Squeezed States

Definition

A squeezed state is a quantum state in which the uncertainty of one quadrature observable is reduced below the vacuum (shot noise) level at the expense of increased uncertainty in the conjugate quadrature, while still saturating the Heisenberg bound.

For a single mode with annihilation operator $a$, define the quadratures $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$ satisfying $[X_1,X_2]=i/2$. The vacuum has $\Delta X_1=\Delta X_2=1/2$. A squeezed vacuum with squeezing parameter $r\ge 0$ and squeezing angle $\varphi$ satisfies

$\Delta X_1=\frac{1}{2}{e}^{-r},\Delta X_2=\frac{1}{2}{e}^r,$

with $\Delta X_1\Delta X_2=1/4$ (still minimum uncertainty). Squeezed states are generated by the squeeze operator

$S(r,\varphi )=\exp \left(\frac{r}{2}(e^{-2i\varphi }{a}^2-e^{2i\varphi }{a}^{†2})\right).$

In the language of Bogoliubov transformations, the squeezing operation is

$S^{†}aS=a\cosh r-e^{2i\varphi }{a}^{†}\sinh r,$

which mixes creation and annihilation operators. This is the hallmark of a Bogoliubov (Bogoliubov-Valatin) transformation.

In QFT, two-mode squeezed states involve two field modes $k$ and $-k$:

$|r{⟩}_{k,-k}\propto {\sum }_{n=0}^{\infty }(\tanh r{)}^n|n{⟩}_k\otimes |n{⟩}_{-k},$

which is an entangled state of the two momentum modes. The Minkowski vacuum is itself a two-mode squeezed state in the Rindler basis (this is the algebraic content of the Unruh effect).

The Weyl operator expectation value in a squeezed state keeps the Gaussian form of the vacuum with a dressed inner product,

$⟨\text{sq}|W(f)|\text{sq}⟩=\exp \left(-\frac{1}{2}⟨f,{\Sigma }_{r}f⟩\right),$

where ${\Sigma }_r$ is the covariance of the squeezed vacuum — the one-particle inner product with the Bogoliubov coefficients $\cosh r$ and $\sinh r$ inserted, reducing to the vacuum norm at $r=0$.

Role in Research

Squeezed states appear in the quantum information program as a natural generalization of the vacuum for Bell inequality and relative entropy computations:

• Bell-CHSH violations for squeezed states: The Gaussian expectation value structure (modified relative to the vacuum by Bogoliubov coefficients) allows explicit computation of the CHSH correlator. Squeezed states can exhibit larger Bell violations than coherent states for optimized test functions.

• Relative entropy for squeezed states: The Araki-Uhlmann relative entropy $S({\omega }_{\text{sq}}\Vert {\omega }_0)$ is computable in closed form using the Bogoliubov coefficients and the one-particle Hilbert space structure. The result involves $\cosh r$ and $\sinh r$ factors from the squeezing.

• Connection to Unruh effect: the Minkowski vacuum restricted to the right Rindler wedge is a two-mode squeezed state in the Rindler particle basis, mode by mode, with $\tanh r_{\omega }=e^{-\pi \omega /a}$ for Rindler frequency $\omega$ and proper acceleration $a$ — the Boltzmann factor at half the Unruh temperature $a/2\pi$. The squeezing is a function of the mode rather than a single number, which is what makes the reduced state thermal.

Relations

• coherent-states — coherent states are displaced vacua; squeezed states are Bogoliubov-transformed vacua; both are Gaussian states with exactly computable expectation values
• cat-states — cat states are superpositions of coherent states; squeezed states are qualitatively different (Bogoliubov transformation rather than displacement); they play complementary roles in the program
• weyl-operators — Gaussian expectation values of Weyl operators in squeezed states are computable in closed form; the Bogoliubov transformation modifies the effective one-particle norm
• araki-uhlmann-relative-entropy — relative entropy between squeezed states and the vacuum is closed-form, involving the squeezing parameters; provides benchmarks for entanglement measures
• bell-chsh-inequality — CHSH correlators for squeezed states are computed explicitly; they generally exceed the vacuum correlator for appropriate test functions
• rindler-wedges — the Minkowski vacuum restricted to one Rindler wedge is a squeezed state in the Rindler basis; squeezing and Unruh temperature are directly related

Papers

See bell-inequalities-qft and relative-entropy-qft for full paper lists.

Notes

• Squeezed states are experimentally realized in quantum optics (optical parametric oscillators) and have applications in gravitational wave detection (LIGO uses squeezed light).
• In QFT, the two-mode squeezed structure of the vacuum across the Rindler horizon is exact and model-independent, following from the Bisognano-Wichmann theorem and the algebraic structure.
• Common confusion: a squeezed state is not the same as an entangled state in general. A single-mode squeezed state is not entangled (it is a pure state of one mode). A two-mode squeezed state is entangled across the two modes. In the QFT/Rindler context, it is always the two-mode version that is relevant.
• The Bogoliubov transformation relating Minkowski and Rindler particles connects squeezed-state language to the algebraic one: the mode-dependent $r_{\omega }$ is fixed by the spectrum of the wedge modular operator, which Bisognano–Wichmann identifies with $e^{-2\pi K}$ for the boost generator $K$.