Injected External Squeezing (IES)

• Injected External Squeezing (IES) is a technique that employs externally prepared squeezed vacuum to influence cavity dynamics and measurement precision.
• It leverages phase-sensitive input correlations and designed symplectic maps to enhance performance in quantum sensing and state engineering.
• IES finds applications in dispersive readout, optomechanical sensing, and dynamic optical gating, improving signal-to-noise ratio and mitigating losses.

Searching arXiv for recent and foundational papers on Injected External Squeezing (IES). Injected external squeezing (IES) denotes squeezing supplied from outside the target system. In dispersive readout and optomechanical sensing papers, IES means that a squeezed vacuum reservoir is injected into the cavity input port, so the cavity is driven by squeezed input noise rather than by an intracavity parametric process (Qin et al., 2024). In broader dynamical formulations, closely related work treats squeezing as injected by externally controlled fields through the unitary evolution itself, with the emphasis on transformations of canonical observables rather than on a specially prepared squeezed state (Mielnik et al., 2014, Mielnik et al., 2024). Across these usages, the common structure is external control of the squeezing resource—either as an input bath, a feed-forward-controlled ancilla, or a time-dependent Hamiltonian—and the main technical questions are how that resource is matched to the system dynamics, how it interacts with loss and back-action, and whether it yields gains in sensing, state preparation, or gate synthesis.

1. Terminology and scope

A central distinction in the literature is between injected external squeezing and intracavity squeezing (ICS). In the dispersive-readout model, IES means that squeezed noise enters through $\hat a_{\rm in}$, whereas ICS means that the cavity itself is parametrically squeezed by the term $$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$ (Qin et al., 2024). In cavity optomechanical weak-force sensing, the same distinction is written as follows: for IES the cavity is not internally pumped by a parametric drive, one sets $\Lambda=\phi_d=0$, and the only squeezing resource is the externally prepared squeezed vacuum characterized by squeezing parameter $r_e$ and squeezing angle $\phi_e$ (Mahana et al., 26 Jun 2026).

The injected resource is usually specified by phase-sensitive input correlations. One common parametrization is

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$

$$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$

with analogous expressions for $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$ and $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$ (Xie et al., 2024). Equivalent notation uses

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$

so that the phase $$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$0 controls how the injected squeezing is rotated in quadrature space (Wu et al., 11 Dec 2025).

The broader, operation-focused literature places less emphasis on a squeezed input state and more on squeezing as a transformation of observables. In that formulation, the question is whether an externally driven quadratic Hamiltonian can realize

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$1

or, more generally, a symplectic map that squeezes one canonical direction while amplifying its conjugate (Mielnik et al., 2014, Mielnik et al., 2024). This broader usage does not always employ the acronym IES, but it explicitly supports the idea that squeezing can be injected by external control rather than only by state preparation.

2. Dynamical and symplectic formulations

The canonical dynamical model is the time-dependent quadratic Hamiltonian

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$2

with

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$3

and equations of motion

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$4

The induced evolution is written as

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$5

with

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$6

For quadratic Hamiltonians, the unitary evolution operator is determined, up to a phase, by the classical symplectic matrix $$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$7; the design of the classical symplectic map is therefore equivalent to the design of the quantum operation modulo phase (Mielnik et al., 2014, Mielnik et al., 2024).

The main squeezing criterion is hyperbolic monodromy. The dynamics are classified by

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$8

and the squeezing regime is

$$\Omega(e^{i\theta}\hat a^{\dagger2}+e^{-i\theta}\hat a^2)$$9

where the evolution matrix has real reciprocal eigenvalues

$\Lambda=\phi_d=0$0

In this regime one observable expands while its conjugate contracts (Mielnik et al., 2024).

