Vivishek Sudhir

Frequency combs enable precision measurements across timekeeping, spectroscopy, ranging and astronomy, and are now extending to integrated and field-deployable platforms. Realizing their full performance demands a comprehensive account of the quantum noise that arises when broadband optical fields are converted into finite-bandwidth electrical signals. Here we present a rigorous first-principles quantum-mechanical framework for optical frequency-comb metrology based on continuous-mode field quantization and a comb-line-resolved description of quantum fluctuations. The theory describes how quantum fluctuations of the comb field are transduced into electrical measurement noise. Formulated at the level of the comb field, the framework unifies the standard quantum limits of optical frequency division (OFD) and dual-comb spectroscopy (DCS) within a single treatment, and provides a general recipe for other comb-based measurements. On this footing, we identify practical, resource-efficient routes to quantum enhancement through engineered comb states, laying a foundation for the design of next-generation frequency combs operating at and beyond standard quantum limits.

Frequency combs enable precision measurements across timekeeping, spectroscopy, ranging and astronomy, and are now extending to integrated and field-deployable platforms. Realizing their full performance demands a comprehensive account of the quantum noise that arises when broadband optical fields are converted into finite-bandwidth electrical signals. Here we present a rigorous first-principles quantum-mechanical framework for optical frequency-comb metrology based on continuous-mode field quantization and a comb-line-resolved description of quantum fluctuations. The theory describes how quantum fluctuations of the comb field are transduced into electrical measurement noise. Formulated at the level of the comb field, the framework unifies the standard quantum limits of optical frequency division (OFD) and dual-comb spectroscopy (DCS) within a single treatment, and provides a general recipe for other comb-based measurements. On this footing, we identify practical, resource-efficient routes to quantum enhancement through engineered comb states, laying a foundation for the design of next-generation frequency combs operating at and beyond standard quantum limits.

We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.

We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.

We study a class of feedback oscillators realized by a phase-insensitive amplifier in positive feedback, where either the amplifier or the feedback element may determine the oscillator's linewidth. The spectral purity of the output of such a device originates from basic demands of quantum mechanics and causality. The resulting expression generalizes the Schawlow-Townes limit, which is itself one component of a standard quantum limit for feedback oscillators. Recently realized bad-cavity oscillators, such as superradiant lasers and solid-state masers, can saturate this generalized Schawlow-Townes limit. This limit can be surpassed through appropriate quantum engineering, for example, by atomic spin squeezing in a superradiant laser.

The extraction of signals from noise is a common problem in all areas of science and engineering. A particularly useful version is that of forecasting: determining a causal filter that estimates a future value of a hidden process from past observations. Current techniques for deriving the filter require that the noise be well described by rational power spectra. However, scale-free noises, whose spectra scale as a non-integer power of frequency, are ubiquitous in practice. We establish a method, together with performance guarantees, that solves the forecasting problem in the presence of scale-free noise. Via the duality between estimation and control, our technique can be used to design control for distributed systems. These results will have wide-ranging applications in neuroscience, finance, fluid dynamics, and quantum measurements.

The extraction of signals from noise is a common problem in all areas of science and engineering. A particularly useful version is that of forecasting: determining a causal filter that estimates a future value of a hidden process from past observations. Current techniques for deriving the filter require that the noise be well described by rational power spectra. However, scale-free noises, whose spectra scale as a non-integer power of frequency, are ubiquitous in practice. We establish a method, together with performance guarantees, that solves the forecasting problem in the presence of scale-free noise. Via the duality between estimation and control, our technique can be used to design control for distributed systems. These results will have wide-ranging applications in neuroscience, finance, fluid dynamics, and quantum measurements.

In idealized treatments of the Unruh effect, a two-level atom is accelerated in a prescribed classical trajectory through the vacuum of a quantum field---the Unruh bath---which causes the atom's internal state to thermalize to a temperature proportional to the acceleration. This happens via emission and absorption of quanta by the atom, leading to a detailed balance between fluctuations in the Unruh bath and associated dissipation of the atom's internal energy. In any physical manifestation of the Unruh effect, the center-of-mass (c.m.) degree of freedom of the atom is dynamical and is therefore coupled to the Unruh bath via momentum recoil during the absorption/emission process. We study this scenario and show how the fluctuation-dissipation theorem fails for the c.m. degree of freedom, while still being upheld for the internal energy. That is, the c.m. of an accelerated atom is not in thermal equilibrium with the Unruh bath, while its internal level can be.

