Squeezed States: First Encounter

A squeezed state is an oscillator state whose uncertainty is reduced in one quadrature below the ground-state value while the uncertainty in the conjugate quadrature is increased. It is a controlled reshaping of quantum noise, not a violation of the uncertainty principle.

For a first encounter, the essential picture is:

• the vacuum or ground state has equal uncertainty in two conjugate quadratures;
• squeezing narrows one quadrature distribution;
• the conjugate quadrature broadens;
• pure ideal squeezed states can still be minimum-uncertainty states;
• Squeezed Light explains optical generation and measurement.

Helpful background. Ladder-Operator Solution: First Encounter to act with oscillator raising and lowering operators.

Dimensionless Quadratures

For the harmonic oscillator, define dimensionless quadrature operators

$$\hat Q = \frac{\hat a+\hat a^\dagger}{\sqrt2}, \qquad \hat P = \frac{\hat a-\hat a^\dagger}{i\sqrt2}.$$

They obey

$$[\hat Q,\hat P]=i.$$

In terms of the physical position and momentum,

$$\hat Q=\frac{\hat x}{\ell}, \qquad \hat P=\frac{\ell\hat p}{\hbar}, \qquad \ell=\sqrt{\frac{\hbar}{m\omega}}.$$

The oscillator ground state has

$$(\Delta Q)^2=(\Delta P)^2=\frac12,$$

or, in physical units,

$$\Delta x=\frac{\ell}{\sqrt2}, \qquad \Delta p=\frac{\hbar}{\sqrt2\,\ell}.$$

The word “quadrature” here means a phase-space component of the oscillator, not a numerical integration rule.

The Basic Squeeze Operator

With one common phase convention, a real squeezing parameter $r$ defines

$$\hat S(r) = \exp\!\left[ \frac r2 \left( \hat a^2-(\hat a^\dagger)^2 \right) \right].$$

The squeezed vacuum is

$$\lvert0;r\rangle = \hat S(r)\lvert0\rangle.$$

The sign of the squeezing follows from the generator, rather than from a convention-independent rule about positive $r$. Write $\hat S(r)=e^{r\hat G}$ with $\hat G=[\hat a^2-(\hat a^\dagger)^2]/2$, and define $\hat A(r)=\hat S^\dagger(r)\hat a\hat S(r)$. Since $[\hat a,\hat G]=-\hat a^\dagger$,

$$\frac{d\hat A}{dr}=-\hat A^\dagger, \qquad \frac{d\hat A^\dagger}{dr}=-\hat A, \qquad \hat A(0)=\hat a.$$

Solving these coupled equations gives

$$\hat S^\dagger(r)\hat a\hat S(r) =\hat a\cosh r-\hat a^\dagger\sinh r.$$

Adding and subtracting its adjoint produces the quadrature transformations

$$\hat S^\dagger(r)\hat Q\hat S(r) = e^{-r}\hat Q, \qquad \hat S^\dagger(r)\hat P\hat S(r) = e^{r}\hat P.$$

Therefore the squeezed-vacuum variances are

$$(\Delta Q)^2 = \frac12e^{-2r}, \qquad (\Delta P)^2 = \frac12e^{2r}.$$

Equivalently,

$$\Delta Q=\frac{e^{-r}}{\sqrt2}, \qquad \Delta P=\frac{e^{r}}{\sqrt2}.$$

The product remains

$$\Delta Q\,\Delta P=\frac12,$$

which corresponds to

$$\Delta x\,\Delta p=\frac{\hbar}{2}.$$

Thus the squeezed vacuum is still a minimum-uncertainty state in this ideal pure case. It is not a state with less uncertainty in every direction.

Phase-Space Picture

A useful first picture is an uncertainty ellipse in the $Q,P$ plane. The ground state is circular in these dimensionless coordinates. Squeezing turns the circle into an ellipse.

A schematic phase-space picture of single-mode squeezing. For $r\gt 0$ in the convention used here, the $Q$ width decreases while the $P$ width increases. A phase rotation can squeeze a different quadrature.

