Jonginn Yun

These notes organize the device-physics background for traveling-wave parametric amplifiers, microwave readout constraints, and slowly varying envelope equations.

TWPA Terminology

A traveling-wave parametric amplifier uses a large pump tone to amplify a weak signal through a nonlinear transmission line.

• pump: a large tone supplying the energy,

  $$\omega_p,\qquad k_p\approx \omega_p/v,$$

  with example current waveform

  $$I_p(x,t)=I_0\cos(k_px-\omega_pt).$$

• signal: the weak tone to be amplified,

  $$\omega_s,\qquad k_s .$$

• idler: the companion frequency component necessarily generated by the nonlinear parametric process,

  $$\omega_i,\qquad k_i .$$

Nonlinear Telegrapher Equations

Consider a nonlinear transmission line where the nonlinearity enters only through the inductance:

$$\frac{\partial I}{\partial x} = -C\frac{\partial V}{\partial t}, \qquad \frac{\partial V}{\partial x} = -\frac{\partial}{\partial t}[L(I)I].$$

Assume the nonlinear inductance is

$$L(I)=L_0(1+\alpha I^2).$$

Then the telegrapher equations become

$$\begin{aligned} \frac{\partial V}{\partial x} &= -L_0(1+3\alpha I^2)\frac{\partial I}{\partial t},\\ \frac{\partial I}{\partial x} &= -C\frac{\partial V}{\partial t}. \end{aligned}$$

Decompose current and voltage into a large pump and a small signal:

$$\begin{aligned} I(x,t)&=I_p(x,t)+i(x,t),\\ V(x,t)&=V_p(x,t)+v(x,t). \end{aligned}$$

The linearized equations are

$$\begin{aligned} \frac{\partial v}{\partial x} &= -L_0(1+3\alpha I_p^2)\frac{\partial i}{\partial t} -6\alpha L_0 I_p(\partial_t I_p)i\\ &= -\frac{\partial}{\partial t} \left[L_{\mathrm{eff}}(x,t)i(x,t)\right],\\ \frac{\partial i}{\partial x} &= -C\frac{\partial v}{\partial t}, \end{aligned}$$

where

$$L_{\mathrm{eff}} = L_0(1+3\alpha I_p^2(x,t)).$$

With

$$I_p(x,t)=I_0\cos(k_px-\omega_pt),$$

the effective inductance is of the form

$$L_{\mathrm{eff}}(x,t) = \bar L+\delta L\cos(2k_px-2\omega_pt).$$

The next step is to derive the equations of motion using pump, signal, and idler fields together with the slowly varying envelope approximation.

Readout Cavity Photon-Number Constraint

The readout cavity should contain only a small number of photons for two reasons.

First, to remain in the dispersive regime, the Schrieffer-Wolff generator

$$S=\frac{g}{\Delta}(a\sigma_+-a^\dagger\sigma_-)$$

must be small. Since

$$a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle,$$

one needs approximately

$$n\ll \frac{\Delta^2}{g^2}.$$

Equivalently, the qubit states should not be strongly mixed by the cavity coupling.

Second, if the cavity is coherently populated, then

$$\mathrm{var}(n)=\bar n.$$

As the mean photon number grows, dispersive coupling shifts the qubit frequency according to

$$\omega_q \to \omega_q+\chi+2\chi n,$$

which produces dephasing. Suppressing dephasing also requires low cavity photon number.

If the cavity linewidth is $\kappa$, the output power for $n$ photons is approximately

$$n\hbar\omega\kappa,$$

because $b_{\mathrm{out}}=\sqrt{\kappa_{\mathrm{ext}}}a$.

Useful scale estimates:

• transmon frequencies are typically a few GHz to suppress thermal population,

• $\kappa$ should be large enough for measurement speed but small enough to resolve $\chi$,

• the readout cavity mean photon number is often on the order of $1$ to $10$ photons, with the exact rationale marked for checking,

• room-temperature thermal-noise power spectral density is approximately

  $$kT\approx 4\times 10^{-21}\,\mathrm{W/Hz}.$$

Using dBm,

$$P_{\mathrm{dBm}} = 10\log_{10}\frac{P_{\mathrm{mW}}}{1\,\mathrm{mW}}.$$

For room-temperature noise over bandwidth $B$,

$$P_N \approx -174+10\log_{10}(B/\mathrm{Hz})\ \mathrm{dBm}.$$

Even for $B=1\,\mathrm{MHz}$, this gives roughly $-114\,\mathrm{dBm}$. Typical weak microwave readout powers can be around this scale, so amplification cannot be treated as an afterthought.

Thus improving SNR by simply increasing readout power is limited by qubit dephasing and by breakdown of the dispersive approximation. The amplification chain must provide large forward gain and low added noise.

The amplifier also produces noise in principle. If that noise propagates backward into the cavity, it injects unwanted photons and increases qubit dephasing. In practice, circulators and isolators are inserted to enforce directionality and suppress backaction.

Phase-Preserving Quantum-Limited Amplifiers

A phase-preserving quantum-limited amplifier has the same input-output structure as a parametric amplifier. The reason is that the minimal-noise mathematical structure is a Bogoliubov transformation.

