Microwave Measurement beyond the Quantum Limit with a Nonreciprocal Amplifier

The measurement of a quantum system is often performed by encoding its state in a single observable of a light field. The measurement efficiency of this observable can be reduced by loss or excess noise on the way to the detector. Even a quantum-limited detector that simultaneously measures a second noncommuting observable would double the output noise, therefore limiting the efficiency to $50\mathrm{\%}$. At microwave frequencies, an ideal measurement efficiency can be achieved by noiselessly amplifying the information-carrying quadrature of the light field but this has remained an experimental challenge. Indeed, while state-of-the-art Josephson-junction-based parametric amplifiers can perform an ideal single-quadrature measurement, they require lossy ferrite circulators in the signal path, drastically decreasing the overall efficiency. In this paper, we present a nonreciprocal parametric amplifier that combines single-quadrature measurement and directionality without the use of strong external magnetic fields. We extract a measurement efficiency of ${62}_{-9}^{+17}\mathrm{\%}$ that exceeds the quantum limit and that is not limited by fundamental factors. The amplifier can be readily integrated with superconducting devices, creating a path for ideal measurements of quantum bits and mechanical oscillators.

Figure 1

(a) A schematic of a generic nonreciprocal phase-sensitive amplifier. A signal incident at the input is phase-sensitively amplified toward the output, while in the reverse direction, a signal is transmitted from the output to the input with unity transmission. (b) A typical physical implementation of (a) using a JPA, providing phase-sensitive reflection gain, and a ferrite circulator, ensuring signal flow from the input to the output. (c) A frequency-space diagram showing the density of states (DOS) for the JPA. A strong pump of frequency ${\omega }_{aa}^P=2{\omega }_a$ amplifies an input signal at the frequency ${\omega }_a^S={\omega }_a+{\delta }_a$ and generates an idler at the frequency $-{\omega }_a^I=-{\omega }_a+{\delta }_a$. (d) An all-parametric physical implementation of (a). The ferrite circulator in (b) is replaced by a parametric circulator based on a FPJA. This device has three resonant modes: an “input” mode of frequency ${\omega }_c$, a “gain” mode of frequency ${\omega }_b$, and an ‘output” mode of frequency ${\omega }_a$. The input and output signals are routed to different physical ports by low-pass and high-pass filters (LPF and HPF, respectively). (e) A frequency-space diagram showing the DOS for the FPJA. Similarly to (c), a strong pump of frequency ${\omega }_{bb}^P=2{\omega }_b$ amplifies a signal and idler at the frequency ${\omega }_b^S={\omega }_b+{\delta }_b$ and $-{\omega }_b^I=-{\omega }_b+{\delta }_b$. Parametric circulation is obtained using three frequency-conversion pumps with frequencies ${\omega }_{ab}^P={\omega }_b-{\omega }_a$, ${\omega }_{bc}^P={\omega }_c-{\omega }_b$, and ${\omega }_{ac}^P={\omega }_c-{\omega }_a$, satisfying the condition ${\omega }_{ab}^P+{\omega }_{bc}^P={\omega }_{ac}^P$ and connecting signals and idlers at ${\omega }_j^S={\omega }_j+{\delta }_j$ and $-{\omega }_j^I=-{\omega }_j+{\delta }_j$ ($j\in \{a,b,c\}$).

Figure 2

The measured scattering parameters of the FPJA programmed as a phase-sensitive directional amplifier (dots) as a function of the drive detuning ${\delta }_{a,c}=({\omega }_{a,c}^S-{\omega }_{a,c})$. The solid lines are the predictions from Eq. (B3). The device exhibits high gain from mode $c$ to mode $a$, over a $7$-$\mathrm{MHz}$ bandwidth, and unity transmission in the reverse direction. The device is impedance matched, with the return loss exceeding $10\mathrm{dB}$ over a $4$-$\mathrm{MHz}$ bandwidth.

Figure 3

(a) The quadrature gain and noise rise measured for the same pump parameters as in Fig. 2, as a function of the LO phase. We observe a quadrature gain of $24\mathrm{dB}$ and an antigain of $-11\mathrm{dB}$, in good agreement with theoretical predictions from Eq. (7) (line). The measurement noise floor rises much slower than the gain, with up to $14\mathrm{dB}$ of signal-to-noise improvement. (b) The estimated system efficiency (dots), error bars (shaded area, dominated by the uncertainty of the calibration of $n_{\mathrm{add}}^{\mathrm{chain}}$), and theoretical predictions from Eq. (8) (lines), as a function of the quadrature gain.

