Squeezed States of Light

A single mode of the quantized electromagnetic field is, formally, a harmonic oscillator: field quantization hands us a pair of conjugate quadratures $\hat X_1$ and $\hat X_2$ — the in-phase and out-of-phase amplitudes — obeying $[\hat X_1,\hat X_2]=\tfrac{i}{2}$ , the position–momentum structure carried over from mechanics. From that one commutator the Heisenberg relation follows, $\Delta X_1\,\Delta X_2\ge\tfrac14$ . The vacuum and every coherent (laser) state sit at the floor of this bound with equal noise $\Delta X_1=\Delta X_2=\tfrac12$ — the isotropic "ball" of zero-point fluctuation that fills phase space even with no photon present. A squeezed state deforms that ball without inflating it: one quadrature is pushed below $\tfrac12$ at the exact cost of the conjugate rising above it, the noise circle flattened into an ellipse of the same area. The deformation is intrinsically quantum and pairwise. Its generator is quadratic in $\hat a$ and $\hat a^\dagger$ , making and destroying photons two at a time, which is why the natural hardware is a parametric process: in degenerate down-conversion one pump photon at $2\omega$ splits into twins at $\omega$ , whose phase correlation drives one quadrature's variance below the vacuum. The result is a genuinely nonclassical resource — sub-vacuum noise no classical field can mimic — and now a working technology, as Advanced LIGO and Virgo inject squeezed vacuum into their dark ports to beat the shot-noise limit and hear more of the sky.

Key takeaways

A squeezed state redistributes a mode's quantum noise between its quadratures $X_1$ and $X_2$ : one variance drops below the vacuum floor $\tfrac14$ while the conjugate rises to compensate.
It is generated by the unitary squeeze operator $\hat S(\xi)=\exp[\tfrac12(\xi^*\hat a^2-\xi\hat a^{\dagger2})]$ , enacting a Bogoliubov transformation $\hat a\to\hat a\cosh r-\hat a^\dagger e^{i\theta}\sinh r$ — quadratic, so photons come in pairs.
The uncertainty product stays at its minimum $\Delta X_1\,\Delta X_2=\tfrac14$ with unequal factors $e^{-2r},e^{+2r}$ ; the state has no positive Glauber–Sudarshan $P$ function, the mark of nonclassicality.
It is made by phase-sensitive amplification — parametric down-conversion in a $\chi^{(2)}$ crystal or four-wave mixing — and read out by homodyne detection, whose local-oscillator phase selects the quadrature.
Optical loss is the enemy: $V=\eta V_{\text{sq}}+(1-\eta)\tfrac14$ , which is why LIGO fields about $6$ dB and lab records reach only about $15$ dB.
Key idea: squeezing is the unitary, photon-pair-generating deformation of the vacuum noise circle into an ellipse of the same area — $\Delta X_1\,\Delta X_2=\tfrac14$ is preserved, but $\Delta X_1=\tfrac12 e^{-r}$ falls below the vacuum while $\Delta X_2=\tfrac12 e^{+r}$ rises: a redistribution, not a violation, of the uncertainty principle.

The core insight

Squeezing is the unitary, photon-pair-generating deformation of the vacuum noise circle into an ellipse of the same area: $\Delta X_1\,\Delta X_2=\tfrac14$ is preserved, but $\Delta X_1=\tfrac12 e^{-r}$ falls below the vacuum while $\Delta X_2=\tfrac12 e^{+r}$ rises — a nonclassical redistribution, not a violation, of the uncertainty principle.

Where this lives today

The clearest home of squeezed light is the global network of gravitational-wave detectors. Advanced LIGO, Virgo, and KAGRA now run with squeezed vacuum injected into the antisymmetric (dark) port through a Faraday isolator, the ellipse oriented to suppress photon shot noise at the high frequencies where they hear merging black holes and neutron stars. The observed $6$ dB surveys a larger volume of the universe and yields more events per year. The frontier subtlety is that squeezing the phase quadrature increases amplitude noise, which pushes on the mirrors as radiation-pressure back-action and hurts low-frequency sensitivity. The deployed fix is frequency-dependent squeezing: the beam first reflects off a long filter cavity that rotates the ellipse with frequency, giving phase squeezing where shot noise dominates and amplitude squeezing where back-action does — broadband enhancement across the whole band. Beyond gravitational waves, squeezed and two-mode-squeezed light is the raw material of continuous-variable quantum information: below-threshold optical parametric oscillators pumped by $\chi^{(2)}$ crystals feed teleportation, Gaussian cluster-state computing, and quantum key distribution, while amplitude-squeezed beams sharpen magnetometry and biological imaging where a bright probe would damage the sample.

