Squeezing More Out of Quantum Sensors

Author: Denis Avetisyan

Researchers demonstrate a powerful new optimization technique for creating complex quantum states, paving the way for more sensitive and precise measurements.

The study demonstrates that Monte Carlo optimization of the quantum Fisher information ($QFI$) scales predictably with system parameters-specifically, exhibiting an asymptotic relationship of $\Delta\phi\sim 1/\sqrt{F}$ between the expected error and the $QFI$-and reveals that insufficient entanglement for shorter evolution times limits optimization performance, a phenomenon validated across varying system lengths ($L$) and particle numbers ($N$).

Monte Carlo optimization efficiently prepares multi-mode bosonic squeezed states, achieving intermediate scaling between standard and Heisenberg limits for quantum sensing applications.

Demonstrating quantum enhancement in sensing beyond two-mode systems remains a significant challenge due to the difficulty in preparing complex multi-mode squeezed states. This work, ‘Engineering multi-mode bosonic squeezed states using Monte-Carlo optimization’, introduces a Monte Carlo-based optimization technique to efficiently engineer Hamiltonian control sequences for multi-mode bosonic systems, specifically a Bose-Einstein condensate in an optical lattice. We demonstrate the generation of metrologically useful squeezed states exhibiting an intermediate scaling of the quantum Fisher information between the standard quantum and Heisenberg limits, achievable with experimentally accessible parameters. Could this approach offer a scalable pathway towards realizing the Heisenberg limit in quantum gravimetry and other precision sensing applications?

Transcending Classical Limits: The Quantum Imperative

The pursuit of increasingly precise measurements across diverse fields, from gravitational wave detection to medical diagnostics, frequently encounters a fundamental barrier known as the Standard Quantum Limit (SQL). This limit arises from the inherent quantum noise present in all measurement devices – fluctuations that stem from the wave-like nature of matter and energy. Specifically, the SQL dictates that the precision of a measurement is fundamentally constrained by the square root of the signal strength; essentially, weaker signals become proportionally harder to discern accurately. This poses a significant challenge because it restricts the ability to detect subtle changes or faint signals, effectively hindering the sensitivity of many crucial technologies. For example, attempting to measure a weak magnetic field or a tiny acceleration is limited by these unavoidable quantum fluctuations, impacting the performance of sensors used in navigation, materials science, and fundamental physics research. Overcoming the SQL is therefore a central goal in the development of next-generation sensing technologies.

The relentless drive toward technological innovation is creating an increasing demand for sensors with capabilities that surpass existing limitations. Fields like medical diagnostics, materials science, and environmental monitoring are now routinely confronted with signals that are incredibly faint or require measurements of extraordinarily subtle changes. Traditional sensors, bound by the Standard Quantum Limit, struggle to detect these nuances, creating a bottleneck in progress. Consequently, researchers are actively pursuing sensors capable of resolving previously undetectable phenomena, effectively redefining the boundaries of metrology – the science of measurement – and opening doors to breakthroughs in diverse scientific and engineering disciplines. This push isn’t merely about incremental improvement; it represents a fundamental need for sensors that can operate at the very edge of what’s physically possible.

Quantum sensing represents a paradigm shift in measurement science, offering the potential to break through the constraints of traditional sensors. These innovative devices exploit quantum phenomena – such as superposition and entanglement – to achieve sensitivities previously deemed impossible. Unlike classical sensors which are fundamentally limited by the Standard Quantum Limit – a consequence of inherent noise and uncertainty – quantum sensors can, in principle, surpass these boundaries. By carefully manipulating and measuring quantum states, these sensors can detect incredibly weak signals and subtle changes in physical quantities like magnetic fields, gravity, and time. This capability stems from the ability of quantum systems to exist in multiple states simultaneously, allowing for enhanced signal detection and a reduction in noise, ultimately promising advancements in fields ranging from medical diagnostics and materials science to navigation and fundamental physics research.

Quantum sensing precision scales between the standard quantum limit and the Heisenberg limit by leveraging the collective behavior of bosons in a lattice, with sensitivity increasing as the number of lattice sites grows, as demonstrated for systems with two and one hundred sites.

