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We are in midst of the second quantum revolution moving from merely computing quantum properties of systems to exploiting them. Researchers are developing new capabilities in secure communication, ultra-sensitive and high signal to noise physical sensing of the environment and Quantum Information Science (QIS). Yet many scientists believe that quantum will enjoy its first real commercial success in sensing. That’s because sensing can take advantage of the very characteristic that makes building a quantum computer so difficult: the extraordinary sensitivity of quantum states to the environment.
Quantum sensing uses some nonintuitive properties of nature to measure things like time, magnetic fields, gravity, or acceleration. Quantum sensors are measuring device that takes advantage of quantum correlations, such as states in a quantum superposition or entanglement, for better sensitivity and resolution than can be obtained by classical systems. We can make quantum sensors to measure acceleration, gravity, rotation, time, pressure, temperature and electric and magnetic fields.
“Quantum sensing” is typically used to describe one of the following:
I. Use of a quantum object to measure a physical quantity (classical or quantum). The quantum object is characterized by quantized energy levels. Specific examples include electronic, magnetic or vibrational states of superconducting or spin qubits, neutral atoms, or trapped ions.
II. Use of quantum coherence (i.e., wave-like spatial or temporal superposition states) to measure a physical quantity.
III. Use of quantum entanglement to improve the sensitivity or precision of a measurement, beyond what is possible classically.
The field of quantum sensing deals with the design and engineering of quantum sources (e.g., entangled) and quantum measurements that are able to beat the performance of any classical strategy in a number of technological applications. Quantum sensing utilizes properties of quantum mechanics, such as quantum entanglement, quantum interference, and quantum state squeezing, which have optimized precision and beat current limits in sensor technology and evade the Heisenberg uncertainty principle.
Quantum sensors reach extreme levels of precision by exploiting the quantum nature of matter—using the difference between, for example, electrons in different energy states as a base unit. Atomic clocks illustrate this principle. The world time standard is based on the fact that electrons in cesium 133 atoms complete a specific transition 9,192,631,770 times a second; this is the oscillation that other clocks are tuned against. Other quantum sensors use atomic transitions to detect minuscule changes in motion and tiny differences in gravitational, electric and magnetic fields.
Quantum sensors can be based on photonic systems or solid state systems.
Solid state systems
In solid-state physics, a quantum sensor is a quantum device that responds to a stimulus. At a basic level, they operate by putting the sensor into a known quantum state and allowing it to interact with the object of interest. The interaction changes the state of the quantum system, and measuring the new state of the system reveals information about the object that could not be obtained with conventional approaches. Usually this refers to a sensor that, which has quantized energy levels, uses quantum coherence to measure a physical quantity, or uses entanglement to improve measurements beyond what can be done with classical sensors.
There are 4 criteria for solid-state quantum sensors:
The system has to have discrete, resolvable energy levels.
You can initialize the sensor and you can perform readout (turn on and get answer).
You can coherently manipulate the sensor (quantum coherence- wave-like spatial or temporal superposition states).
The sensor interacts with a physical quantity and has some response to that quantity. The interaction with V leads to a shift of the quantum system’s energy levels or to transitions between energy levels
Quantum sensing in solid state systems can utilize spin qubits, trapped ions, and flux qubits. These systems can be compared by physical characteristics to which they respond, for example, trapped ions respond to electrical fields while spin systems will respond to magnetic fields. Trapped Ions are useful in their quantized motional levels which are strongly coupled to the electric field. They have been proposed to study electric field noise above surfaces, and more recently, rotation sensors. Some quantum sensors may respond to several physical parameters
Quantum sensing protocol
Generic scheme where a measurement consists of three elementary steps: the initialization of the quantum sensor, the interaction with
the signal of interest, and the readout of the final state. Phase estimation and parameter estimation techniques are then used to reconstruct the physical quantity from a series of measurements. Experimentally, the protocol is typically implemented as an interference measurement using pump-probe spectroscopy, although other schemes are possible. The key quantity is then the quantum phase picked up by the quantum sensor due to the interaction with the signal. The protocol can be optimized for detecting weak signals or small signal changes with the highest possible sensitivity and precision.
