Complex structures and quantization: squeezed states

Intuition [Beginner]

The quantum harmonic oscillator has equally spaced energy levels, like rungs on a ladder. You can climb the ladder one rung at a time using the "creation operator," which adds one quantum of energy. Descending uses the "annihilation operator." The ground state (bottom rung) is the vacuum, a Gaussian bell curve centered at the origin.

A coherent state is what you get by "pushing" the vacuum away from the origin without distorting its shape. It slides the Gaussian bell curve to a new center point in phase space while keeping the same width in all directions. Coherent states are the quantum states that behave most like classical particles: they follow the classical trajectory and minimise the uncertainty product.

A squeezed state goes further. Instead of just sliding the Gaussian, you stretch it in one direction and compress it in the perpendicular direction, like squeezing a balloon. The total uncertainty product stays at the minimum, but the individual uncertainties in position and momentum become unequal: you know one very precisely at the cost of knowing the other less precisely.

Visual [Beginner]

A phase-space diagram with position $q$ on the horizontal axis and momentum $p$ on the vertical axis. Three uncertainty blobs are shown: (1) a circular Gaussian (the vacuum/coherent state), (2) the same circle shifted to a new center (coherent state), and (3) an ellipse stretched along the $q$-axis and compressed along the $p$-axis (a squeezed state). Each blob has the same area (representing minimum uncertainty).

The squeezing transformation rotates and stretches the uncertainty circle into an ellipse while preserving phase-space area, which is the quantum analogue of a symplectic transformation preserving area.

Worked example [Beginner]

The vacuum state as a Gaussian. The ground state of the one-dimensional harmonic oscillator is ${\psi }_0(x)=(\alpha /\pi {)}^{1/4}{e}^{-\alpha x^2/2}$ where $\alpha =m\omega /\hslash$. Its position uncertainty is $\Delta x=1/\sqrt{2\alpha }$ and its momentum uncertainty is $\Delta p=\hslash \sqrt{\alpha /2}$. The product is $\Delta x\Delta p=\hslash /2$, which saturates the Heisenberg uncertainty bound.

When you squeeze this state by a factor $r>1$ in position, the position uncertainty shrinks to $\Delta x_{\mathrm{sq}}=\Delta x/r$ while the momentum uncertainty grows to $\Delta p_{\mathrm{sq}}=r\Delta p$. The product remains $\hslash /2$. The squeezed wave function becomes ${\psi }_{\mathrm{sq}}(x)=(\alpha r^2/\pi {)}^{1/4}{e}^{-\alpha r^2{x}^2/2}$, a narrower and taller Gaussian.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Complex structure on phase space). A complex structure on the symplectic vector space $({\mathbb{R}}^{2n},{\omega }_0)$ is a linear map $J:{\mathbb{R}}^{2n}\to {\mathbb{R}}^{2n}$ with $J^2=-\mathrm{id}$ that is compatible with ${\omega }_0$ (i.e., $g_J(u,v):={\omega }_0(u,Jv)$ defines a positive-definite inner product) and ${\omega }_0$-tame (i.e., ${\omega }_0(Ju,Jv)={\omega }_0(u,v)$). The standard choice is $J_0=\left(\begin{array}{cc}0& -I\\ I& 0\end{array}\right)$.

Definition (Creation and annihilation operators). In the complex coordinates $a=\frac{1}{\sqrt{2\hslash }}(\sqrt{m\omega }x+ip/\sqrt{m\omega })$, the annihilation operator $a$ and creation operator $a^{†}$ satisfy $[a,a^{†}]=1$. The vacuum $|0⟩$ is the unique state annihilated by $a$.

Definition (Squeezed state). A squeezed state is obtained by acting on the vacuum with a unitary operator from the metaplectic representation: $|z,S⟩=U_{S}D(z)|0⟩$, where $D(z)=\exp (z{a}^{†}-\bar{z}a)$ is the displacement operator (producing coherent states) and $U_S$ is the metaplectic operator corresponding to a squeezing symplectic matrix $S\in \mathrm{Sp}(2,\mathbb{R})$.

Key theorem with proof [Intermediate+]

Theorem (Bargmann transform and holomorphic representation). The Bargmann transform $\mathcal{B}:L^2(\mathbb{R})\to {\mathcal{H}}_{\mathrm{hol}}(\mathbb{C})$ defined by $(\mathcal{B}\psi )(z)={\pi }^{-1/4}{\int }_{\mathbb{R}}{e}^{-(z^2+x^2-2\sqrt{2}xz)/2}\psi (x)dx$ is a unitary isomorphism onto the Segal-Bargmann space of holomorphic functions on $\mathbb{C}$ square-integrable against the Gaussian measure $d\mu (z)={\pi }^{-1}{e}^{-|z{|}^2}{d}^{2}z$. Under this isomorphism, $a^{†}$ acts as multiplication by $z$ and $a$ acts as $d/dz$.