Continuous-wave (CW) interferometry has stood at the frontier of precision measurement science since its inception, where it was used to search for the luminiferous ether, to the present day, where it forms the basis of interferometric gravitational-wave detection. Quantum theory predicts that this frontier can be expanded more rapidly by employing certain quantum resources, compared with the case of using only classical resources. In the quantum case, we can achieve ``Heisenberg scaling'', which manifests as a quadratic improvement over the best possible classical precision scaling. Although Heisenberg scaling has been demonstrated in pulsed operation, it has not been demonstrated for continuous operation. The challenge in doing so is two-fold: continuous measurements capable of Heisenberg scaling were previously unknown, and the requisite CW quantum states are fragile. Here we overcome these challenges and demonstrate the first CW interferometer exhibiting resource efficiency approaching Heisenberg scaling. Our scheme comprises a Mach-Zehnder interferometer illuminated with a pair of squeezed light sources, followed by a nonlinear estimator of the output homodyne record to estimate a differential phase modulation signal that drives the interferometer. We observe that this signal can be extracted with a precision that scales faster than what is allowed classically, and approaches the Heisenberg scaling limit.

Buccal drug delivery has emerged as one of the most promising and patient-friendly approaches for systemic and local therapy due to its ability to bypass first‑pass metabolism, offer rapid absorption, and improve bioavailability. The oral mucosa, consisting of keratinized and non‑keratinized epithelium, provides a unique environment for mucoadhesive formulations, making it ideal for buccal films. These films utilize polymers, plasticizers, penetration enhancers, and other excipients to achieve an optimal balance between mechanical strength, adhesive properties, and drug release characteristics. Compared with traditional dosage forms, buccal films offer numerous benefits, such as enhanced patient compliance, quick disintegration, and suitability for pediatric and geriatric patients. However, the formulation and manufacture of buccal films present significant challenges, including moisture sensitivity, limited surface area for drug absorption, and constraints on the quantity of active pharmaceutical ingredients. To overcome these, advances in polymer technology, plasticizers, and mucoadhesion strategies have enabled the design of sophisticated delivery platforms, making buccal films highly attractive for the administration of peptides, proteins, and other therapeutic agents. This review provides an in-depth examination of the characteristics of the oral cavity, the design and evaluation of buccal films, their manufacturing methods, and critical evaluation parameters. Additionally, it explores future prospects for extending the application of buccal films to a wider range of therapeutic areas, highlighting their role as a viable and innovative delivery platform for the next generation of pharmaceuticals.

The detection of weak forces is a central problem in physics and engineering, ranging in importance from fundamental pursuits such as precision tests of gravity, gravitational-wave detection, and searches for dark matter to applications such as force microscopy. These pursuits require a low-mass mechanical force transducer with a high quality factor, whose motion can be measured in a quantum-noise-limited manner. Here, we study the ultimate example of such a transducer: a single trapped electron. We propose and analyze in detail a scheme for high-sensitivity continuous force detection using a single trapped electron whose motion is coupled to a microwave cavity field via image currents induced in an antenna. We derive the fundamental and technical limits to the sensitivity of this scheme and show that despite the disparity in size between that of a single electron and the wavelength of the microwave field, it is possible to continuously monitor the charge's zero-point motion and use it as a force detector with a sensitivity as low as $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}\phantom{\rule{0.2em}{0ex}}\mathrm{N}/\sqrt{\mathrm{Hz}}$ in the gigahertz regime. This sensitivity improves on the state of the art by 4 orders of magnitude and thus paves the way to experiments with previously unattainable precision.

In a consistent description of the quantum measurement process, whether the wave or particle-like aspect of a system is revealed depends on the details of the measurement chain, and cannot be interpreted as an objective fact about the system independent of the measurement. We show precisely how this comes to be in the measurement of gravitational radiation. Whether a wave or particle-like aspect is revealed is a property of the detector employed at the end of the quantum measurement chain, rather than of the meter, such as a gravitational-wave (GW) antenna or resonant bar, used to couple the radiation to the detector. A linear detector yields no signal for radiation in a Fock state and a signal proportional to the amplitude in a coherent state -- supporting a wave-like interpretation. By contrast, the signal from a detector coupled to the meter's energy is non-zero only when the incident radiation contains at least a single graviton. Thus, conceptually simple modifications of contemporary GW antennae can reveal wave-particle duality in the measurement of gravitational radiation.

