Multiplying and detecting propagating microwave photons using inelastic Cooper-pair tunneling

The interaction between propagating microwave fields and Cooper-pair tunneling across a DC-voltage-biased Josephson junction can be highly nonlinear. We show theoretically that this nonlinearity can be used to convert an incoming single microwave photon into an outgoing $n$-photon Fock state in a different mode. In this process, the electrostatic energy released in a Cooper-pair tunneling event is transferred to the outgoing Fock state, providing energy gain. The created multiphoton Fock state is frequency entangled and highly bunched. The conversion can be made reflectionless (impedance matched) so that all incoming photons are converted to $n$-photon states. With realistic parameters, multiplication ratios $n>2$ can be reached. By two consecutive multiplications, the outgoing Fock-state number can get sufficiently large to accurately discriminate it from vacuum with linear postamplification and power measurement. Therefore, this amplification scheme can be used as a single-photon detector without dead time.

Figure 1

(a) We investigate microwave scattering in a transmission line connected to two resonators, with frequencies ${\omega }_a$ and ${\omega }_b$, and a Josephson junction with coupling energy $E_{\mathrm{J}}$. An incoming photon from the left interacts with Cooper pair tunneling across the Josephson junction that emits an outgoing field to right. When impedance matched, an incoming photon of frequency ${\omega }_a$ deterministically converts into $n$ outgoing photons of average frequency ${\omega }_b$. (b) The energy diagram of photon tripling with slight frequency down-conversion. Energy is absorbed from the Cooper-pair tunneling event, $\hslash {\omega }_a+2eV=3\hslash {\omega }_b$. Generally, it is possible to up-convert (${\omega }_a>{\omega }_b$) and down-convert (${\omega }_a<{\omega }_b$) incoming microwave photons.

Figure 2

(a) Average number of created photons $N_{\mathrm{out}}$ from a single-photon input as a function of (absolute value of) coupling amplitude ${\epsilon }_{\mathrm{I}}$, Eq. (27), for multiplication factors $n=1,2,3,4$ (which are also the maximum values of $N_{\mathrm{out}}$, correspondingly). Irrespective of the resonator quality factors, one can always achieve a deterministic photon multiplication (impedance matching) by correctly tuning ${\epsilon }_{\mathrm{I}}$. The corresponding value of ${\epsilon }_{\mathrm{I}}$ decreases with $n$. (b) The result of panel (a) plotted as a function of Josephson coupling $E_{\mathrm{J}}^{*}$, Eq. (28), for couplings $g_{a/b}=1$. The optimal value for $E_{\mathrm{J}}^{*}$ increases rapidly with $n$. This ultimately leads to breakdown of the RWA for higher $n$, as discussed in Sec. 7.

Figure 3

The conversion probability $T$ as a function of frequency offset $\delta \omega =\omega -{\omega }_a$ and coupling ${\epsilon }_n$ for $n=1$ as given by Eq. (40). We consider the cases (a) ${\gamma }_a={\gamma }_b$ and (b) ${\gamma }_a={\gamma }_b/3$. We obtain that perfect transmission is also possible for $|{\epsilon }_1|>1$, but only when ${\gamma }_a={\gamma }_b$. The result of case (a) is also valid for arbitrary $n$ when ${\gamma }_a=n{\gamma }_b$. The result of case (b) is also valid for $n=3$ when ${\gamma }_a={\gamma }_b$.

Figure 4

Average conversion probability $T=N_{\mathrm{out}}/n{N}_{\mathrm{in}}$ of coherent-state pulses as a function of identical resonator decay rates ${\gamma }_a={\gamma }_b=\gamma$ and average input photon number $N_{\mathrm{in}}$, when biased at the photon-tripling resonance ($n=3$). We consider an incoming pulse of width ${\gamma }_{\mathrm{in}}$ with wave form of Eq. (57). The reflectionless conversion corresponds to the limit $\gamma /{\gamma }_{\mathrm{in}}\to \infty$, where $N_{\mathrm{out}}/n{N}_{\mathrm{in}}\to 1$.

