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Introduction

The Theory Division started its activities in December 2001. Since then, we carry out theoretical research in Quantum Information Theory, Quantum Optics and Information, and Quantum Many-Body systems.

Our main goals can be summarized as follows. First, we propose and analyze experiments that aim at observing and discovering interesting quantum phenomena in atomic systems. Second, we investigate how atomic systems can be controlled and manipulated at the quantum level using lasers, and how such systems can be scaled up in a controlled way. Third, we participate in the development of a theory of Quantum Information which will be the basis of the applications in the world of communication and computation once microscopic systems can be completely controlled at the quantum level. Fourth, we apply the ideas and concepts developed in the field of Quantum Optics and Quantum Information to other fields, in particular that of Condensed Matter Physics. Our work is done in strong collaboration with several theoretical and experimental groups inside and outside our Institute, and benefits very much from the visits of many scientists from different institutions around the world.

In the following we summarize the work carried out at the MPQ during the last two years.

Quantum Information Theory

Quantum Mechanics holds the promise to revolutionize the world of information. The existence of superposition quantum states and the non-local character of this theory provide us with new laws to describe, manipulate and transmit information. Equipped with these new rules one can devise novel methods for communication and computation which may allow us to perform certain tasks that are not possible with current systems. In particular, there exist already quantum protocols to achieve secure and efficient communication, as well as to perform calculations which we would never be able to do with classical computers. Thus, there is a very strong effort world-wide both to investigate the applications of Quantum Mechanics in the field of information, as well as to tame the microscopic world to build quantum computers and communication systems. In fact, a theory of Quantum Information is being developed to serve as the theoretical basis for those applications, to describe the quantum phenomena emerging in the experiments, and to help us to overcome the problems in taming the microscopic world.

We are developing the following aspects of Quantum Information Theory.

Quantum Memories

A passive quantum memory is a device in which one can store quantum information (i.e. superpositions) for long times without having to perform active quantum error correction. Ideally, one simply has to “switch on” an appropriate interaction among the qubits storing the quantum information such that the encoded state is protected against external perturbations. In fact, this is what occurs with most classical memories, where errors are self-corrected without the need of using error correcting codes. Kitaev proposed a simple model which might achieve this task using topological properties. However, it is not yet clear if such a device would be as powerful as a fault-tolerant device, where active error correction would be carried out at fixed time intervals, and the storing time would grow exponentially with the number of qubits, N, encoding each logical qubit. We have answered this question: the time a general passive quantum memory can store quantum information in the presence of (local) errors scales, at most, logarithmically with N. Thus, it is much less powerful that an active one. Apart from that, we have shown that, in the presence of Hamiltonian perturbations, Kitaev’s proposal in 2D can only store quantum information for a time which does not grow with N. We have also shown that other proposals of quantum memories in 3D have a similar behavior.

Quantum Computation

Figure 1.- Quantum computation by dissipation: In order to perform an arbitrary quantum computation one just has to engineer the coupling of the systems (grey balls) with local environments (green boxes) appropriately as indicated by the arrows. Once the steady-state is reached, one can perform incomplete measurements on the system which retrieve the desired state with a high probability. The time required to reach the steady state, and obtain the right state after the measurement scales as O(poly(M)), where M is the number of basic gates required to perform the same computation in the standard circuit model.
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Figure 1.- Quantum computation by dissipation: In order to perform an arbitrary quantum computation one just has to engineer the coupling of the systems (grey balls) with local environments (green boxes) appropriately as indicated by the arrows. Once the steady-state is reached, one can perform incomplete measurements on the system which retrieve the desired state with a high probability. The time required to reach the steady state, and obtain the right state after the measurement scales as O(poly(M)), where M is the number of basic gates required to perform the same computation in the standard circuit model.

A crucial issue in quantum computing is to understand how and why a quantum computer is more powerful than a classical one. This is the main research line of Maarten Van den Nest in our group. In order to attack this important question, he has devised a common language which allows him to analyze quantum computations that can be simulated classically. In this way, he has discovered a large set of problems where a classical computer is equally powerful than a classical one. The main idea is to express a general quantum computation as a quantum circuit that, in the end, just gives outcomes of measurements with certain probabilities. Thus, if for a given circuit, one is able to sample classically according to those probabilities, one can simulate the quantum computation. An important issue is the accuracy with which one has to sample the probability distribution. He has shown that it is enough to sample with an error that scales as O(poly(N)), where N is the number of qubits in the corresponding quantum circuit. Apart from that, in collaboration with the group of Hans Briegel in Innsbruck, he has mapped the problem of determining partition functions of classical spin models with the performance of quantum computers. That is, if one were able to determine precisely all those partition functions, one could perform arbitrary quantum computations.