The physical picture

Start with the vacuum. Even the empty mode carries irreducible fluctuation, and in the phase space spanned by $X_1$ and $X_2$ that fluctuation is a featureless disk centred on the origin, the same spread $\tfrac12$ in every direction. A coherent state — the quantum analogue of a laser beam — is that identical disk slid off to a point $\alpha$ , its noise still isotropic and at the Heisenberg floor. Squeezing squashes the disk: quieter than vacuum along one direction and, by the exact compensating amount, noisier along the perpendicular one. The area is conserved, so the state stays minimum-uncertainty — it has simply chosen an axis to be quiet on.

Figure

Vacuum noise disk of radius ½ squashed into a squeezed ellipse of equal area. For 6 dB squeezing r = 0.69 the squeezed quadrature narrows to ΔX₁ = ½e⁻ʳ = 0.25 and the anti-squeezed one stretches to ΔX₂ = ½eʳ = 1.00, so the product ΔX₁·ΔX₂ = ¼ is unchanged

Where does the pairwise character come in? The squeeze operator is quadratic in the creation operator, so on the vacuum it makes photons two at a time — never one. Those twins are born with correlated phases, and that correlation lets one quadrature's fluctuations partly cancel, dropping its variance below shot noise. It is why the squeezed vacuum holds only even photon numbers and why its mean $\sinh^2 r$ is nonzero even though $\langle\hat a\rangle$ vanishes: pure paired fluctuation energy, not a classical amplitude. To see the ellipse you use homodyne detection: beat the squeezed field against a bright local oscillator whose phase $\phi$ you dial, measuring $\hat X_\phi=\hat X_1\cos\phi+\hat X_2\sin\phi$ . Sweeping $\phi$ scans the noise around the ellipse — dipping below shot noise near the squeezed axis, swelling above it a quarter-turn later — the fingerprint of a squeezed state.

Figure

Homodyne noise power versus local-oscillator phase φ for an ideal 10 dB squeezed vacuum r = 1.15. The variance dips to −10 dB below the shot-noise floor at the squeezed quadrature φ = 0° and 180°, and rises to +10 dB at the anti-squeezed quadrature φ = 90° and 270°

Because the quiet quadrature carries less noise than any classical field can, squeezing is a true quantum resource: aim its narrow axis along the quantity you measure — a phase, a displacement — and you read it more precisely than the standard quantum limit shot noise imposes on ordinary laser light.

The core equations

Everything begins with the two quadratures and the commutator they obey. They are the Hermitian real and imaginary parts of $\hat a$ , and because $\hat a$ and $\hat a^\dagger$ do not commute, neither do they — that one non-zero commutator is the whole source of the noise floor squeezing works against.

\hat X_1=\tfrac12(\hat a+\hat a^\dagger),\qquad \hat X_2=\tfrac{1}{2i}(\hat a-\hat a^\dagger),\qquad [\hat X_1,\hat X_2]=\tfrac{i}{2} \qquad (1)

where: $\hat a,\hat a^\dagger$ — annihilation and creation operators with $[\hat a,\hat a^\dagger]=1$ ; $\hat X_1,\hat X_2$ — in-phase and quadrature amplitudes, the optical analogues of position and momentum. Reading it: two orthogonal amplitudes of the oscillating field, and no state can sharpen both at once. The commutator forces a Heisenberg bound on the product of the spreads — the wall every state stays behind, and the one squeezing exploits rather than breaks.

