1. Introduction to Squeezed Light
Squeezed states of light are quantum states in which the uncertainty in one quadrature component is reduced below the standard quantum limit (shot noise), at the expense of increased uncertainty in the conjugate quadrature. This fundamental quantum resource enables noise reduction beyond classical limits.
Key Achievements
- • 15 dB squeezing in LIGO detectors
- • 10+ dB on-chip in TFLN platforms
- • Continuous-wave and pulsed generation
- • Broadband and narrowband sources
Applications
- • Gravitational wave detection
- • Quantum key distribution
- • Quantum computing (CV)
- • Precision metrology
2. Quantum States of Light
The electromagnetic field can be described by the bosonic annihilation and creation operators â and â†, satisfying [â, â†] = 1. The quadrature operators are defined as:
X̂ = (â + â†) / √2
P̂ = i(↠- â) / √2
[X̂, P̂] = i → ΔX · ΔP ≥ 1/2
Common Quantum States
| State | ΔX² | ΔP² | Properties |
|---|---|---|---|
| Vacuum |0⟩ | 1/2 | 1/2 | Minimum uncertainty, isotropic |
| Coherent |α⟩ | 1/2 | 1/2 | Displaced vacuum, laser light |
| Squeezed |r,θ⟩ | e⁻²ʳ/2 | e²ʳ/2 | Anisotropic noise ellipse |
| Thermal | (n̄+½) | (n̄+½) | Classical excess noise |
3. Squeezed States in Detail
A squeezed vacuum state is generated by the squeezing operator Ŝ(ξ) = exp[½(ξ*â² - ξ↲)] where ξ = r·e^(iθ) is the complex squeezing parameter:
|ξ⟩ = Ŝ(ξ)|0⟩
r = squeezing parameter, θ = squeezing angle
Quadrature Variances
⟨ΔX²⟩ = ½ e⁻²ʳ (squeezed)
⟨ΔP²⟩ = ½ e⁺²ʳ (anti-squeezed)
Product: ⟨ΔX²⟩⟨ΔP²⟩ = ¼ (minimum uncertainty)
Squeezing in dB
S_dB = -10 log₁₀(e⁻²ʳ)
S_dB ≈ 8.686 × r
3 dB → r ≈ 0.35
10 dB → r ≈ 1.15
15 dB → r ≈ 1.73
Photon Statistics
Squeezed vacuum contains only even photon number states: |ξ⟩ = Σₙ cₙ|2n⟩. The mean photon number is ⟨n⟩ = sinh²(r), indicating more photons at higher squeezing.
4. Gaussian Quantum States
Gaussian states are fully characterized by their first moments (mean field) and second moments (covariance matrix). For an N-mode system with quadratures ξ = (X₁, P₁, X₂, P₂, ...):
σᵢⱼ = ½⟨{Δξᵢ, Δξⱼ}⟩
Covariance matrix with Δξᵢ = ξᵢ - ⟨ξᵢ⟩
Single-Mode Covariance Matrices
Vacuum/Coherent
σ = ½ [1 0]
[0 1]
Squeezed (θ=0)
σ = ½ [e⁻²ʳ 0 ]
[0 e²ʳ]
Thermal
σ = (n̄+½) [1 0]
[0 1]
Wigner Function
The Wigner function for Gaussian states is a 2N-dimensional Gaussian: W(ξ) = (2π)⁻ᴺ (det σ)⁻½ exp[-½(ξ-μ)ᵀ σ⁻¹(ξ-μ)]
5. Kerr Squeezing (χ³ Nonlinearity)
Third-order nonlinearity in materials like silicon nitride (SiN) enables squeezing through self-phase modulation and four-wave mixing (FWM). The Kerr Hamiltonian is:
Ĥ_Kerr = ℏg ↲ â²
g = χ³ nonlinear coupling rate
Four-Wave Mixing Process
In degenerate FWM, two pump photons (ωₚ) are converted to signal and idler photons:
2ωₚ → ωₛ + ωᵢ
The signal and idler modes become correlated, producing two-mode squeezing. In a ring resonator, this creates squeezed vacuum at the output coupler.
SiN Parameters
- • n₂ ≈ 2.4 × 10⁻¹⁹ m²/W
- • Propagation loss: 0.1-1 dB/cm
- • Achievable squeezing: 3-10 dB
- • Typical pump power: 10-100 mW
Advantages
- • CMOS-compatible fabrication
- • High optical damage threshold
- • Broadband phase matching
- • Mature fabrication technology
6. OPA Squeezing (χ² Nonlinearity)
Second-order nonlinearity in materials like thin-film lithium niobate (TFLN) enables parametric down-conversion and optical parametric amplification:
Ĥ_OPA = iℏκ(â²e⁻ⁱᶿ - ↲eⁱᶿ)
κ = parametric coupling rate ∝ χ² × E_pump
Parametric Down-Conversion
A pump photon at 2ω is converted to two signal photons at ω:
ω_pump → ω_signal + ω_idler (degenerate: ω_s = ω_i)
The squeezing parameter grows with interaction length: r ∝ κL. Below threshold, this produces squeezed vacuum with high purity.