Squeezing the Uncertainty: A Path to Enhanced Precision

Squeezing reduces the quantum noise in one observable at the expense of increased noise in its conjugate observable, thereby circumventing the limitations imposed by the Standard Quantum Limit (SQL). The SQL, derived from the Heisenberg uncertainty principle, dictates a minimum noise level for any measurement, scaling with the square root of the number of particles, $N$. Squeezed states, however, achieve noise levels below the SQL for specific measurements by redistributing the uncertainty. This is accomplished by manipulating the quantum state to reduce the variance in one quadrature of an electromagnetic field, or one component of angular momentum, while simultaneously increasing the variance in its complementary quadrature. This technique is critical for enhancing the precision of measurements in areas like gravitational wave detection, quantum metrology, and quantum communication.

Entanglement serves as a fundamental resource for the creation and control of squeezed states. Specifically, generating entangled states, such as Bell states or Greenberger-Horne-Zeilinger (GHZ) states, allows for correlations between quantum particles that can be exploited to reduce quantum noise below the Standard Quantum Limit. These correlations, quantified by entanglement measures like concurrence or negativity, enable the transfer of quantum fluctuations from one quadrature of the electromagnetic field to another, effectively “squeezing” the uncertainty. Manipulation of entangled states via operations like controlled-NOT gates or more complex multi-qubit gates allows for precise control over the squeezing parameter, $r$, which determines the degree of noise reduction in a given quadrature.

The One-Axis Twisting (OAT) Hamiltonian, described by $H = \hbar \chi S_z^2$, provides a mathematical model for generating squeezed states. Here, $\hbar$ is the reduced Planck constant, $\chi$ represents the squeezing parameter, and $S_z$ is the collective spin operator along the z-axis. Applying this Hamiltonian to an initial coherent state induces correlations between photons, reducing the quantum noise in one quadrature at the expense of increased noise in the other. The Bloch sphere representation visually depicts this squeezing process, showing a deformation of the sphere along the z-axis, indicating a reduction in uncertainty along that axis and an increase in the perpendicular axis. Analysis using this framework allows for precise control and prediction of squeezing levels based on the magnitude of the squeezing parameter $\chi$ and the initial state properties.

A carefully designed pulse sequence successfully steers the system towards a GHZ-like state, as evidenced by the evolution of the quantum Fisher information towards the Heisenberg limit and low infidelity, achieved through smooth parameter optimization.

Atomic Systems: Platforms for Quantum Control

Atomic Bose-Einstein condensates (BECs) function as effective quantum sensors due to the macroscopic quantum coherence exhibited by the condensate. This coherence allows for highly precise measurements of external perturbations, including magnetic and gravitational fields, as well as rotations. The sensitivity stems from the collective behavior of atoms occupying the same quantum state, amplifying the signal produced by the measured quantity. Furthermore, the wavefunction of a BEC is readily manipulated using radiofrequency or optical control, enabling the implementation of diverse sensing protocols like atom interferometry. The long coherence times achievable in BECs – often exceeding one second – contribute to increased measurement precision and reduced noise floors, surpassing the capabilities of many classical sensors. Applications include precision measurements of fundamental constants, gravitational wave detection, and the development of advanced inertial sensors.

Optical lattices and tweezer arrays are employed to spatially organize atoms within a Bose-Einstein condensate (BEC) for quantum control. Optical lattices are created by interfering laser beams, forming a periodic potential that confines atoms to discrete lattice sites. Tweezer arrays utilize highly focused laser beams to trap individual atoms at precisely defined locations. These techniques allow for control of interatomic spacing, typically down to the wavelength of light ($\approx$ 400-800 nm), and enable manipulation of atomic interactions. By adjusting laser parameters – intensity, frequency, and polarization – researchers can modify the potential landscape, tune the strength of interactions between atoms, and reconfigure the array geometry, providing a high degree of control over the BEC’s quantum state.

The Hubbard interaction, a key component in many condensed matter systems, describes the on-site Coulomb repulsion between electrons. When applied to ultracold atoms in optical lattices, this interaction effectively governs the strength of electron correlation and can be tuned via lattice parameters and atomic density. Specifically, the Hubbard Hamiltonian, expressed as $H = -t \sum_{\langle i,j \rangle} (a^\dagger_i a_j + h.c.) + U \sum_i n_i^2$, dictates the system’s behavior, where $t$ is the hopping parameter, $U$ is the on-site interaction strength, and $n_i$ is the number of atoms at lattice site $i$. By controlling $U/t$, researchers can transition between various quantum phases, including Mott insulators and superfluid phases, and thereby manipulate the collective quantum state of the atomic ensemble for applications in quantum simulation and sensing.