\Delta X_1\,\Delta X_2\ge\tfrac14 \qquad (2)

where: $\Delta X_i=\sqrt{\langle\hat X_i^2\rangle-\langle\hat X_i\rangle^2}$ — the spread of quadrature $i$ . Reading it: vacuum and coherent states hit this bound with $\Delta X_1=\Delta X_2=\tfrac12$ ; pushing one spread below $\tfrac12$ forces the conjugate above it. To build such a state we need an operator quadratic in $\hat a$ and $\hat a^\dagger$ — the algebraic statement that photons are handled in pairs. Its parameters are a magnitude and a phase.

\hat S(\xi)=\exp\!\Big[\tfrac12\big(\xi^*\hat a^2-\xi\,\hat a^{\dagger 2}\big)\Big],\qquad \xi=r\,e^{i\theta} \qquad (3)

where: $r\ge0$ — squeeze magnitude, how far below vacuum one quadrature drops; $\theta$ — squeeze phase, which axis is squeezed. Reading it: $\hat a^{\dagger2}$ creates a pair, $\hat a^2$ destroys one, and the anti-Hermitian combination makes $\hat S$ unitary. Conjugating the mode with $\hat S$ gives the workhorse identity — a Bogoliubov transformation mixing $\hat a$ with $\hat a^\dagger$ , from which every variance below flows.

\hat S^\dagger(\xi)\,\hat a\,\hat S(\xi)=\hat a\cosh r-\hat a^\dagger e^{i\theta}\sinh r \qquad (4)

where: $\cosh r,\sinh r$ — mixing weights tied by $\cosh^2 r-\sinh^2 r=1$ so $[\hat a,\hat a^\dagger]=1$ survives. Reading it: the $\hat a^\dagger$ admixture is the pair-creation that fills the vacuum with photons. Feed the vacuum through $\hat S$ and the disk becomes an ellipse; with $\theta=0$ the axes align with $X_1$ and $X_2$ .

|\xi\rangle=\hat S(\xi)\,|0\rangle,\qquad \Delta X_1=\tfrac12 e^{-r},\qquad \Delta X_2=\tfrac12 e^{+r}\quad(\theta=0) \qquad (5)

where: $|\xi\rangle$ — the squeezed vacuum. Reading it: the exponentials cancel, $\Delta X_1\Delta X_2=\tfrac14$ — still minimum-uncertainty, just anisotropic. Displacing it, $|\alpha,\xi\rangle=\hat D(\alpha)\hat S(\xi)|0\rangle$ with $\hat D(\alpha)=e^{\alpha\hat a^\dagger-\alpha^*\hat a}$ , slides the ellipse to a mean amplitude $\alpha$ for a bright squeezed beam. Though the mean field is zero, pair-creation leaves real photons behind, on even counts only.

\bar n=\langle\hat a^\dagger\hat a\rangle=\sinh^2 r,\qquad P(2m)\ne0,\quad P(2m+1)=0 \qquad (6)

where: $\bar n$ — mean photon number; $P(n)$ — probability of $n$ photons. Reading it: at $10$ dB, $r=1.15$ and $\bar n\approx2.0$ photons, all born in pairs — no odd count ever occurs. What a homodyne detector reports is the rotated-quadrature variance as the phase $\phi$ turns — the plotted curve above, fit to extract $r$ .

V(\phi)=\tfrac14\big(e^{-2r}\cos^2\phi+e^{+2r}\sin^2\phi\big)\quad(\theta=0) \qquad (7)

where: $\phi$ — local-oscillator phase selecting $\hat X_\phi=\hat X_1\cos\phi+\hat X_2\sin\phi$ ; $V(\phi)$ — measured variance. Reading it: at $\phi=0$ the noise is $\tfrac14 e^{-2r}$ , below vacuum; a quarter-turn later it peaks at $\tfrac14 e^{+2r}$ . Loss degrades this curve, admixing vacuum in proportion to the inefficiency.

V_{\mathrm{det}}=\eta\,V_{\mathrm{sq}}+(1-\eta)\,\tfrac14 \qquad (8)

where: $\eta$ — overall detection efficiency; $V_{\mathrm{sq}}$ — intrinsic squeezed variance. Reading it: each lost photon is replaced by a vacuum fluctuation $\tfrac14$ , pulling measured noise toward shot noise — the chief obstacle to large observed squeezing.