A second important characteristic is a quantum sensor’s “intrinsic sensitivity”. On the one hand, a quantum sensor is expected to provide a strong response to wanted signals, while on the other hand, it should be minimally affected by unwanted noise. Clearly, these are
conflicting requirements. In order to optimize the sensitivity, transduction parameter (γ) should be large (for example, by choice of an appropriate physical realization of the sensor) and the decoherence time Tχ must be made as long as possible
A. Neutral atoms as magnetic field sensors
Alkali atoms are suitable sensing qubits fulfilling the above definitions (Kitching et al., 2011). Their ground state spin – a coupled angular momentum of electron and nuclear spin – can be both prepared and read out optically by the strong spin-selective optical dipole transition
linking their s-wave electronic ground state to the first (p-wave) excited state.
Magnetic field is often the easiest thing for a quantum system to measure. The most common one for quantum sensors is the familiar Zeeman effect, where magnetic field causes one energy level of the quantum system to split into multiple ones, with the energy gap proportional to the applied field. A quantum sensor will typically directly measure this splitting and extract the B-field from the result.
1. Atomic vapor cells
high-performance quantum sensor. In the simplest implementation, a thermal vapor of atoms serves as a quantum sensor for magnetic fields. Held in a cell at or above room temperature, atoms are spin polarized by an optical pump beam. Magnetic field sensing is based on the Zeeman effect due to a small external field orthogonal to the initial atomic polarization. The system is prepared and detected optically with a laser beam.
Despite their superficial simplicity, these sensors achieve sensitivities in the range of 100 aT/√Hz and approach a theory limit of < 10 aT/√Hz, placing them on par with Super conducting Quantum Interference Device (SQUIDs,) as the most sensitive magnetometers to date.
Vapor cells have been miniaturized to few mm3 small volumes and have been used to demonstrate entanglement-enhanced sensing. The most advanced application of vapor cells is arguably the detection of neural activity , which has found use in magnetoencephalography.
2. Cold atomic clouds
Some quantum sensors use atoms to sense changes. This works because the atoms can be controlled and measured precisely. In quantum
physics, particles such as atoms can act like waves, exploring extended areas of space. They can be put in two places or two states at once.
Because this quantum superposition is highly sensitive to the environment, it can be used as the basis for precise sensors. For example, in an atom interferometer atoms are trapped as a tiny cloud. The cloud is then released and falls, with precisely-timed laser pulses used to control it as it goes. The atoms behave as waves, which interfere with one another like ripples crossing on the surface of water. The pattern made by this interference depends on what is influencing them as they go. If that’s simply falling under gravity, it allows for very precise gravity sensing.
The advent of laser cooling in the 1980s spawned a revolution in atomic sensing. The reduced velocity spread of cold atoms enabled sensing with longer interrogation times using spatially confined atoms, freely falling along specific trajectories in vacuum or trapped. Freely falling atoms have enabled the development of atomic gravimeters and gyrometers. In these devices an atomic cloud measures acceleration by sensing the spatial phase shift of a laser beam along its freely falling trajectory.
Establishing the absolute value of gravitational acceleration, g, at any point on the Earth’s surface is in principle very easy—simply measure the time and distance of a free-falling body and plug the results into a standard equation of motion. Classically, a laser beam bounced off a freefalling mirror interferes with a reference beam reflected off of a seismically isolated mirror. With the light’s wavelength providing the distance measurement and an atomic clock tracking the elapsed time, g can be measured with an accuracy of about one part in a billion.
Atom interferometers provide similar accuracy levels, but have the advantage that their test masses—the atoms—do not suffer the wear and tear of a macroscopic reflector. This reduces maintenance and speeds up measurements.
Typically, several million atoms are cooled by lasers inside a trap to just a few millionths of a degree above absolute zero before being released under gravity. During their roughly 1 m of freefall, the atoms are subject to a series of laser pulses that puts their wavefunction in a superposition of two different states; those states are then interfered with one another. The slightly different paths through the gravitational field taken by the two states sets up a phase shift between them, with the resulting interference pattern then used to determine g.
This has become possible thanks to continuing advances in components such as lasers and cooling systems that allow researchers and engineers to manipulate the quantum states of atoms and measure how they are affected by their environment. Some of the large vacuum systems and magnetic traps now used to hold cold atoms are being replaced with chip-scale devices, while researchers are developing compact, low-power lasers that could make quantum sensors much more accessible
B. Trapped ions
Ions, trapped in vacuum by electric or magnetic fields, have equally been explored as quantum sensors. The most advanced applications employ the quantized motional levels as sensing qubits for electric fields and forces. These levels are strongly coupled to the electric field by
dipole-allowed transitions and sufficiently (MHz) spaced to be prepared by Raman cooling and read out by laser spectroscopy. The sensor has a predicted sensitivity of 500 nV/m/√Hz or 1 yN/√Hz for the force acting on the ion.
