Past exam of the nonlinear Optics and Quantum States of Light course of the University of Cambridge / 2014

Original paper: 23 April 2014, physical PDF pages 678–680.

Put $g=\gamma P_0$. For a temporally uniform envelope, $A_z=igA$ and $A_{TT}=0$, so the propagation equation reduces to $-gA+\gamma P_{0}A=0$. Thus the power stays $P_0$ and the accumulated nonlinear wave phase is $gz$. The optical Kerr effect gives an intensity-dependent refractive index, and hence self-phase modulation. For the usual power normalization $\gamma \simeq n_2{\omega }_0/(c{A}_{\mathrm{eff}})$, with effective guided-mode area $A_{\mathrm{eff}}$. Losslessness preserves the background power; it does not remove the self-phase modulation. The carrier wave phase has already been factored out.

Write $A=(\sqrt{P_0}+ϵ)e^{igz}$. To first order,Cancelling the background and its phase derivative leavesThe last term couples a perturbation to its complex conjugate: the pump can transfer energy into a pair of sidebands. It is not an independent wave phase of each Fourier component. This sideband coupling is the linear origin of fibre modulational instability.

For real $K,\Omega ,c_1,c_2$, put $d={\beta }_2{\Omega }^2/2$. Matching the two exponential coefficients givesA nonzero pair requires a zero determinant, henceFor unstable complex $K$, the real-coefficient assumption is too restrictive. The consistent pair is $ϵ=u{e}^{iKz-i\Omega T}+v^{\ast }{e}^{-i{K}^{\ast }z+i\Omega T}$, with complex $u,v$. Its equations give the same determinant and allow the growing and decaying branches. Merely inserting imaginary $K$ while treating the two original coefficients as real would not respect complex conjugation.

For $P_0>0$ and nonzero ${\beta }_2,\gamma$, exponential growth requires ${\beta }_2\gamma <0$ andFor ordinary positive Kerr nonlinearity this means anomalous group velocity dispersion, ${\beta }_2<0$. Define ${\Omega }_c=\sqrt{4g/|{\beta }_2|}$. The growing branch is $K=-iG$, with amplitude growth rateSideband power grows at rate $2G$, rather than $G$. The cutoff and zero-frequency modes have no exponential growth; in a degenerate zero-frequency sector a secular phase response can remain. Positive dispersion with positive $\gamma$ gives real $K$ and oscillatory perturbations.

Initially small ripples grow into pronounced intensity peaks and deplete the uniform background. Dispersion and self-phase modulation can then support localized bright pulses. A fundamental temporal bright soliton obeys $\gamma P_s{T}_0^2=|{\beta }_2|$, so its width is comparable to $1/{\Omega }_c$ when its peak power is comparable to the original power. The dominant ripple spacing is $2\pi /|\Omega {|}_{\max }$, not simply $1/{\Omega }_c$. Propagation distance $z$ is the evolution variable; $T$ measures time within the envelope. An ideal conservative fibre need not settle permanently into a unique pulse train: nonlinear compression, interactions and recurrence depend on the seed and boundary conditions.

A degenerate optical parametric amplifier sends a strong pump at $2\omega$ through a phase-matched second-order nonlinear medium, creating pairs in a selected mode near $\omega$. With the pump treated classically and undepleted, the effective interaction is quadratic in the mode's creation and annihilation operators. Vacuum input evolves into an optical squeezed vacuum; the pump phase fixes the quiet optical quadrature. A cavity enhances the interaction below oscillation threshold. balanced homodyne detection compares this optical quadrature's noise with the blocked-signal vacuum reference; phase locking, detector balance, electronic-noise measurement and loss calibration distinguish squeezing from an instrumental noise reduction. A primary cavity experiment observed below-vacuum noise by degenerate down conversion and balanced homodyne readout.^[ref]

Use $S(r)=\exp [\frac{r}{2}(a^2-a^{†2})]$, so $S^{†}aS=a\cosh r-a^{†}\sinh r$. Write $s={\sinh }^{2}r$, $c=\cosh r$ and $N=a^{†}a$. Vacuum expectations then givePair creation makes the mean field zero while the mean photon number is positive. Expanding the transformed ladder operators in the vacuum gives the squeezed-vacuum photon-number moments: $⟨N(N-1)⟩=3s^2+s$ and $⟨N^2⟩=3s^2+2s$; equivalently $\mathrm{Var}N=2s(s+1)$.

