2024/10/25 Photometry is evolving rapidly to a new and cheap technique for measuring the mass of large to medium-sized molecules, with current bounds on masses reaching down to about 10,000 Dalton, nurturing hopes to replace traditional mass spectrometry. And quantum metrology could help resolve even smaller masses, as we describe in our new preprint “Pushing the Boundaries: Interferometric Mass Photometry at the Quantum Limit of Sensitivity”.

2024/06/18 Recently we realized that the gravitational force from the highly energetic LHC beam, where protons move close to the speed of light, should be close to be measurable with modern quantum opto-mechanical sensors.  This would test general relativity in an entirely new parameter regime, where the source of gravity is almost entirely the kinetic energy of the protons rather than their rest mass.  If such experiments become possible, would they also allow to distinguish and possibly eliminate alternative theories of gravitation? Find out in our new preprint  “Gravitational attraction of ultra-relativistic matter: A new testbed for modified gravity at the Large Hadron Collider”.  

2024/06/12 We have a new preprint on “Simulation of the Dissipative Dynamics of Strongly Interacting NV Centers with Tensor Networks”.  NV Centers can be used for building highly sensitive quantum sensors for mangetic fields and other quantities, with the additional benefit of large spatial resolution.  These two properties lead to the demand of sensors with high densities of NV centers, which, however, lead to a kind of intrinsic decoherence due to the complicated interactions between the NV centers that become important when their mutual distance becomes small.  Accurately simulating them then becomes a challenge, even with modern techniques such as tensor networks.  Here we benchmark several methods against exact diagonalization and then investigate the benefits of real decoherence, which can limit the entanglement arising due to the interactions and hence make simulations easier again.  But that is only part of the full story.

2024/05/21 We have a new preprint on “Machine learning of quantum channels on NISQ devices”.  Characterizing quantum devices is notoriously difficult due to the number of degrees of freedom that increases exponentially with the number of qubits. Here we show that machine learning techniques can help.  See here: https://arxiv.org/abs/2405.12598

2023/03/17 Originally motivated by the extra-ordinary precision with which resistiance plateaus in the quantum Hall effect are quantized, we asked ourselves how precisely one can measure in principle magnetic fields using the Hall effect, or possibly the quantum Hall effect. The Hall effect is used technologically on a very large scale - probably billions of them are built in all those smartphones being schlepped around in the world. It is clear that making them smaller and smaller, they become less sensitive - but just how so?  And does it help to use graphene as new material?
To find out, we had to dig a bit deeper into theory and developed, over the last few years, a rather general framework for precision sensing using fermionic systems based on fermionic quantum field theory.  Besides the main results, we found a few other goodies on the way, such as a chain rule, when applying two consecutive unitary transformations.  And in the end, we figured out that super-Heisenberg scaling might be possible, and funnily, the power law found depends on the geometry of the sensor!

15/03/2023 Rotations about different rotation axis do not commute in general, not even classically. Quantum mechanically, rotations are generated by components of the angular momentum operator, and different components do not commute either.  But there is a subtle difference between measuring components of the angular momentum and estimating the rotation angles corresponding to two different rotations imprinted on a quantum state.  Inspired by seminal work on amplication without added noise of both quadratures of micro-waves, we now found out that a similar feast is possible for rotations of a spin: rotation angles of non-commuting rotations can be measured simultaneously with Heisenberg-limited sensitivity!  This has immediate application for measuring magnetic field components in different directions.

12/01/2023 How does a light pulse propagate in a medium that is dispersive, non-linear, possibly inhomogeneous and elastic, and on top of that embedded in a curved space-time?  If you think that these are a lot of complications at a time you are definitely right! But it turns out that understanding such a complex system is key in several areas: distributed remote sensing of Earth quakes, analogue gravity experiments, and quantum metrology of space-time.  So we had to find out. And, building on previous famous work that allows one to describe light propagation in a curved space-time by an effective medium in flat space time, we found a neat formalism how to describe this complex situation. 