Independently, the ground state spin sublevels of ions are magnetic-field-sensitive qubits analogous to neutral atoms discussed above Being an extremely clean system, trapped ions have demonstrated sensitivities down to 4.6 pT/√Hz and served as a testbed for advanced sensing protocols such as dynamical decoupling and entanglement-enhanced sensing . Recently, trapped ions have also been proposed as rotation sensors, via matter-wave Sagnac interferometry . Their use in practical applications, however, has proven difficult. Practically all sensing demonstrations have focused on single ions, which, in terms of absolute sensitivity, cannot compete with ensemble sensors such as atomic vapors. Their small size could compensate for this downside in applications like microscopy, where high spatial resolution is required.
Trapped atoms have been employed to detect and image magnetic fields at the microscale, by replicating mor precession spectroscopy on a trapped Bose-Einstein condensate and by direct driving of spin-flip transitions by microwave currents or thermal radio frequency samples. Arguably the most advanced demonstrations of entanglement-enhanced quantum sensing (“Definition III”) have been implemented in trapped cold atoms and vapor cells. It has improved the sensitivity of magnetometry devices beyond the shot noise limit and has increased their bandwidth.
However, operation of ion traps in close proximity to surfaces remains a major challenge. Recent work on large ion crystals opens however
the potential for novel applications to precise clocks and spectroscopy.
C. Rydberg atoms
Rydberg atoms – atoms in highly excited electronic states – are remarkable quantum sensors for electric fields for a similar reason as trapped ions: In a classical picture, the loosely confined electron in a highly excited orbit is easily displaced by electric fields. In a quantum picture, its motional states are coupled by strong electric dipole transitions and experience strong Stark shifts. Preparation and readout of states is possible by laser excitation and spectroscopy.
As their most spectacular sensing application, Rydberg atoms in vacuum have been employed as single-photon detectors for microwave photons in a cryogenic cavity in a series of experiments that has been highlighted by the Nobel prize in Physics in 2012. Recently, Rydberg states have become accessible in atomic vapour cells . They have been applied to sense weak electric fields, mostly in the GHz frequency range , and have been suggested as a candidate for a primary traceable standard of microwave power.
D. Atomic clocks
At first sight, atomic clocks – qubits with transitions so insensitive that their level splitting can be regarded as absolute and serve as a frequency reference – do not seem to qualify as quantum sensors since this very definition violates criterion (4). Their operation as clocks, however, employs identical protocols as the operation of quantum sensors, in order to repeatedly compare the qubit’s transition to the frequency of an unstable local oscillator and subsequently lock the latter to the former. Therefore, an atomic clock can be equally regarded as a quantum sensor measuring and stabilizing the phase drift of a local oscillator. Vice versa, quantum sensors discussed above can be regarded as clocks that operate on purpose on a bad, environment-sensiitive clock transition in order to measure external fields. Today’s most advanced atomic clocks employ optical transitions in single ions or atomic clouds trapped in an optical lattice.
E. Solid state spins – Ensemble sensors
1. NMR ensemble sensors
Some of the earliest quantum sensors have been based on ensembles of nuclear spins. Magnetic field sensors have been built that infer field strength from their Larmor precession, analogous to neutral atom magnetometers described above . Initialization of spins is achieved by thermalization in an externally applied field, readout by induction detection. Although the sensitivity of these devices (10 pT/√Hz) is
inferior to their atomic counterparts, they have found broad use in geology, archaeology and space missions thanks to their simplicity and robustness. More recently, NMR sensor probes have been developed for in-situ and dynamical field mapping in clinical MRI systems
2. NV center ensembles
Much excitement has recently been sparked by ensembles of nitrogen-vacancy centers (NV centers) – electronic spin defects in diamond that can be optically initialized and read out. Densely-doped diamond crystals promise to deliver “frozen vapor cells” of spin ensembles that combine the strong (electronic) magnetic moment and efficient optical readout of atomic vapor cells with the high spin densities achievable in the solid state. Although these advantages are partially offset by a reduced coherence time (T2 < 1 ms at room temperature, as compared
to T2 > 1 s for vapor cells), the predicted sensitivity of diamond magnetometers (250 aT/√Hz/cm−3/2) or gyroscopes (10−5rad/s/√Hz/mm3/2 ) would be competitive with their atomic counterparts.
