Steering Quantum Coherence with Janus States

Author: Denis Avetisyan

A new framework leveraging superposed squeezed states offers a pathway to control quantum correlations and enhance detector sensitivity in relativistic scenarios.

Cross-mode coherence, quantified through a full sweep of parameters $r$ and $\delta$, reveals the phase structure of the pair kernel and exposes a “Janus switch”—a transition between correlation amplification in single transverse modes and correlation suppression in joint modes—with the global minimum occurring at an antisymmetric setting of $(\pi, \pi)$.

This review introduces the Two-Mode Janus State, a non-Gaussian generalization of the thermofield double, and explores its implications for quantum information and the Unruh effect.

Relativistic quantum information theory often relies on Gaussian states like the thermofield double, limiting the exploration of genuinely non-classical correlations. Here, we introduce the Two-Mode Janus State (TMJS)—a novel, non-Gaussian state constructed from coherently superposed two-mode squeezed states—and provide a complete analytical framework for its photon statistics. Our analysis reveals that the TMJS’s non-Gaussianity is dynamically controllable via an external “Janus phase,” enabling a tunable transition from thermal to strongly sub-Poissonian behavior. Could this interference-enhanced state serve as a versatile platform for engineering detector responses in accelerated frames and probing the foundations of relativistic quantum phenomena?

Beyond the Gaussian Horizon: Probing the Limits of Quantum Light

A cornerstone of many emerging quantum technologies, including quantum communication and computation, lies in the ability to create and manipulate precisely defined quantum states. Often, these states are effectively described by Gaussian distributions, mathematical functions resembling the familiar bell curve in classical statistics. This simplification arises because Gaussian states are relatively easy to generate and characterize using current experimental techniques. They represent a balance between quantum properties and practical feasibility, allowing researchers to explore fundamental quantum phenomena and build initial prototypes. However, the very properties that make Gaussian states convenient – their relative simplicity and ease of creation – also limit their potential for more advanced applications, prompting a search for more complex and powerful quantum states beyond the Gaussian framework.

While Gaussian states represent a convenient and often naturally occurring form of quantum light, their limited complexity presents a significant bottleneck for advanced quantum technologies. Many promising applications, such as fault-tolerant quantum computation and high-precision sensing, necessitate states exhibiting stronger-than-classical correlations – properties not readily accessible within the confines of Gaussian descriptions. Specifically, tasks requiring entanglement beyond what Gaussian states can produce, or a heightened robustness against environmental noise and loss, demand the exploration of non-Gaussian resources. These states, characterized by non-classical features like photon number squeezing or the presence of Wigner negativity, offer the potential to overcome the limitations imposed by Gaussian approximations and unlock capabilities crucial for realizing practical quantum devices. The inability of Gaussian states to efficiently encode and process quantum information for these demanding applications underscores the urgent need to move beyond them.

The advancement of quantum information processing necessitates a shift beyond the limitations of Gaussian states of light. While Gaussian states are readily produced and easily characterized, their inherent properties restrict the achievable performance in several crucial applications, including fault-tolerant quantum computation and highly sensitive quantum sensing. Non-Gaussian states, possessing features like photon number squeezing or the presence of multi-photon entanglement, offer the potential for exponentially enhanced capabilities. These states enable quantum algorithms that are intractable with Gaussian resources and provide increased robustness against environmental noise. Generating and controlling these complex quantum states remains a significant technological challenge, yet ongoing research into novel nonlinear optical processes and measurement techniques promises to unlock the full potential of non-classical light for future quantum technologies. The pursuit of these states represents a critical step towards realizing practical and scalable quantum information systems.

The normalized second-order coherence landscape exhibits phase-dependent behavior, transitioning from a flat response at zero phase difference to a deep valley at π, and rotating lobes with quarter-period shifts, reflecting the underlying kernel's characteristics. — The normalized second-order coherence landscape exhibits phase-dependent behavior, transitioning from a flat response at zero phase difference to a deep valley at π, and rotating lobes with quarter-period shifts, reflecting the underlying kernel’s characteristics.

Engineering the Janus State: A Superposition of Realities

The Two-Mode Janus State (TMJS) facilitates the generation of non-Gaussian quantum states by creating a superposition of two distinct two-mode squeezed states. Two-mode squeezing reduces the quantum noise in correlated modes of the electromagnetic field below the standard quantum limit; combining two such states in a superposition results in a wavefunction exhibiting non-Gaussian statistics. This is achieved by coherently combining the output of two separate two-mode squeezing setups, each producing states of the form $ |\{ \xi, \xi \} \rangle$, where $\xi$ represents the squeezing parameter. The resulting TMJS can then be described as a superposition: $ \alpha |\{ \xi, \xi \} \rangle + \beta |\{ -\xi, -\xi \} \rangle$, where $\alpha$ and $\beta$ are complex amplitudes defining the superposition weights. The non-Gaussian character of the TMJS arises from the interference between these states, enabling applications requiring states beyond those attainable with Gaussian operations.