Experimental tests of gravity’s fundamental nature call for mechanical systems in the quantum regime while being sensitive to gravity. Torsion pendula, historically vital in studies of classical gravity, are ideal for extending gravitational tests into the quantum realm due to their inherently high mechanical quality factor, even when mass-loaded. Here, we demonstrate laser cooling of a centimeter-scale torsional oscillator to a temperature of 10 mK (average occupancy of 6000 phonons) starting from room temperature. This is achieved by optical radiation pressure forces conditioned on a quantum-noise-limited optical measurement of the torsional mode with an imprecision 9.8 dB below its peak zero-point motion. The measurement sensitivity is the result of a “mirrored” optical lever that passively rejects extraneous spatial-mode noise by 60 dB. The high mechanical quality (1.4×10 7 ) and quantum-noise-limited measurement imprecision demonstrate the necessary ingredients for realizing the quantum ground state of torsional motion—a prerequisite for mechanical tests of gravity’s alleged quantum nature.

What if gravity is classical? If true, a consistent coexistence of classical gravity and quantum matter requires that gravity exhibit irreducible fluctuations. These fluctuations can mediate classical correlations, but not quantum entanglement, between the quantized motion of the gravitationally interacting matter. We use a consistent theory of quantum-classical dynamics in the Newtonian limit of gravity to show that experimentally relevant observables can conclusively test the hypothesis that gravity is classical. This can be done, for example, by letting highly coherent source masses interact with each other gravitationally, and performing precise measurements of the cross-correlation of their motion. Theory predicts a characteristic phase response that distinguishes classical gravity from quantum gravity, and from naive sources of decoherence. Such experiments are imminently viable.

We present an experiment to reach the fundamental quantum limit of interferometry. We outline the theory and demonstrate it experimentally with phase-locked squeezed light, double homodyne readout, and an optimal phase estimator.

Linear time-translation-invariant (LTI) models offer simple, yet powerful, abstractions of complex classical dynamical systems. Quantum versions of such models have so far relied on assumptions of Markovianity or an internal state-space description. We develop a general quantization scheme for multimode classical LTI systems that reveals their fundamental quantum noise, is applicable to non-Markovian scenarios, and does not require knowledge of an internal description. The resulting model is that of an open quantum LTI system whose dilation to a closed system is characterized by elements of the conjugate symplectic group. Using Lie group techniques, we show that such systems can be synthesized using frequency-dependent interferometers and squeezers. We derive tighter Heisenberg uncertainty bounds, which constrain the ultimate performance of any LTI system, and obtain an invariant representation of their output noise covariance matrix that reveals the ubiquity of "complex squeezing" in lossy systems. This frequency-dependent quantum resource can be hidden to homodyne and heterodyne detection and can only be revealed with more general "symplectodyne" detection. These results establish a complete and systematic framework for the analysis, synthesis, and measurement of arbitrary quantum LTI systems.

Experimental tests of gravity's fundamental nature call for mechanical systems in the quantum regime while being sensitive to gravity. Torsion pendula, historically vital in studies of classical gravity, are ideal for extending gravitational tests into the quantum realm due to their inherently high mechanical quality factor, even when mass-loaded. Here, we demonstrate laser cooling of a centimeter-scale torsional oscillator to a temperature of 10 mK (average occupancy of 6000 phonons) starting from room temperature. This is achieved by optical radiation pressure forces conditioned on a quantum-noise-limited optical measurement of the torsional mode with an imprecision 9.8 dB below its peak zero-point motion. The measurement sensitivity is the result of a novel `mirrored' optical lever that passively rejects extraneous spatial-mode noise by 60 dB. The high mechanical quality ($1.4\times 10^7$) and quantum-noise-limited measurement imprecision demonstrate the necessary ingredients for realizing the quantum ground state of torsional motion -- a pre-requisite for mechanical tests of gravity's alleged quantum nature.

Temperature fluctuations over long time scales (≳ 1 h) are an insidious problem for precision measurements. In optical laboratories, the primary effect of temperature fluctuations is drifts in optical circuits over spatial scales of a few meters and temporal scales extending beyond a few minutes. We present a lab-scale environment temperature control system approaching 10 mK-level temperature instability across a lab for integration times above an hour and extending to a day. This is achieved by passive isolation of the laboratory space from the building walls using a circulating air gap and an active control system feeding back to heating coils at the outlet of the laboratory's Heating-Ventilation-Air-Conditioning (HVAC) unit. These techniques together result in 20 dB suppression of the temperature power spectrum across the lab at 10-4 Hz-approaching the limit set by statistical coherence of the temperature field-and 10 mK Allan deviation around 15 °C after an hour of averaging, which is an order of magnitude better than any previous report for a full laboratory.