While large-scale sensing of homogeneous fields remains a challenge, micrometer-sized ensembles of NV centers have found application in imaging applications, serving as detector pixels for microscopic mapping of magnetic fields. Most prominently, this line of research
has enabled imaging of magnetic organelles in magnetotactic bacteria and microscopic magnetic inclusions in meteorites , as well
as contrast-agent-based magnetic resonance microscopy.
F. Solid state spins – Single spin sensors
Readout of single spins in the solid state – a major milestone on the road towards quantum computers – has been achieved both by electrical and optical schemes. Electrical readout has been demonstrated with phosphorus dopants in silicon and electrostatically-defined semiconductor quantum dots . Optical readout was shown with single organic molecules, optically active quantum dots, and defect centers in crystalline materials including diamond and silicon carbide.
Among all solid state spins, NV centers in diamond have received by far the most attention for sensing purposes. This is in part due to the convenient room temperature optical detection, and in part due to their stability in very small crystals and nanostructures. The latter permits use of NV centers as sensors in high resolution scanning probe microscopy , as biomarkers within living organisms, or as stationary probes close to the surface of diamond sensor chips.
Single NV centers have been employed and/or proposed as sensitive magnetometers , electrometers , pressure sensors and thermometers ,
using the Zeeman, Stark and temperature shifts of their spin sublevels. NV centers in ∼10-nm-sized nanodiamonds have also been inserted into living cells. They have been employed for particle tracking and in vivo temperature measurements and could enable real-time monitoring of metabolic processes.
G. Superconducting circuits
1. SQUIDs
The Superconducting Quantum Interference Device (SQUIDs) is simultaneously one of the oldest and one of the most sensitive type of magnetic sensor These devices – interferometers of superconducting conductors – measure magnetic fields with a sensitivity down
to 10 aT/√Hz. Their sensing mechanism is based on the Aharonov-Bohm phase imprinted on the superconducting wave function by an encircled magnetic field, which is read out by a suitable circuit of phase-sensitive Josephson junctions.
From a commercial perspective, SQUIDs can be considered the most advanced type of quantum sensor, with applications ranging from materials characterization in solid state physics to clinical magnetoencephalography systems for measuring tiny (∼ 100 fT) stray fields of electric currents in the brain. In parallel to the development of macroscopic (mm-cm) SQUID devices, miniaturization has given birth to sub-micron sized “nanoSQUIDs” with possible applications in nanoscale magnetic, current, and thermal imaging.
SQUIDs have been employed to process signals from the DC up to the GHz range , the upper limit being set by the Josephson frequency.
2. Superconducting qubits
Temporal quantum superpositions of supercurrents or charge eigenstates have become accessible in superconducting qubits. Being associated with large magnetic and electric dipole moments, they are attractive candidates for quantum sensing. Operating qubits as magnetic field sensors, very promising sensitivities (3.3 pT/√Hz for operation at 10 MHz) were demonstrated . Extending
these experiments to the study of extrinsic samples appears simultaneously attractive and technically challenging, since superconducting qubits have to be cooled to temperatures of only few tens of millikelvin.
H. Elementary particle qubits
Interestingly, elementary particles have been employed as quantum sensors long before the development of atomic and solid state qubits. This somewhat paradoxical fact is owing to their straightforward initialization and readout, as well as their targeted placement in relevant samples by irradiation with a particle beam.
Muons are frequently described as close cousins of electrons. Both particles are leptons, carry an elementary charge and have a spin that can be employed for quantum sensing. Sensing with muons has been termed “muon spin rotation” (µSR). Slow beams of thermal neutrons can be spin-polarized by Bragg reflection on a suitable magnetic crystal. Spin readout is feasible by a spin-sensitive Bragg analyzer and subsequent detection.
I. Other sensors
In addition to the many implementations of quantum sensors already discussed, three further systems deserved special attention for their future potential or for their fundamental role in developing quantum sensing methodology.