Figure 4

The device picture and layout. (a) An optical micrograph of the device. (b) The circuit equivalent of the device. The center element is a FPJA [19] and consists of a SQUID shunted by a set of inductors and capacitors. The FPJA is capacitively coupled to ground and to an on-chip frequency diplexer built using a nine-pole low-pass filter (LPF), on the left, and a nine-pole high-pass filter (HPF), on the right, with a cutoff frequency around 7.6 GHz.

Figure 5

Process spectroscopy. (a) The measured mode frequencies ${\omega }_j$ and the predicted first-order modulation frequencies $∣{\omega }_j\pm {\omega }_k∣$ ($j,k\in \{a,b,c\}$) as a function of the flux. The shaded areas represent a bandwidth of $180\mathrm{MHz}\approx 3\times ({\kappa }_a+{\kappa }_c)/2$, which is necessary to ensure a good rotating-wave approximation. The cutoff of the diplexer at 7.6 GHz is shown as the horizontal dashed line. The circuit is biased at $\mathrm{\Phi }/{\mathrm{\Phi }}_0\approx 0.26$, shown as a vertical dotted line.

Figure 6

Mode-coupling graphs. (a) The full graph for the phase-sensitive directional amplifier. (b) A reduced graph evaluated at zero pump detuning. This graph is equivalent to that in (a), with the upper triangle of the original graph included as an effective self-loop of mode $b$.

Figure 7

The measured region of instability as a function of the amplifier direct gain, for the same pump parameters as in Fig. 2. Below a gain of approximately 6 dB, the amplifier is stable for all loop phases. At higher gain, the region of instability grows and approaches the loop phases ${\varphi }_{\mathrm{loop}}=\pm \pi /2$, at which the amplifier is directional.

Figure 8

The input power at the 1-dB compression point, as a function of the gain, for two modes of operation of the FPJA. In green, the FPJA is programmed using a single pump at ${\omega }_{aa}^P=2{\omega }_a$, leading to phase-sensitive gain around mode $a$. In red, the FPJA is programmed as a directional phase-sensitive amplifier, as described in the main text. The gain is defined as $G_S(r)-G_S(0)$, to account for internal losses and frequency-conversion losses at finite $s$. The solid green and red lines are linear fits, with slopes of $-1.5$ dBm/dB and $-1.3$ dBm/dB, respectively.

Figure 9

The directional phase-sensitive amplifier is tuned in four steps. (a) The frequency converter ($s=0$, $r=0$, ${\varphi }_{\mathrm{loop}}=na$). A pump at ${\omega }_{ac}^P=∣{\omega }_c-{\omega }_a∣$ enables near-ideal frequency conversion between modes $a$ and $c$. (b) The isolator ($s=0.5$, $r=0$, ${\varphi }_{\mathrm{loop}}=\pi /2$). Two additional pumps at, ${\omega }_{ab}^P=∣{\omega }_b-{\omega }_a∣$ and ${\omega }_{bc}^P=∣{\omega }_c-{\omega }_b∣$, couple modes $a$ and $c$ to mode $b$ at a rate matching its dissipation. The device behaves as an isolator, with the loop phase set to have unity transmission from mode $a$ to mode $c$ and high isolation in the reversed direction. (c) The mismatched isolator ($s=0.8$, $r=0$, ${\varphi }_{\mathrm{loop}}=\pi /2$). The coupling rates to mode $b$ are increased to overwhelm its dissipation, leading to almost full reflection off mode $b$. Impedance matching and unity transmission from mode $a$ to mode $c$ are maintained but isolation in the reversed direction is reduced. This will ensure low added noise when operated as an amplifier. (d) The directional phase-sensitive amplifier at high gain ($s=0.8$, $r=0.91$, ${\varphi }_{\mathrm{loop}}=\pi /2$). A pump at ${\omega }_{bb}^P=2{\omega }_b$ induces gain from mode $c$ to mode $a$.