A distinctive exact construction uses symmetric evolution intervals. If

$\Lambda=\phi_d=0$1

and

$\Lambda=\phi_d=0$2

then the external field is obtained from the inverse formula

$\Lambda=\phi_d=0$3

The matrix entries obey

$\Lambda=\phi_d=0$4

This is presented as an exact design rule: choose $\Lambda=\phi_d=0$5, and the required symmetric external field $\Lambda=\phi_d=0$6 follows (Mielnik et al., 2024). The same line of work describes these constructions as “soft control techniques,” explicitly preferring smooth, slowly varying, non-abrupt controls over $\Lambda=\phi_d=0$7-pulses and sudden jumps, and formulates symmetric sequences in a Toeplitz-type structure rather than with Ermakov-Milne invariants (Mielnik et al., 2014, Mielnik et al., 2024).

3. Measurement-induced optical realization

The experimental realization most directly aligned with externally controlled Gaussian processing is the dynamic squeezing gate for traveling optical modes (Miyata et al., 2014). It is described as a physically implemented form of externally controlled squeezing: the squeezing strength and squeezing axis are not fixed in hardware, but are instead set in real time by an external electronic driving signal. At the conceptual level, the operation is a quadratic Gaussian gate with time-dependent strength,

$\Lambda=\phi_d=0$8

which generates

$\Lambda=\phi_d=0$9

In the supplemental material, the transformation is decomposed into phase shifts and squeezing by introducing

$r_e$0

so the gate is not a fixed squeezer but a dynamically reconfigurable Gaussian transformation whose squeezing angle depends on $r_e$1 (Miyata et al., 2014).

The implementation uses the standard measurement-induced feed-forward architecture. The input signal beam is combined on a balanced beamsplitter with an ancillary squeezed vacuum state. One output port is measured by homodyne detection, and the measurement result is then used to displace the unmeasured output beam by electro-optic feed-forward. The ancillary squeezed state is generated by an optical parametric oscillator: a 300 mm bow-tie cavity containing periodically poled KTiOPO$r_e$2 (PPKTP), pumped at 430 nm with 120 mW from a second-harmonic generation stage. The OPO squeezing bandwidth is reported as 12.5 MHz HWHM, and the typical observed squeezing level is $r_e$3 dB from DC to 10 MHz. The feed-forward displacement is implemented with an EOM, an auxiliary beam, and a weakly transmitting beamsplitter (99:1), and a 13 m free-space optical delay line is inserted so that the electronic processing time matches the optical propagation delay (Miyata et al., 2014).

The dynamic control parameter is supplied externally and changes in real time. In the reported experiment, $r_e$4 is a 1 MHz sinusoidal control signal. Nonlinear electronic circuits produce

$r_e$5

The local oscillator phase for homodyne detection is actively set to $r_e$6, so the measured quadrature is

$r_e$7

and the measured signal is amplified by $r_e$8 before driving the feed-forward displacement. The supplemental material states that the nonlinear circuits approximate $r_e$9 and $\phi_e$0 using high-speed analog clamp circuits that act like an analog lookup table, operate with high precision over $\phi_e$1 at MHz rates, and have latency below 10 ns (Miyata et al., 2014).

Because the ancilla is a finite-squeezed vacuum rather than an ideal $\phi_e$2 eigenstate, the realized transformation includes additional noise terms, and in the ideal limit $\phi_e$3 it reduces to the desired dynamic Gaussian gate up to a constant 3 dB squeezing factor, which the authors note can be compensated using existing methods. The output remains Gaussian, and the reconstructed covariance matrix is used to extract the squeezed and anti-squeezed quadratures and the squeezing angle. The reported behavior is that the squeezing angle exhibits a square-wave-like dependence on the sign of $\phi_e$4, the maximal anti-squeezing grows from about 3 dB at $\phi_e$5 to about 7 dB at $\phi_e$6, and the maximal squeezing improves from about $\phi_e$7 dB to about $\phi_e$8 dB (Miyata et al., 2014). The gate is framed not merely as a standalone squeezer but as a feed-forward-ready Gaussian resource that can be immediately employed as the feed-forward needed for the deterministic implementation of the quantum cubic phase gate.