The ellipse orientation depends on the squeezing phase. A more general squeezed state can have its narrow axis at an angle in phase space. This page keeps the axes aligned with $Q$ and $P$ to make the uncertainty tradeoff visible.

Relation To Coherent States

A coherent state is a displaced ground-state Gaussian:

$$\lvert\alpha\rangle = \hat D(\alpha)\lvert0\rangle.$$

It has nonzero mean quadratures but ground-state variances:

$$(\Delta Q)^2=(\Delta P)^2=\frac12.$$

A squeezed state changes the variances. A displaced squeezed state combines both operations:

$$\lvert\alpha;r\rangle = \hat D(\alpha)\hat S(r)\lvert0\rangle.$$

Its center can follow classical-like oscillator motion while its noise ellipse is not circular. This distinction is why coherent states and squeezed states should not be used as synonyms. Coherence describes a special displacement and phase relation; squeezing describes anisotropic quadrature uncertainty.

Coherent-State Dynamics shows why rotations and linear drives preserve ordinary coherent states, whereas parametric quadratic terms generally generate squeezed coherent states.

Number-State Content

For $r\ne0$, the squeezed vacuum is neither a number state nor an ordinary coherent state. At $r=0$ it is the vacuum, which is both. In the number basis it contains only even occupation numbers:

$$\lvert0;r\rangle = \frac{1}{\sqrt{\cosh r}} \sum_{n=0}^{\infty} (-\tanh r)^n \frac{\sqrt{(2n)!}}{2^n n!} \lvert2n\rangle.$$

The absence of odd number states is a signature of pair structure in this ideal single-mode squeezed vacuum. In quantum-optics language, squeezing is often produced by interactions that create or annihilate excitations in pairs. That more detailed mechanism belongs in the quantum-optics and continuous-variable treatments.

The mean occupation number is

$$\langle\hat N\rangle = \sinh^2 r.$$

Consequently, the squeezed-vacuum mean energy is

$$\langle H\rangle = \hbar\omega \left( \sinh^2 r+\frac12 \right).$$

Reducing one quadrature noise below the vacuum value costs energy by increasing excitation content and anti-squeezing the conjugate quadrature.

Minimum Uncertainty, But Not Always

The ideal squeezed vacuum above satisfies

$$\Delta Q\,\Delta P=\frac12.$$

So do displaced squeezed states of the form $\hat D(\alpha)\hat S(r)\lvert0\rangle$. They are Gaussian pure states with a reshaped covariance ellipse.

Even an ideal pure squeezed Gaussian rotated away from the $Q,P$ axes can have $\Delta Q\Delta P>1/2$. In that case the covariance $C_{QP}=\langle\Delta Q\Delta P+\Delta P\Delta Q\rangle/2$ is nonzero, while $\det\Sigma=(\Delta Q)^2(\Delta P)^2-C_{QP}^2=1/4$. It is the covariance-corrected inequality that remains saturated.

Realistic or mixed squeezed states need not saturate that stronger bound either. Loss, thermal noise, imperfect mode matching, and phase noise can make

$$\Delta Q\,\Delta P\gt \frac12.$$

The phrase “squeezed below vacuum” therefore requires a specified quadrature, a specified reference vacuum, and a specified measurement convention.

Applications Preview

Squeezed states matter because many measurements are limited by fluctuations in a particular quadrature. If the signal is encoded in that quadrature, reducing its variance can improve sensitivity, provided the anti-squeezed quadrature does not leak back into the measurement.

Common settings include:

• optical and microwave parametric amplifiers;
• homodyne detection of field quadratures;
• gravitational-wave interferometry and precision metrology;
• continuous-variable quantum information;
• generation of two-mode entanglement from pair-correlated modes.

These applications are powerful but conditional. Useful squeezing depends on phase control, loss, detector efficiency, mode matching, and the full noise budget. The first oscillator lesson is the geometry of uncertainty; the engineering of useful squeezed light or squeezed motion is a later subject.