Derivation AFrom the squeeze operator to the Bogoliubov transformation

The goal is to prove equation (4) — that conjugating $\hat a$ by the squeeze operator gives the linear mixture $\hat a\cosh r-\hat a^\dagger e^{i\theta}\sinh r$ — since this one identity underlies every quantitative claim about squeezed light: the ellipse variances, the mean photon number, the homodyne curve. We use the Baker–Hausdorff (Hadamard) expansion, computing the ladder of nested commutators and showing they resum into the hyperbolic cosine and sine series. Take $\hat G=\tfrac12(\xi\hat a^{\dagger2}-\xi^*\hat a^2)$ so $\hat S=e^{\hat G}$ and $\hat S^\dagger\hat a\hat S=\hat a+[\hat G,\hat a]+\tfrac{1}{2!}[\hat G,[\hat G,\hat a]]+\cdots$ .

Step 1. Evaluate the first commutator with $[\hat a,\hat a^{\dagger2}]=2\hat a^\dagger$ and $[\hat a,\hat a^2]=0$ : $[\hat G,\hat a]=-\xi\,\hat a^\dagger$ and, by conjugation, $[\hat G,\hat a^\dagger]=-\xi^*\hat a$ . Why: the quadratic generator turns $\hat a$ into $\hat a^\dagger$ and back, so the ladder never leaves the span of $\{\hat a,\hat a^\dagger\}$ — that closure makes the transformation exactly linear.

Step 2. Nest once more: $[\hat G,[\hat G,\hat a]]=|\xi|^2\hat a=r^2\hat a$ , so the third commutator returns $-\xi r^2\hat a^\dagger$ , and so on. Why: even-order terms point along $\hat a$ with coefficients $r^{2k}/(2k)!$ and odd-order along $\hat a^\dagger$ with $r^{2k+1}/(2k+1)!$ — the exact Taylor series of $\cosh r$ and $\sinh r$ .

Step 3. Resum: the $\hat a$ terms give $\hat a\cosh r$ , the $\hat a^\dagger$ terms $-e^{i\theta}\hat a^\dagger\sinh r$ , so $\hat S^\dagger\hat a\hat S=\hat a\cosh r-e^{i\theta}\hat a^\dagger\sinh r$ . Why: this is equation (4); the phase $e^{i\theta}$ rides entirely on the pair-creation term, which is why $\theta$ rotates the ellipse without changing its shape.

Step 4. Check the map is canonical: $[\hat S^\dagger\hat a\hat S,\hat S^\dagger\hat a^\dagger\hat S]=\cosh^2 r-\sinh^2 r=1$ . Why: only a commutator-preserving map is physical; the hyperbolic identity keeps the squeezed mode a legitimate oscillator — the algebraic shadow of "area preserved in phase space".

Go deeper · Derivation B · 1 minThe squeezed-vacuum ellipse and its preserved area

Derivation BThe squeezed-vacuum ellipse and its preserved area

The goal is to turn the operator identity (4) into the measurable statement (5): the vacuum's isotropic noise $\tfrac12$ becomes $\tfrac12 e^{-r}$ along one quadrature and $\tfrac12 e^{+r}$ along the other, product locked at $\tfrac14$ . We work in the Heisenberg picture — transform the quadratures with $\hat S$ , then take vacuum expectation values, legitimate because $\hat S^\dagger\hat X\hat S$ on $|0\rangle$ equals $\hat X$ on $|\xi\rangle=\hat S|0\rangle$ . Set $\theta=0$ so the squeezed axes coincide with $X_1$ and $X_2$ .

Step 1. Transform the mode with $\theta=0$ : $\hat S^\dagger\hat a\hat S=\hat a\cosh r-\hat a^\dagger\sinh r$ , conjugate $\hat S^\dagger\hat a^\dagger\hat S=\hat a^\dagger\cosh r-\hat a\sinh r$ . Why: both transformed ladder operators let us assemble the quadratures directly, since $\hat X_1,\hat X_2$ are their symmetric and antisymmetric combinations.

Step 2. Add and subtract: $\hat S^\dagger\hat X_1\hat S=\hat X_1(\cosh r-\sinh r)=\hat X_1 e^{-r}$ and $\hat S^\dagger\hat X_2\hat S=\hat X_2 e^{+r}$ . Why: the quadratures are the eigen-directions of the squeeze — each rescaled by $e^{\mp r}$ , a pure stretch-and-squeeze with no rotation when $\theta=0$ .