4. Dispersive readout and thermometry

In dispersive qubit readout, IES is modeled as a squeezed vacuum reservoir injected into the cavity input port, while the cavity output is measured by homodyne detection (Qin et al., 2024, Xie et al., 2024). For the standard dispersive Hamiltonian

$\phi_e$9

the output field obeys

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$0

and the measured quadrature is

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$1

The signal-to-noise ratio is written as

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$2

and in favorable limits the noise takes the form

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$3

(Xie et al., 2024).

The 2024 dispersive-readout analysis makes a sharp distinction between IES alone, ICS alone, and the matched combined protocol (Qin et al., 2024). IES alone is said not to give a practically interesting SNR gain because the qubit-state-dependent cavity response rotates the squeezing ellipse differently for the two qubit states, increasing the overlap of the two pointer states. ICS alone is described as even negligible because it also suffers from qubit-state-dependent squeezing rotation and is built up gradually from zero. The key result is that under the matching conditions

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$4

the transformed input noise becomes vacuum in a Bogoliubov mode, the effective dispersive coupling satisfies

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$5

and the measurement noise becomes

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$6

In the short-time limit,

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$7

whereas in the long-time limit the enhancement remains exponential as $$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$8 (Qin et al., 2024).

That paper also gives representative finite-time numbers. For

$$\langle A_{\rm in}(t)A_{\rm in}(t')\rangle=\frac{1}{2}e^{i\phi}\sinh(2r)\delta(t-t'),$$9

at $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$0 and $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$1, the combined protocol gives

$$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$2

compared with no squeezing $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$3, IES only $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$4, and ICS only $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$5 (Qin et al., 2024).

The thermometry analysis uses the same dispersive architecture but emphasizes that improved readout SNR does not automatically imply improved temperature estimation (Xie et al., 2024). The temperature uncertainty is

$$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$6

and for a fully thermalized qubit isolated from the bath the derived expression contains a thermal-fluctuation term $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$7. The stated conclusion is that exponential improvement from IES is possible only in special limits, namely when $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$8, or $$\langle A_{\rm in}(t)A_{\rm in}^\dagger(t')\rangle=\cosh^2(r)\delta(t-t'),$$9, or the input amplitude / photon number $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$0. Outside these limits, thermal fluctuations prevent squeezed light from producing exponential gains in temperature precision, even though the SNR may still improve. In the $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$1-qubit extension, independent qubits isolated from the bath yield the standard quantum limit scaling $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$2, while qubits that remain in contact with the bath can achieve

$$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$3

in the regime

$$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$4

with an additional exponential factor $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$5 from IES (Xie et al., 2024).

5. Optomechanical sensing and state engineering

In cavity optomechanical weak-force sensing, IES means that the cavity input port is driven by a squeezed-vacuum state injected from outside the cavity, with no internal parametric amplifier: $$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$6 The generalized homodyne readout is

$$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$7

and the force sensitivity is defined by

$$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$8

The paper concludes that variational homodyne detection can beat the SQL off resonance and that IES further improves this by injecting squeezed vacuum into the cavity input, which suppresses shot noise and reshapes the optimal noise balance. It explicitly states that both variational homodyne readout and quantum squeezing induce quantum correlation between the amplitude and phase quadratures of the cavity's output field, thereby improving the force sensitivity, and that IES is more favorable than ICS in terms of probe-power requirement for off-resonant weak-force sensing (Mahana et al., 26 Jun 2026).

A different optomechanical line combines externally injected squeezed vacuum with two-tone cavity driving (Wu et al., 11 Dec 2025). The system is driven simultaneously by a blue-detuned tone and a red-detuned tone, and a squeezed vacuum field is injected into the cavity input. The input bath is characterized by

$$\langle A_{\rm in}^\dagger A_{\rm in}^\dagger\rangle$$9

while the Bogoliubov-mode picture uses

$\langle A_{\rm in}^\dagger A_{\rm in}\rangle$0

The two-tone drive creates a squeezed mechanical steady state by cooling a Bogoliubov mode, and the externally injected squeezed vacuum enhances that state by reducing optical noise and by imposing a phase-dependent quadrature correlation (Wu et al., 11 Dec 2025).