04/01/2023 Remote sensing of Earth via satellites has become one of the most important sources of information about our Planet, notably for monitoring its surface water.  The state of the art in the micro-wave regime are interferometric measurement schemes that numerically synthesize a large antenna out of an antenna array, see e.g. the SMOS mission of ESA.  Helas, the spatial resolution of the images obtained is diffraction limited by  and for a given wavelength can´t be improved much using this technique without requiring unreasonably large satellites.  Following our previous work that focused on estimating positions of source points, we now went a step further and showed that actual multi-pixel images can be reconstructed with an order of magnitude better resolution than with the classical interferometers by using quantum-enhanced multi-parameter estimation.  Instead of about 35 km resolution, 3 km appear to be achievable with the same size of the satellite!  Learn how in our recent preprint...

2022/08/12  If we think of chemical reactions, we typically have two atoms or molecules in mind that bang into each other and then form a new molecule.  But the very idea of atoms following trajectories is, of course, a very classical and ultimately wrong way of describing particles in the microscopic world: atoms are quantum objects and their motion in space is described by a wave function whose dynamics is governed by Schrödinger's equation.  This becomes particularly important at low temperatures, where the atomic wave functions  spread over large distances, and so - in a certain sense - the atoms "see" all the other atoms at the same time.  This has profound consequences for chemical reactions and leads to what has been called "superchemistry", where even simple reactions like A+A -> A₂ that would classically be essentially one-way, can lead to coherent oscillations between an atom-rich and a molecule-rich state. The reaction itself is described by a quantized vector-field with a new type of quasi-particle, a "reacton".  In a new preprint we now show that this "superchemistry" harbors great potential for quantum metrology!  It combines a large number of atoms with almost perfect quantum coherence over long time-scales with an exquisite detection scheme, namely the spectroscopic detection of a few coherently created molecules due to an external perturbation...

2022/01/28 The good old Mach-Zehnder interferometer is one of the most important tools in optics and quantum optics.  A light beam is split into two beams, one of them undergoes a phase shift that one wants to measure, and then the two beams are brought back together in a second beamspitter.  At its exit one measures an interference pattern, i.e. an oscillation of the intensity as function of the phase shift, that can arise e.g. due to passage through another medium or changes in the length of the path.  Besides this direct implementation, there are more abstract versions based on precessing spins that allow one to measure very precisely magnetic fields.  Inspired by that latter realization and our previous work on quantum chaotic sensors, we now show that in certain parameter regimes the sensitivity of the Mach-Zehnder interferometer can be enhanced drastically by periodically "kicking" the light via reapeated passage through a non-linear medium. This is particularly useful if dissipation and decoherence arise through photon-loss, as is inevitably the case in real life.  And another realization holds great promise.

2021/12/10 Passive remote sensing of Earth by satellite provides precious data for weather and flood predictions, the salinity of oceans, monitoring vegetation, and many others.  State of the art satellites such as SMOS use interferometric techniques, where many relatively small antennas synthesize one large antenna.  Each antenna measures in real time the fluctuating electric fields in a narrow bandwidth in the GHz range that arise from the thermal noise emission from the surface of Earth, and then the measured signals are electronically interfered.  The achieved resolution is about 35km.  Improving the resolution is highly desirable, but this interferometric technique quickly leads to unrealistically large antenna arrays.
Now we showed that quantum metrology can come to a rescue: Instead of directly measuring the electric fields, it pays to first interfere the signals from different antennas with well defined phases and amplitude ratios whose optimal values we calculate.  And another trick from quantum optics that has just been brought to the microwave domain in the context of superconducting quantum processors is crucial.

2021/09/16 In his famous article "Simulating Physics with Computers" , now considered to have triggered the development of quantum information theory, Richard Feynman asks "Can quantum systems be probabilistically simulated by a classical computer?"   And answers a few lines later: "If you take the computer to be the classical kind I´ve described so far (...) and there´re no changes in any laws, and there´s no hocus-pocus, the answer is certainly, No!".  He justifies his answer with the observation that quantum mechanics appears to require negative probabilities, which, however, a classical computer cannot generate. 
Now we proved that the answer is "Yes!" nevertheless.  Admittedly, it does require a bit of hocus-pocus: the main trick is to code a quantum mechanical wave function not directly in a probability distribution, but in a (higher-order) derivative of a probability distribution.   In fact, one derivative per particle.  That object obviously can go negative, and this way we gain access to the same kind of interference in an exponentially large (tensor-product) Hilbert space as we are used to in quantum mechanics.  And that derivative of the probability distribution can be propagated via a stochastic process on a classical computer in such a way that up to a prefactor it follows exactly the evolution of the quantum mechanical state propagated by an arbitrary quantum algorithm: each quantum gate from a universal gate set is translated to a stochastic gate, where you flip one or two bits with certain probabilities.  Sounds intriguing?  It is!  But there is catch... 