Another line of research in our group is to look for alternative ways of performing quantum computations. For instance, we have shown, in collaboration with Frank Verstraete, that one can do that by simply engineering the coupling of a set of qubits to an environment, so that the outcome of the quantum computation is obtained in steady state, independent of the initial state (see Fig. 1). We have also shown that it is possible to perform arbitrary quantum computations by preparing an initial product state and let it evolve for certain time under the action of a universal Hamiltonian. This Hamiltonian acts on a 1D spin chain, acts on near neighbors only, and is translationally invariant. Thus, this results in a programmable quantum computer where the interaction among the qubits is fixed and one only has to change the “software” (the initial state) to carry out different computations.

Additionally, in collaboration with Frank Verstraete and Ignacio Latorre, we have constructed quantum circuits which are able to efficiently prepare quantum states that naturally appear in many-body quantum problems. An example is the ground or thermal state of an Ising Hamiltonian in a transverse field in one dimension. For particular values of the parameters, the state is critical and thus possesses non-trivial correlations.

Quantum Networks

Figure 2.- Quantum random network: each node (green dot) is composed of several qubits, which are in a non-maximally entangled state with their neighbors as indicated by the dashed lines. An appropriate joint measurement in each node results, with certain probability, in a set of maximally entangled states among certain nodes. If the entanglement of each qubit pairs scales as 1/Nz  with z<2, then in the limit of large N one can obtain, by selecting the measured observable in each node, any desired entangled states between a fixed number of nodes with probability one. This is in contrast to what occurs in classical random networks, which corresponds to individual measurements in each qubit. In that case, depending on the value of z, only certain entanglement patterns can be obtained.
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Figure 2.- Quantum random network: each node (green dot) is composed of several qubits, which are in a non-maximally entangled state with their neighbors as indicated by the dashed lines. An appropriate joint measurement in each node results, with certain probability, in a set of maximally entangled states among certain nodes. If the entanglement of each qubit pairs scales as 1/Nz with z<2, then in the limit of large N one can obtain, by selecting the measured observable in each node, any desired entangled states between a fixed number of nodes with probability one. This is in contrast to what occurs in classical random networks, which corresponds to individual measurements in each qubit. In that case, depending on the value of z, only certain entanglement patterns can be obtained.

Complex networks describe a wide variety of systems in nature and society, as chemical reactions in a cell, the spreading of diseases in populations or communications using the Internet. Their study has traditionally been the territory of graph theory, which initially focused on regular graphs, and was extended to random graphs by the Hungarian mathematicians Paul Erdos and Alfred Renyi in a series of seminal papers in the 1950s and 1960s. With the improvement of computer power and the emergence of large databases, those theoretical models have become increasingly important. In the past few years, they have also given rise to new models that describe properties that seem universal in real networks, as a small-world or a scale-free behavior. One expects that in the near future quantum networks will be developed in order to achieve, for instance, perfectly secure communications. Those networks will be based on the laws of Quantum Mechanics, and thus, will offer us new opportunities and phenomena as compared to their classical counterpart. In fact, in the previous period we showed that quantum phase transitions may occur in the entanglement properties of regular quantum networks and that the use of joint strategies may be beneficial in order to establish the entanglement which is useful, for example, for quantum teleportation between different nodes. In the last year we have introduced a simple model to describe complex quantum networks and show that they exhibit some totally unexpected properties (see Fig. 2). In fact, we obtained a completely different classification of the behavior as compared to what one would expect from a complex classical network. In particular, in a network of N nodes, if the entanglement present between any pairs of nodes scales as 1/N2, any arbitrary quantum state (and therefore correlation subgraph) among a subset of nodes can be generated by local operations and classical communication. This result opens new perspectives in the study of quantum networks and their applications. This work was done in collaboration with Antonio Acin and Maciej Lewenstein.