The observables of a noisy quantum system can be estimated by appropriately filtering the records of their continuous measurement. Such filtering is relevant for state estimation, and if the filter is causal, also relevant for measurement-based feedback control. It is therefore imperative that a pair of conjugate observables estimated causally satisfy the Heisenberg uncertainty principle. In this article, we prove this fact---without assuming Markovian dynamics or Gaussian noises, in the presence or absence of feedback control of the system, and where in the feedback loop (inside or outside) the measurement record is accessed. Indeed, causal estimators using the in-loop measurement record can be as precise as those using the out-of-loop record. These results clarify the role of causal estimators to non-Markovian quantum systems, restore the equanimity of in-loop and out-of-loop measurements in their estimation and control, and simplify future experiments on measurement-based quantum feedback control.

Exceptional-point (EP) sensors exhibit a square-root resonant frequency bifurcation in response to external perturbations, making them appear attractive for sensing applications. However, there is an open debate as to whether or not this sensitivity advantage is negated by additional noise in the system. We settle this debate by showing that increased fundamental noises of quantum and thermal origin in EP sensors, and in particular self-excited (or PT-symmetric) EP sensors, negate the sensitivity benefit. Accordingly, EP sensing schemes are only beneficial either with further quantum enhancement or if compared to sensors limited by technical noise. As many modern sensors are limited by technical noise, EP sensors may still find practical uses despite their lack of a fundamental advantage. Alternatively, we propose a quantum-enhanced EP sensor that achieves a sensing advantage even when limited by quantum or thermal fluctuations.

Temperature fluctuations over long time scales ($\gtrsim 1\,\mathrm{h}$) are an insidious problem for precision measurements. In optical laboratories, the primary effect of temperature fluctuations is drifts in optical circuits over spatial scales of a few meters and temporal scales extending beyond a few minutes. We present a lab-scale environment temperature control system approaching $10\, \mathrm{mK}$-level temperature instability across a lab for integration times above an hour and extending to a few days. This is achieved by passive isolation of the laboratory space from the building walls using a circulating air gap and an active control system feeding back to heating coils at the outlet of the laboratory HVAC unit. The latter achieves 20 dB suppression of temperature fluctuations across the lab, approaching the limit set by statistical coherence of the temperature field.

Exceptional-point (EP) sensors are characterized by a square-root resonant frequency bifurcation in response to an external perturbation. This has lead numerous suggestions for using these systems for sensing applications. However, there is an open debate as to whether or not this sensitivity advantage is negated by additional noise in the system. We show that an EP sensor's imprecision in measuring a generalized force is independent of its operating point's proximity to the EP. That is because frequency noises of fundamental origin in the sensor -- due to quantum and thermal fluctuations -- increase in a manner that exactly cancels the benefit of increased resonant frequency sensitivity near the EP. So the benefit of EP sensors is limited to the regime where sensing is limited by technical noises. Finally, we outline an EP sensor with phase-sensitive gain that does have an advantage even if limited by fundamental noises.

We present a theory of quantum noise in exceptional point sensors. We find that these sensors have no sensing advantage relative to traditional sensors limited by fundamental quantum or thermal noise.

Feedback oscillators, consisting of an amplifier whose output is partially fed back to its input, provide stable references for standardization and synchronization. Notably, the laser is such an oscillator whose performance can be limited by quantum fluctuations. The resulting frequency instability, quantified by the Schawlow-Townes formula, sets a limit to laser linewidth. Here, we show that the Schawlow-Townes formula applies universally to feedback oscillators beyond lasers. This is because it arises from quantum noise added by the amplifier and out-coupler in the feedback loop. Tracing the precise origin of quantum noise in an oscillator informs techniques to systematically evade it: we show how squeezing and entanglement can enable sub-Schawlow-Townes linewidth feedback oscillators. Our analysis clarifies the quantum limits to the stability of feedback oscillators in general, derives a standard quantum limit (SQL) for all such devices, and quantifies the efficacy of quantum strategies in realizing sub-SQL oscillators.

What if gravity is classical? If true, a consistent co-existence of classical gravity and quantum matter requires that gravity exhibit irreducible fluctuations. These fluctuations can mediate classical correlations, but not quantum entanglement, between the quantized motion of the gravitationally interacting matter. We use a consistent theory of quantum-classical dynamics in the Newtonian limit of gravity to show that experimentally relevant observables can conclusively test the hypothesis that gravity is classical. This can be done for example by letting highly coherent source masses interact with each other gravitationally, and performing precise measurements of the cross-correlation of their motion. Theory predicts a characteristic phase response that distinguishes classical gravity from quantum gravity, and from naive sources of decoherence. Such experiments are imminently viable.