2021/09/07 We live in an expanding universe whose expansion rate is governed by the Hubble parameter.  That, at least, is the picture we see on very large length scales of the universe, beyond the super-galaxy clusters, and it agrees perfectly well with very precise measurements of the cosmic microwave background and redshifts of distant galaxys.  Whether this cosmic expansion should be visible also on a much smaller scale, notably within in our solar system, is another question.   Einstein and Strauß answered it to the negative already in the mid 1940s, but their work was criticized later as making too restrictive assumptions.  Over the last few decades the question has been studied again many times, but no consenus has emerged even from the theoretical side.  It appears just too difficult to solve Einstein´s field equations with sources of gravity clustering on length scales that differ by many orders of magnitude.

So could experiments answer the question?

Interestingly, the current value of the Hubble parameter H is, in SI units, of order 10 ^-18 /s --- which means that the proper distance between two objects that follow the Hubble flow grows in 1 second by a relative amount of order 10^-18.  That is definitely a small number, but on the other hand, if you compare it with the relative changes of length on the order 10^-22 detected by LIGO when hit by gravitational waves from far away mergers of astronomical objects, it is roughly a factor 10,000 larger!  Moreover, modern optical clocks have reached a precision of about 10^-18 Hz ^-1/2, i.e. after averaging for 1s they reach a relative uncertainty of order 10^-18.   So shouldn´t that make it possible to measure the expansion of the universe even with a table-top experiment using an optical clock, maybe in an interferometric measurement scheme?  Or at least with some satellite experiment?

The answer, so we found out, hinges on that condition "follow the Hubble flow"...  What does that mean, and can the condition be satisfied?  Learn more about it in our new preprint.

2021/05/17 Detecting and quantifying causal influences are fundamental to all of science.  But where does causality come from?  If one looks closely, one realizes that even in our most fundamental and well-established descriptions of Nature, such as quantum mechanics and quantum field theory, causality is built in by hand.  E.g. by selecting retarded Green´s functions as the physical ones, where both retarded and advanced Green´s functions (meaning: forward and backward in time causal influences) are equally valid mathematically.

In recent years, there has been increasing interest in the role of causality in quantum mechanics, motivated in part by the hope to find more fundamental theories that would e.g. allow quantum mechanical superpositions of forward and backward causal influences, out of which in a classical limit only the forward in time causal influence might get selected.  Or there is the more mundane motivation of finding quantum advantages in causal analysis - something we also worked on before - and which might become relevant once quantum computers learn machine learning...

In a new preprint we stepped a bit back and just asked (and answered ;-) the very simple question: how to actually measure causal influence in standard quantum mechanics?  We follow Judea Pearl´s lead from classical analysis, according to which causal influence means that one random variable "listens" to another.  We replace the random variables by quantum states in quantum mechanics and build a measure that quantifies how much a given quantum state depends on another.  Applied to various physical situations, it leads to immediate surprises: E.g. if you have two atoms just interacting with the electromagnetic field fluctuations that propagate the causal influence from one atom to the other, the causal influence arrives way behind the light cone that is normally thought of deliminating the causal influence!  On the other hand, causal influence is almost perfectly in sync with the reservoir induced entanglement generated in this process!

Many other things can be analyzed with this measure, e.g. how causal influence propagates in a quantum circuit.  In short, a new tool is available now, waiting to be used! Find out more here.

2021/04/19 All energy has a mass equivalent, according to Einstein´s most famous formula. Even if there is no rest mass at all, as for light. So if you take a really powerful laser beam, would you actually get attracted to it due to its gravitational field? The gravitational field of light has been calculated already in 1931, and a few years ago we pushed these calculations to a new level by considering real laser beams - namely laser beams with a finite opening angle and depth of focus.  Things are a bit more complicated than just calculating a mass equivalent, as not just the single quantity "mass" determines gravity according to General Relativity, but a 4x4 energy-momentum tensor.  Interestingly, for light it has an entry in an off-diagonal matrix element that is just as big as the one corresponding to the equivalent mass density, whereas - as far as we can tell - all lab experiments on gravity so far only tested Newtonian gravity with mass as its source.  So measuring the gravitational effect of light would be extremely interesting, as it would allow one to test General Relativity in a controlled lab experiment in so far uncharted territory.