Step 3. Take vacuum expectation values with $\langle0|\hat X_i|0\rangle=0$ , $\langle0|\hat X_i^2|0\rangle=\tfrac14$ : then $\langle\xi|\hat X_1^2|\xi\rangle=\tfrac14 e^{-2r}$ and $\langle\xi|\hat X_2^2|\xi\rangle=\tfrac14 e^{+2r}$ . Why: the vacuum supplies the isotropic seed noise $\tfrac14$ ; the squeeze only reshapes the fluctuation already there, it does not create it.

Step 4. Read off $\Delta X_1=\tfrac12 e^{-r}$ , $\Delta X_2=\tfrac12 e^{+r}$ , so $\Delta X_1\Delta X_2=\tfrac14$ . Why: the exponentials cancel exactly — the state stays minimum-uncertainty, the ellipse keeps the vacuum circle's area, and squeezing redistributes noise rather than removing it.

Go deeper · Derivation C · 1 minWhy squeezed vacuum beats the standard quantum limit

Derivation CWhy squeezed vacuum beats the standard quantum limit

The goal is to show how injecting squeezed vacuum into the unused ("dark") port of an interferometer sharpens a phase measurement from the standard quantum limit $\Delta\phi_{\text{SQL}}=1/(2\sqrt N)$ to $\Delta\phi=e^{-r}/(2\sqrt N)$ — the principle behind squeezing-enhanced gravitational-wave detection. The key insight, due to Caves, is that an interferometer's phase noise comes not from the laser but from vacuum leaking through the empty port; if that is the problem, replace it with something quieter.

Step 1. Baseline: a coherent beam of $N=|\alpha|^2$ photons splits and recombines, the differential photocurrent measuring the phase with resolution $\Delta\phi_{\text{SQL}}=1/(2\sqrt N)$ . Why: this $1/\sqrt N$ scaling is the benchmark laser interferometry cannot beat, and what more power buys — slowly, since halving the noise costs four times the power.

Step 2. Locate the noise: track the vacuum entering the dark port and find that its fluctuations, not the laser's amplitude noise, dominate the output phase uncertainty. Why: the limiting noise is a property of an empty port we control, so the fix is to engineer that input, not build a brighter laser.

Step 3. Substitute squeezed vacuum with its quiet quadrature $\Delta X_{\text{dark}}=\tfrac12 e^{-r}$ aligned to the signal; the fluctuation feeding the phase quadrature drops by $e^{-r}$ . Why: the signal rides one quadrature, so squeezing exactly that one and dumping the cost on the signal-free axis is a free lunch tailored to the measurement.

Step 4. Collect: $\Delta\phi=e^{-r}/(2\sqrt N)$ . For $10$ dB, $e^{-r}=0.32$ , a threefold improvement. Why: matching that with brighter light alone would take about ten times the power, which is why LIGO injects squeezing instead of cranking up the laser.

Limiting cases

The single knob is the squeeze magnitude $r$ ( $\theta$ fixes orientation); its limits map the landscape.

Case	Result	Meaning
No squeezing	$r\to0$	$\hat S(\xi)\to\hat I$ ; reduces to vacuum or coherent state, isotropic $\Delta X_1=\Delta X_2=\tfrac12$ , $\bar n\to0$ .
Strong squeezing	$r\to\infty$	one variance $\to0$ (quadrature eigenstate), while the conjugate and $\bar n=\sinh^2 r$ diverge — an infinite-energy idealization.
Decibel scale	$S_{\mathrm{dB}}=-10\log_{10}e^{-2r}$	$3$ dB $\to r\approx0.35$ ; $6$ dB $\to r\approx0.69$ ; $10$ dB $\to r\approx1.15$ ; records reach about $15$ dB ( $r\approx1.73$ ).
Quadrature vs number	$\Delta n^2\ \lessgtr\ \bar n$	distinct: an amplitude-squeezed beam can be sub-Poissonian ( $Q<0$ ), yet the squeezed vacuum is super-Poissonian.
Loss-limited	$\eta\to1$ needed	with $V_{\text{sq}}\to0$ the floor is $V_{\text{det}}=(1-\eta)\tfrac14$ : seeing $10$ dB needs $\eta\ge0.90$ whatever the source.