1. Single electron transistors
Single electron transistors (SET’s) sense electric fields by measuring the tunneling current across a submicron conducting island sandwiched between tunneling source and drain contacts. In the “Coulomb blockade regime” of sufficiently small (typically ≈ 100 nm) islands, tunneling
across the device is only allowed if charge eigenstates of the island lie in the narrow energy window between the Fermi level of source and drain contact.
SETs have been employed as scanning probe sensors to image electric fields on the nanoscale, shedding light on a variety of solid-state-phenomena such as the fractional quantum Hall effect or electron-hole puddles in graphene.
3. Photons
Squeezing of light – the creation of partially-entangled states with phase or amplitude fluctuations below those of a classical coherent state of the light field – has been proposed and achieved long before squeezing of spin ensembles. Vacuum squeezed states have meanwhile been employed to improve the sensitivity of gravitational wave detectors.
Squeezed states are promising for quantum-limited sensing as they can reach sensitivity beyond the standard quantum limit. Squeezed states of light have been introduced by Caves, 1981, as a mean to reduce noise in interferometry experiments. One of the most impressive application of squeezed states of light has been the sensitivity enhancement of the LIGO gravitational wave detector , obtained by injecting vacuum squeezed states in the interferometer.
In addition, quantum correlations between photons have proven to be a powerful resource for imaging. This has been noted very early on in the famous HanburyBrown-Twiss experiment, where bunching of photons is employed to filter atmospheric aberrations and to perform “super-resolution” measurements of stellar diameters smaller than the diffraction limit of the telescope employed.
Vice versa, multi-photon correlations have been proposed and employed to create light patterns below the diffraction limit for superresolution lithography. They can equally improve image contrast rather than resolution by a scheme known as “quantum illumination”. Here, a beam of photons is employed to illuminate an object, reflected light being detected as the imaging signal. Entangled twins of the illumination photons are conserved at the source and compared to reflected photons by a suitable joint measurement. In this way, photons can be certified to be reflected light rather than noise, enhancing imaging contrast. In simpler schemes, intensity correlations between
entangled photons have been employed to boost contrast in transmission microscopy of weakly absorbing objects and the reduced quantum fluctuations of squeezed light have been used to improve optical particle tracking.
The key difficulty with using entangled states for sensing is that they are less robust against noise. Thus, the advantage in sensitivity is compensated by a concurrent reduction in coherence time. In particular, it has been demonstrated that for frequency estimation, any nonzero value of uncorrelated dephasing noise cancels the advantage of the maximally entangled state over a classically correlated state . An analogous result can be proven for magnetometry.
The most advanced demonstrations of entanglement enhanced sensing have been performed with single photons or carefully assembled few-photon Fock states. Most prominently, these include Heisenberg-limited interferometers. In these devices, entanglement between photons
or adaptive measurements are employed to push sensitivity beyond the 1/√N scaling of a classical interferometer where N is the number of photons
Qubits on a tip
In one of the technologies under development, the scanning qubit microscope, the quantum sensor would consist of one or more qubits placed on the tip of a probe and moved over the surface of a material. A qubit is a basic unit of quantum information, like the bits of ordinary computer memory that flip back and forth between zero and 1. But a qubit exists as a superposition of both zero and 1 states at once. The scanner’s qubit might consist of a single hydrogen atom, for instance, with the spin of its single electron simultaneously existing as up, down and all possible states in between.
“You can try to entangle the qubit sensor with the quantum state of the material you’re studying so you can actually sense the entanglement of quantum states within the material,” said Kathryn Moler, Stanford’s vice provost and dean of research. “If we can do that, it will be really cool.” Entanglement – in the form of spin squeezing – has been produced by optical non-destructive measurements of atomic population and atomic interactions.
Electrons and Atoms embedded in materials
Alternative types of quantum sensor make use of atoms embedded in materials such as diamond and silicon. These are particularly suited for magnetic sensors.
Nitrogen-vacancy (NV) centers in diamond — probably the smallest magnetometer in the world. An NV center is an artificially synthesised defect that can be thought of as a single spin frozen inside diamond. This is essentially the product already available from Swiss start-ups Qnami and QZabre.