Unfortunately, the gravity of light is extremely small - the upside being that you don´t have to worry that a laser will ever pull you into it by its gravitational force... One needs extremely powerful lasers combined with extremely sensitive sensors to even hope that one might measure that gravitational pull.  What counts is (time-)averaged power - so it does not help to have a Peta-Watt laser if the laser pulses are only femto-seconds long.  In a new preprint, we tried to develop ideas from a theoretical side, how one might nevertheless be able to measure the gravity of laser beams.  Or of high-energy particle beams, which from a gravitational perspective are essentially equivalent to light beams, as the rest mass of the particles can be completely neglected compared to the kinetic energy.  We looked at a combination of the most powerful sources known with the most sensitive quantum sensors.  And we found that in principle we (as humanity...) are getting close to feasibility of such an experiment: with the planned upgrade of the Large Hadron Collider to high luminosity LHC, a monolithique pendulum cooled down to mK temperature as detector, and one week of measurement time, one should achieve a signal-to-noise ratio barely of order one! Alternatively, a single mode UV laser with some 10KW continous power and focused down to a spot-size of order micro-meter might work as a source. Here´s such a beast that was pointed out to us...

The experiment is extremely challenging, but it would also open up whole new perspectives, including squeezing the light and hence creating a quantum-mechanically non-trivial source of gravity - for which so far no one has a theoretical framework!  Find out more in our new preprint.

2021/03/04 A while ago we introduced the Majorana representation for mixed states (or in fact any hermitian operator).  Now, in a collaboration with colleagues in Mexico and Krakow, we figured out a Majorana representation of symmetric multi-qudit states.  These are pure states of several spins-j (or pseudo-spins-j, i.e. just quantum systems with finite dimensional Hilbert space), with j arbitrary, generalizing Majorana's original work for j=1/2.  Central to the construction is again a decomposition into SU(2) irreps, where now, however, typically several values of the total angular momentum J are needed (whereas Majorana's only needs the case of J=N/2, for N spins-1/2).  As for the case of mixed quantum states, the relative weights and phases for these different SU(2) irreps are coded in a "spectator state".  We use the new tool to explore a large number of interesting quantum states, including states that lead to optimal sensitivity for sensing rotations, either about a fixed axis or averaged over all axes. 

2021/03/02 Some of the most spectacular effects of quantum mechanics appear when a large number of subsystems are found in the same quantum state.  We wrote previously about this effect in the context of quantum-phase synchronization.  Together with colleagues from Krakow, we now solved the full problem of synchronizing the (reduced) quantum states of an arbirtary number of quantum systems with finite-dimensional Hilbert space, in the sense of finding necessary and sufficient conditions for such "quantum-state synchronization", proving tight lower bounds on the environment's Hilbert space dimension, and providing a construction for the joint unitary that propagates the subystem and the environment.  And we show experimentally on an IBM quantum computer that this quantum-state synchronizer really works for the simplest case of two qubits with a single qubit as environment!

2021/02/25 Our patent on quantum chaotic sensors was finally granted! See here.

2020/09/01 We have a new paper in Phys. Rev. Research on how to best measure small gravitational fields using optomechanical systems.  It  builds on our previous work that allows us to solve the non-linear dynamics of a cavity with a movable mirror whose motion is light-pressure coupled to the electro-magnetic field inside the cavity, and which is driven by an external force.  Based on our results we estimate that it should be possible in the near future to detect the gravitational field of an oscillating nanogram mass.  We also identify the experimental parameter regime necessary for gravitational wave detection with a quantum optomechanical sensor.

2020/06/26 With the availability of first small-scale and yet imperfect quantum computers the question whether these devices are "really quantum" has become center stage.  Now, if there is one thing that most people would agree upon is "really quantum", it is quantum entanglement. Indeed, it has been known for a long time that a pure-state quantum algorithm that does create only limited amounts of entanglement can be efficiently simulated classically.  So the bare minimum one would request from a quantum computer is that it can entangle two qubits - and keep that entanglement indefinitely.  While the existing quantum computers have no problem with the first part, they typically loose the entanglement again after a relatively short time (in fact, even keeping a single qubit "alive", i.e. in the chosen coherent superposition of its basis states over arbitrarily large time is still not possible at the time or writing!).  Hence, finding out whether a quantum channel can create entanglement (the so-called quantum-channel separability problem) over a certain time is still an important task - and one that had no general solution so far.  Now we provide a full solution of this problem by mapping it to a problem of "truncated moment sequences" - following the succssful recipe for quantum states that we developed earlier.  The approach leads to a semi-definite algorithm that provides a definite "yes" or "no" answer in finite time.  And while in practice the algorithm is very heavy on computational resources we provide examples where it succeeds where other simpler criteria have failed...  Find out more in our new preprint!

