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The inverse of the $(v,\lambda )$ metric tensor block isThe normal covector $d\lambda$ has squared norm ${\lambda }^2/m^2$. ThereforeMore explicitly, the induced metric tensor is $-{\lambda }_0^{2}d{v}^2+m^{2}d{\Omega }_2^2$; at zero its sole zero direction is ${\partial }_v$. The coefficients are independent of $v$, so $K={\partial }_v$ is a Killing vector field. On $\lambda =0$,Hence $K$ is both tangent and normal to this null hypersurface, proving it is a Killing horizon. The definition of surface gravity is ${\nabla }_{K}K=\kappa K$ on the Killing horizon, or equivalently $d(K^2)=-2\kappa K^{♭}$. Since $d(K^2)=-2\lambda d\lambda$ vanishes there while $K^{♭}\ne 0$,Thus this is a degenerate Killing horizon, and $v$ is an affine parameter on each of its null geodesics.

For a concrete horizon frequency comparison, fix the same angular position for a narrow high-frequency light beam and two freely falling radial observers. Work in geometric optics, with the beam's affinely transported wavevector tangent to the Killing horizon, $k=qK$. Here $q>0$ is constant because ${\nabla }_{K}K=0$. Choose the observers as translates of one timelike geodesic by the isometry generated by $K$, crossing at $v_1$ and $v_2$. Give them the same positive Killing energy per unit rest mass, $E=-K\cdot W$. At their crossings their normalized four-velocities havewhich satisfy $g(W,W)=2m{W}^v{W}^{\lambda }=-1$. They can be realized by local geodesics with precisely these initial data. The measured angular frequency isTo connect this measurement to surface gravity, if ${\nabla }_{K}K=\kappa K$ with constant $\kappa$, affine transport of $k=q(v)K$ would give $dq/dv=-\kappa q$. For observers related by the same Killing vector field translation, $-K\cdot W$ agrees at the crossings, giving ${\omega }_2/{\omega }_1=e^{-\kappa (v_2-v_1)}$. Here there is no accumulated redshift along the Killing horizon for these equally prepared observers. Arbitrary observers with different four-velocities can still measure different frequencies through their different Killing energies.

Define the unperturbed orbital angular frequency vector byIt is the rate of advance of the angle variables along the unperturbed Hamiltonian flow; the action variables are constant. Expand the Collisionless Boltzmann equation to first order in the perturbations. Since $f_0$ and $H_0$ depend only on the action variables, $[f_0,H_0]=0$, andThe omitted Poisson bracket $[\delta f,\delta \Phi ]$ is quadratic in the perturbations. Use the Fourier series conventionFor each harmonic the linear ordinary differential equation isMultiplication by the integrating factor $e^{i\mathbf{k}\cdot \mathbf{\Omega }t}$ and use of $\delta f_{\mathbf{k}}(0)=0$ give the causal linear response in action-angle variables:The zero harmonic stays zero at this order. This formula retains the entire forcing history; replacing it immediately by a resonant Dirac delta function would discard the transient response.

A universal quantum gate set can implement any unitary operator on any finite number of qubits, exactly when arbitrary continuous gate parameters are allowed, or to arbitrary accuracy with a suitable finite gate alphabet. More precisely, for every target $V$ and $ϵ>0$ there is a finite quantum circuit $W$ with ${\min }_{\gamma }\Vert W-e^{i\gamma }V\Vert <ϵ$ in operator norm. This ignores only an overall global phase of the complete operation. Universality lets a fixed collection of controllable elementary operations run arbitrary quantum algorithms rather than requiring a different physical interaction for each algorithm.

All single-qubit unitary gates, together with the controlled-NOT gate between selectable pairs, form an exact universal quantum gate set. One way to see sufficiency is to eliminate entries of a finite unitary matrix by two-level rotations between computational basis vectors; these rotations decompose into controlled single-qubit rotations, which in turn decompose into single-qubit gates and controlled-NOT gates. Single-qubit gates alone preserve product states and cannot supply general quantum entanglement. A finite approximate universal example is the Hadamard gate, the T gate $T=\mathrm{diag}(1,e^{i\pi /4})$, and the controlled-NOT gate; finite exact words over a finite alphabet cannot represent every continuously parameterized unitary operator exactly.

