## Kicking an interferometer

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, 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.

## Quantum-enhanced remote sensing

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.

## Emulation of quantum algorithms on a classical computer

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...

## Can one measure the expansion of the universe in the lab?

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.

## Nadia Milazzo successfully defended her Ph.D.

2021/07/30 Another Ph.D. defense under Corona conditions, this time in hybrid format. Well done, Nadia! Congratulations!

## How to measure causal influence in quantum mechanics?

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.

## Can you weigh a laser beam?

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.

## Majorana representation of symmetric multiqudit states

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.

## Completely synchronizing quantum channels

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!

## Optimal estimation of time-dependent gravitational fields with quantum optomechanical systems

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.

## Entangling - or not?

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!

## New research project approved

2020/06/05 Our new research project "Experimental demonstration of quantum chaotic sensors" together with Prof. Fortagh and Dr. Günther was approved for funding by the Federal Ministery of Economy and Energy.

## Lukas Fiderer successfully defends his PhD thesis

2020/05/13 Corona obliging, we had an online viva, but Lukas defended his PhD thesis with flying colors! Congratulations!!

## New research project approved

2020/03/27 Our new research project "Towards local measurements of space-time geometries with optical clocks" was approved for funding by the DFG. A post-doc position is available, see here!

## Quantum router of light

2020/03/09 We have a new preprint out on a quantum-router of light. It allows one to deflect weak light pulses with a single photon (or a coherent state) into an arbitrary direction in the plane that can be set fast and purely electronically, i.e. without using any mechanical elements such as moving mirrors. It works by generalizing existing quantum memories of light based on absorption in and re-emission from atomic clouds in the sense that after the storage process, when the light has been transformed into a spin wave in the atomic ensemble, a magnetic field gradient is applied that changes the phases of the components of the quantum state in Hilbert space. This leads effectively to a deflection of the spin-wave, and when it is transformed back into a pulse of light by switching back on a control-beam, the photon is re-emitted into a new direction in a controled way. This is a nice new example that shows how phases in Hilbert space control the dynamics of quantum objects in real space. After optimization the energy efficiency is comparable to the one in forward direction.

## Neural-Network Heuristics for Adaptive Bayesian Quantum Estimation

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.

## Switching the condensate magnetization in a dipolar spinor gas

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.

## Quantum metrology with opto-mechanical systems

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!

## Majorana representation for hermitian operators

2019/10/17 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...

## More sensitive quantum sensors with reinforcement learning

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!

## Trade-off relations for operation entropy of complementary quantum channels

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...

## Time-evolution of nonlinear optomechanical systems

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.

## Outcoupling from a Bose-Einstein condensate in the strong-field limit

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.

## Quantum parameter-estimation of frequency and damping of a harmonic-oscillator

2019/05/20 The harmonic oscillator is one of the most important systems in all of physics, be it classical or quantum. Its importance arises from the fact that a.) for small amplitudes most stable systems can be approximated by a set of harmonic oscillators, and b.) that it can be solved exactly. Moreover, it has only two free parameters: its frequency and its damping. How precisely can one determine these in principle? While this question was answered many years ago for an undamped oscillator in this paper, in reality there is always some damping, and taking it into account can change the result a lot. In a new preprint, we solve the problem for Gaussian states, which are a broad class of experimentally relevant states. We predict that with existing carbon nanotube resonators it should be possible to achieve a mass sensitivity of the order of an electron mass per square root of Hz, orders of magnitude lower than the current record, due to the shift of frequency from an adsorbed mass.

## Maximal sensitivity for mixed states

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.

## Fast entanglement detection

13/2/2019 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.

## Farday effect and optical activity in the gravitational field of a laser beam

11/12/2018 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.