Quantum sensors are emerging as a revolutionary technology poised to transform navigation, meteorology, and various other fields. Leveraging the principlesof quantum mechanics, these sensors offer unparalleled precision and reliability, even in environments where traditional sensing systems are compromised.

The phenomena of entanglement which is purely a quantum construct, adds to the ability of sensors with no counter in the classical world. This article specifically explores the technology behind quantum sensors, their mathematical foundations with the understanding of governing physics, and their demand in industry. These devices have many industry applications but the most profound use cases are found in military applications. Thus, some applications from military are also provided.

**Quantum Sensing: Principles and Mathematics**

Any quantum attribute such as spin, momentum, position, quantized energy levels, etc can be characterized and tuned for sensing. These devices have to satisfy sensors the following conditions, Discrete energy levels

- Initialization into known state
- Coherent control via external driving and evolution of system.
- Energy levels shift in response to environment (temperature, magnetic field, electrical field, time, position, gravity. . . ).
- Readout from quantum system to extract information and entanglement can be exploited to achieve higher sensitivity and precision than classical sensors. The operational principle of quantum sensors can be described using the Hamiltonian formalism. Which is a standard approach for the study of most quantum systems in physics and quantum optics. The total

Hamiltonian H of the system is written as:

where:

- Hint is the internal Hamiltonian describing the intrinsic properties of the quantum system.
- Hsig represents the interaction with the external signal to be measured.
- Hctrl denotes the control Hamiltonian used for manipulating the quantum state during measurements.
- For example, in a Ramsey measurement scheme, the signal-induced phase shift φ is given by:

where ω is the angular frequency of the external signal and t is the evolution time. The precision of the measurement is limited by the quantum projection noise, quantified by the Cramer-Rao bound, which is the lower bound on quantum sensors.

Where m is the number of measurements and FQ is the Quantum Fisher Information.

Here, by varying the time interval between the two pulses and measuring the resulting probability of finding the system in the excited state, much information about the frequency shift is extracted. Thus, further directly allowing us to measure the frequency of the external field i.e. the signal from the environment.

The most basic understanding of a quantum sensor can be taken by considering an atomic interferometer. These devices use the wave-like nature of atoms, cooled to near absolute zero, to form an ’optical ruler’ with laser pulses.

The phase shift ∆φ observed in an interferometer can be expressed as:

where λ is the wavelength of the laser light and ∆x is the displacement of the atoms. Similarly, for acceleration a, the relationship is given by:

Where ħ is the reduced Planck’s constant, k is the wave number, m is the mass of the atom, and ∆t is the time interval. In photonic quantum sensors, the phase estimation can be made more precise by using quantum resources like squeezed states and entangled photons.

The current aim of quantum sensing is to surpass the Standard Quantum Limit (SQL) and achieve the Heisenberg Limit (HL) where the precision scales as 1/N, where N is the total number of entangled qubits. Some basic devices that fall in the category of quantum sensors are, Atomic Clocks, Magnetometers, LADARS etc. The industry demand for quantum sensors is based on some key

application given below:

**Biological Systems:** Enhanced sensitivity in quantum sensors can improve measurements in biological systems, leading to advancements in medical diagnostics and imaging.

**Gravitational Wave Detection:** Quantum sensors can increase the sensitivity of detectors like LIGO, allowing for the detection of weaker gravitational waves.

**Telecommunications:** Quantum sensors can improve the precision of timekeeping and synchronization, which is crucial for data transmission and network reliability.

**Geophysical Exploration:** Quantum magnetometers and accelerometers can enhance the accuracy of measurements in remote sensing and exploration activities.

The most profound applications come from the military sector, where the significant interest in quantum sensors is for several reasons, some are listed below:

- GPS-Denied Environments: Quantum sensors provide reliable navigation solutions in environments where satellite signals are blocked or degraded, such as urban canyons or underwater.
- Resilience to Jamming and Spoofing: Unlike GNSS, quantum sensors do not rely on external signals, making them less susceptible to electronic warfare tactics.
- Unmanned Vehicles: Quantum sensors can ensure accurate navigation for unmanned aerial and underwater vehicles in contested environments.
- Submarine Navigation: Quantum-enabled navigation systems can maintain high precision over long durations without the need for periodic calibration.

For example, the Imperial College London and the Royal Navy are working on quantum sensor technology for the above-mentioned reasons. These sensors will be utilized on Navy frigates i.e. lighter warships that are fully loaded but also have edge on speed.

Thus, the quantum sensors represent a significant leap forward in measurement technology, offering unparalleled precision and robustness. The mathematical foundations and physical principles underlying these sensors enable them to surpass the limitations of traditional systems. As research and development continue, we can expect quantum sensors to revolutionize various fields.

Where, by leveraging the principles of quantum mechanics, quantum sensorspromise to overcome the vulnerabilities of traditional GNSS-based systems, offering enhanced accuracy, reliability, and resilience. This technology’s successful integration into real-world applications will herald a new era of precision and reliability in navigation and beyond.