Distinguish the Hamiltonian operator $H_{\mathrm{int}}$ from the Hadamard gate $H_{\mathrm{Had}}$. For the printed collisional phase gate,As written, $\Omega$ has units of energy. The Schrödinger equation givesbecause only $|01⟩$ has a nonzero energy eigenvalue. In the ordered computational basis $(00,01,10,11)$, defineIf an interaction strength is instead quoted as an angular frequency ${\Omega }_{\mathrm{f}}$, the Hamiltonian operator is $\hslash {\Omega }_{\mathrm{f}}|01⟩⟨01|$ and $\varphi ={\Omega }_{\mathrm{f}}\tau$. An alternative definition with $e^{+i\varphi }$ would have $\varphi =-\Omega \tau /\hslash$. The minus sign above follows the usual time-evolution operator convention. At $\varphi =\pi$ these conventions give the same $U_{\pi }$.

For the quantum circuit, use $X$ on the first qubit to move the selective minus sign from $01$ to $11$:Conjugating the target of this Controlled-Z gate by the Hadamard gate gives the required controlled-NOT gate:The rightmost matrix acts first. Thus the network applies $X_1,H_2$, then $U_{\pi }$, then $X_1,H_2$, and maps $|a,b⟩$ to $|a,b\oplus a⟩$, where $\oplus$ is exclusive or.

Use the ordered computational basis $(|00⟩,|01⟩,|10⟩,|11⟩)$, with $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$. Write $X,Y,Z$ for the Pauli matrices, $I$ for the identity operator, $J={\omega }_{12}$, $\Sigma ={\omega }_1+{\omega }_2$, and $\Delta ={\omega }_1-{\omega }_2$. As in the question, the Hamiltonian operators have angular frequency units, so the time-evolution operator is $e^{-iHt}$. The Heisenberg coupling matrix for two spins isConsequently the full Heisenberg model Hamiltonian operator and its proposed Ising coupling approximation are

For nondegenerate perturbation theory, choose $H_{\mathrm{I}}$ as the unperturbed Hamiltonian operator andas the perturbation. Only $|01⟩$ and $|10⟩$ are coupled, and their unperturbed energy levels differ by $\Delta$. All first-order energy corrections vanish. The first-order eigenvectors and second-order energy corrections areThe states $|00⟩,|11⟩$ are exact eigenvectors throughout. Hence the weak coupling condition for two NMR spins isA large common Zeeman energy alone does not ensure this condition. Crossings involving the uncoupled parallel-spin states do not invalidate the argument: the relevant denominator is the separation of the two states connected by $V$.

As an exact check, the middle $2\times 2$ matrix has eigenvaluesFor $\Delta \ne 0$, labeling by the limiting computational basis states reproduces the shifts above. The quantum transition probability from $|01⟩$ to $|10⟩$ isThus unwanted exchange has a uniformly small amplitude when $|J/\Delta |\ll 1$. If accumulated phases must also be approximated over a duration $t$, neglecting the second-order shifts additionally requires $J^{2}t/|\Delta |\ll 1$, or those shifts must be retained in an improved effective Hamiltonian operator.

The same result is the secular approximation for scalar-coupled spins. In the interaction picture of the Zeeman energy terms,These terms average rapidly on the coupling timescale $1/|J|$ when $|\Delta |\gg |J|$, while $JZ\otimes Z/2$ survives.

For fixed nuclei and electronic environments, the individual NMR angular frequencies scale with the static magnetic field $B_0$: schematically ${\omega }_i={\kappa }_i{B}_0$, where ${\kappa }_i$ includes the gyromagnetic ratio and magnetic shielding. The NMR scalar coupling $J$ in angular frequency units is approximately independent of $B_0$. Therefore a nonzero separation $|\Delta |=|{\kappa }_1-{\kappa }_2|B_0$ increases with magnetic field, improving the approximation. In particular, a fixed difference of chemical shifts gives a larger separation in hertz at higher magnetic field. If the two resonances remain exactly equal, increasing their common frequency does not suppress their exchange.