This experimental design proposes creating a tunable optical lattice using repulsive barriers within an attractive dipole trap, formed by independently controlling the intensity and spacing of a tweezer array.

Refining Control: Optimizing Quantum Sequences

Quantum Optimal Control (QOC) represents a significant advancement in the manipulation of quantum systems, offering precise methodologies for crafting control sequences that dramatically enhance squeezing and entanglement. By treating the control parameters – such as pulse amplitudes and durations – as variables to be optimized, QOC algorithms can sculpt the system’s evolution to achieve specific target states with high fidelity. This is particularly crucial for applications in quantum information processing, where maximizing entanglement is essential for tasks like quantum computation and communication. The ability to engineer highly squeezed states – reducing quantum noise below the standard quantum limit – further improves the sensitivity of quantum sensors and enables the exploration of fundamentally new quantum phenomena. Through sophisticated numerical techniques, QOC effectively navigates the complex landscape of quantum dynamics, unlocking the full potential of quantum systems for technological innovation.

The pursuit of precise quantum state manipulation relies heavily on sophisticated algorithms capable of navigating the vast parameter space of control sequences. Algorithms such as GRAPE (Gradient-based Robust Algorithm for Pulse Engineering) and Q-PRONTO employ iterative refinement techniques to sculpt control pulses, optimizing them for specific quantum tasks. GRAPE, for instance, utilizes gradients of the objective function to efficiently adjust pulse parameters, while Q-PRONTO leverages a pruning strategy to focus computational effort on the most impactful parameters. These methods aren’t simply about finding a solution, but rather discovering solutions that are robust to experimental imperfections and noise. Through repeated optimization cycles, these algorithms effectively ‘learn’ the ideal control parameters to reliably prepare desired quantum states, enabling advancements in areas like quantum computation and sensing, and pushing the boundaries of quantum control precision.

Recent advancements in quantum trajectory optimization leverage the combined power of Monte Carlo methods and Quantum Fisher Information (QFI) analysis to navigate the complex landscape between fundamental quantum limits. This approach efficiently sculpts control sequences, moving beyond the constraints of the Standard Quantum Limit – where precision is limited by classical noise – while stopping short of the computationally intensive demands required to fully reach the Heisenberg Limit. By utilizing Monte Carlo simulations, the algorithm explores a vast parameter space, identifying control strategies that maximize information gain as quantified by the QFI. This allows for a practical intermediate scaling, offering significant improvements in precision and measurement accuracy without requiring prohibitive computational resources, and paving the way for more robust and efficient quantum technologies.

Towards Heisenberg-Limited Sensing and Beyond

Precision sensing benefits from a fundamental quantum advantage achievable through the strategic use of entanglement and optimized control. By preparing systems in highly correlated, multi-particle entangled states – such as the Greenberger-Horne-Zeilinger (GHZ) state – and carefully tailoring the interaction between the system and the parameter being measured, it becomes possible to surpass the limitations of classical sensing. This approach aims to approach the Heisenberg Limit, a benchmark where precision scales inversely with the number of particles, offering a substantial improvement over standard quantum limit techniques. The optimization of control sequences is crucial for effectively harnessing these quantum resources, allowing researchers to manipulate the system’s quantum state and maximize the information gained from each measurement, ultimately enhancing the sensitivity of the sensor.

Recent research highlights the power of Monte Carlo optimization in crafting multi-mode bosonic squeezed states – a crucial step towards surpassing classical limits in precision measurement. This technique efficiently navigates the complex parameter space required to generate these highly non-classical states, resulting in an intermediate scaling of the Quantum Fisher Information (QFI). For a system involving four modes and four bosons, the achieved QFI reaches 143.73, remarkably close to the theoretical Heisenberg Limit of 144. This near-achievement demonstrates a viable pathway toward Heisenberg-limited precision, offering significant potential for advancements in fields demanding exquisite sensitivity, such as the detection of gravitational waves and high-resolution biological imaging. The success of this optimization approach suggests that further refinement of both multi-mode bosonic systems and the algorithms used to control them will continue to push the boundaries of sensing capabilities.