This type of quantum technology has simple subsystems and can be built using standard materials-science techniques: nitrogen–vacancy (NV) centers. These are atom-like defects in a synthetic diamond crystal consisting of a nitrogen atom and a gap in place of two carbon atoms. NV centers can emit red light when excited by green, but the probability of doing so depends on the spin states of their electrons. By placing the spin states in a superposition, microwaves with just the right frequency can change the emission intensity.
Crucially, this quantum state can persist for up to a millisecond at room temperature, thanks to the stiff diamond lattice that shields the NV centers from vibrations. And, because changes in the local magnetic field will change the spacing of the spin states and knock the microwave frequency off-resonance—with the change proportional to the field strength—NV centers could potentially make extremely sensitive magnetometers. The system is also sensitive to variations in electric field, strain and temperature, as these change the distance between atoms and, again, shift the resonance.
By allowing extremely sensitive magnetic-field measurements, diamond nitrogen–vacancy (NV) sensors could yield new insights into the workings of the brain and body. Ronald Walsworth and colleagues at Harvard University have shown how such sensors can be used to measure the magnetic field from the electrical current flowing in individual neurons. Using a diamond wafer measuring just 4×4×0.5 mm that contained trillions of NV centers, the researchers were able to detect the fields generated by single neurons inside live worms.
The work by the team marks the development of the first quantum sensor to be based on the spin of a single electron, which in this case, was trapped in a diamond nitrogen-vacancy center. It is so sensitive that it is able to measure the strength of a magnetic field to the very limits of that described by quantum physics.
Photonic Quantum Sensors
Other devices use photons, the particles of light. For example, photonic sensors can detect the optical properties of molecules to measure faint chemical traces. Photonic quantum sensing leverages entanglement, single photons and squeezed states to perform extremely precise measurements.
In photonics and quantum optics, quantum sensors are often built on continuous variable systems, i.e., quantum systems characterized by continuous degrees of freedom such as position and momentum quadratures. The basic working mechanism typically relies on optical states of light, often involving quantum mechanical properties such as squeezing or two-mode entanglement. These states are sensitive to physical transformations that are detected by interferometric measurements.
Optical sensing makes use of continuous variable quantum systems such as different degrees of freedom of the electromagnetic field, vibrational modes of solids, and Bose-Einstein condensates. These quantum systems can be probed to characterize an unknown transformation between two quantum states. Several methods are in place to improve photonic sensors such quantum illumination of targets which has been used to improve detection of weak signals by the use of quantum correlation.
While cold-atom and NV sensors rely on superposition, photonic sensors are designed to exploit entanglement. One of these technologies is quantum radar. Based on a theoretical scheme by physicist Seth Lloyd, this involves bouncing one half of a series of entangled-photon pairs off an object and then comparing the returning photons with those held back. The idea is to distinguish the radiation originally sent out from strong sources of noise, to spot otherwise undetectable objects such as stealth aircraft, and to keep the radar operators hidden.
Jonathan Baugh of the University of Waterloo in Canada is developing a near-infrared device to produce pairs of entangled photons on demand, which he says should be ready for field testing “a few years” from now. Although this will require an additional device to convert the output to microwave frequencies, it might in the end only need quantum statistics rather than entanglement per se. “If you always produce two photons at almost exactly the same time, you can use the timing correlation,” he says.
We can also use quantum technology to improve the read out of classical devices such as MEMS (micro electromechanical sensors) technology.
Radiation that drives photosynthesis is called photosynthetically active radiation (PAR) and is typically defined as total radiation across a range of 400 to 700 nm. PAR is often expressed as photosynthetic photon flux density (PPFD): photon flux in units of micromoles per square meter per second (µmol m-2 s-1 , equal to microEinsteins per square meter per second) summed from 400 to 700 nm (total number of photons from 400 to 700 nm). While Einsteins and micromoles are equal (one Einstein = one mole of photons), the Einstein is not an SI unit, so expressing PPFD as µmol m-2 s-1 is preferred.
Sensors that measure PPFD are often called quantum sensors due to the quantized nature of radiation. A quantum refers to the minimum quantity of radiation, one photon, involved in physical interactions (e.g., absorption by photosynthetic pigments). In other words, one photon is a single quantum of radiation.
The Defense Advanced Research Projects Agency has launched a research program in optical quantum sensors that seeks to exploit ideas from quantum metrology and quantum imaging, such as quantum lithography and the NOON state, in order to achieve these goals with optical sensor systems such as lidar.