For a vertical interband transition, the crystal momentum is effectively unchanged because the photon momentum is small. A stimulated downward transition needs an occupied upper state and an empty lower state, with factor $f_c(E_2)[1-f_v(E_1)]$; upward photon absorption has factor $f_v(E_1)[1-f_c(E_2)]$. The two matrix elements agree by microscopic reversibility. Their difference is thereforeA common slowly varying transition-strength factor is implicit. Broadening replaces the sharp transition by a convolution; multiple valence bands contribute separate terms.

For one isotropic pair of parabolic energy bands, writeThus $E_2-E_1=E_g+{\hslash }^2{k}^2/(2m_r)$. Including the usual twofold spin degeneracy, the number of paired states per volume inside a sphere of radius $k$ is $k^3/(3{\pi }^2)$. Differentiating with respect to transition energy $E$ gives the three-dimensional parabolic interband joint density of statesFor density per angular frequency, $g_j^{(\omega )}(\omega )=\hslash g_j^{(E)}(\hslash \omega )$. Stating the density convention avoids a missing factor of $\hslash$; other degeneracies multiply the formula.

At a common carrier temperature, the Fermi-Dirac distribution is a strictly decreasing function of $E-F$. The direct-gap semiconductor gain condition follows:The lower inequality denotes a transition above the ideal band edge, where the displayed joint density of states is nonzero. Thus quasi-Fermi-level splitting must exceed the transition photon energy. Equality is transparency, rather than net amplification; at zero temperature an allowed transition must also lie within the occupied-electron and empty-valence windows.

Write the optical system's spectral response as $A(\omega )e^{i\varphi (\omega )}$. Expanding its spectral phase around the pulse's central angular frequency givesThe group delay is ${\varphi }^{(1)}=d\varphi /d\omega$, with units of time. The linear spectral phase term delays the pulse envelope by this amount. The group delay dispersion is ${\varphi }^{(2)}=d^2\varphi /d{\omega }^2$, with units of time squared. It makes the group delay frequency dependent: $t_g(\omega )\simeq {\varphi }^{(1)}+{\varphi }^{(2)}(\omega -{\omega }_0)$. Positive group delay dispersion delays higher frequencies more and gives positive chirp. A nonzero quadratic spectral phase broadens an initially transform-limited pulse; applied to an already chirped pulse, a suitable opposite sign can compress it.

In chirped pulse amplification, an ultrashort seed pulse first passes through a dispersive stretcher, often a grating-based system. This correlates frequency with arrival time and increases pulse duration while approximately preserving pulse energy, thereby lowering peak intensity. An optical amplifier can then increase pulse energy without exposing the gain medium to the original short pulse's enormous peak intensity, reducing optical damage and unwanted nonlinear phase accumulation. A final dispersive compressor, typically using gratings, cancels the accumulated spectral phase to recover a short, high-energy pulse. The essential components are therefore a short-pulse source, stretcher, amplifier and matched compressor; spectral gain bandwidth and higher-order dispersion limit the achievable compression.

The temporal intensity is proportional to the squared field modulus:Its half-maximum points satisfy $2a{t}^2=\ln 2$, so the intensity full width at half maximum isWith the chosen $e^{-i\omega t}$ convention, the instantaneous angular frequency is minus the time derivative of the field's argument. HenceA localized Gaussian requires $\mathrm{Re}\Gamma =a>0$. It is positively chirped precisely when $d\omega /dt=2\mathrm{Im}\Gamma >0$, or $b>0$; $b=0$ gives no chirp and $b<0$ gives negative chirp.

A time delay $d$ multiplies the spectral field by $e^{i\omega d}$. The bright-port transfer function is thereforeThe spectral transmission maxima have angular frequency spacing $\Delta \omega =2\pi /|d|$. For a strongly chirped pulse the local frequency-time mapping is $\omega (t)={\omega }_0+2bt$. Thus the filter's adjacent transmitted frequency bands emerge at times separated byas obtained directly from the temporal fields. The spectral transfer function itself is exact; mapping its features directly onto time requires a sufficiently strong chirp and overlapping envelopes.

As $|d|$ approaches the pulse duration, the two envelopes overlap only partly and the small-delay formula fails. For $|d|\gg \tau$, the selected port contains two separate replicas of field amplitude $E_0/2$, each with one quarter of the input pulse energy. Their combined energy is half the input; the other output port carries the remainder. There is no temporal overlap or corresponding time-resolved interference. A sufficiently resolving spectrometer can still reveal fine spectral fringes, whose spacing decreases as $1/|d|$.