The phase dependence is central. The paper states that the squeezed vacuum field can generate both position squeezing and momentum squeezing, depending on $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$1: $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$2 gives optimal position squeezing, while $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$3 gives optimal momentum squeezing. The squeezing is strictly $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$4-periodic in $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$5, and the total mechanical squeezing is characterized by the smallest eigenvalue $\langle A_{\rm in}^\dagger A_{\rm in}\rangle$6 of the reduced covariance matrix,

$\langle A_{\rm in}^\dagger A_{\rm in}\rangle$7

One numerical result is that the maximum mechanical squeezing can reach as high as 22.26 dB under optimal conditions. The same study reports that the system can still surpass the 3 dB strong-squeezing threshold even at

$\langle A_{\rm in}^\dagger A_{\rm in}\rangle$8

and interprets this as strong robustness against cavity dissipation and environmental thermal noise (Wu et al., 11 Dec 2025).

Related optomechanical control papers are explicitly distinguished from IES proper. The optimal-control study of two-tone sideband driving states that its mechanism is externally driven optomechanical squeezing via two-tone sideband control, not injected squeezed-vacuum squeezing (Halaski et al., 2024). The periodically modulated three-mode optomechanical system likewise supports the broader idea that externally applied driving can be engineered to enhance squeezing, but it does not treat true injected squeezed input noise or a squeezed reservoir (1803.02004).

6. Loss, hybrid architectures, and broader physical roles

The cavity-enhanced interferometer analysis with external and internal squeezing treats IES as the usual Caves-style squeezing in which a squeezed vacuum is prepared externally and injected into the interferometer or cavity (Korobko et al., 2023). The paper distinguishes injection loss, readout loss, and internal loss, models external squeezing by

$\langle A_{\rm in}^\dagger A_{\rm in}\rangle$9

and shows that internal squeezing is the tool that mitigates readout loss, while injected external squeezing reduces the injected quadrature noise. In the lossless reference case,

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$0

but in the realistic optimized limit the central asymptotic result is

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$1

The stated conclusion is that IES is very useful, but once readout loss is compensated the remaining bottleneck is the decoherence that happens inside the cavity itself (Korobko et al., 2023).

A related readout issue is that standard homodyne detection can be blind to squeezing spectra in which the correlation between amplitude and phase fluctuations is complex. The synodyne-detection analysis is not an IES paper in the sense of sending in a pre-prepared squeezed vacuum from an external source, but it is relevant because it proposes a generalized two-tone local-oscillator readout that can reveal complex squeezing and account for measurement back-action (Buchmann et al., 2016). This suggests that squeezing source engineering and readout engineering are distinct problems: IES can provide the resource, but the measurement basis must match the covariance structure if the full advantage is to be recovered.

Beyond optical sensing, externally injected squeezing has also been used as a many-body control concept. In the superconductivity study, phonons are both linearly and parametrically driven,

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$2

and the steady-state coherent amplitude is

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$3

The paper identifies a phase-sensitive enhancement mechanism: the effective coupling is maximized for

$$N=\sinh^2(r), \qquad M=e^{-i\theta}\sinh(r)\cosh(r),$$4

which corresponds to coupling electrons to the anti-squeezed quadrature, and the strongest enhancement of superconductivity is shown to be on the boundary with the dynamical lattice instabilities caused by driving (Grankin et al., 2020).

Taken together, these works present IES as a heterogeneous but technically coherent family of methods. In its narrow cavity-input meaning, IES is a squeezed-vacuum resource injected from outside and used to reduce imprecision noise, reshape quadrature correlations, or cooperate with intracavity squeezing. In its broader dynamical meaning, it is squeezing injected by external time-dependent fields into the evolution operator itself. A plausible implication is that the unifying object is not a single hardware implementation but an externally controlled, phase-sensitive Gaussian resource whose usefulness depends on matching between source, dynamics, and readout (Qin et al., 2024, Miyata et al., 2014, Mielnik et al., 2024).