TFLN Parameters
- • d_eff ≈ 27 pm/V (d₃₃)
- • Propagation loss: 0.03-0.1 dB/cm
- • Achievable squeezing: 10-15+ dB
- • Typical pump power: 1-10 mW
Advantages
- • Stronger nonlinearity than χ³
- • Higher squeezing levels possible
- • Lower pump power requirements
- • Electro-optic tuning capability
7. Ring Resonator Physics
Microring resonators enhance nonlinear interactions by confining light in a small mode volume and providing resonant intensity buildup:
Key Parameters
Quality Factor
Q = ω₀ / κ_total
Free Spectral Range
FSR = c / (n_g × L)
Finesse
F = FSR / Δν = π√(r) / (1-r)
Coupling Regimes
Over-coupled
κ_ex > κ_i: Maximum escape efficiency
Critical coupling
κ_ex = κ_i: Maximum extinction
Under-coupled
κ_ex < κ_i: High internal buildup
η_escape = κ_ex / (κ_ex + κ_i)
Escape efficiency determines output squeezing
8. Squeezing Detection Methods
Homodyne Detection
Measures single quadrature by interfering with a strong local oscillator (LO):
î_diff = |α_LO| (X̂ cos θ + P̂ sin θ)
- • LO phase θ selects quadrature
- • Balanced detection cancels classical noise
- • Limited to one quadrature per measurement
Heterodyne Detection
Measures both quadratures simultaneously with added noise:
X̂_m, P̂_m with ΔX² = ΔP² = 1
- • 3 dB noise penalty vs homodyne
- • Full state tomography possible
- • Used in CV-QKD receivers
Detection Efficiency
The total detection efficiency η_det determines measured squeezing: V_measured = η_det × V_squeezed + (1 - η_det) × V_vacuum. High-efficiency InGaAs photodiodes achieve η > 95%.
9. Loss & Decoherence
Optical loss is the primary challenge for maintaining squeezed states. Any loss η < 1 degrades squeezing by mixing with vacuum noise:
V_out = η × V_in + (1 - η)
V = 1 for vacuum (shot noise level)
Loss Sources in PIC Squeezers
| Source | Typical Value | Mitigation |
|---|---|---|
| Propagation loss | 0.1-1 dB/cm | High-quality film deposition |
| Coupling loss | 1-3 dB/facet | Mode converters, tapers |
| Filter insertion loss | 0.5-2 dB | Integrated filter design |
| Detector efficiency | 90-98% | High-QE photodiodes |
10. Material Platforms Comparison
| Property | SiN (χ³) | TFLN (χ²) | Si (χ³) |
|---|---|---|---|
| Nonlinearity | n₂ ≈ 2.4×10⁻¹⁹ | d₃₃ ≈ 27 pm/V | n₂ ≈ 4.5×10⁻¹⁸ |
| Prop. loss (dB/cm) | 0.1-1 | 0.03-0.1 | 1-3 |
| Max squeezing | 3-10 dB | 10-15+ dB | 1-5 dB |
| Pump power | 10-100 mW | 1-10 mW | 50-200 mW |
| Fab maturity | High | Emerging | Very High |
| EO tuning | No | Yes | No |
11. Applications of Squeezed Light
Gravitational Wave Detection
LIGO and Virgo inject squeezed vacuum to reduce quantum noise, improving sensitivity by up to 3 dB. Future detectors target 10+ dB squeezing with frequency-dependent rotation for broadband improvement.
Quantum Key Distribution
CV-QKD uses squeezed or coherent states for secure key generation. Squeezing improves the secret key rate and extends transmission distance by reducing excess noise relative to shot noise.
Quantum Computing
Continuous-variable quantum computing uses squeezed states as resources for measurement-based computation. Cluster states are built from entangled squeezed modes for universal quantum gates.
Precision Metrology
Squeezed light enables sub-shot-noise measurements in interferometry, spectroscopy, and microscopy. Applications include biological imaging, atomic clocks, and magnetometry.
References
1. Walls, D. F. & Milburn, G. J. "Quantum Optics" (Springer, 2008)
2. Andersen, U. L. et al. "30 years of squeezed light generation" Phys. Scr. 91, 053001 (2016)
3. Zhao, Y. et al. "Near-Degenerate Quadrature-Squeezed Vacuum Generation on a Silicon-Nitride Chip" PRL 124, 193601 (2020)
4. Nehra, R. et al. "Few-cycle vacuum squeezing in nanophotonics" Science 377, 1333 (2022)
5. Kashiwazaki, T. et al. "Continuous-wave 6-dB-squeezed light with 2.5-THz-bandwidth from single-mode PPLN waveguide" APL Photonics 5, 036104 (2020)
6. Vahlbruch, H. et al. "Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency" PRL 117, 110801 (2016)
7. Aasi, J. et al. "Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light" Nature Photon 7, 613 (2013)