Displaced Squeezed States in Quantum Optics

• Displaced squeezed states are quantum states achieved by applying a squeezing operation followed by a displacement, resulting in minimized uncertainty and nonzero field amplitudes.
• They can be generated via linear optical mixing, parametric amplification, microwave circuits, and mode-selective engineering, each balancing between brightness and squeezing depth.
• These states serve as essential resources in quantum information, metrology, and computation, enabling robust entanglement, precision measurement, and error-correcting capabilities.

Displaced squeezed states are quantum states of a bosonic mode (typically of light or microwave fields) formed by sequential application of a squeezing transformation followed by a phase-space displacement. These states occupy a central position in quantum optics, quantum information processing, metrology, and the paper of nonclassical electromagnetic fields due to their unique blend of quadrature squeezing (sub-vacuum noise in one observable) and nonzero field amplitudes. Displaced squeezed states generalize fundamental Gaussian states and serve as resources for encoding, transmitting, and storing quantum information, as well as for noise reduction in precision measurement contexts.

1. Mathematical Structure and Fundamental Properties

A single-mode displaced squeezed state is constructed as

$|\alpha, \xi\rangle = D(\alpha) S(\xi) |0\rangle$

where $D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a)$ is the displacement operator (with $\alpha \in \mathbb{C}$) and $$S(\xi) = \exp\left[\frac{1}{2} (\xi^* a^2 - \xi a^{\dagger 2})\right]$$ is the squeezing operator (with $\xi = r e^{i\theta} \in \mathbb{C}$). The action of $S(\xi)$ reduces noise in one quadrature at the expense of amplifying noise in the conjugate quadrature, while $D(\alpha)$ shifts the state in phase space, imparting a nonzero mean field value.

For multimode scenarios, the generalization is

$$|\{\alpha_k\}, \{\xi_k\}\rangle = \bigotimes_k D_k(\alpha_k) S_k(\xi_k) |0\rangle$$

with independent (or correlated) squeezing and displacement applied to each spectral or spatial mode.

Displaced squeezed states are minimum-uncertainty Gaussian states with a covariance matrix and a nonzero first moment. Their Wigner functions are centered Gaussians, elongated or compressed along rotation axes governed by the squeezing phase.

2. Generation Methods and Physical Realizations

Displaced squeezed states can be generated by several methods, each presenting distinct structure–performance tradeoffs (Young et al., 2023):

• Linear Optical Mixing: A squeezed vacuum is generated (for example via an optical parametric amplifier) and then mixed with a strong coherent field on a beam splitter. The quadrature variances in the output mode are

$(\Delta X)^2 = e^{-2r} \cos^2\theta + \sin^2\theta$

with overall uncertainty increasing monotonically with mixing ratio.

• Direct Parametric Amplification with Seeding: The squeezed state is generated inside a nonlinear crystal or cavity, and the field is seeded with a coherent amplitude. The process is governed by cubic Hamiltonians, resulting in unavoidable mixing of pump noise with the squeezed signal as soon as the field is displaced. This induces tradeoffs between squeezing depth and displacement (i.e., brightness).
• Microwave Circuits and Optomechanics: Squeezing is generated using flux-driven Josephson parametric amplifiers or optomechanical coupling, with displacement achieved via directional couplers or local drives (Fedorov et al., 2016).
• Mode-Selective Engineering: In high-harmonic generation in solids, displaced squeezed states naturally arise in multiple spectral modes due to nonlinear light–matter interaction, with modal structure characterized through Schmidt decomposition (Theidel et al., 4 Nov 2024).
• Waveguide Arrays (classical analog): The evolution of optical amplitudes in a waveguide lattice with engineered nearest- and next-nearest-neighbor coupling and a linear site gradient mimics the evolution of photon number distributions in displaced squeezed number states (Villegas-Martínez et al., 2021).

3. Quantum Properties, Correlations, and Measurement

Displaced squeezed states exhibit the following salient quantum features:

• Quadrature Squeezing: Reduction of noise below the vacuum level in one quadrature. For pure squeezed states, the minimum quadrature variance is $(\Delta X)^2 = e^{-2r}/2$.
• Nonclassical Correlations and Entanglement: Two-mode or multimode generalizations yield entangled states exploitable for continuous-variable quantum information (Jensen et al., 2010, Theidel et al., 4 Nov 2024).
• Photon Statistics: The photon-number distribution can be oscillatory and sub-Poissonian, especially when constructed over Fock or superposed number states (El-Orany, 2011).
• Wigner Functions and Nonclassicality: Squeezed and displaced states display negative regions or fringes in their Wigner quasiprobability distributions, signaling nonclassicality (El-Orany et al., 2011).
• Temporal Coherence: The second-order coherence function $g^{(2)}(\tau)$ for displaced squeezed thermal states reflects rich temporal structure, with power-law or exponential decay regimes dictated by the squeezing parameter and state preparation time (Alexanian, 2015, Alexanian, 2015).
• Robustness under Displacement: Experimental evidence demonstrates that strong displacement operations do not degrade squeezing or path entanglement, confirming theoretical predictions that critical quantum resources are preserved under coherent displacements (Fedorov et al., 2016).