2020/03/04 We have a new preprint out which shows that machine learning can be used to efficiently design experiments that are optimized for maximal information gain. The setup is Bayesian quantum parameter estimation, where knowledge about parameters to be estimated is updated with every new result of a quantum measurement, and choosing the optimal next experiment depends on the knowledge already accumulated.   One would like to have fast design rules with which one can calculate the optimal parameters for the next run, so-called "heuristics".  We demonstrate that neural networks, once sufficiently trained (we use reinforcement learning, with the cross-entropy method and trusted-region-policy optimization) can provide such heuristics.  We examine the approach in detail for the fundamental and technologically important problem of estimating precession frequency and damping of a single spin-1/2, for which we show that the trained neural network can outperform substantially the previously known heuristics.  Commited to open science, we provide the source codes, which should allow experimentalists to easily integrate these heuristics into their measurement setup.

2020/02/20 Dipolar spinor gases of magnetic atoms carry a magnetization and can behave effectively as a ferro-magnet.  Due to the relatively large distance between the atoms, the spins interact via the long-range dipole interaction rather than the usual Heisenberg interaction familiar in solid ferro-magnets.  For sufficiently small sample sizes the sample behaves like a single domain, i.e. all magnetic dipoles point in the same direction.  In a long cigar-shaped sample, they point preferably in the direction of the long axis, but can be switched by applying an external magnetic field.  In a new preprint we show that the dynamics of the magnetization of a dipolar spinor Bose Einstein condensate is governed by a modified Landau-Lifshitz Gilbert equation, where the external field is modified by an effective dipole field arising from the magnetic dipoles themselves - if one makes the standard assumption of a scalar damping parameter. In a suitable parameter regime the dipole field accelerates the switching substantially.  That´s great news for measuring the damping constant in the gas, of which so far only little is known, both theroetically and experimentally, as the typical damping time scale of density fluctuations is too long compared to the life-time of the gas in a typical experiment.  We also show that in the limit of very slow switching one recovers the celebrated Stoner-Wolfarth model, one of the pilars of our understanding magnetization switching in small ferromagnets.

2020/03/04 We have a new preprint out which shows that machine learning can be used to efficiently design experiments that are optimized for maximal information gain. The setup is Bayesian quantum parameter estimation, where knowledge about parameters to be estimated is updated with every new result of a quantum measurement, and choosing the optimal next experiment depends on the knowledge already accumulated.   One would like to have fast design rules with which one can calculate the optimal parameters for the next run, so-called "heuristics".  We demonstrate that neural networks, once sufficiently trained (we use reinforcement learning, with the cross-entropy method and trusted-region-policy optimization) can provide such heuristics.  We examine the approach in detail for the fundamental and technologically important problem of estimating precession frequency and damping of a single spin-1/2, for which we show that the trained neural network can outperform substantially the previously known heuristics.  Commited to open science, we provide the source codes, which should allow experimentalists to easily integrate these heuristics into their measurement setup.

2020/02/20 Dipolar spinor gases of magnetic atoms carry a magnetization and can behave effectively as a ferro-magnet.  Due to the relatively large distance between the atoms, the spins interact via the long-range dipole interaction rather than the usual Heisenberg interaction familiar in solid ferro-magnets.  For sufficiently small sample sizes the sample behaves like a single domain, i.e. all magnetic dipoles point in the same direction.  In a long cigar-shaped sample, they point preferably in the direction of the long axis, but can be switched by applying an external magnetic field.  In a new preprint we show that the dynamics of the magnetization of a dipolar spinor Bose Einstein condensate is governed by a modified Landau-Lifshitz Gilbert equation, where the external field is modified by an effective dipole field arising from the magnetic dipoles themselves - if one makes the standard assumption of a scalar damping parameter. In a suitable parameter regime the dipole field accelerates the switching substantially.  That´s great news for measuring the damping constant in the gas, of which so far only little is known, both theroetically and experimentally, as the typical damping time scale of density fluctuations is too long compared to the life-time of the gas in a typical experiment.  We also show that in the limit of very slow switching one recovers the celebrated Stoner-Wolfarth model, one of the pilars of our understanding magnetization switching in small ferromagnets.