Take ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. In the resonant Jaynes-Cummings model, the interaction connects only $|e,n⟩$ and $|g,n+1⟩$. Both have free energy $\hslash \omega (n+1/2)$, and their interaction matrix is $\hslash g\sqrt{n+1}{\sigma }_x$. ThereforeExponentiating this two-state block and applying it to $|e,n⟩$ yieldsThe populations undergo Rabi oscillations at angular frequency $2g\sqrt{n+1}$. The common phase in this fixed-$n$ block does not affect its probabilities, but it must be retained when blocks with different photon numbers are superposed.

For $\bar{n}=|z{|}^2\gg 1$, the Poisson distribution of photon number has width $\sqrt{\bar{n}}\ll \bar{n}$. Initially the atom therefore undergoes nearly classical Rabi oscillations at angular frequency approximately $2g\sqrt{\bar{n}}$. However, the atomic inversion is a sum over slightly different frequencies:The components dephase, causing collapse of the visible Rabi oscillations on a time scale of order $1/g$; a Gaussian photon-number approximation gives an initial envelope approximately $e^{-g^2{t}^2/2}$. Their discrete phases later rephase, giving revivals nearThe revival estimate follows by requiring the phase difference of neighbouring photon-number components to advance by $2\pi$. These Jaynes-Cummings collapse and revival effects arise from atom-field entanglement and photon-number discreteness. Large amplitude makes the initial classical description accurate over many Rabi periods, but does not eliminate quantum collapse and revival at all later times.

Place the optical lens at $z=0$. For a sufficiently broad incident Gaussian beam, the beam waist positions approach the focal lengths frequency by frequency:Add the group delay at the optical lens to propagation through vacuum:Here an arbitrary common time origin has been omitted. Differentiating both equations with respect to angular frequency yieldsFor $\beta >0$, define $u=2\eta c/\beta$. The velocity graph in the preceding figure has horizontal asymptote zero and a vertical asymptote at $u=-1$. For $\eta >0$, $0<v_f<c$: the higher-frequency components both enter later and travel farther. At $\eta =0$, $v_f=c$. For $-\beta /(2c)<\eta <0$, $v_f>c$: the farther-focusing components leave earlier, partly compensating their extra travel. At $\eta =-\beta /(2c)$ every component focuses at the same time, so the trajectory is not a finite-velocity function of time. For $\eta <-\beta /(2c)$ the farther positions focus first, giving $v_f<0$; it tends to zero from below as $\eta \to -\infty$.

A flying focus is a moving pattern assembled from different portions of an optical pulse. Each portion propagates causally at $c$ in this model. The superluminal or backward motion of the focus therefore does not describe a material object or information overtaking light. The calculation assumes negligible additional frequency-dependent group delay from the optical lens; any such delay must be included in $\psi$.

Quantize a single electromagnetic normal mode of angular frequency ${\omega }_c$ and represent one isolated atomic transition as a two-level system with raising operator ${\sigma }_{+}=|e⟩⟨g|$. The bare Hamiltonian operator isThe first main approximation is the electric-dipole approximation: the field varies negligibly across an atom, $k{a}_{\mathrm{atom}}\ll 1$, and the leading coupling is $H_d=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$. For states of definite opposite parity, the relevant dipole part is off-diagonal. After choosing operator phases, this gives $H_d=\hslash g(a+a^{†})({\sigma }_{+}+{\sigma }_{-})$, with $g$ fixed by the transition dipole and the one-photon field wave amplitude.

In the interaction picture, $a{\sigma }_{+}$ and $a^{†}{\sigma }_{-}$ oscillate at the difference ${\omega }_a-{\omega }_c$, whereas $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$ oscillate at the sum ${\omega }_a+{\omega }_c$. The second main approximation is the rotating-wave approximation: near resonance and for $g\sqrt{n+1}\ll {\omega }_a+{\omega }_c$, the latter terms average away on the interaction time scale. The resulting Jaynes-Cummings model hasThe two-level system and single-normal mode truncations are additional assumptions: other atomic transitions and optical normal modes must be sufficiently far off resonance or weakly coupled. Dissipation is neglected over the interaction time. The retained terms exchange one atomic excitation for one photon, conserving $a^{†}a+{\sigma }_{+}{\sigma }_{-}$.