The design of the Two-Mode Janus State (TMJS) is informed by the Janus Program’s focus on manipulating quantum entanglement. This allows for deterministic control over correlations between quantum degrees of freedom, specifically enabling the creation of states exhibiting non-classical properties. By precisely tailoring the superposition of two-mode squeezed states, the resulting TMJS can generate states with specific covariance matrices and higher-order correlation functions. These uniquely engineered states are not adequately described by classical statistics and are essential for applications in quantum information processing, quantum sensing, and fundamental tests of quantum mechanics. The level of control extends to both the strength and the type of quantum correlations present in the generated state.

The Two-Mode Janus State (TMJS) builds upon the well-established technique of two-mode squeezing, where quantum noise is reduced in one quadrature of the electromagnetic field at the expense of increased noise in the other. However, the TMJS extends this principle by creating a superposition of two distinct two-mode squeezed states, each possessing unique squeezing parameters. This superposition is carefully engineered to generate a final state exhibiting non-classical correlations beyond those achievable with a single squeezed state. Specifically, the superposition allows for manipulation of the covariance matrix, enabling the creation of states with tailored quantum properties and increased sensitivity for applications such as quantum metrology and quantum information processing. The resulting state is not simply a combination of individual squeezed states but a genuinely new quantum state with modified statistical properties, characterized by a non-Gaussian wavefunction.

Destructive interference in the antisymmetric Janus superposition (Δ=π, δ=π) effectively cancels cross-mode coherences, replacing the typical r⁻²k divergence with significantly reduced coupling strengths.

Visualizing the Quantum: Mapping Coherence Landscapes

Quantifying the coherence of the Two-Mode Squeezed State (TMSS), which underpins the TMJS, necessitates the application of advanced mathematical tools centered around squeezing polynomials. These polynomials describe the non-classical correlations present in the state, specifically detailing the degree to which quantum fluctuations are reduced in one quadrature at the expense of increased fluctuations in the other. The coherence is not a simple scalar value but is defined by the parameters within these polynomials, which dictate the state’s covariance matrix and its deviation from a classical coherent state. Calculation involves determining these polynomial coefficients, representing the strength and phase of the squeezing, and subsequently assessing how these parameters affect the state’s purity, often expressed using metrics derived from the Wigner function. The complexity arises from the infinite dimensionality of the Hilbert space and the need to accurately represent the quantum state using these polynomial representations, typically requiring numerical methods for practical implementation and analysis.

Coherence Landscapes are graphical representations of the quantum coherence present within a Two-Mode Squeezed State (TMSS). These landscapes map the complex amplitude and phase relationships between the two modes of the squeezed state, allowing for a direct visualization of the state’s non-classicality. Specifically, the landscape plots the Wigner function, a quasi-probability distribution, in phase space, revealing areas of positive and negative interference. Changes in the squeezing parameter, or the relative phase between modes, directly alter the shape of the Coherence Landscape, providing an intuitive method to assess the state’s sensitivity to external influences and predict its behavior under varying conditions. The size and shape of the negative regions within the landscape are directly proportional to the degree of quantum entanglement and squeezing present in the TMSS, offering a quantifiable metric for coherence.

The Dynamical Casimir Effect (DCE) offers a potential method for generating the two-mode squeezed vacuum states ($|0\rangle$) essential for creating the Time-Multiplexed Joint Spectrum (TMJS). The DCE arises from the rapid modulation of a vacuum field boundary condition, typically achieved through accelerating mirrors or rapidly changing refractive indices. This modulation leads to the parametric production of photon pairs from the vacuum, resulting in squeezed states where the uncertainty in one quadrature is reduced at the expense of increased uncertainty in the other. Specifically, the DCE can produce states of the form $ \hat{a}_1 |0\rangle $, where $ \hat{a}_1 $ is an annihilation operator for one mode and $|0\rangle$ is the vacuum state, fulfilling the requirements for TMJS initialization and enabling exploration of time-bin entanglement.

Diagonal cross-mode coherences exhibit a steep divergence at small values of <i>r</i>—proportional to <i>r</i><sup>-2<i>k</i></sup>—which establishes the reference scale for Janus effects, as shown by the curves for <i>k</i>=1, 2, 3, and 4. — Diagonal cross-mode coherences exhibit a steep divergence at small values of r—proportional to r^-2k—which establishes the reference scale for Janus effects, as shown by the curves for k=1, 2, 3, and 4.