Common trapSqueezing does not beat the Heisenberg limit — it only moves the noise

The misconception is that a sub-vacuum quadrature beats the uncertainty principle. It does not: the product $\Delta X_1\,\Delta X_2$ stays at or above $\tfrac14$ , and the ideal squeezed vacuum sits exactly on that bound, trading $\Delta X_1=\tfrac12 e^{-r}$ against $\Delta X_2=\tfrac12 e^{+r}$ . You buy quiet in one quadrature only by paying at least as much in the conjugate. A second, practical trap: the squeezing you observe is never the squeezing you made, because every beamsplitter, window, and imperfect photodiode mixes in vacuum. Since $V_{\text{det}}=\eta V_{\text{sq}}+(1-\eta)\tfrac14$ , even small loss pins the result near the vacuum floor — the reason pushing past $10$ dB demands efficiencies above $90\%$ and is experimentally brutal.

Worked examples

1 · Reading 6 dB of LIGO squeezing (confidence builder)

A detector reports $6$ dB of injected squeezing. Find $r$ , the two quadrature spreads, their product, the mean photon number, and the amplitude-noise reduction.

Step 1. Decibels refer to noise power relative to shot noise, $S_{\text{dB}}=-10\log_{10}e^{-2r}$ , so $e^{-2r}=10^{-0.6}=0.251$ , giving $2r=1.382$ and $r=\mathbf{0.691}$ .

Step 2. Spreads from (5): $\Delta X_1=\tfrac12 e^{-r}=\tfrac12(0.501)=\mathbf{0.251}$ and $\Delta X_2=\tfrac12 e^{+r}=\tfrac12(1.995)=\mathbf{0.998}$ — squeezed just below the vacuum $0.50$ , anti-squeezed twice it.

Step 3. Product: $\Delta X_1\,\Delta X_2=0.251\times0.998=\mathbf{0.250}=\tfrac14$ — still minimum-uncertainty, only reshaped.

Step 4. Mean photons from (6): $\bar n=\sinh^2(0.691)=(0.747)^2=\mathbf{0.56}$ per mode — nonzero though $\langle\hat a\rangle=0$ , pure paired fluctuation energy. The amplitude-noise reduction is $e^{-r}=0.50$ : noise halved, the factor $2$ that $6$ dB in power ( $10^{0.6}\approx4$ ) implies. r = 0.69, ΔX₁ = 0.25, ΔX₂ = 1.00, product = ¼, n̄ = 0.56, noise halved.

2 · How optical loss eats 10 dB of squeezing

A source makes $10$ dB of pure squeezing ( $V_{\text{sq}}=\tfrac14\times0.1$ ), detected with efficiency $\eta=0.80$ . What is observed, and what efficiency would show the full $10$ dB even from a perfect source?

Step 1. Intrinsic variance: $V_{\text{sq}}=\tfrac14\times0.1=0.025$ , one-tenth of the shot-noise floor $0.25$ .

Step 2. Loss formula (8): $V_{\text{det}}=0.80(0.025)+0.20(0.25)=\mathbf{0.070}$ — the vacuum term $0.050$ now dwarfs the surviving squeezed term $0.020$ .

Step 3. Back to dB: $V_{\text{det}}/0.25=0.28$ , so $-10\log_{10}(0.28)=\mathbf{5.5}$ dB — twenty percent loss threw away nearly half the decibels.

Step 4. For the ceiling, take $V_{\text{sq}}\to0$ : then $V_{\text{det}}=(1-\eta)\tfrac14$ , and reaching $10$ dB needs $\eta\ge\mathbf{0.90}$ — no source shows $10$ dB through more than $10\%$ loss. Observed 5.5 dB at 80% efficiency; even infinite squeezing needs η ≥ 0.90 to display 10 dB.

3 · The power a squeezer replaces in an interferometer

An interferometer runs at the standard quantum limit with $N$ photons. Inject $6$ dB of squeezing aligned to the signal quadrature. By what factor does phase resolution improve, and how much extra laser power would match it without squeezing?