We have also studied quantum repeaters in the context of quantum networks. We have shown that, by placing the nodes of the repeaters according to a 2D or a 3D lattice, instead of aligning them in 1D, one can establish a secure channel between any two nodes in the presence of arbitrary errors (as long as they do not trespass certain threshold). Whereas in 2D the requirement to achieve this task for arbitrary number of nodes, N, is that the number of qubits per node scale as log(N), in 3D this number is constant, independent of N. This can be understood by noting that in 3D there are many more paths that lead from one node to another one, such that the probability of finding an error-free path is larger. This last result solves a fundamental question for quantum repeaters.

Entanglement Theory

Bell inequalities distinguish the correlations that may appear in quantum systems with respect to those that can appear in classical ones. Certain quantum states violate those inequalities, indicating that they possess certain correlations which do not have classical counterpart. There is a way of quantifying the amount of violation of Bell inequalities. For bipartite systems, this amount is always smaller than certain number (of the order of 2), independent of the dimension of the Hilbert space of the particles in place. For quantum systems, Michael Wolf and collaborators were able to show that, unexpectedly, the amount of violation may be unbounded. The method used to proof such a result is very sophisticated, and utilizes recent developments in the context of Functional Analysis. This, in turn, connects the field of Bell inequalities (and communication complexity, which is related to the first), to a particular branch of that area of mathematics. In fact, very recently, the group of Harry Buhrman has managed to proof an open conjecture in that area using tools developed in the context of quantum communication complexity, something which has attracted the attention in pure mathematicians.

While the notion of pairing in condensed matter physics is widely used, a precise definition of pairing has been missing so far. For example, there exist Fermionic states that despite the fact that they present two-particle correlations, they can be written as mixtures of product states (Slater determinants), and thus cannot be paired. This situation is very reminiscent of the one that occurs in the theory of entanglement, where states possessing correlations are not necessarily entangled (since they can be expressed as mixtures of product states). This motivates of using theoretical tools taken from the theory of entanglement to define, characterize, and quantify the notion of pairing in Fermionic systems. We have done that which has allowed us to give a precise definition of that property, as well as a way of quantifying it. We have also given a physical interpretation as a resource: whenever we have pairing, we can use those states to measure certain quantities with a higher precision than with product states; and the more pairing, the better is the precision.

Theoretical Techniques

Members of our group, led by Michael Wolf, and in collaboration with other scientists, have developed several theoretical tools to describe quantum systems and processes. For example, a new decomposition of Gaussian mixed states of multipartite systems has been developed, which allows one to express them in a standard form. In particular, one can use it in combination with other techniques in order to generalize the de Finneti theorem to that family of states. Such a theorem states that whenever we have a multipartite state of N subsystems which is invariant under permutations, the reduced density operator of k subsystems can be approximated by a separable state, with an error scaling as k/N in the limit of N>>1. The proportionality constant scales like the square of the dimension of the Hilbert space of the systems, and thus for Gaussian states the error would diverge. However, using these new techniques one can derive upper bounds to the errors which do not diverge. These results have lead to an unconditionally secure proof of quantum cryptography with continuous variables, and thus closed one important problem in quantum information. Apart from that, new properties of unital quantum channels have been investigated. Quantum channels can be understood as the most general physical actions (i.e. transform density operators onto density operators). Unital means that the completely depolarized state is mapped into itself. Those channels are a very important class since they encompass a large set of physical actions. An active area of research is to understand the structure of this class of channels. Examples of unital channels are those which can be understood as a random unitary operation (i.e. that may be expressed as convex combinations of unitary operations). It is well know that there exist unital channels that do not have that form. Michael Wolf has found is that, for some channels, the action of two of them can be expressed as random unitary operations, despite this is not possible with each channel alone. This indicates that in the asymptotic limit, when one has many identical channels, the action of all of them can be expressed as random unitary operation. If this was true, it would have important consequences in quantum information theory; in particular it would imply certain properties of the capacity of unital channels.

