$\xi=r e^{i\phi_s}$2

so that $\xi=r e^{i\phi_s}$3. The uncertainty ellipse therefore follows a fully predictable rotation-and-decay law even though the state center diffuses according to the observed Wiener process (Dabrowska et al., 2012).

This produces a sharp geometric interpretation. In phase space, the Wigner function remains elliptical; its axes and orientation are determined by $\xi=r e^{i\phi_s}$4, $\xi=r e^{i\phi_s}$5, and $\xi=r e^{i\phi_s}$6. Because $\xi=r e^{i\phi_s}$7 as $\xi=r e^{i\phi_s}$8, the ellipse becomes more circular and the state approaches vacuum uncertainty. A geometrically squeezed state in this sense is a state whose phase-space shape evolves deterministically, while only its displacement is measurement dependent (Dabrowska et al., 2012).

3. Hyperbolic state manifolds and information geometry

A more formal geometric description identifies single-mode squeezed states with points of the Poincaré disk. Up to phase, a squeezed state can be characterized by

$\xi=r e^{i\phi_s}$9

and the complex parameter $r$0 gives a bijection between pure Gaussian squeezed states and the open unit disk $r$1. The disk carries the hyperbolic metric

$r$2

and the projective Hilbert-space geometry of these states is isometric to that hyperbolic geometry (Chi et al., 2024).

Quadratic Hamiltonians preserve this manifold and act as Möbius transformations. For

$r$3

the squeezed-state parameter evolves as

$r$4

The resulting trajectories are hyperbolic circles, horocycles, or hypercycles depending on whether the Hamiltonian is in the stable, free, or unstable regime. This turns evolution and control of squeezing into explicit geometry on the disk (Chi et al., 2024).

A related but distinct information-geometric construction regards coherent-squeezed states

$r$5

as a real $r$6-dimensional surface in Fock space. On the minimal-uncertainty locus $r$7, the metric simplifies dramatically and its determinant becomes

$r$8

This links squeezing geometry directly to the uncertainty principle: the metric becomes especially simple precisely on the submanifold যেখানে minimal uncertainty is saturated (Fujii et al., 2013).

4. Multimode squeezing, relative phase, and entanglement geometry

For two-mode squeezed states, geometry enters through both phase-space correlations and geometric phase. A two-mode squeezed vacuum is written

$r$9

and under the Hamiltonian considered in the cited work it evolves as

$\phi_s$0

The geometric phase is defined kinematically by

$\phi_s$1

For cyclic evolution, $\phi_s$2, and the entanglement entropy can be written directly as a function of $\phi_s$3 (Yang et al., 2015).

A separate geometric effect concerns entanglement generation by optical elements. For squeezed-vacuum inputs of equal squeezing strength, the relative orientation of the phase-space ellipses is decisive. Minimal entanglement occurs when

$\phi_s$4

whereas maximal entanglement occurs when

$\phi_s$5

The cited analysis emphasizes that the angle needed to rotate one phase-space distribution to overlap with the other determines how much entanglement the pair of states will generate. Squeezed-vacuum states therefore both minimize and maximize the same operational notion of quantumness, depending on phase alignment (Goldberg et al., 2021).

Reservoir-engineered multimode squeezing makes this geometry operational for metrology. In a trapped-ion realization, a stabilized two-mode squeezed state is generated along the collective quadratures

$\phi_s$6

with

$\phi_s$7

The reported EPR variance is $\phi_s$8, and simultaneous estimation of two collective displacements surpasses the classical limit by up to $\phi_s$9 and $\phi_s$0 decibels. Here the phrase geometrically squeezed refers to simultaneous squeezing along multiple orthogonal axes in a multi-mode phase space (Li et al., 2023).

5. Higher-symmetry, many-body, and field-theoretic generalizations

In spin-1 quantum gases, squeezing is not exhausted by the SU(2) Bloch-sphere picture. A spin-1 condensate is an SU(3) system requiring both the spin vector $\phi_s$1 and the rank-2 nematic tensor $\phi_s$2 for state characterization. The squeezed quadratures are linear combinations of spin and quadrupole operators, notably the pairs $\phi_s$3 and $\phi_s$4, satisfying

$\phi_s$5

Following a quench through a nematic to ferromagnetic quantum phase transition, squeezing was observed up to $\phi_s$6, or $\phi_s$7 corrected for detection noise. The cited work explicitly interprets the state as a vacuum squeezed state for the nearly empty $\phi_s$8 modes, analogous to optical two-mode vacuum squeezing (Hamley et al., 2011).

A geometric definition of squeezing in collective qutrit systems generalizes the same idea to SU(3) coherent states on the coset space $\phi_s$9. There, coherent states possess isotropic fluctuations of magnitude $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$0 in a tangent hyperplane, and a state is squeezed when the variance of a suitable transformed operator $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$1 drops below $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$2. Nonlinear Hamiltonians in the generators of $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$3 deform the initially isotropic Wigner-function blob, producing anisotropic fluctuations that constitute geometric squeezing in the higher-symmetry setting (Klimov et al., 2011).

Field-theoretic constructions display a different spatial geometry. In the D1D5 CFT, the twist operator $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$4 acting on the double Ramond vacuum generates a squeezed state with explicit bosonic and fermionic bilinears. In the continuum limit, the state acquires a position-space kernel

$$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$5

which encodes nonlocal correlations between fields at positions $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$6 and $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$7 relative to the twist insertion $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$8. In this usage, the geometry is the geometry of spatial correlation generated by the twist, rather than only the geometry of quadrature ellipses (Burrington et al., 2014).