Proof sketch. The kernel $A(z,x)={\pi }^{-1/4}{e}^{-(z^2+x^2-2\sqrt{2}xz)/2}$ is holomorphic in $z$ for each $x$, and the generating function identity $\int |A(z,x){|}^2{e}^{-|z{|}^2}{d}^{2}z=|(m\omega /\pi \hslash {)}^{1/4}{e}^{-m\omega x^2/2\hslash }{|}^2$ shows $\mathcal{B}$ sends the vacuum to the constant function $1$. The creation and annihilation operators are verified by differentiating under the integral: $(\mathcal{B}(a^{†}\psi ))(z)=z(\mathcal{B}\psi )(z)$ and $(\mathcal{B}(a\psi ))(z)=d/dz(\mathcal{B}\psi )(z)$. Unitarity follows from the resolution of identity $\int |A(z,x)⟩⟨A(z,x^{\prime })|d\mu (z)=\delta (x-x^{\prime })$. $\square$

Bridge. The Bargmann transform implements the passage from the position representation to the holomorphic representation, converting the real symplectic structure of phase space into the complex-analytic structure of the Segal-Bargmann space ^{[Bargmann 1961]}; this is the analytic counterpart of the complex structure $J$ on $({\mathbb{R}}^{2n},{\omega }_0)$, where multiplication by $i$ in the holomorphic picture encodes the same data as $J$ in the symplectic picture, just as the Fourier transform converts position-space uncertainty into momentum-space uncertainty in the free-particle analysis of 09.03.03 pending. The creation operator becoming multiplication by $z$ is the holomorphic incarnation of the ladder structure, paralleling how the normal mode decomposition in classical mechanics separates the Hamiltonian into independent oscillators.

Exercises [Intermediate+]

Advanced results [Master]

Metaplectic representation. The symplectic group $\mathrm{Sp}(2n,\mathbb{R})$ has no faithful finite-dimensional unitary representations. The metaplectic group $\mathrm{Mp}(2n,\mathbb{R})$, the unique connected double cover of $\mathrm{Sp}(2n,\mathbb{R})$, admits a faithful unitary representation on $L^2({\mathbb{R}}^n)$ called the metaplectic representation (Segal-Shale-Weil). This representation sends the squeeze $S=\mathrm{diag}(r,r^{-1})$ to the integral operator $(U_S\psi )(x)=\sqrt{r}\psi (rx)$, and rotations in $\mathrm{Sp}(2n,\mathbb{R})$ to fractional Fourier transforms.

Weyl quantisation and the uncertainty principle. The Weyl quantisation map ${\mathrm{Op}}_W:C^{\infty }({\mathbb{R}}^{2n})\to \mathrm{Op}(L^2({\mathbb{R}}^n))$ sends the position observable $q$ to $\hat{q}=$ multiplication by $x$ and the momentum $p$ to $\hat{p}=-i\hslash d/dx$. The Robertson-Schrodinger uncertainty relation $\Delta A\Delta B\ge \frac{1}{2}|⟨[\hat{A},\hat{B}]⟩|$ is saturated precisely by Gaussian states, and squeezed states are the orbit of the vacuum under the metaplectic action, giving all minimum-uncertainty states up to displacement.

Synthesis. Complex structures and squeezed states lie at the intersection of symplectic geometry, complex analysis, and quantum physics; the Bargmann transform provides the bridge from the real symplectic picture to the holomorphic picture ^{[Bargmann 1961]}, converting the symplectic form ${\omega }_0$ into the Kähler form of the Segal-Bargmann space just as the complex structure $J$ on a symplectic manifold makes it Kähler in the Hodge theory of 03.02.12, while the metaplectic representation lifts the symplectic group action from classical phase space to the quantum Hilbert space, paralleling how the spin double cover lifts the rotation group from SO(3) to its quantum representation in 07.01.01. The squeezed states themselves are the Gaussian orbit of the vacuum under this metaplectic action, forming a family of minimum-uncertainty states parametrised by the coset space $\mathrm{Sp}(2n,\mathbb{R})/U(n)$, which is the Siegel upper half-space appearing in the Riemann bilinear relations of the Jacobian variety, connecting this quantum-mechanical construction back to the algebraic geometry of 04.09.01.