2019/10/11 Opto-mechanical systems couple optical and mechanical degrees of freedom.  One examples is a thin membrane that serves as mirror on one side of an optical cavity.  The light pressure can, at sufficiently high intensity, push the elastic membrane a little bit to the side, modifying at the same time slightly the resonance frequency of the cavity, and so the light field and the mechanical object couple to each other.  Even though a mode of the light and the elastically suspended membrane can be desrcibed by harmonic oscillators, the coupling between them is non-linear, which makes it difficult to solve the dynamics.  Owing to our previous work leading to the exact analytical solution of the quantum dyamics of the system that makes use of a finite Lie-algebra created by the commutators of different parts of the hamiltonian, we now calculated the full expressions for the quantum Fisher information for all parameters of the system, including the possibility of linear or parametric driving with external forces.  This allows us to predict what are the most precise measurements possible with such systems.  For certain types of measurements, resonant driving can lead to a large improvement of sensitivity.  Find out more in this preprint!

2019/09/17  In 1932, Ettore Majorana published a famous paper in which he showed how to visualize quantum states of any quantum system with finite-dimensional Hilbert space by a configuration of stars on the unit-sphere - the stellar representation was born.  This was 6 years before he disappeared mysteriously on a boat trip from Palermo to Naples, where he had become a professor of theoretical physics just a year before.  But his stellar representation lived on, owing to its beauty and invariance properties. E.g. when the physical system is a spin or angular momentum and you rotate the lab and then look at the state, its stellar configuration has rigidly rotated just in the same way as the lab, even though they exist in completely differen spaces, of course.  For the most classical states, all the stars coincide.  The more "quantum" the state, the more the stars are spread out.  We have used the stellar representation earlier to visualize for example the "queens of quantum", the spin-states with the most "quantumness" possible.

Now we have generalized the stellar representation to mixed quantum states, i.e. states with a certain classical uncertainty, as for example in thermal states, and in fact even to any hermitian operator on a finite dimensional Hilbert space!  We show how important operations such as taking the partial trace of the density matrix, or products of operators can be expressed in terms of the polynomials that determine the stars of the representation, compare representations of Schrödinger cat-states with their decohered mixtures, and show the connection to the tensor-representation of spin-states, as well as the Husimi- and P-functions, i.e. quasiprobability distributions familiar mostly from quantum optics that also describe the state of the system. Read more and discover some nice examples of stellar representations in this preprint...

2019/08/22 At least since Alpha Go made headlines by beating a human world champion in the board game of Go, machine learning, and in particular reinforcement learning has been in the limelight of public and scientific interest. How we can profit from machine learning in all disciplines of science is a major long-term research topic, one that is highly relevant for society, and also part of the scientific program that won the University of Tübingen again its place in the nobel class of "Excellence Universities".  In this new preprint we show that reinforcement learning can strongly improve the sensitivity of quantum sensors. These are devices with which one can measure e.g. magnetic fields already now extremely precisely - the current world record is below a femto Tesla per square root of Hertz (roughly hundred thousand million times smaller than the magnetic field of Earth at its surface) - and this by using a tiny little gas cell with about 10 billion atoms!  Such sensors are already used for monitoring more precisely heart and brain activities than was possible before.  Recently, we had found that these sensor can be improved substantially by "kicking" them periodically with laser pulses, rendering them chaotic.  Now our analysis shows that even more sensitivity can be gained by adjusting the laser pulses via re-inforcement learning. In addition, the sensor becomes more robust against unavoidable decoherence: the machine learns by itself how to optimally shoot the laser pulses to maximize the sensitivity, and can even adapt to changing environments. We were stunned!