Quantum Optics and Information

Quantum optical systems can nowadays be manipulated and measured in a very controlled way. In particular, experiments with single or few photons, atoms, and ions allow us to observe quantum phenomena in a very clean environment. Those systems also provide us with the basic elements to build quantum devices that have applications in the field of quantum information. During the last two years we have continued our theoretical investigations on quantum optical systems. Our main goal is to propose and analyze new ways to manipulate the quantum state of those systems. The systems we investigate are: trapped ions, atoms in optical lattices, atomic ensembles interacting with laser light; and electrons and nuclear spins in quantum dots. We have been mainly interested in proposing experiments where one can observe collective quantum phenomena, new phases of matter, or effects that have been predicted in other areas of physics. We have also developed some methods to extract information from experiments, or specific experimental designs.

Analysis of Quantum Phases

Figure 3.- Dynamical creation of a super-solid: One starts with a superlattice where heavy atoms (blue balls) are in a Mott state and light ones (red balls) delocalized. Then, the lattice potential is quenched. The state evolves towards the one in which both species  still possess diagonal long-range order and an off-diagonal quasi-long range order is created.
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Figure 3.- Dynamical creation of a super-solid: One starts with a superlattice where heavy atoms (blue balls) are in a Mott state and light ones (red balls) delocalized. Then, the lattice potential is quenched. The state evolves towards the one in which both species still possess diagonal long-range order and an off-diagonal quasi-long range order is created.

We have studied different situations involving trapped ions or atoms in optical lattices, where novel phases of matter emerge. For trapped ions, we have investigated the phase diagram at zero temperature when they are stored in an optical lattice. In that case, each ion occupies a lattice site which is separated from the other ones by several empty ones. Coulomb interactions may excite phonons in the system, which can be understood as excitations of each ion in the local confining lattice potential. If one considers only two states per lattice sites, the system of ions is described by an XY model in a triangular lattice, where the interactions along different directions can be independently tuned by varying the lattice potential. As one does that, one encounters different phases, including a spin liquid. In collaboration with Jan von Delft and Florian Marquardt we have shown that a very natural situation, namely a set of ions in a linear trap, in which one of the ions is driven by a laser, is described by a very well studied model in condensed matter physics, the so-called spin boson model. By changing the laser intensity one can thus explore the different regimes of that model, and in particular fascinating phenomena like localization. In contrast to the situations studied in the literature, in our case we have a finite set of bosonic modes, which gives rise to recurrences and other interesting phenomena.

Figure 4.- Creation of Pfaffian states using dissipation: For high atomic densities, the three-body decay rate is enhanced, which effectively induces a three-particle repulsive interaction. For a rotating harmonic, by increasing the density and changing the trap parameters, one can adiabatically create a Pfaffian state out of a Laughlin one. The figure displays a contour plot of the energy gap (which determines the time scale of the process) as a function of the trap parameters (δω, the mismatch between the trap and the rotation frequency, and ε, the anisotropy of the trap), as well as the path along which the desired state can be achieved.
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Figure 4.- Creation of Pfaffian states using dissipation: For high atomic densities, the three-body decay rate is enhanced, which effectively induces a three-particle repulsive interaction. For a rotating harmonic, by increasing the density and changing the trap parameters, one can adiabatically create a Pfaffian state out of a Laughlin one. The figure displays a contour plot of the energy gap (which determines the time scale of the process) as a function of the trap parameters (δω, the mismatch between the trap and the rotation frequency, and ε, the anisotropy of the trap), as well as the path along which the desired state can be achieved.



Matteo Rizzi, in collaboration with his former group at Pisa has analyzed the quantum phases appearing with fermionic atoms, or mixtures of fermions and bosons, in an optical lattice at zero temperature. For instance, using DMRG techniques, he has certified that the Fulde-Ferrell-Larking-Oychinnikov phase should be stable in 1D lattices. He has analyzed the possibility of pairing in 1D Bose-Fermi mixtures, where the role of the different masses of the atoms determines the physical properties of the phases. Some members of our group, in collaboration with two former members, Tommaso Roscilde and Juan José García Ripoll have also analyzed the phases appearing in 1D optical lattices in the presence of correlated hopping, disorder, or a quench (see Fig. 3). In this last case, for instance, one can create dynamically a super-solid phase, which does not occur in thermal equilibrium.