Recent investigations into quantum sensing have yielded a Fisher Information (QFI) of 143.73 within a specifically designed four-mode, four-boson system utilizing Quantum Optimal Control (QOC). This value signifies a remarkable approach to the Heisenberg Limit – a theoretical benchmark of precision sensing equal to 144 – demonstrating the potential to dramatically enhance measurement accuracy. The proximity to this limit suggests that, with further refinement of control techniques and system parameters, surpassing classical precision boundaries is increasingly achievable. Such advancements promise significant improvements in technologies reliant on highly sensitive measurements, paving the way for innovations in areas like gravitational wave detection and advanced microscopy, where even marginal gains in precision can unlock new discoveries.

The advancements in precision sensing, facilitated by techniques like quantum optimization and squeezed states, extend far beyond fundamental physics research. Gravitational wave detection, a field striving to capture ripples in spacetime, stands to benefit immensely from increased sensitivity, allowing for the observation of more distant and fainter events. Simultaneously, biological imaging could be revolutionized; enhanced precision would enable the visualization of cellular structures and processes with unprecedented detail, potentially leading to earlier disease detection and a deeper understanding of life itself. These quantum-enhanced sensing methods promise not only to refine existing technologies but also to unlock entirely new possibilities in diverse scientific disciplines, offering a pathway to observe and understand the universe – and life within it – with greater clarity than ever before.

The pursuit of enhanced sensing capabilities is poised for significant advancement through continued investigation of multi-mode bosonic systems. Current research indicates that manipulating the quantum properties of multiple bosons – particles that obey Bose-Einstein statistics – offers a pathway toward surpassing classical limits in precision measurement. Specifically, optimized control sequences and sophisticated algorithms, such as those employing quantum optimal control (QOC), are crucial for tailoring these bosonic states to maximize the quantum Fisher information, a key metric of sensing precision. While recent studies have demonstrated near-Heisenberg-limited performance in systems with a few modes, further development of both the quantum hardware to support increasingly complex multi-mode states and the computational techniques used to design optimal control sequences promises to unlock even greater sensitivity. This ongoing exploration is expected to yield substantial benefits across a range of applications, from detecting faint gravitational waves to enhancing the resolution of biological imaging techniques, pushing the boundaries of what is measurable in the quantum realm.

The population is equally distributed between the initial Fock states |4000⟩ and |0004⟩, as evidenced by the corresponding density matrices, indicating a superposition characteristic of a GHZ-like state.

The pursuit of squeezed states, as detailed in this work, exemplifies a dedication to minimizing uncertainty – a concept resonating with the very foundations of quantum mechanics. Werner Heisenberg himself articulated this principle succinctly: “The very act of observing alters that which we seek to measure.” This alteration, inherent in any measurement process, necessitates techniques like Monte Carlo optimization to navigate the trade-offs between minimizing noise and maximizing signal, ultimately pushing the boundaries of quantum sensing towards the Heisenberg limit. The article’s demonstration of intermediate scaling underscores a methodical approach to achieving precision, recognizing that perfect knowledge is asymptotically approached, not instantaneously obtained. The efficacy of this technique lies in its ability to address the inherent limitations of observation, as Heisenberg keenly observed.

Beyond the Limits

The demonstrated capacity to sculpt multi-mode bosonic squeezed states via Monte Carlo optimization represents a departure from the traditionally rigorous, yet computationally constrained, landscape of gradient-based quantum control. While achieving intermediate scaling towards the Heisenberg limit is, in itself, a noteworthy progression, the true significance lies in the method’s potential scalability. The question remains, however, whether this approach will ultimately reveal a fundamental barrier to truly realizing the Heisenberg limit-or merely shift the computational burden elsewhere. A satisfying solution must, of course, be provably optimal, not simply empirically successful.

Future investigations should address the interplay between optimization landscape complexity and the inherent symmetries of the target states. A deeper understanding of this relationship could yield algorithms that exploit these symmetries, significantly reducing the computational cost. Furthermore, extending this methodology to systems with more complex interactions-those beyond the relatively clean bosonic framework-will necessitate a re-evaluation of the underlying assumptions and potentially demand entirely novel optimization strategies.

The pursuit of quantum sensing with demonstrable advantage over classical techniques is, at its core, a search for elegant solutions. This work offers a promising, if imperfect, path toward that goal. The challenge now is not merely to find squeezed states, but to understand their optimal configuration within the broader context of quantum information processing-a task that demands both computational power and mathematical precision.

Original article: https://arxiv.org/pdf/2511.15805.pdf

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