Full proof set [Master]

Proposition (Minimum uncertainty characterisation). A state $\psi \in L^2(\mathbb{R})$ saturates the Heisenberg uncertainty relation $\Delta x\Delta p=\hslash /2$ if and only if $\psi$ is a Gaussian, i.e., $\psi (x)=C\exp (-(a{x}^2+bx+c))$ with $\mathrm{Re}(a)>0$.

Proof. For observables $\hat{A},\hat{B}$ with $[\hat{A},\hat{B}]=iC$, the Cauchy-Schwarz inequality on the inner product $⟨(\hat{A}-⟨\hat{A}⟩)\psi ,(\hat{B}-⟨\hat{B}⟩)\psi ⟩$ gives $(\Delta A{)}^2(\Delta B{)}^2\ge \frac{1}{4}|⟨C⟩{|}^2+\frac{1}{4}|⟨\{\hat{A}-⟨\hat{A}⟩,\hat{B}-⟨\hat{B}⟩\}⟩{|}^2$. Equality requires $(\hat{B}-⟨\hat{B}⟩)\psi =\lambda (\hat{A}-⟨\hat{A}⟩)\psi$ for some $\lambda \in \mathbb{C}$. Setting $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$, this becomes $(-i\hslash d/dx-⟨p⟩)\psi =\lambda (x-⟨x⟩)\psi$, a first-order ODE whose unique $L^2$ solutions are Gaussians with $\lambda =i\hslash /(2(\Delta x{)}^2)$ and $\mathrm{Re}(\lambda )>0$. $\square$

Connections [Master]

• The complex structure $J$ on phase space is the same structure that makes a symplectic manifold into a Kähler manifold in 03.02.09; the Segal-Bargmann space is the quantised incarnation of the Kähler metric, with the Gaussian measure encoding the symplectic volume.

• The metaplectic double cover $\mathrm{Mp}(2n)\to \mathrm{Sp}(2n)$ is the symplectic analogue of the spin double cover $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ appearing in 03.09.02; both arise because quantisation requires passing to a double cover of the classical symmetry group.

• Squeezed states appear as Gaussian wave packets in the semiclassical limit of the Schrodinger equation, and their evolution under quadratic Hamiltonians is governed by the same Williamson normal form that classifies quadratic Hamiltonians in symplectic geometry 09.05.01 pending, connecting quantum squeezing back to classical symplectic linear algebra.

Bibliography [Master]

@article{bargmann1961,
  author = {Bargmann, Valentine},
  title = {On a Hilbert space of analytic functions and an associated integral transform},
  journal = {Comm. Pure Appl. Math.},
  volume = {14},
  pages = {187--214},
  year = {1961}
}

@book{de-gosson2006,
  author = {de Gosson, Maurice A.},
  title = {Symplectic Geometry and Quantum Mechanics},
  publisher = {Birkh{\"a}user},
  year = {2006}
}

@book{folland1989,
  author = {Folland, Gerald B.},
  title = {Harmonic Analysis in Phase Space},
  publisher = {Princeton University Press},
  year = {1989}
}

@book{hall2013-quantum,
  author = {Hall, Brian C.},
  title = {Quantum Theory for Mathematicians},
  publisher = {Springer},
  year = {2013}
}

Historical & philosophical context [Master]

The complex-variable approach to quantum mechanics was pioneered by Bargmann in 1961 ^{[Bargmann 1961]}, who constructed the holomorphic representation space now named after him. Bargmann recognised that the creation and annihilation operators take their simplest form in the holomorphic picture: creation is multiplication and annihilation is differentiation. This representation made rigorous the informal "complex wave function" methods that physicists had used since Dirac and Fock in the 1930s.

Squeezed states were introduced in quantum optics by Stoler in 1970 and developed by Yuen, Hollenhorst, and Caves in the late 1970s and early 1980s. They arise naturally in the metaplectic representation studied by Weil (1964), Shale (1962), and Segal (1963). The experimental generation of squeezed light by Slusher et al. in 1985 opened the field of quantum-enhanced measurement, culminating in the use of squeezed states in the LIGO gravitational-wave detector to improve sensitivity beyond the standard quantum limit.

Philosophically, squeezed states demonstrate that quantum uncertainty is not a single number but has directional structure in phase space. The Heisenberg bound constrains the product of uncertainties, but not their individual values. By squeezing uncertainty into an unwanted direction, one can achieve precision in the desired direction beyond what an isotropic (coherent) state allows, embodying the principle that quantum limits are resource bounds rather than absolute barriers.