Microwave Photon-Number Amplification

We experimentally demonstrate a microwave photon-multiplication scheme which combines the advantages of a single-photon detector and a power meter by multiplying the incoming photon number by an integer factor. Our first experimental implementation achieves an $n=3$-fold multiplication with 0.69 efficiency in a 116 MHz bandwidth up to an input photon rate of 400 MHz. It loses phase information but does not require any dead time or time binning. We expect an optimized device cascading such multipliers to achieve number-resolving measurement of itinerant photons with low dark count, which would offer new possibilities in a wide range of quantum-sensing and quantum-computing applications.

Popular Summary

The workhorse for quantum measurements in the microwave domain is the Josephson parametric amplifier, which can linearly amplify microwave signals with noise very close to the minimal amount permitted by quantum mechanics: one photon. Such an amplifier can even reach below this limit for homodyne measurements, when the phase of the signal of interest is well defined and signals with conjugate phase may be deamplified. For many quantum-sensing applications, however, the important information is in the signal power, and the phase is irrelevant or unknown. Here, we experimentally demonstrate a new photon-number amplification scheme able to linearly amplify the photon number by an integer factor.

Our amplification scheme is not subject to the quantum limit of amplification because, like a phase-sensitive amplifier, it deamplifies the conjugate variable, in this case phase. It fills the gap between linear amplifiers and nonlinear single-photon detectors, which sense just the presence or absence of photons. Our implementation can multiply incoming microwave photons by a factor of up to 3 with high efficiency.

We expect an optimized device cascading such multipliers to achieve number-resolving measurements of itinerant photons, which would offer new possibilities in a wide range of quantum-sensing and quantum-computing applications.

Figure 1

Setup, sample, and working principle. (a) The sample consists of two buffer resonators at frequencies ${\nu }_{\mathrm{in}}=4.8\mathrm{GHz}$ and ${\nu }_{\mathrm{out}}=6.13\mathrm{GHz}$. The resonators are nonlinearly coupled by a SQUID biased at a voltage $V$ via a heavily filtered bias line. Its source impedance is modeled by a $R_{\mathrm{B}}=5\mathrm{\Omega }$ resistor at an effective temperature of 90 mK. An on-chip capacitor $C_{\mathrm{B}}\approx 100\mathrm{pF}$ shunts $R_{\mathrm{B}}$ at the operation frequency (see Appendix pp1 for more details). (b) Schematics of the energy balance for the $n=2$ conversion process. (c) Optical micrograph of the photomultiplier device. The input and output transmission lines are highlighted in yellow, the SQUID in red, the flux bias line in purple, and the input (output) resonator in green (blue). The large spirals provide the inductance of the input and output modes, thus allowing for high characteristic impedance modes.

Figure 2

Spontaneous emission. (a) Power spectral density (PSD) as a function of emission frequency and bias voltage. The PSD for the input (output) resonator is plotted in shades of green (blue). The flux in the SQUID loop is set to ${\mathrm{\Phi }}_0/2$ to minimize the Josephson energy. (b),(c) Photon emission rate from the input and output resonator, respectively, integrated over a 400 MHz bandwidth centered at 4.8 GHz (input) and 6.13 GHz (output) for 5 values of flux in the SQUID loop. These frequencies are represented by horizontal dashed gray lines in (a). The flux values of $0.47{\mathrm{\Phi }}_0$, $0.39{\mathrm{\Phi }}_0$, and $0.28{\mathrm{\Phi }}_0$ correspond to the dashed vertical lines in Fig. 3 and the flux used in Fig. 4. The three vertical green dashed lines correspond, from left to right, to the voltages used for the conversions of Figs. 3 and 4 with, respectively, $n=1$, 2, and 3.

Figure 3

Reflection, conversion, transmission, and inelastic reflection probabilities as well as spontaneous emission for $n=1$ (a), 2 (b), and 3 (c) as a function of flux in the SQUID loop. The vertical dashed lines represent the flux values at which the conversion probabilities are highest. In (a), the input frequency is 4.773 GHz and the bias voltage is 1.30 GHz. In (b), the input frequency is 4.71 GHz and the bias voltage is 7.37 GHz. In (c), the input frequency is 4.74 GHz and the bias voltage is 13.37 GHz. For these three plots, the input power of $-127\mathrm{dBm}$ is chosen to have less than one photon on average in the input resonator.