Step 1. From Derivation 3, squeezing scales the phase uncertainty by $e^{-r}$ . For $6$ dB, $e^{-r}=0.50$ , so $\Delta\phi$ halves — a twofold improvement.

Step 2. By power alone sensitivity improves only as $1/\sqrt N$ , so halving $\Delta\phi$ needs $N'=4N$ : four times the circulating power.

Step 3. Scaling up, $10$ dB ( $e^{-r}=0.32$ ) would take $(1/0.32)^2\approx\mathbf{10\times}$ the power to match — why squeezing, not brute-force power, is the tool of choice when circulating power is capped by thermal mirror distortion. 6 dB = 2× resolution = 4× power; 10 dB = 10× power.

Physics of the portrait

Squeezed light is a rare corner of physics where a single 1981 paper reframed both the theory and the technology. The portrait belongs to the theorist who showed that the quantum noise limiting an interferometer comes from vacuum entering the dark port — and that squeezed vacuum could defeat it.

Origins · 1 minCarlton Caves and how this was discovered

Carlton Caves

b. 1950 · quantum optics and the quantum limits of measurement

Carlton M. Caves is an American theoretical physicist whose work on the quantum limits of measurement turned squeezed light from a curiosity into a tool. In his 1981 paper Quantum-mechanical noise in an interferometer (Phys. Rev. D 23, 1693), Caves resolved a long-standing puzzle: the shot noise limiting a laser interferometer originates not in the laser's amplitude fluctuations but in the vacuum leaking through the unused port. The decisive corollary — that injecting squeezed vacuum into that port pushes phase sensitivity below the standard quantum limit — is the exact scheme Advanced LIGO adopted four decades later. Caves also helped formalize the theory of squeezed and two-photon states alongside Horace Yuen and clarified quantum non-demolition measurement. His framing — that the vacuum itself is the adjustable input you engineer — remains how physicists think about quantum-enhanced metrology.

History

1927

— Heisenberg's uncertainty relation fixes the vacuum quadrature noise floor at ΔX₁ΔX₂ ≥ ¼

1963

— Glauber and Sudarshan formulate coherent states and the P representation

1976

— Yuen introduces two-photon coherent states, the theory of squeezed light

1981

— Caves shows squeezed vacuum in a dark port beats the standard quantum limit

1985

— Slusher and co-workers observe squeezing via four-wave mixing in sodium vapour

1986

— Kimble's group generates strong squeezing with a sub-threshold optical parametric oscillator

2019

— Advanced LIGO and Virgo run their observing campaigns with squeezed-vacuum injection

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 5 and 7 · Walls and Milburn, Quantum Optics, 2nd ed., ch. 2 and 7 · Gerry and Knight, Introductory Quantum Optics, ch. 7 (squeeze operator and Bogoliubov transformation) · Saleh and Teich, Fundamentals of Photonics, 2nd ed., ch. 13 · Bachor and Ralph, A Guide to Experiments in Quantum Optics (homodyne detection and loss).

Provenance. A verifier should recompute: (1) the Bogoliubov coefficients from the Hadamard series for $\hat S^\dagger\hat a\hat S$ (Loudon §7.4; Gerry and Knight §7.1); (2) the variances $\tfrac12 e^{\mp r}$ and their product $\tfrac14$ ; (3) $\bar n=\sinh^2 r$ (Walls and Milburn §2.7); (4) the decibel conversions, $3$ dB $\leftrightarrow r=0.35$ , $6$ dB $\leftrightarrow r=0.69$ , $10$ dB $\leftrightarrow r=1.15$ ; (5) the loss formula with the worked figures $5.5$ dB at $\eta=0.8$ and $\eta\ge0.90$ for a $10$ dB floor. Reported experimental numbers (LIGO about $6$ dB, lab record about $15$ dB) are cited, not derived, and the interferometry bound is Caves, Phys. Rev. D 23, 1693 (1981).

Prerequisites. Quadrature Operators and the Uncertainty Relation · Coherent States · Vacuum State and Zero-Point Energy · Field Quantization. Leads to. Spontaneous Parametric Down-Conversion · Phase-Space Quasiprobability Distributions · Photon Statistics and the Mandel Q Parameter.