For a general phase-preserving linear amplifier,

$$\hat a_{\mathrm{out}} = \sqrt G\,\hat a_{\mathrm{in}} + \hat F .$$

For the output to remain a valid bosonic mode, the canonical commutation relation must be preserved. This implies

$$[F,F^\dagger]=1-G<0.$$

Therefore $F$ must be proportional to a creation operator of another mode:

$$F=e^{i\phi}\sqrt{G-1}\,\hat b_{\mathrm{in}}^\dagger .$$

Conventional resistive amplifiers involve dissipative degrees of freedom and cannot generally reach this ideal limit. Parametric amplifiers modulate device parameters rather than relying on resistance, so their native interaction can realize the two-mode-squeezing Hamiltonian that produces this Bogoliubov transformation.

TWPA As A Distributed Nonlinear Medium

Compared with a JPA, which uses discrete standing-wave cavity modes, a TWPA is a distributed nonlinear medium.

If nonlinearity enters through the inductance, the inductive energy density is

$$u(I)=\int_0^I l(I')\,dI'.$$

For

$$l(I)=l_0(1+\epsilon I+\xi I^2+\cdots),$$

the energy density is

$$u(I) = \frac12 l_0I^2 + \frac13 l_0\epsilon I^3 + \frac14 l_0\xi I^4+\cdots .$$

Thus the nonlinear Hamiltonian has the schematic form

$$H_{\mathrm{nl}} = \int dx\, \left( \chi_3 I^3(x,t)+\chi_4 I^4(x,t)+\cdots \right).$$

Decompose the positive-frequency current into pump, signal, and idler slowly varying envelopes:

$$I^{(+)} = I_p\alpha_p(x,t)e^{i(k_px-\omega_pt)} + I_s\psi_s(x,t)e^{i(k_sx-\omega_st)} + I_i\psi_i(x,t)e^{i(k_ix-\omega_it)} .$$

Under the RWA, keeping the resonant process $\omega_p=\omega_s+\omega_i$ with

$$\Delta k=k_p-k_s-k_i,$$

one obtains an interaction of the form

$$H_{\mathrm{int}} = \hbar\int dx\, \left[ g(x)e^{i\Delta kx}\psi_s^\dagger\psi_i^\dagger + \mathrm{c.c.} \right].$$

For a JPA, the signal and idler are standing-wave cavity modes, so one can replace $\psi_i(x)$ by a fixed mode function times a single annihilation operator. For a TWPA, the fields remain extended along the transmission line, so the local field $\psi(x)$ must be kept.

Slowly Varying Envelope Equations

Applying the Heisenberg equation and the slowly varying envelope approximation gives

$$\begin{aligned} \partial_x \psi_s &= \frac{g(x)}{v_{gs}}e^{i\Delta kx}\psi_i^\dagger(x),\\ \partial_x \psi_i &= \frac{g^*(x)}{v_{gi}}e^{-i\Delta kx}\psi_s^\dagger(x). \end{aligned}$$

The approximation assumes a locally linear dispersion,

$$\omega(k)\approx \omega_0+v_g(k-k_0),$$

so higher derivatives in $q=k-k_0$ are neglected. In real space this corresponds to neglecting higher spatial derivatives of the envelope.

The free Hamiltonian for the envelope is

$$H_0=\hbar v_g\int dq\,q\,b^\dagger(q)b(q).$$

Define

$$\psi(x)=\frac{1}{\sqrt{2\pi}}\int dq\,e^{iqx}b(q).$$

Then

$$H_0=-i\hbar v_g\int dx\,\psi^\dagger\partial_x\psi.$$

This envelope description follows from the full field:

$$\begin{aligned} \Phi(x,t) &= \frac{1}{\sqrt{2\pi}} \int dq\,e^{i(kx-\omega(k)t)}a(k,t)\\ &= \frac{1}{\sqrt{2\pi}} \int dq\,e^{i(k_0+q)x-i\omega(k_0+q)t}a(k_0+q,t)\\ &= e^{i(k_0x-\omega_0t)} \left[ \frac{1}{\sqrt{2\pi}} \int dq\,e^{i(qx-(\omega(k_0+q)-\omega_0)t)}b(q,t) \right]. \end{aligned}$$

Thus

$$\psi(x,t) = \frac{1}{\sqrt{2\pi}} \int dq\,e^{i(qx-(\omega(k_0+q)-\omega_0)t)}b(q,t).$$

The envelope is governed by the same Hamiltonian as the Fourier components $b(q,t)$. With the SVEA Hamiltonian,

$$\begin{aligned} H_{\mathrm{exact}} &= \int dq\,\hbar[\omega(k_0+q)-\omega(k_0)]b^\dagger(q)b(q),\\ H_{\mathrm{SVEA}} &= \int dq\,\hbar v_gq\,b^\dagger(q)b(q). \end{aligned}$$

The commutator gives

$$\begin{aligned} [\psi(x),H_0] &= \frac{1}{\sqrt{2\pi}}\int dq\,e^{iqx}[b(q),H_0]\\ &= \hbar v_g\int dq\,q\,e^{iqx}b(q)\\ &= -i\hbar v_g\partial_x\psi(x). \end{aligned}$$

Together with the interaction term,

$$\begin{aligned} [\psi,H_0] &= -i\hbar v_g\partial_x\psi,\\ [\psi_s,H_{\mathrm{int}}] &= i\hbar g(x)e^{i\Delta kx}\psi_i^\dagger . \end{aligned}$$

This is the route to the steady-state propagation equations above.