Figure 4

Bandwidth and saturation power. (a) Reflection, conversion, transmission, and inelastic reflection probabilities for the $n=3$ process, taken at the flux and voltage values maximizing the conversion probability indicated by a dashed vertical line in Fig. 3. The input power is $-127\mathrm{dBm}$. (b) Conversion probability for the $n=3$ conversion for various input powers. The inset shows the conversion probability at 4.74 GHz (maximum of conversion at low input power) as a function of input power.

Figure 5

Schematics of the experimental setup in the dilution refrigerator.

Figure 6

Voltage noise. PSD of the input resonator at 4.78 GHz (green) and of the output resonator (blue) at 6.1 GHz, both taken at $\mathrm{\Phi }={\mathrm{\Phi }}_0/2$.

Figure 7

Measured conversion probability for the $n=1$ conversion as a function of input frequency and bias voltage for several SQUID flux values. $0.47{\mathrm{\Phi }}_0$ corresponds to the optimal operation point represented by a dashed vertical line in Fig. 3.

Figure 8

Fit of the conversion probabilities for $n=1$ from Fig. 7 to theory [17]. The fit is performed considering $0.47{\mathrm{\Phi }}_0$ to be the ideal matching value of the system (see text).

Figure 9

Bandwidth and saturation power for the $n=1$ conversion. (a) Reflection, conversion, transmission, and inelastic reflection probabilities taken at the flux value maximizing the conversion probability, indicated by a dashed vertical line in Fig. 3. The input power is $-127\mathrm{dBm}$. (b) Conversion probability for various input powers. The inset shows the conversion probability at 4.84 GHz (maximum of conversion at low input power) as a function of input power.

Figure 10

Bandwidth and saturation power for the $n=2$ conversion. (a) Reflection, conversion, transmission, and inelastic reflection probabilities taken at the flux value maximizing the conversion probability, indicated by a dashed vertical line in Fig. 3. The input power is $-127\mathrm{dBm}$. (b) Conversion probability for various input powers. The inset shows the conversion probability at 4.71 GHz (maximum of conversion at low input power) as a function of input power.

Figure 11

Partial spontaneous emission from the input or output resonator of our device as a function of the bias voltage, at $T=10\mathrm{mK}$, calculated using $P(E)$ theory with $f_{\mathrm{in}}=4.8\mathrm{GHz}$, $f_{\mathrm{out}}=6.13\mathrm{GHz}$, $Z_{\mathrm{c}}^{\mathrm{in}}=Z_{\mathrm{c}}^{\mathrm{out}}\approx 400\mathrm{\Omega }$, ${\gamma }_{\mathrm{in}}=96\mathrm{MHz}$, and ${\gamma }_{\mathrm{out}}=226\mathrm{MHz}$. $E_{\mathrm{J}}$ is chosen to be the theoretical value for perfect matching for each conversion process. The dashed lines indicate the nonapplicability of the theoretical model (see main text). The vertical dashed red lines indicate the voltage bias used for the given conversion process, and the rates written in gray correspond to the expected total spontaneous emission rate at the operating point.

Figure 12

Partial spontaneous emission from the input or output resonator of a hypothetical device as a function of the bias voltage, at $T=10\mathrm{mK}$, calculated using $P(E)$ theory with $f_{\mathrm{in}}=4.8\mathrm{GHz}$, $f_{\mathrm{out}}=6.13\mathrm{GHz}$, $Z_{\mathrm{c}}^{\mathrm{in}}=600\mathrm{\Omega }$ $Z_{\mathrm{c}}^{\mathrm{out}}=800\mathrm{\Omega }$, ${\gamma }_{\mathrm{in}}=96\mathrm{MHz}$, and ${\gamma }_{\mathrm{out}}=100\mathrm{MHz}$. $E_{\mathrm{J}}$ is chosen to be the theoretical value for perfect matching for each conversion process. The dashed lines indicate the nonapplicability of the theoretical model (see main text). The vertical dashed red lines indicate the voltage bias used for the given conversion process, and the rates written in gray correspond to the expected total spontaneous emission rate at the operating point.