Josephson-Photon Multiplication (JPM)

• Josephson-Photon Multiplication (JPM) is an engineered nonlinear process that converts single microwave photons into higher-order Fock states using inelastic Cooper-pair tunneling.
• Device designs employ voltage-biased Josephson junctions coupled to resonators to achieve deterministic photon amplification with efficiencies up to 95% and low dark count rates.
• JPM plays a crucial role in circuit QED by enhancing quantum detection, enabling precise number-resolved measurements, and supporting scalable quantum information protocols.

Josephson-Photon Multiplication (JPM) refers to the engineered nonlinear conversion of microwave photons into higher-order Fock states via inelastic Cooper-pair tunneling in Josephson circuits. Devices based on JPM enable deterministic, number-resolved amplification of itinerant microwave photons and have become pivotal for high-efficiency detection, photon-number amplification, and the generation of nonclassical states in circuit QED and microwave quantum optics.

1. Fundamental Principles and Hamiltonian Formalism

Josephson-photon multiplication utilizes superconducting circuits where two or more distinct microwave modes are coupled by a voltage-biased Josephson junction. The energy associated with a tunneling Cooper pair (set by the dc bias $V_{\rm dc}$) is precisely matched to the energy difference between an incoming photon and a set of outgoing photons in a separate mode. The generic lab-frame Hamiltonian for a two-mode device is (Zeller et al., 17 Mar 2026, Danner et al., 9 Oct 2025):

$$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$

where $a$, $b$ are the annihilation operators of the incoming and outgoing resonators (frequencies $\omega_a$, $\omega_b$); $E_J$ is the Josephson energy; $\omega_{\rm dc} = 2e V_{\rm dc}/\hbar$ is the Josephson frequency; $\alpha_0$, $\beta_0$ are the zero-point phase fluctuations.

Expanding the cosine and retaining only resonant terms under the rotating-wave approximation (RWA), for $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$0-photon multiplication ($$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$1 conversion), the effective Hamiltonian is

$$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$2

The resonance condition is $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$3, ensuring that each Cooper-pair tunneling event "consumes" one photon in mode $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$4 and emits $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$5 photons in mode $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$6 (Zeller et al., 17 Mar 2026, Danner et al., 9 Oct 2025, Leppäkangas et al., 2016).

2. Device Architectures and Multiplication Mechanisms

Typical JPM circuits consist of a Josephson junction (often SQUID-tunable) in series with two lumped-element or distributed LC resonators (or transmission lines) (Danner et al., 9 Oct 2025, Leppäkangas et al., 2016, Albert et al., 2023). The input resonator is coupled to the signal to be detected or multiplied; the output resonator collects the multiplied photon emission. Multiplication factors $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$7 (and higher via cascading) are readily achievable with realistic Josephson energies and resonator parameters.

The junction provides the highly nonlinear interaction; by tuning EJ and the impedance (via $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$8, $$H_{\rm JPD} = \hbar\,\omega_a\,a^\dagger a + \hbar\,\omega_b\,b^\dagger b - E_J \cos\left[\omega_{\rm dc} t + \alpha_0(a + a^\dagger) + \beta_0(b + b^\dagger)\right]$$9), optimal coupling and impedance matching are achieved. Multiplication can be staged—either as a single high-$a$0 process or via cascaded stages of moderate gain—e.g., two consecutive $a$1 multipliers for a total gain of 16 (Danner et al., 9 Oct 2025, Albert et al., 2023).

Different platforms have demonstrated single- and multi-photon multiplication, including:

• Two-mode resonant circuits with Josephson coupling for $a$2 conversion (Leppäkangas et al., 2016).
• High-impedance resonator-terminated junctions emitting $a$3-photon bunches at bias-tuned resonances up to $a$4 (Ménard et al., 2021).
• Multinode transmission lines with single-junction boundaries for spontaneous $a$5 triplet emission (Fraudet et al., 2024).
• Three-mode Josephson mixers (parametric multipliers) for coherent multiplication and entanglement protocols (Roy et al., 2015).

3. Performance Metrics: Efficiency, Noise, and Dark Counts

The crucial figures of merit for JPM devices include conversion efficiency ($a$6), bandwidth, and dark-count probability. Optimal operation occurs when the internal $a$7-photon coupling matches the external decay rates:

$a$8

Experimentally, single-stage multipliers achieve efficiencies up to 81% ($a$9), while cascaded two-stage devices with $b$0 reach 88.5% with dark count rates $b$1, and simulations project $b$2 for higher $b$3 (Zeller et al., 17 Mar 2026, Danner et al., 9 Oct 2025, Leppäkangas et al., 2016, Albert et al., 2023). Experimentally, photon triplet emission was observed with internal conversion rates comparable to total loss rates and efficiency up to $b$4 (Fraudet et al., 2024). For $b$5, conversion probability of $b$6 over a $b$7 MHz bandwidth was reported (Albert et al., 2023).