In mode locking, fixed relative phases between many longitudinal normal modes produce constructive interference during a short part of each optical resonator round trip. The round-trip time sets the repetition period. The available coherent gain bandwidth, the number of locked normal modes, the spectral phase, and residual group velocity dispersion determine the optical pulse duration.

For $M$ consecutive equally weighted normal modes of angular frequency spacing $\Delta \omega$, their wave amplitude sum is proportional toIts central peak has width of order $1/(M\Delta \omega )$, while its repetition period is $2\pi /\Delta \omega$. A transform-limited pulse therefore becomes shorter as more coherent gain bandwidth is used. This mechanism can give picosecond or femtosecond optical pulses: it does not require depletion of the whole stored population inversion during each peak, as Q-switching does. Mode locking generally accesses a much shorter time-bandwidth product scale than the gain-discharge time of Q-switching.

Put $b=\Delta {\omega }_b>0$ and $s=t-t_m$. At the peak time, $e^{-ip\Delta \omega t_m}=1$, so the exact comb sum becomesFor many occupied normal modes, $b\gg \Delta \omega$, replace the sum by an integral with density $1/\Delta \omega$. Setting $x=\omega -{\omega }_0$ and evaluating the Fourier transform of a Gaussian givesThus the isolated-optical pulse approximation hasThe optical pulse has a Gaussian function envelope and no chirp. Its carrier angular frequency is ${\omega }_0$, and its light intensity full width at half maximum is $2\sqrt{2\ln 2}/b$. Successive envelope maxima have different carrier-envelope phases, consistent with b.

Approximation in the printed assertion. A discrete optical frequency comb gives a train, not one exact isolated Gaussian pulse. Applying the Poisson summation formula to the ideal sum over all integers instead givesNear $t_m$, the $j=m$ term dominates when $bT\gg 1$. The other terms are exponentially small but not identically zero. A single Gaussian function used globally would decay between $0$ and $T$, whereas the exact light intensity has equal maxima there; this supplies a concrete reason the isolated form cannot be an exact global identity. The extension to negative frequencies is the usual narrow-positive-band analytic-field approximation when ${\omega }_0\gg b$.

In second-harmonic generation, the quadratic nonlinear polarization contains products of pairs of comb components. Their angular frequencies areincluding $2{\omega }_p$ when $p=q$. Original comb components have angular frequencies ${\omega }_r={\omega }_{\mathrm{ce}}+r\Delta \omega$, henceWhen the doubled low-frequency region overlaps the original high-frequency region, choosing $r=p+q$ gives the beat at $|{\omega }_{\mathrm{ce}}|$. More generally the nearest positive beat has angular frequencywhich is $|{\omega }_{\mathrm{ce}}|$ for the stipulated small offset. For $0\le {\omega }_{\mathrm{ce}}<\Delta \omega$ it is $\min ({\omega }_{\mathrm{ce}},\Delta \omega -{\omega }_{\mathrm{ce}})$, provided the corresponding teeth are in the overlap.

In comb self-referencing, overlap the doubled red part and the original blue part on a photodiode. The beat measures the carrier-envelope offset frequency. Feedback on an intracavity dispersion control or pump setting can set ${\omega }_{\mathrm{ce}}=0$ modulo $\Delta \omega$, giving $\boxed{{\varphi }_{\mathrm{slip}}=0(\mathrm{mod}2\pi )}$. A phase-sensitive lock or calibrated approach to zero resolves the sign ambiguity of an unsigned beat measurement.^[ref]

The coherent optical spectrum must span an octave, ${\omega }_{\max }\gtrsim 2{\omega }_{\min }$, with sufficient overlapping spectral strength after second-harmonic generation. This calls for very short broadband optical pulses, or coherent external spectral broadening. The two branches must preserve optical coherence, overlap in time and polarization, and have adequate phase matching; bandwidth alone does not guarantee a detectable beat.