We have also continued our investigations in collaboration of Gerhard Rempe and Stephan Dürr of the dissipative Tonks gas. As explained in the previous report, one can induce strong interactions by using dissipation in a system. If two-body loses are very strong, then atoms tend to be separated from each other which can be viewed as an effective interaction among them. In this period we have developed a theory based on Bethe Ansatz to describe this situation in 1D, and performed numerical calculations in order to describe the process of dynamical creation of strong correlations in that system. We have also considered the case in which three-body loses are important, which gives rise to a strong three-body interaction. In the presence of a rotating trap, such interactions result in Pfaffian states, which posses interesting non-abelian anyonic excitations (see Fig. 4). We have studied specific ways of creating those states with few atoms.

Design and Methods

Figure 5.- Mapping the state of nuclear spins in a quantum dot to that of a light field: The nuclear spins are coupled to an electronic spin by hyperfine interaction. The electron is coupled to a trion state |X> with a laser and to the cavity field (red and right arrow in Fig. b). By selecting the laser frequency and detuning, one can effectively map the state of the nuclear spins to the cavity mode and vice versa. The light in the cavity mode then leaves the cavity.
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Figure 5.- Mapping the state of nuclear spins in a quantum dot to that of a light field: The nuclear spins are coupled to an electronic spin by hyperfine interaction. The electron is coupled to a trion state |X> with a laser and to the cavity field (red and right arrow in Fig. b). By selecting the laser frequency and detuning, one can effectively map the state of the nuclear spins to the cavity mode and vice versa. The light in the cavity mode then leaves the cavity.

Roman Schmied, in collaboration with other scientists, has designed surface electrodes to create trap lattices of trapped ions. They can be then used to perform quantum simulations in two spatial dimensions, attending at different geometries. This would us give the possibility of designing different kinds of spins and phonon Hamiltonians, and simulating them, which is one of the most interesting perspectives offered by trapped ions at the moment and which was proposed several years ago in our group. Those traps are currently being built and tested at NIST. The group led by Geza Giedke has proposed a new way of interfacing nuclear spins in a quantum dot with light (see Fig. 5). The idea is to embed the dot in a cavity, and to drive with a laser an electron in the dot. The hyperfine interaction couples the nuclear with the electron spin, which can be transferred to the cavity mode with the help of the laser, then leaving the cavity. This mechanism can be used to: (i) map the state of the nuclear spins into light; (ii) transfer the state of light into the nuclear spins. The latter can be achieved by entangling the nuclear spins with the light leaving the cavity, which can in turn be used to teleport the state of another light mode to the spins.

Figure 6.- Creation of quantum superpositions of dielectric objects: This figure illustrates a protocol for this task. The object is trapped with optical tweezers, and interacts with a cavity mode. First, the center of mass motion of the object is cooled by a laser beam coupled to the cavity. Then, a single photon is sent to the cavity which is partially reflected, and partially converted into an excitation of the motion of the object, thus obtaining the desired superposition.
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Figure 6.- Creation of quantum superpositions of dielectric objects: This figure illustrates a protocol for this task. The object is trapped with optical tweezers, and interacts with a cavity mode. First, the center of mass motion of the object is cooled by a laser beam coupled to the cavity. Then, a single photon is sent to the cavity which is partially reflected, and partially converted into an excitation of the motion of the object, thus obtaining the desired superposition.

We have introduced a novel way of performing quantum simulations based on measurements and feed-back. Together with Peter Zoller, we showed many years ago that if we detect one photon emitted by one out of two atoms sitting at different locations, this creates an entangled state among them. By performing this procedure in a continuous way, one can induce atom-atom interactions. The inherent randomness in the measurement process results in a random interaction between the atoms. However, by a proper (single qubit) feed-back mechanism one can correct for this and implement the desired interaction. We have devised a way of performing this in the presence of photon losses and in a many-atom configuration. In collaboration with Jerry Gabriels, we have proposed a method to entangle electrons in a Penning trap. The main idea is to use the collective cyclotron mode, which possesses a very high frequency and thus can be cooled to its ground state, in order to mediate the interaction between the electrons. We have also proposed a new setup consisting of a dielectric object of around hundreds of nanometers levitated by optical tweezers and interacting with a cavity mode (see Fig. 6). We have analyzed the prospects of cooling its center-of-mass motion to the ground state, and of preparing non-trivial quantum superpositions with it. The main advantage with respect to standard nano-mechanical oscillators is that the first ones are not in thermal contact with a bulk, and thus may possess a much higher quality factor. The dielectric object could be any object with little absorption and a sufficiently high index of refraction. In particular, we have investigated the possibility of performing this kind of experiments with living objects, which would open the door to test the quantum superposition principle with living beings. This work has been done in collaboration with several experimentalists, who are currently trying to implement those ideas.