Dark counts arise predominantly from off-resonant terms in the full Josephson Hamiltonian and spontaneous emission channels. Mitigation strategies include optimal filtering of environmental modes, high-impedance engineering, and flux-pulse control (Zeller et al., 17 Mar 2026, Albert et al., 2023, Danner et al., 9 Oct 2025). For detection purposes, the dominant noise source is spontaneous emission in the absence of input, which can be effectively reduced by proper circuit design and cascading (Albert et al., 2023).

4. Implementation in Photon Detection and Number-Resolving Protocols

By cascading JPM stages, it is possible to amplify a single itinerant photon into a large photon-number state, enabling detection via heterodyne or power measurement with conventional quantum-limited amplifiers (Danner et al., 9 Oct 2025, Leppäkangas et al., 2016, Albert et al., 2023). Detection is accomplished by mode-matched integration of the output and thresholding, discriminating between vacuum and multi-photon Fock state (Danner et al., 9 Oct 2025). Number resolution is limited by photon relaxation in the resonator, JPM intrinsic relaxation, and dark counts.

Specialized JPM-based detectors, such as two-photon threshold detectors, operate via nonlinear coupling between resonator dimers and the JPM, realizing high-fidelity threshold detection for specific photon numbers ($b$8 discrimination for two-photon states in under $b$9 ns) (Stolyarov et al., 22 Apr 2025). Repetition and statistical analysis allow extraction of full photon-number statistics via generalized binomial or geometric counting protocols, with resolution for up to $\omega_a$0 photons and $\omega_a$1 for $\omega_a$2–7 (Stolyarov et al., 2023).

In some architectures, the JPM concept is generalized for threshold detection in dispersive qubit readout, where the presence of a bright cavity pointer state triggers a macroscopic JPM switching event (Opremcak et al., 2020).

5. Experimental Demonstrations and Observed Phenomena

A range of experiments has directly demonstrated JPM-based multiplication and detection with various figures of merit:

Reference	Multiplier Type	Gain $\omega_a$3	Efficiency	Bandwidth	Dark Count Rate	Comments
(Albert et al., 2023)	Two-resonator, SQUID	3	0.69	116 MHz	~4e8 s$\omega_a$4	First 3-photon exp.
(Ménard et al., 2021)	Single-mode, JJ	1–6	up to 0.35	100 MHz	—	k-resolved emission
(Costa et al., 2015)	Cascaded multiplier	16	0.845	1/T	$\omega_a$5	2-stage $\omega_a$6
(Zeller et al., 17 Mar 2026)	Sequential cavity pairs	2,3	0.885–0.98	$\omega_a$7	$\omega_a$8	Pulsed det. protocol
(Stolyarov et al., 22 Apr 2025)	Dimer+JPM threshold	2 (pair)	$\omega_a$90.99	$\omega_b$050 ns	—	Two-photon selectivity

Photon bunch statistics and frequency-resolved emission (Fano factors $\omega_b$1 at low drive) confirm the deterministic generation of multiphoton Fock states (Ménard et al., 2021, Fraudet et al., 2024). The concept extends to triplet and higher-order spontaneous parametric emission in ultra-strongly coupled boundary-impurity devices (Fraudet et al., 2024).

6. Extensions: Parametric Multiplication and Quantum Networking

JPM architectures include Josephson parametric multipliers where a ring of junctions mediates trilinear (three-wave) interactions, enabling coherent multiplication of concurrent quantum signals and parity measurement in remote entanglement protocols (Roy et al., 2015). These devices are essential for networked quantum information processing, supporting fast, quantum-limited entanglement generation between spatially separated qubits.

The device operation is governed by Hamiltonians of the form:

$\omega_b$2

after pump-induced frequency matching, enabling time-independent effective interaction in the rotating frame (Roy et al., 2015). High-fidelity parity measurements and entanglement have been theoretically verified under realistic loss and noise conditions.

7. Limitations, Optimization, and Outlook

The main limitations of JPM arise from breakdown of the RWA at large $\omega_b$3, competing off-resonant processes, voltage/flux noise, and junction parasitics (e.g., direct capacitive coupling). Fidelity is highest for moderate $\omega_b$4 ($\omega_b$5–$\omega_b$6); higher $\omega_b$7 multiplication is possible but subject to increasingly stringent parameter constraints due to factorial suppression and dark count growth. Cascading moderate gain stages is preferred for scalable, number-resolved detection (Danner et al., 9 Oct 2025, Albert et al., 2023).

Optimization of impedance (to approach $\omega_b$8), minimization of environmental noise, and active filtering are key to suppressing unwanted emissions and reaching near-unity efficiency (Albert et al., 2023, Danner et al., 9 Oct 2025). High-impedance resonator design is essential for achieving large zero-point phase fluctuations and strong nonlinearity, critical for multi-photon emission (Ménard et al., 2021, Fraudet et al., 2024).

JPM-based photomultiplication enables lossless, dead-time-free, projective detection of itinerant microwave photons, number-state engineering, and scalable quantum information protocols in the microwave domain (Zeller et al., 17 Mar 2026, Roy et al., 2015, Leppäkangas et al., 2016). The ongoing development of high-fidelity, number-resolving JPMs is anticipated to further advance quantum sensing, error correction, and hybrid quantum networks in superconducting hardware.