Ask the tutor: Curious how the filter cavity rotates the squeezing ellipse to fight shot noise and radiation-pressure noise at once — ask me to walk through frequency-dependent squeezing.

Does squeezed light break the Heisenberg uncertainty principle?

No — it obeys it exactly. The principle constrains the product $\Delta X_1\,\Delta X_2$ , not either factor alone, and squeezing keeps that product at its minimum $\tfrac14$ . It only changes the balance: $\Delta X_1$ drops to $\tfrac12 e^{-r}$ , below the vacuum's $\tfrac12$ , at the reciprocal cost of $\Delta X_2$ rising to $\tfrac12 e^{+r}$ — a redistribution of noise, not its removal. The trick pays off because most measurements read only one quadrature, so the extra noise is dumped onto the signal-free axis.

What physically makes light squeezed — where does the ellipse come from?

It comes from a nonlinear process that makes photons in correlated pairs — usually degenerate parametric down-conversion in a $\chi^{(2)}$ crystal or four-wave mixing. A pump photon at $2\omega$ splits into two at $\omega$ , and because the pair shares one parent its members have tightly correlated phases — exactly the content of the squeeze operator's $\hat a^{\dagger2}$ term. Those correlated pairs interfere so that noise along one quadrature partly cancels (dropping below shot noise) while it adds along the other, and the pump phase fixes $\theta$ , setting which axis ends up quiet.

Why does loss destroy squeezing so easily, when it barely affects a laser?

Because squeezing lives in the absence of noise, and loss can only add noise back. Model any loss as a beamsplitter of transmission $\eta$ : it passes a fraction $\eta$ of the squeezed field and lets in $1-\eta$ of vacuum, so $V_{\text{det}}=\eta V_{\text{sq}}+(1-\eta)\tfrac14$ . A bright laser has huge signal, so admixed vacuum is negligible; but a squeezed state's whole value is a variance below $\tfrac14$ , and the $(1-\eta)\tfrac14$ term pulls it back toward the floor. This is why every optic in a squeezing setup is chosen for the last fraction of a percent of efficiency.

What is the difference between quadrature squeezing and photon-number (sub-Poissonian) squeezing?

They are distinct nonclassical properties that do not imply each other. Quadrature squeezing means one quadrature's variance is below $\tfrac14$ ; photon-number squeezing (sub-Poissonian, Mandel $Q<0$ ) means the count variance is below the mean, $\Delta n^2<\bar n$ . The squeezed vacuum shows the gap sharply: quadrature-squeezed by construction yet strongly super-Poissonian, its counts a broad even-only distribution with $\Delta n^2>\bar n$ ; conversely an amplitude-squeezed bright beam can be sub-Poissonian. "Squeezed light" almost always means quadrature squeezing; number squeezing is a separate claim, checked by a direct intensity measurement.

How is squeezing measured, and what does "6 dB" actually refer to?

By balanced homodyne detection: the squeezed field is combined on a 50/50 beamsplitter with a strong local oscillator of adjustable phase $\phi$ , and the difference of the two photocurrents gives $\hat X_\phi$ ; sweeping $\phi$ traces $V(\phi)$ below and above shot noise (calibrated by blocking the input so only vacuum enters). "6 dB" is that dip in decibels of noise power, $S_{\text{dB}}=-10\log_{10}(V/V_{\text{vac}})$ : the variance is $10^{-0.6}\approx0.25$ of vacuum — power four times smaller, amplitude twice smaller.

Why inject squeezed vacuum into an interferometer instead of using a brighter laser?

Because sensitivity from laser power grows only as $\sqrt N$ , while the limiting noise is set by the vacuum in the dark port — an input you can replace outright. Doubling resolution by power alone means quadrupling the circulating power, capped in a real detector by thermal lensing and mirror distortion long before the needed level. Squeezed vacuum sidesteps this: it sits at the empty dark port and quiets the exact quadrature carrying the signal, an $e^{-r}$ gain with no extra heat in the mirrors, and a filter cavity makes it frequency-dependent to tame back-action at low frequencies too.