Observation of Quantum Phenomena

Figure 7.- Superradiance and other collective effects with matter waves: Bosonic atoms in internal level are subjected by a periodic dipole potential, whereas atoms in level b are free. At a given site, one can denote by |0> and |1> the states with zero or one atom. Strong atom-atom interaction inhibits double occupation of a lattice site. A laser beam in Raman configuration couples the two internal states. Thus, transition |1> to |0> is accompanied by the emission of a boson in the continuum. This situation is analog to the one in which many two-level atoms interact with the free electromagnetic field, and thus one can expect similar phenomena: for instance, collective spontaneous emission or superradiance.
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Figure 7.- Superradiance and other collective effects with matter waves: Bosonic atoms in internal level are subjected by a periodic dipole potential, whereas atoms in level b are free. At a given site, one can denote by |0> and |1> the states with zero or one atom. Strong atom-atom interaction inhibits double occupation of a lattice site. A laser beam in Raman configuration couples the two internal states. Thus, transition |1> to |0> is accompanied by the emission of a boson in the continuum. This situation is analog to the one in which many two-level atoms interact with the free electromagnetic field, and thus one can expect similar phenomena: for instance, collective spontaneous emission or superradiance.

We have made and analyzed several proposals to observe, within quantum optical systems, phenomena that have been predicted in other areas of Physics. For instance, we have proposed to observe Hawking radiation in an ion trap setting. Consider a set of ions in a ring trap, circulating around it, and assume that in some region a static potential is present. As the ions reach that region, they will have to climb up the potential: they will move more slowly and they will accumulate. Thus, the speed of the ions will go down, whereas the speed of sound will increase. In stationary conditions there may be a point where those velocities are equal, which defines a horizon for the phonons. In one side of the horizon, phonons can move in both directions. However, in the other side, they can only circulate in the direction the ions move. This situation resembles the one of photons near a black hole, and thus the physics is very similar. In particular, the presence of the horizon generates Hawking radiation, which should be measurable in an experiment. On a different setup, we have proposed to observe the Unruh effect: an accelerated detector should observe thermal radiation in the vacuum field. The idea is to take a Bose-Einstein condensate of atoms and to introduce a particle in it which may change the internal state in the presence of phonons. At zero temperature, i.e. in the absence of phonons, if the particle is held by a dipole trap which is then moved appropriately, it may change its internal state, which gives an indication that, in its reference frame, is seeing phonons, as predicted by the Unruh effect. These works were done in collaboration with Benni Reznik.

We have also been interested in observing phenomena that typically appears in quantum optics but with other systems. For instance, we have proposed to observe the analogous of spontaneous emission and superradiance but with matter waves (see Fig. 7). Assume we have atoms with two internal levels, one trapped by a periodic optical potential and the other one not. We can identify the state with one atom at a given site with one excitation, and the one with no atom at that site with no excitation. Thus, sites act as if they were two-level atoms, which can be in their ground or excited states. The atoms in the other internal state form a continuous of modes, and thus they can be viewed as an electromagnetic field. One can use a laser or microwave field to couple the two internal levels: this transforms an excitation of a site into a boson, and thus its effect is analogous to the interaction of atoms with light. In summary, we have that this set-up is very close to the one in which one has atoms interacting with the electromagnetic field, and thus they should give rise to similar phenomena. We have also investigated a different set up where similar phenomena could be observed. Led by Geza Giedke and in collaboration with Mikhail Lukin and Susanne Yelin, we have studied a quantum dot setup for this purpose. In it, nuclear spins are coupled through the hyperfine interaction to one electronic one, which in turn is driven by a laser exciting a trion state. This is analogous to the situation where one has many atoms coupled to a cavity mode, where collective phenomena are present. Here, the role of the atoms in played by the nuclear spins, and that of the cavity mode by the electron spin. We have predicted that the intensity coming out of the dot should display the typical phenomena associated to superradiance, and studied the possibility of observing phase transitions in the stationary regime.

Quantum Many-Body Systems

Quantum many-body systems appear very naturally in several fields of Physics, like Condensed Matter or High Energy Physics. They also play a predominant role in Quantum Information. During the last years we have started an effort in our group to use some of the theoretical tools developed in the context of Quantum Information Theory, in order to describe many-body quantum systems.