Figure 5

Cascaded photomultiplication. (a) Two photomultiplication stages as in Fig. 1 are cascaded with a shared cavity mode at frequency ${\omega }_c$, acting as output mode for the first stage and as input mode for the second stage. This mode is assumed to have negligible loss. (b) If one photon leaves the output mode, the full process becomes irreversible and all photons have to leave via the output mode. Therefore, like in the one-stage case, an incoming photon is either reflected or fully converted.

Figure 6

The conversion probability $T$ in the cascaded setup for $n_{\mathrm{in}}=n_{\mathrm{out}}=1$, as given by Eq. (67). (a) For ${\gamma }_a={\gamma }_b$, ${\epsilon }_{\mathrm{in}}={\epsilon }_{\mathrm{out}}$, and as a function of frequency offset $\delta \omega =\omega -{\omega }_a$, the (deterministic) conversion peak splits into three when the resonator couplings are increased. (b) For $\omega ={\omega }_a$ and asymmetric parameters ${\gamma }_a\ne {\gamma }_b$ (here ${\gamma }_b=9{\gamma }_a$), deterministic conversion is possible when $\sqrt{{\gamma }_b}{\epsilon }_{\mathrm{in}}=\sqrt{{\gamma }_a}{\epsilon }_{\mathrm{out}}$.

Figure 7

(a) The conversion probability $T$ in the cascaded photomultiplication when $n_{\mathrm{in}}=n_{\mathrm{out}}=3$, $g_c=1.0$, $g_b=1.41$, and $\omega ={\omega }_a$ (the result does not depend on $g_a$). The conversion can be deterministic (very close to one) when ${\epsilon }_{\mathrm{out}}\gg {\epsilon }_{\mathrm{in}}$. (b) The frequency dependence of the transmission probability for ${\epsilon }_3^{\mathrm{in}}=1/3$ (narrow Lorentzian), ${\epsilon }_3^{\mathrm{in}}=1/2$ (rectangular shape), and ${\epsilon }_3^{\mathrm{in}}=1$ (wide Lorentzian) with ${\epsilon }_3^{\mathrm{out}}$ that provides deterministic conversion when $g_c=0.25$ and $g_b=1.41$.

Figure 8

Error probabilities for single-photon detection using a linear quantum-limited phase-preserving amplifier at the output of a reflectionless $1\to n$ photon multiplier. The detector clicks whenever the effective photon number measured by the amplifier in the output mode of multiplier (see text) exceeds a threshold $N_{\mathrm{th}}$. The black line labeled $|0\rangle$ indicates the dark-count probability $P_{\mathrm{dark}}$ (false click) within an inverse bandwidth due to the amplifier noise. The lines labeled $|n\rangle$ with $n>0$ show the probability $P_{\mathrm{miss}|n\rangle }$ of a $|n\rangle$ state in the output mode of the photomultiplier not triggering a click.

Figure 9

Spontaneous emission by single-junction photomultiplier triggered by vacuum fluctuations at frequencies other than ${\omega }_{a/b}$. We consider a bias point for converting single ${\omega }_a/2\pi =7$ GHz photon into three ${\omega }_b/2\pi =5$ GHz photons. We plot the numerically evaluated total photon flux (in units of $\gamma ={\gamma }_a={\gamma }_b$) at output frequency ${\omega }_b$ within bandwidth 1 GHz as a function of resonator bandwidths $\gamma /2\pi$. We consider emission for parameters $g_a=0.25$, $g_b=1.41$ (top) and $g_a=g_b=1$ (second from the top). The emission with an antiresonance at $\delta \omega ={\omega }_{\mathrm{J}}-{\omega }_b=3$ GHz is plotted for $g_a=0.25$, $g_b=1.41$ (third from the top), and $g_a=g_b=1$ (bottom). The form of the antiresonance is given by Eq. (92).

Figure 10

Lumped-element model of the considered microwave circuit.