The vectors produced by photon addition and photon subtraction in the paper are not normalized. Their squared norms are $s+1$ and $s$, respectively. A particularly direct evaluation isBoth successful operations therefore prepare the same squeezed one-photon state, up to an overall phase, whenever subtraction has nonzero probability. Since $⟨1|S^{†}NS|1⟩=c^2+2s=1+3s$, the unnormalized moments areThe printed photon-added formula equals $M_{+}$, not a normalized mean. With its literal unnormalized convention, the corresponding subtraction result is $M_{-}$ andThis difference tends to one as $r\to 0$ and grows as $\frac{3}{4}{e}^{2r}$ for large positive $r$. Keeping this convention recovers the paper's requested expression without misidentifying it as a physical conditional mean.

For the normalization of photon-added and photon-subtracted states, divide by the squared norms to obtain actual conditional means:These equalities for subtraction require $s>0$. As $r\to 0$ the normalized successful subtraction tends to $|1⟩$, but its success probability vanishes; exactly at $r=0$, $a|0⟩=0$ gives no conditional state. At large $r$, both normalized means grow as $3e^{2r}/4$, while their difference remains zero.

For the photon statistics of a squeezed vacuum, the even-number probabilities areAfter subtraction the normalized distribution is $p_n^{-}=(n+1)p_{n+1}/s$. It has only odd occupation and no vacuum component. photon addition gives $p_n^{+}=n{p}_{n-1}/(s+1)$, identical here to $p_n^{-}$. The diagram uses normalized distributions with an explicitly chosen $r$, rather than treating the unnormalized moments as means.

A subtraction event samples initial occupation with weight $n$, preferentially selecting the larger components before removing one photon. In general ${\bar{n}}_{-}=⟨N(N-1)⟩/⟨N⟩=⟨N⟩-1+\mathrm{Var}N/⟨N⟩$. Here the large variance gives ${\bar{n}}_{-}-⟨N⟩=1+2s>0$. This conditional photon-number bias does not create energy by removing a photon: the unconditional ensemble also includes the runs without a heralded event.

An optical soliton maintains a localized profile because nonlinear refraction compensates linear spreading. A temporal optical soliton balances group velocity dispersion against self-phase modulation; a spatial optical soliton balances diffraction against self-focusing. The relevant coordinate and sign of nonlinearity must be specified.

For anomalous group velocity dispersion and positive $\gamma$, a fundamental temporal bright soliton isThe identity $(\mathrm{sech}u{)}^{\prime \prime }=\mathrm{sech}u-2{\mathrm{sech}}^{3}u$ verifies both the width relation and propagation phase. Its dispersion length $L_D=T_0^2/|{\beta }_2|$ equals its nonlinear length $L_{\mathrm{NL}}=1/(\gamma P_s)$. Bright pulses decay to zero background. dark solitons are intensity depressions on a nonzero background with a phase change across the dip; black dips reach zero intensity, while grey dips do not. For positive $\gamma$ the corresponding temporal defocusing regime has normal group velocity dispersion.

A higher-order temporal soliton has soliton order $N_s=\sqrt{\gamma P_s{T}_0^2/|{\beta }_2|}$ equal to an integer greater than one for the special initial $N_s\mathrm{sech}$ profile in normalized units. It breathes through compression and splitting/recombination, rather than keeping one fixed shape. Separated fundamental solitons can interact and acquire collision shifts. An arbitrary pulse with noninteger $N_s$ is not automatically a pure higher-order soliton; dispersive radiation can carry part of its energy.

In ideal one-dimensional cubic propagation, dispersion and nonlinearity support stable fundamental bright pulses. Loss, higher-order dispersion, delayed nonlinear response and finite bandwidth change real-fibre propagation. A one-dimensional spatial bright solution also does not guarantee stability against two-dimensional collapse. Profile preservation, finite-energy background conditions and robustness should be checked rather than inferred from any solitary-looking peak.