The main problem in describing the states of N particles is that the number of parameters increases exponentially with N. One may try to devise new ways of representing many-particle states so that physical quantities can be efficiently calculated. The rationale behind these representations is that most of the states that appear in Nature at low temperatures are ground states of translationally invariant Hamiltonians with short-range interactions. Thus, they have a very special character and thus, the usual description in which each (product) state is treated on equal footing, may not be appropriate to describe them. Most of our effort has been concentrated in introducing efficient representations of quantum states using techniques borrowed from Quantum Information Theory, proving that such a representation indeed describes all physical states corresponding to systems with short range interactions in thermal equilibrium, and applying the new algorithms to systems for which no other method works efficiently. The states underlying such a description are called projected entangled-pair states (PEPS) and in one spatial dimension they coincide with the well known matrix product states (MPS). In addition, we have also characterized mathematically all those descriptions. We have also written two review papers about this topic. This work has been carried out in close collaboration with two former postdocs of our group, F. Verstraete and D. Perez-Garcia.

Mathematical Description

Figure 8.- Fermionic PEPS: PEPS can be understood as follows: four auxiliary spins (blue balls) in each of the nodes of a lattice are entangled with their nearest neighbors. The state of the physical spins (red balls) is obtained by applying an operator which maps the space of the auxiliary spins to the one of the physical spins. Fermionic PEPS can be understood in the same way if one replaces the auxiliary and physical spins by fermions, and expresses the maps in terms of fermionic creation and annihilation operators.
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Figure 8.- Fermionic PEPS: PEPS can be understood as follows: four auxiliary spins (blue balls) in each of the nodes of a lattice are entangled with their nearest neighbors. The state of the physical spins (red balls) is obtained by applying an operator which maps the space of the auxiliary spins to the one of the physical spins. Fermionic PEPS can be understood in the same way if one replaces the auxiliary and physical spins by fermions, and expresses the maps in terms of fermionic creation and annihilation operators.

In previous reports we introduced a family of states, the PEPS, which efficiently describe spin states in lattices in thermal equilibrium and whenever interactions are short range. In this period we have further characterized those states. For translationally invariant systems, PEPS are characterized by a single tensor, A, with (Z+1) indices, where Z is the coordination number. The state can be obtained by associating such a tensor to the nodes of the lattice, and contracting the indices according to the vertices. Thus, all physical properties of that state are encapsulated in that tensor, and emerge when we contract many of those tensors. The first question we have addressed is how one can extract physical properties from the tensor. In particular, how symmetries in the state, which in turn classify the different phases of the spins, are displayed in the tensor. We have also shown that for each PEPS fulfilling a special property (we called injectivity) one can find a local Hamiltonian for which it is the unique ground state. This last result was known in 1 spatial dimension, but the extension to higher dimensions was not trivial. We have also introduced Gaussian PEPS (for bosonic systems in lattices) and characterized them. Additionally, we have shown how operators, also expressed in terms of tensors, can be exploited to build Hamiltonians using the same language, and further used in algorithms. With all that, together with the existing and new algorithms to determine the PEPS for any given problem, we now have a set of powerful tools to characterize spin systems in lattices.

In the last years, we have extended the family of PEPS to other systems. First, we have introduced the fermionic PEPS, which efficiently describe fermionic systems in lattices. Note that one can always describe Fermionic systems in terms of spins; however, in spatial dimensions higher than one, local interactions or hopping translate into non-local interactions between the spins, where it is not clear that usual PEPS give an efficient description. Fermionic PEPS provide the solution to this problem (see Fig. 8). Furthermore, for one dimensional lattices, together with Germán Sierra, we have introduced the infinite MPS, where two of the indices of the tensor have infinite dimensions. Those states can be related to a conformal field theory acting as ancillas, and can describe critical systems; in particular, a power law decay of correlations, something which finite MPS cannot do. We have also defined continuous MPS, which no longer describe systems in lattices but in the continuum. We have used this formulation to approximate the ground state of the Lieb-Liniger model, obtaining very satisfactory result, which indicates that this description may prove very useful in Quantum Field Theory.