4. Applications in Quantum Information, Sensing, and Networks

Displaced squeezed states are instrumental in various quantum technologies:

• Quantum Memories: Storage of displaced two-mode squeezed states in atomic ensembles enables faithful node operations in continuous-variable quantum networks, provided the memory preserves both squeezing and mean-field displacements (Jensen et al., 2010).
• Bosonic Quantum Error Correction: Cat codes constructed from superpositions of displaced-squeezed states (squeezed Schrödinger cats) enhance tolerance to both photon loss and dephasing noise, outperforming standard cat codes in the regime of moderate squeezing (Schlegel et al., 2022).
• Quantum Computation: Use in continuous-variable quantum circuits includes encoding of logical qubits (in the displacement degree of freedom), realization of elementary gates (Hadamard, CZ) via manipulation and measurement in a displaced-number-state basis (Podoshvedov, 2011, Podoshvedov, 2015).
• Quantum Metrology: The phase-sensitive properties of displaced squeezed vacuum states allow parameter estimation at the ultimate quantum limit, with achievable accuracy and optimal probe state design governed by the relative phases of displacement and squeezing (Tao et al., 2021).
• Quantum Communications: Their invariance of squeezing and entanglement under arbitrary displacement is crucial in quantum teleportation and hybrid quantum repeater protocols (Fedorov et al., 2016).
• Quantum Kernel Methods in Learning: Displaced squeezed vacuum states underpin the construction of quantum Gaussian kernels with tunable hyperparameters, enabling enhanced learning capacity in quantum-inspired support vector machines (Mehta et al., 18 Mar 2024).

5. Multimode Structure, Mode Decomposition, and Quantum Networks

Realistic sources, especially in high-harmonic generation and network scenarios, generate displaced squeezed states extending over multiple modes:

• Schmidt Decomposition characterizes the effective mode number, with typical experimental findings indicating nearly single-mode or low-rank structure, which is highly desirable for quantum information protocols (Theidel et al., 4 Nov 2024).
• Correlation Functions: Experimental access to $g^{(2)}$ and $g^{(3)}$ provides tomographic data for reconstructing multimode structure and identifying the joint nonclassicality.
• Entanglement Verification: Violation of the Cauchy–Schwarz-type inequalities in multimodal systems provides operational evidence of two- and multi-mode squeezing, i.e., genuine multipartite entanglement (Theidel et al., 4 Nov 2024).

6. Advanced Theoretical Developments: Multidimensional Squeezing and SU(1,1) Displacement

• Superpositions and Multidimensional Squeezing: More sophisticated displaced squeezed number states, including those constructed via superpositions (even/odd or Yurke–Stoler-type) or generalized squeeze operators acting on three or more modes, furnish new families of nonclassical states with engineered photon statistics and multimode entanglement (El-Orany, 2011).
• SU(1,1) Displaced Coherent States: A broader class of displacement/squeezing operations based on noncompact Lie algebras (e.g., SU(1,1)), implemented by unitary transformations constructed from noncanonical ladder operators, gives rise to states with distinctive photon-counting statistics and quadrature uncertainty scaling (Gazeau et al., 2023).

7. Fundamental Limitations and Tradeoffs

• Tradeoffs in Generation: There exist inescapable tradeoffs between output brightness (displacement amplitude), squeezing strength (noise suppression), and total uncertainty. Parametric amplification methods inherently mix pump noise into the output squeezed mode when displacement is present, while linear mixer-induced displacements unavoidably dilute squeezing with classical noise. No known method circumvents these tradeoffs; application-driven optimization thus defines the chosen operational regime (Young et al., 2023).
• Robustness to Decoherence: The temporal and environmental stability of displaced squeezed states, including their dynamics in non-Markovian (strongly coupled) environments, depends on both the system–bath coupling and thermal background, with strong coupling giving rise to coherence revivals due to memory effects (Ali et al., 2022).

8. Quantum-Engineered Metamaterials and Mesoscopic Devices

• Microwave and Metamaterial Platforms: Preparation of transmission line modes in displaced squeezed Fock states modulates effective physical parameters, including the negative refractive index, via engineered quantum fluctuations in current and field operators (Zhao et al., 2016, Guo et al., 17 Mar 2024). This controllability opens possibilities for miniaturized, adaptive, and quantum-programmable microwave devices.

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Displaced squeezed states represent a unifying thread through modern quantum science and technology, embodying both the nonclassical/probabilistic and deterministic/classical features of electromagnetic fields. They can be tailored in single mode or engineered in multimode structures, leveraged as robust carriers of quantum information, generators of entanglement and squeezing, versatile probes for high-precision tasks, and as building blocks for quantum-enhanced circuits, quantum networks, and emerging photonic quantum machine learning paradigms.