2019/08/09 Quantum channels map the quantum mechanical state of a system to another one. Non-unitary maps arise from interactions with an environment, which leads typically to decoherence and the destruction of quantum mechanical interference effects. Less studied is what happens to the environment in such a case. The map that propagates the system´s initial state to the environment´s final state is called the "complementary channel".  In the case of a quantum communication channel you can think of it as propagating quantum information from the system to an evesdropper.  The loss of quantum information measured by the entropy of the channel is zero if the system evolves unitarily, but can become large in the case of strong interaction.  It turns out that there is a trade-off relation between between the entropies of the channel and the complementary channel: The sum of the entropies of the original channel and its complementary channel is bounded from below.  Hence, if the original system evolves unitarily, its channel entropy is zero, whereas the entropy of the complementary channel is maximal, meaning that an eavesdropper gets no information, and vice versa. We prove the exact lower bound in the case of two qubits, find the channels that saturate it, and conjecture the bound for higher dimensional systems based on extensive numerical evidence. Apart from the mathematical insight into quantum information processing, this work also led to graphs of stunning beauty...

2019/08/02 Non-linear systems are notoriously difficult to solve.  However, there are exceptions.  In this new preprint we describe how we can exactly solve the dynamics of an optomechanical system consisting of a cavity mode and a movable mirror, or any other mechanical oscillator such as a trapped micro-bead, to which the light couples through its light pressure.  The solution is based on realizing that the operators in the Hamiltonian form a closed Lie algebra, which allows one decouple it into several parts. This works in principle even for arbitrary time-dependent parameters, with the caveat that the resulting differential equations that need to be solved may only be solvable numerically and might be unstable. Nevertheless, this method marks real progress compared to previous analyses of such a systems that approximated the dynamics by assuming that Gaussian initial states remain approximately Gaussian.  This is something we can check now via our exact solutions, and we also investigate the interplay between non-Gaussianity and squeezing.  This is a milestone towards rigorously investigating the usefulness of this system for quantum metrology, in particular the measurement of extremely weak forces.

2019/06/12 Have you ever seen a laser that emmits not light, but cold, coherently propagating atoms? Have a look at our new preprint that investigates such matter waves both theoretically and experimentally!  The bose laser works essentially by producing a Bose-Einstein condensate of atoms trapped in a harmonic potential, and then cutting a hole into the trap, such that atoms simply fall out.  While none of this is new, lead author Caroline Arnold developed full 3D simulations and was able to probe the regime of strong outcoupling.  She found that the intensity saturates in that regime -  a purely theoretically finding first, that was then confirmed by experiments here in Tübingen.  

2019/05/15 More often than not, quantum states come with a certain amount of uncertainty about themselves, which means in quantum mechanics jargon that they are "mixed states", rather than pure ones.  This uncertainty reduces on average the nice quantum mechanical effects such as quantum interference that we want to use for enhancing e.g. sensitivity of sensors.  The question is then, how to optimally prepare an initial mixed state through unitary transformations to obtain maximal sensitivity in a subsequent evolution that encodes the parameter to be measured in the  state.  This is an old open problem that we resolved completely in recent work.  Here is how.

2019/2/13 With the increasing size of existing quantum processors, the question of how to test their functionality as efficiently as possible has become a challenge.  In a new preprint, "Optimal measurement strategies for fast entanglement detection" we show that one of the core goodies of quantum information processing, namely quantum entanglement, can be verified much more rapidly than through full state tomography.   The approach is based on our work on truncated moment sequences, which deals very naturally with missing data.  One set of measurements turns out to be particularly efficient.

2018/12/11 Another chapter in our exploration of the gravitational interaction of light with light: Our preprint "Rotation of polarization in the gravitational field of a laser beam - Faraday effect and optical activity" is now finally out.  It turns out that the gravitational field of a circularly polarized laser beam leads to the rotation of the polarization of a probe beam in its vicinity - a gravitational effect that is clearly beyound the scope of Newtonian gravity, but that can be precisely calculated using General Relativity.  The rotation of polarization mixes effects reminiscent of the Faraday effect and optical activity in media which have been known for a long time in optics. In the latter, the effect is undone when the probe-beam propagates back, in the former, it is doubled. Unfortunately, the magnitude of the effect is very very small: of the order of the power of the laser beam divided by the Planck power, where the latter is the Planck energy divided by the Planck time. Optimistic estimates for the current most powerful lasers let us expect rotation angles of order 10^-32 - and that includes already an amplification using an optical cavity.  Clearly not an effect that will be measured in the lab any time soon.  Yet, the effect is of substantial fundamental interest, as it demonstrates gravitational spin-spin coupling in the well-defined and tested frame-work of General Relativity, while the effect is also predicted by certain quantum gravity theory candidates.