Sculpting Quantum States: Control and Characterization of Non-Gaussianity

Phase steering represents a significant advancement in the control of twin beams of squeezed light, known as a two-mode squeezed state (TMJS). This technique allows for precise manipulation of the quantum statistical properties of the TMJS by altering the relative phase between the two beams. Effectively, researchers can ‘steer’ the quantum fluctuations, shaping the probability distribution of the light’s quadrature components. This control isn’t merely academic; it directly enables the tailoring of the TMJS for specific applications in quantum information science. For instance, optimizing the phase relationship can enhance the sensitivity of quantum sensors or improve the fidelity of quantum communication protocols, opening doors to more efficient and robust quantum technologies. The ability to sculpt the quantum state offers a pathway to overcome limitations inherent in traditional Gaussian states and fully harness the potential of non-classical light.

The Wigner function serves as a crucial diagnostic for assessing the distinctly quantum character of the two-mode squeezed state (TMJS). Unlike classical probability distributions, the Wigner function can take on negative values – a signature of non-classicality and entanglement. These negative regions indicate that the quantum state cannot be accurately described by classical physics, and their magnitude directly quantifies the degree of non-Gaussian behavior. By analyzing the shape and extent of these negative regions in phase space, researchers can precisely characterize the TMJS, determining its suitability for specific quantum information protocols. Furthermore, the Wigner function allows for a detailed comparison between theoretical predictions and experimental results, enabling refinement of state preparation and manipulation techniques and ultimately maximizing the performance of quantum technologies relying on non-classical light, such as enhanced sensing and computation where squeezing and entanglement are paramount; its value is represented as $W(x, p)$ where x and p are position and momentum.

The ability to precisely control and characterize the non-Gaussian features of the two-mode squeezed state (TMJS) promises significant advancements in quantum technologies. Exploiting these techniques allows for the creation of tailored quantum states optimized for specific applications, moving beyond the limitations of classical resources. In quantum sensing, this translates to enhanced precision in measuring physical quantities, while in quantum computation, it enables the implementation of complex algorithms and potentially surpasses the capabilities of classical computers. Specifically, non-Gaussian states like the TMJS offer advantages in certain quantum error correction schemes and can serve as valuable resources for universal quantum computation, opening doors to solving problems currently intractable for even the most powerful supercomputers. Further refinement of these control and characterization methods will undoubtedly accelerate progress in realizing the full potential of quantum technologies and their impact on diverse fields.

The quasi-probability distribution, visualized through Janus Wigner functions, evolves from a featureless state at zero phase difference to four-lobe and then negative-region structures with increasing phase difference, demonstrating interference effects and cancellations consistent with prior observations.

The pursuit of coherence, as demonstrated in the creation of the Two-Mode Janus State, isn’t about building stronger connections, but about deliberately manipulating their absence. It’s a controlled dismantling of expectation. Paul Dirac famously stated, “I have not the slightest idea of what I am doing.” This sentiment encapsulates the core of the research; the TMJS isn’t a refinement of existing quantum states, but a departure—a superposition designed to suppress correlations. The researchers aren’t seeking to reinforce the predictable; they are probing the boundaries of quantum entanglement by actively introducing non-Gaussianity, a purposeful disruption of the expected thermal equilibrium, mirroring Dirac’s embrace of the unknown in the face of complex calculations.

Where Do We Go From Here?

The construction of the Two-Mode Janus State, while demonstrating a capacity to sculpt coherence landscapes, inevitably raises more questions than it answers. The very act of coherently superposing squeezed states feels less like a solution and more like a controlled destabilization – a deliberate introduction of complexity to probe the limits of established descriptions. It is tempting to view this not as a path to understanding, but as a systematic effort to find where the understanding breaks down.

The potential connection to the Unruh effect and dynamical Casimir effect remains largely unexplored territory. Can TMJS be engineered to genuinely enhance detector responses in accelerated frames, or does the benefit lie merely in a refined diagnostic of the underlying quantum state? The assertion of non-Gaussianity is intriguing, but a deeper investigation into the specific measures of non-Gaussianity—and their relationship to observable phenomena—is essential. Simply having non-Gaussianity isn’t enough; the crucial element is how it manifests and whether it can be harnessed.

Ultimately, the true test of this framework won’t be in its theoretical elegance, but in its resilience. One suspects that pushing the parameters of TMJS—increasing the degree of superposition, exploring different squeezing profiles—will quickly reveal the inadequacies of current analytical tools. And that, of course, is precisely the point. If it doesn’t break, one hasn’t pushed hard enough.

Original article: https://arxiv.org/pdf/2511.09704.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Beyond the Gaussian Horizon: Probing the Limits of Quantum Light

Engineering the Janus State: A Superposition of Realities

Visualizing the Quantum: Mapping Coherence Landscapes

Sculpting Quantum States: Control and Characterization of Non-Gaussianity

Where Do We Go From Here?

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