Measure the car's vertical displacement $y$ from its static equilibrium, so gravity's constant contribution is already balanced. Both the spring and the viscous damper act on motion relative to the wheel. The base-excited oscillator therefore satisfiesFor dimensional clarity write the road profile as $B\sin (qx)$; the printed notation sets the implicit wave number $q=1$ in the chosen length units. Constant forward speed gives forcing angular frequency $\omega =qv$. With complex amplitudes $Y,Y_0$, the steady response isSet $q=1$ for the expression in the question's notation. Assuming $M,k>0$ and $\gamma >0$, the transient response decays, leaving this steady-state response.

For a useful sketch define ${\omega }_n=\sqrt{k/M}$, $r=\omega /{\omega }_n$ and damping ratio $\zeta =\gamma /(2\sqrt{kM})$. The displacement transmissibility isIt starts at $1$, rises to a finite maximum for finite positive damping, crosses $1$ at $r=\sqrt{2}$, then tends to zero. Differentiation with respect to $r^2$ gives the maximum at $r^2=2/[1+\sqrt{1+8{\zeta }^2}]$.

Very small damping permits large resonant oscillations near ${\omega }_n$, producing discomfort and potentially large suspension travel. At exactly zero damping the assumed decay of transients fails, and resonant forcing produces secular growth. Very large damping makes the car follow the bumps over a wide range of speeds, sacrificing vibration isolation: at each fixed forcing frequency, $Y/Y_0\to 1$ as $\gamma \to \infty$. For any fixed finite damping the response still eventually decreases at sufficiently high frequency. These two limits should not be interchanged.

In the steady periodic response, the cycle average of the mechanical energy derivative vanishes. The applied average power therefore equals the power dissipated by the viscous force:Using solution givesThe same result follows from averaging the instantaneous input power $F(t)\dot{z}(t)$, using the response phase relative to the drive.

For positive $m,k,\gamma$ and $\omega >0$, rewrite this asThe last factor is at most one, with equality precisely when $k-m{\omega }^2=0$. Thus the power peak occurs at ${\omega }_0=\sqrt{k/m}$ andThis power-resonance angular frequency is exactly ${\omega }_0$; it need not equal the angular frequency maximizing displacement amplitude. Positive damping is essential: at $\gamma =0$ and exact resonance there is no bounded steady periodic response to nonzero forcing, so the divergent expression is not a finite-power solution.

Use ${\omega }_0^2=k/m$ and the power formula from solution. Its normalized form isIt is zero at $\omega =0$, rises to one at ${\omega }_0$, and decays to zero as $\omega \to \infty$. The two positive half-power angular frequencies satisfyOn the low-frequency side, ${\omega }_0^2-{\omega }_1^2=\gamma {\omega }_1$; on the high-frequency side, ${\omega }_2^2-{\omega }_0^2=\gamma {\omega }_2$. Solving these quadratic equations givesConsequently the frequency bandwidth of viscous oscillator power absorption obeys the stronger exact resultFor a sharp resonance, $\gamma \ll {\omega }_0$, expansion gives ${\omega }_{1,2}={\omega }_0\mp \gamma /2+O({\gamma }^2/{\omega }_0)$, which recovers the requested approximation. The exact width above assumes the constant viscous damping coefficient in this model. Although the curve is approximately symmetric near a sharp peak, the exact half-power points satisfy ${\omega }_1{\omega }_2={\omega }_0^2$ rather than exact symmetry about ${\omega }_0$.

Let $x_1,x_2$ be displacements from equilibrium, with positive directions chosen alike. Hooke's law and Newton's second law giveFor a normal mode $x_j=a_j\cos \Omega t$, the determinant condition isSolving the quadratic for ${\Omega }^2$ givesThe angular frequencies are the positive square roots. Positive masses and spring constants make the stiffness quadratic form $k_1{x}_1^2+k_2(x_2-x_1{)}^2+k_3{x}_2^2$ positive definite, ensuring both squared frequencies are positive. Distinct spring constants do not require equal masses or any symmetry approximation.

The transmitted and incident waves have the same angular frequency and tension, but the transmitted wavenumber is twice as large. The time-average string-power formula therefore givesThe extra factor $k_2/k_1$ is essential. Squaring the displacement transmission coefficient alone would give $4/9$, which is not the transmitted power fraction in strings with different linear mass densities.