Common Mistakes

• Saying squeezing beats the uncertainty principle. It redistributes uncertainty between conjugate quadratures.
• Forgetting the anti-squeezed quadrature.
• Confusing a squeezed state with a coherent state.
• Calling a single-mode squeezed state bipartite entangled without specifying a bipartition.
• Quoting a squeezing parameter without stating the quadrature convention or phase.
• Assuming all states with reduced noise are pure minimum-uncertainty states.
• Treating “quadrature” in this context as numerical quadrature.

Exercises

1. Show that the ideal squeezed vacuum remains a minimum-uncertainty state.

Solution

For the convention used on this page,

$$\Delta Q=\frac{e^{-r}}{\sqrt2}, \qquad \Delta P=\frac{e^{r}}{\sqrt2}.$$

Therefore

$$\Delta Q\,\Delta P = \frac{e^{-r}e^{r}}{2} = \frac12.$$

In physical units, $\Delta x=\ell\Delta Q$ and $\Delta p=\hbar\Delta P/\ell$, so

$$\Delta x\,\Delta p=\frac{\hbar}{2}.$$

2. If $r=\ln 2$, what happens to $\Delta Q$ and $\Delta P$ relative to the ground-state values?

Solution

The ground-state values are

$$\Delta Q_0=\Delta P_0=\frac{1}{\sqrt2}.$$

For $r=\ln 2$,

$$e^{-r}=\frac12, \qquad e^r=2.$$

Thus

$$\Delta Q=\frac12\Delta Q_0, \qquad \Delta P=2\Delta P_0.$$

One quadrature width is halved, and the conjugate quadrature width is doubled.

3. In the squeezed vacuum with $r=\ln2$, measure the rotated quadratures $\hat Q_\theta=\hat Q\cos\theta+\hat P\sin\theta$ and $\hat P_\theta=-\hat Q\sin\theta+\hat P\cos\theta$ at $\theta=\pi/4$. Find their variances and symmetrized covariance. Does an uncertainty product greater than $1/2$ imply that this state is mixed?

Solution

The original variances are $v_Q=1/8$ and $v_P=2$, with zero covariance. The means vanish before and after the rotation. Consequently,

$$\begin{aligned} (\Delta Q_\theta)^2&=v_Q\cos^2\theta+v_P\sin^2\theta,\\ (\Delta P_\theta)^2&=v_Q\sin^2\theta+v_P\cos^2\theta,\\ C_\theta&=(v_P-v_Q)\sin\theta\cos\theta. \end{aligned}$$

At $\theta=\pi/4$, both variances equal $17/16$ and $C_\theta=15/16$. Thus $\Delta Q_\theta\Delta P_\theta=17/16>1/2$, even though the state is pure. The covariance-corrected relation gives

$$(\Delta Q_\theta)^2(\Delta P_\theta)^2-C_\theta^2 =\left(\frac{17}{16}\right)^2-\left(\frac{15}{16}\right)^2 =\frac14.$$

The state still saturates the stronger bound. Both measured widths exceed the vacuum width because these axes lie between the narrow and broad principal axes; rotating the measurement back recovers the squeezed quadrature.

4. Explain why a squeezed state with smaller $\Delta Q$ than the vacuum does not imply a smaller total quantum uncertainty.

Solution

For the ideal squeezed vacuum,

$$\Delta Q=\frac{e^{-r}}{\sqrt2}, \qquad \Delta P=\frac{e^r}{\sqrt2}.$$

If $r\gt 0$, then $\Delta Q$ is smaller than the vacuum value, but $\Delta P$ is larger by the reciprocal factor. The product remains $\Delta Q\,\Delta P=1/2$. Squeezing has redistributed uncertainty between conjugate quadratures rather than removing it.

References

• C. M. Caves, “Quantum-mechanical noise in an interferometer,” Physical Review D 23, 1693-1708, 1981.
• C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005.
• J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific, 1985.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• H. P. Yuen, “Two-photon coherent states of the radiation field,” Physical Review A 13, 2226-2243, 1976.
• B. Zwiebach, “Quantum Dynamics”, 8.05 Quantum Physics II, MIT OpenCourseWare, Fall 2013, §5.