Contemporary gravitational-wave detectors are fundamentally limited by thermal noise-due to dissipation in the mechanical elements of the test mass-and quantum noise-from the vacuum fluctuations of the optical field used to probe the test-mass position. Two other fundamental noises can in principle also limit sensitivity: test-mass quantization noise due to the zero-point fluctuation of its mechanical modes and thermal excitation of the optical field. We use the quantum fluctuation-dissipation theorem to unify all four noises. This unified picture shows precisely when test-mass quantization noise and optical thermal noise can be ignored.

We propose an adaptive phase technique for the parametric cooling of mechanical oscillators. Our scheme calls for a sequence of periodic adjustments of the phase of a parametric modulation of the mechanical oscillator that is conditioned on measurements of its two quadratures. The technique indicates an exponential loss of thermal energy at initial high occupancies, similar in performance to other optomechanical techniques such as cold-damping or cavity self-cooling. As the quantum ground state is approached, the phase adaptive scheme leads to residual occupancies at the level of a few phonons owing to the competition between parametric amplification of quantum fluctuations and the feedback action.

The observables of a noisy quantum system can be estimated by appropriately filtering the records of their continuous measurement. Such filtering is relevant for state estimation and measurement-based quantum feedback control. It is therefore imperative that the observables estimated through a causal filter satisfy the Heisenberg uncertainty principle. In the Markovian setting, prior work implicitly guarantees this requirement. We show that any causal estimate of linear observables of a linear, but not necessarily Markovian, system will satisfy the uncertainty principle. In particular, this is true irrespective of any feedback control of the system and of where in the feedback loop -- inside or outside -- the measurement record is accessed. Indeed, causal estimators using the in-loop measurement record can be as precise as those using the out-of-loop record. These results clarify the role of causal estimators to a large class of quantum systems, restores the equanimity of in-loop and out-of-loop measurements in their estimation and control, and simplifies future experiments on measurement-based quantum feedback control.

We study an abstract model of an oscillator realized by an amplifier embedded in a positive feedback loop. The power and frequency stability of the output of such an oscillator are limited by quantum noise added by two elements in the loop: the amplifier, and the out-coupler. The resulting frequency instability gives the Schawlow-Townes formula. Thus the applicability of the Schawlow-Townes formula is extended to a large class of oscillators, and is shown to be related to the Haus-Caves quantum noise limit for a linear amplifier, while identifying the role of quantum noise added at the out-coupler. By illuminating the precise origin of amplitude and frequency quantum noise in the output of an oscillator, we reveal several techniques to systematically evade them.

Crystalline materials are increasingly employed to construct precision optical instruments because of their reduced mechanical dissipation and consequent reduction of thermal Brownian noise. However, the anisotropy of the crystalline state implies a fundamental source of thermal noise; depolarization induced by thermal fluctuations of its birefringence. We establish the theory of this effect, which is a generalization of prior treatments of thermo-optic noises in amorphous materials. This theory---in conjunction with poorly understood anisotropic thermal stress coefficients of crystalline coatings---predict that thermo-refringent noise in crystalline mirror coatings may be lurking within an order of magnitude of Brownian noise (below 100 Hz). Thus, in order to appreciate the full promise of crystalline optical materials, a more precise understanding of their anisotropic material constants is necessary. Barring that, we elucidate measurement techniques that can affect partial coherent cancellation of thermorefringent noise. In passing, our general formalism also predicts the existence of thermal beam-pointing noise.

Contemporary gravitational-wave detectors are fundamentally limited by thermal noise -- due to dissipation in the mechanical elements of the test mass -- and quantum noise -- from the vacuum fluctuations of the optical field used to probe the test mass position. Two other fundamental noises can in principle also limit sensitivity: test-mass quantization noise due to the zero-point fluctuation of its mechanical modes, and thermal excitation of the optical field. We use the quantum fluctuation-dissipation theorem to unify all four noises. This unified picture shows precisely when test-mass quantization noise and optical thermal noise can be ignored.

We develop a general quantum theory of feedback oscillators and use it to study the origin of quantum noise in such systems, the limitations posed by quantum noise, and systematic techniques to evade these limitations. For a laser, the quantum noise limit is the Schawlow-Townes limit, which can be evaded using squeezing or entanglement.