A detector too slow to resolve a femtosecond field can measure its integrated nonlinear response instead. Split the pulse, delay one replica, overlap both in a thin phase-matched crystal, and select the sum-frequency signal. For negligible depletion and sufficiently broad phase matching, the cross-generated field is proportional to $A(t)A(t-\tau )$; integrated signal gives the second-harmonic intensity autocorrelationA noncollinear geometry can reject each replica's individual second harmonic. The optical response supplies the fast gate while the detector integrates slowly. Crystal thickness, group-velocity mismatch, spectral acceptance, beam overlap and delay calibration limit resolution.

For a Gaussian pulse specified by its intensity profile, the autocorrelation's full width at half maximum is $\sqrt{2}$ times the intensity full width at half maximum. For a sech-squared intensity profile the ratio is about $1.543$. These are model-dependent conversion factors. An intensity autocorrelation is symmetric in delay and loses phase information; different pulses can share it. An interferometric autocorrelation retains rapid carrier fringes but still does not generally reconstruct an arbitrary complex field uniquely.

frequency-resolved optical gating also resolves the gated signal's spectrum, giving a two-dimensional delay-frequency record. For second-harmonic frequency-resolved optical gating,where $\omega$ is measured relative to the signal's carrier. An iterative retrieval seeks an envelope whose predicted record matches the measurement. Alternatively, polarization-gating frequency-resolved optical gating uses the optical Kerr effect to gate the probe by a delayed replica's intensity, with signal envelope proportional to $A(t)|A(t-\tau ){|}^2$. The original FROG work introduced nonlinear optical gating plus spectral resolution to characterize arbitrary femtosecond pulses.^[ref]

Absolute wave phase and absolute time origin remain undetermined by these intensity records. The SHG record also cannot distinguish $A(t)$ from $A^{\ast }(-t)$, so its time-direction ambiguity remains. Retrieval agreement, an independently measured spectrum, adequate signal dynamic range and gate-response calibration are practical consistency checks. A pulse train that changes between shots can yield an averaged record that no single pulse faithfully represents. Thus nonlinear gating solves the detector-speed problem but does not remove inverse problem ambiguities or experimental limitations.

For $X=(a+a^{†})/2$ and $Y=(a-a^{†})/(2i)$, $[X,Y]=i/2$, so $\Delta X\Delta Y\ge 1/4$. Vacuum and coherent states have variance $1/4$ in either optical quadrature. quadrature squeezing means that some rotated optical quadrature has variance below this reference, with compensating noise elsewhere. A reduced second moment about zero is not the correct test when the field's mean is displaced.

For $S(\zeta )=\exp [({\zeta }^{\ast }{a}^2-\zeta a^{†2})/2]$ and $\zeta =r{e}^{i\theta }$, the quiet principal axis of the optical squeezed vacuum is at angle $\theta /2$ under the squeeze-operator convention used above, andThe pure Gaussian quantum state saturates the uncertainty product. Displacement moves its centre without changing its covariance ellipse; squeezed thermal states instead have a larger uncertainty product. The optical squeezed vacuum has zero mean field, even photon-number parity and mean occupation ${\sinh }^{2}r$; it need not have sub-Poissonian photon statistics. number squeezing and quadrature squeezing are different conditions.

Degenerate parametric amplification generates single-mode squeezing; a two-mode squeezed vacuum instead has correlated occupations and joint quadrature squeezing. Either individual mode of an ideal two-mode state is thermal after the other is ignored, not individually squeezed. balanced homodyne detection measures the noise versus local oscillator phase. Reference phase drift admixes the anti-squeezed optical quadrature, while electronic noise and detector imbalance need separate checks.

For an optical loss channel with efficiency $\eta$ and vacuum entering the unused port, optical squeezing degradation by loss givesThe linear-loss limit follows directly by writing the output field as $\sqrt{\eta }a+\sqrt{1-\eta }v$ with independent vacuum mode $v$. Loss therefore places a floor under measured squeezing. Squeezed input can improve phase-sensitive measurements when aligned with the measured optical quadrature; it does not reduce every noise source or both conjugate uncertainties at once.