Chapter 18: Classical Treatment of Quantum Noise and Squeezed States
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 18. Use each problem number with the book; the original prompts are not reproduced. Each entry supplies the governing model, a decisive solution route, and an independent consistency check.
End-of-chapter problems
Problem 18.1 — field quadratures, coherent states, and uncertainty: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\).
3. Verification.
The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\). The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Problem 18.2 — field quadratures, coherent states, and uncertainty: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\).
3. Verification.
The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\). The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Problem 18.3 — field quadratures, coherent states, and uncertainty: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\).
3. Verification.
The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\). The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Problem 18.4 — field quadratures, coherent states, and uncertainty: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\).
3. Verification.
The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\). The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Problem 18.5 — field quadratures, coherent states, and uncertainty: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\).
3. Verification.
The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write the field quadratures in terms of \(a\) and \(a^\dagger\), use \([a,a^\dagger]=1\), and normal-order products before taking number or coherent-state expectations. Relate phase uncertainty to time through \(\Delta\phi=\omega\Delta t\). The two quadrature variances of a coherent state must be equal and saturate the uncertainty bound; coherent-state overlaps must approach one for identical amplitudes and zero as their separation grows.