Hyperbolic many-mode extensions make this explicit at the group level. The $$X=\frac{a+a^\dagger}{2}, \qquad Y=\frac{a-a^\dagger}{2i},$$9 squeezed vacuum on the Bloch four-hyperboloid $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$0 is a four-mode squeezed state parameterized by $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$1, where $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$2 is the radial squeezing parameter and $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$3 are angular coordinates on the $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$4 latitude. It is interpreted as a superposition of maximally entangled spin pairs of all integer spins, with angularly dependent uncertainty relations and tunable spin correlations. A related $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$5 construction on the split-signature four-hyperboloid $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$6 identifies concurrence with a hyperboloid coordinate, giving entanglement a direct geometric meaning (Hasebe, 2020, Hasebe, 2019).

6. Spatially engineered and structured-light forms

In quantum optics proper, geometric squeezing can refer to simultaneous control of quadrature noise and transverse mode geometry. One implementation begins with a squeezed fundamental $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$7 mode generated by an optical parametric amplifier and then reshapes it with cascaded phase-only spatial light modulators in a $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$8 optical configuration. Using the Gerchberg–Saxton algorithm, the method produces squeezed higher-order Hermite–Gauss modes, Laguerre–Gauss modes such as $$\hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right),$$9, and arbitrary complex amplitude distributed modes. The reported output includes up to $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$0 squeezing in $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$1, mode-conversion efficiency as high as $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$2 for lower order, and $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$3 for complex arbitrary modes. In this context, a geometrically squeezed state is one in which the spatial intensity and phase geometry, as well as quadrature noise, are engineered together (Ma et al., 2020).

The phrase is also used in trapped-ion mechanics for squeezed oscillator wavepackets. Starting from a squeezed vacuum with $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$4, corresponding to about $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$5 reduction in quadrature variance, state-dependent forces generate superpositions of displaced-squeezed motional states. The overlap

$$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$6

shows explicitly that distinguishability depends on whether the displacement is aligned with the squeezed or anti-squeezed axis. Coherent revivals were observed after separating the wavepackets by more than $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$7 times the ground-state root mean squared extent, corresponding to $$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$8 times the root mean squared extent of the squeezed wavepacket along the displacement direction (Lo et al., 2014).

A distinct but mathematically parallel line of work constructs classical analogies. The “classical analogy of squeezed coherent state” applies squeezed and displacement operators to structured spatial modes, creating classical wave-packets with tunable squeezed degree and displacement degree and a direct correlation between quadrature operator space and real space (Wang et al., 2021). The later “structured light analogy of squeezed state” introduces

$$\langle(\Delta \hat X_\theta)^2\rangle=\frac{1}{4}e^{-2r}$$9

and builds a squeezed-vacuum analogue as a superposition of modes $\theta=\phi_s/2$0. In that setting, the standard quantum limit is mapped to a standard spatial limit, and a simulated example with $\theta=\phi_s/2$1 and $\theta=\phi_s/2$2 reaches a full width at half maximum of $\theta=\phi_s/2$3, below the classical limit of $\theta=\phi_s/2$4 for a tightly focused Gaussian beam (Wang et al., 2022).

7. Control, quantisation, and alternative Hilbert-space organizations

The geometric picture is especially powerful in control theory. On the Poincaré disk, bang-bang control of squeezing is represented by concatenated Möbius transformations, and adiabatic control follows curves traced by the instantaneous ground-state parameter

$\theta=\phi_s/2$5

This makes reachable sets, cyclic protocols, and adiabatic paths geometric objects on a negatively curved manifold rather than abstract unitary evolutions (Chi et al., 2024).

Squeezed states also define quantisation schemes. In two-dimensional squeezed-state quantisation, separable and non-separable two-mode squeezed states satisfy completeness relations and induce a quantisation map

$\theta=\phi_s/2$6

The squeezing parameters set the Gaussian regularisation scale of the semiclassical portraits, smoothing sharp boundaries and allowing trajectories to penetrate regions that are classically forbidden. The cited work emphasizes that non-separable squeezing produces anisotropic regularisation not available in coherent-state quantisation alone (Gazeau et al., 2022).

Digital simulation and condensed-matter reorganization provide two recent extensions. A fully digital simulation of single-mode squeezed states on the Zuchongzhi-2 superconducting platform uses Gray-code-based encoding, restriction to even-photon-number Fock states, and a variational quantum simulation protocol; the implementation covers a parameter sweep from $\theta=\phi_s/2$7 to $\theta=\phi_s/2$8, with tomography and Wigner-function analysis confirming high-fidelity state preparation (Li et al., 16 May 2025). In three-dimensional twisted crystals, conventional Bloch bands are replaced by squeezed coherent states in a Fock space of free-particle vortex states. There the Coriolis term induces number-nonconserving couplings of the form $\theta=\phi_s/2$9, so squeezing reorganizes the Hilbert space and produces unconventional phase-space dynamics and edge-state structure (Phong et al., 2024).

Taken together, these works suggest that a geometrically squeezed state is not a single standardized object but a family of constructions in which squeezing is best understood through geometry: the deterministic evolution of uncertainty ellipses, the hyperbolic structure of state manifolds, the alignment of multimode correlations, the deformation of many-body or field-theoretic phase-space distributions, or the deliberate shaping of spatial modes in quantum and classical optics.