Algorithms

Figure 9.- Time-evolution algorithm: A matrix product state (represented by the blue dots in the bottom) evolves according to some Hamiltonian (re-expressed as a matrix product operator, and represented by the brown boxes). The resulting tensor represents the ket |Φ(t)>. In order to determine the expectation value of an observable (represented by the green dot), one has to sandwich it between the bra and the ket, resulting in a tensor contraction. We perform the contraction along the space direction, i.e. horizontally. In order to perform the contraction more efficiently, the tensor is folded as shown in the figure, and the tensors corresponding to the bras and the kets are considered as a single tensor.
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Figure 9.- Time-evolution algorithm: A matrix product state (represented by the blue dots in the bottom) evolves according to some Hamiltonian (re-expressed as a matrix product operator, and represented by the brown boxes). The resulting tensor represents the ket |Φ(t)>. In order to determine the expectation value of an observable (represented by the green dot), one has to sandwich it between the bra and the ket, resulting in a tensor contraction. We perform the contraction along the space direction, i.e. horizontally. In order to perform the contraction more efficiently, the tensor is folded as shown in the figure, and the tensors corresponding to the bras and the kets are considered as a single tensor.

We have introduced new algorithms to describe many-body quantum systems based on MPS and PEPS. Together with the group of Guifré Vidal, we have put forward a new method to simulate infinite-size quantum lattice systems in two spatial dimensions using PEPS. This algorithm is now widely used by other groups and has, for instance, allowed studying the phase diagram of the t-J model in two dimensions. Members of our group have also introduced a method to calculate the time evolution using another family of states, the multi-scale entanglement renormalization ansatz (MERA). Additionally, in collaboration with Rosario Fazzio and his group, they have devised a novel and precise way of determining the critical exponents of 1D theories using this ansatz. Furthermore, together with Matt Hastings, under the leadership of Mari Carmen Bañuls, we have presented a new algorithm to determine the time evolution of an infinite spin chain (see Fig.9). The most remarkable property of this new method is that it allows us to extend the simulations of spin systems to longer times, and thus to study physical phenomena which could not be studied before with previous methods. In particular, we are currently studying the thermalization of non-integrable systems.

We have combined the power of all the tensor methods with that of Monte Carlo simulation. First of all, in collaboration with Massimo Bonisegni, we have introduced a new numerical algorithm to describe 2D lattice systems based on a set of variational wave functions that we called entangled plaquette states. With this method we have studied the J1-J2-J3 frustrated model, obtaining the most precise results to date. We have also performed calculations in 3D lattices using the method introduced in the previous report period, and which was based on string bond states. In particular, we have been able to capture the phase transition in a transverse Ising and in a frustrated model.

We have also used techniques coming from quantum information theory to assess the computational complexity of solving certain many-body problems. Norbert Schuch, in collaboration with Frank Verstraete, has shown that as long as the complexity classes QMA and P are different (which is the standard assumption in computer science), solving problems of interacting electrons in molecular systems is terribly hard (i.e. the computational time scales exponentially with the number of electrons), something which give a fundamental limitation to the density functional theory. We have also shown that there exist problems for which an efficient description in terms of MPS exists, but that cannot be found in practice unless the above conjecture is incorrect. On the positive side, we have proven that mean field theory in 1D lattices can be obtained efficiently, as well as any approximation in terms of MPS with a fixed bond dimension. This last result, together with previous results for the description of gapped systems in terms of MPS, establishes the problem of finding the ground state of a gapped 1D Hamiltonian with short range interactions can be efficiently solved with a classical computer.

Topological Phases

In our group we have also been interested in the description of topological phases of matter in 2D in terms of PEPS and MERA. Under the leadership of Miguel Aguado, several results have been obtained. First of all, a connection between two (apparently) different classes of topological models (Kitaev’s double and Levin-Wen string-nets) has been established. Furthermore, a tensor network representation for the ground state of those models has been explicitly constructed. All this shows that those topological phases can be exactly (and not only approximately) described in terms of PEPS. The scaling of the area law for such systems at finite temperature has been obtained, showing that the topological correction vanishes at any finite temperature in 2D. Studies of thermal states of anionic systems have been also carried out. From the point of view of implementations, a proposal using multiphoton states in order to observe interferences of non-abelian anyons has been put forward, and a method to create, braid, fuse and manipulate non-abelian anyons in optical lattices has been presented.