Degree of Quantumness in Quantum Synchronization
We introduce the concept of degree of quantumness in quantum synchronization, a measure of the quantum nature of synchronization in quantum systems. Following techniques from quantum information, we propose the number of non-commuting observables that synchronize as a measure of quantumness. This figure of merit is compatible with already existing synchronization measurements, and it captures different physical properties. We illustrate it in a quantum system consisting of two weakly interacting cavity-qubit systems, which are coupled via the exchange of bosonic excitations between the cavities. Moreover, we study the synchronization of the expectation values of the Pauli operators and we propose a feasible superconducting circuit setup. Finally, we discuss the degree of quantumness in the synchronization between two quantum van der Pol oscillators.
Synchronization is a process in which two or more self-sustained oscillators evolve to swing in unison. The original intrinsic frequencies are modified by the interaction between the oscillators, and a common effective frequency is observed Pikovsky2003 (); luo (). It is a rich phenomenon manifested in a variety of disciplines that was typically studied in classical settings. Examples beyond pendula include heart beats Cai1993 (), neural networks Rosin2013 () and beating of flagella Goldstein2009 ().
During the last decades, a significant progress has been achieved in quantum technologies, which has allowed the search of synchronizing behaviors in quantum platforms Goychuk2006 (); Zhirov2006 (). Since then, the most studied case consists of chains of quantum harmonic oscillators with driving fields, dissipative mechanics, and nonlinearities Lee2013 (); Manzano2013 (); Walter2014 (); Lee2014 (). These models can be straightforwardly compared with the classical case, which corresponds to the presence of many quanta in the model. Quantum mechanics, on the other hand, introduces two main effects, namely, quantum noise and quantum correlations Giorgi2012 (). Quantum correlations have been reported to be strong in quantum synchronization Hush2015 () and, indeed, a synchronization between micromasers stronger than expected by semiclassical models has been recently discussed Davis2016 (). Additionally, measures to quantify the synchronization of continuous variable quantum systems, such as two coupled optomechanical systems, have also been proposed Mari2013 ().
The study of quantum synchronization has also been extended to quantum systems without classical analogue, e.g. two-level systems Zhirov2008 (); Zhirov2009 (); Giorgi2013 (). The lack of a classical counterpart makes the definition of synchronization non-trivial, and it has been addressed by studying periodically oscillating observables. This approach has been further validated by measures of quantum correlations, such as quantum mutual information Ameri2015 (), and the first practical applications in qubits has been recently presented giorgi2016 (); bellomo2017 (). However, some of the aforementioned results might be considered as classical synchronization processes in quantum setups, as we will explain below. Moreover, we will show that synchronization can occur even when there are no quantum correlations between the synchronized parts in the steady state Ameri2015 (), which has been also recently noted in Ref. zam2016 (). This rises the question about the quantumness of quantum synchronization processes.
In this Article, we address the problem of how quantum a quantum synchronization process is from the point of view of quantum information. To this aim, we introduce the concept of degree of quantumness of quantum synchronization. Afterwards, we illustrate it in two composed cavity-qubit systems, showing that one can tune internal parameters to achieve all possible degrees of quantumness between the qubits. Then, to exemplify it, we propose a feasible circuit quantum electrodynamics (cQED) setup. Finally, we discuss the extension of the concept of degree of quantumness to infinite-dimensional quantum systems, as is the case of quantum van der Pol oscillators. It is worth stressing that we are not interested in developing another method to quantify how much two observables of two different systems are synchronized. Our goal here is the identification and quantification of the true quantum nature of a synchronization process between two quantum systems. For the sake of clarity, hereafter, when we assert that two observables are synchronized, we are considering the general case in which their dynamics converge to periodic oscillations with the same frequency. In this manner, these processes can be classified in any type of synchronization, e.g, in-phase, anti-phase or complete synchronization, and so on Pikovsky2003 (); luo (). To characterize and to quantify the type of synchronization between two observables, one can make use of the already known measures of synchronization Pikovsky2003 (); luo (); zam2016 ().
A natural language to deal with synchronization is information theory, since the parties share out information during this process. In this sense, the mutual information was proposed as an order parameter for signaling the presence or absence of quantum synchronization Ameri2015 (). However, this quantifier is not sufficient to answer the question of how quantum this process actually is and, indeed, one could straightforwardly engineer quantum dynamics in which only one observable is synchronized. However, from the point of view of quantum information, this synchronization may be considered classical, since there exists an equivalent classical dynamics describing the same synchronization process. This approach has already been followed in the context of partial cloning of quantum information Ferraro2005 (); Lindblad1999 (); Sanz2010 () or bio-inspired quantum processes Alvarez-Rodriguez2014 (), and could be useful to quantify the quantumness of quantum operations with respect to the environment Meznaric2013 (). Along these lines, we extend this idea to quantum synchronization, constructing a quantifier of the quantumness of the process.
We will formally define the concept of degree of quantumness of quantum synchronization for a bipartite system, since the extension to multipartite cases is straightforward. Let us consider a bipartite quantum system , which can also be coupled to a complex environment. Let be a set of all linearly independent operators which simultaneously synchronize in both subsystems, with , that we call cardinality of quantum synchronization, and the sets and . Then, the degree of quantumness of quantum synchronization is given by . Notice that and that is a set but not a vector space since, if , it does not necessarily mean that . The reason is that even though and synchronize, they could do it with different frequencies and phases, so that linear combinations do not synchronize in principle. Therefore, linear independence removes the redundancies when more than one operator synchronize with the same frequency.
Let us remark that, if the degree of quantumness of a given synchronization process is , i.e. every operator synchronized in both subsystems can be diagonalized in the same basis, then this is just classical synchronization from the point of view of information theory. The reason is that an equivalent classical dynamics synchronizing for the same operators can be constructed Sanz2010 (). On the other hand, if the degree of quantumness is maximum, i.e. , then all non-commuting observables are synchronized, which in the case of synchronization with the same frequency, phase and amplitude is equivalent to the synchronization of the reduced density matrices. From the point of view of quantum information, we call this a complete quantum synchronization.
Continuous-variable systems, i.e. infinite dimensional quantum systems, deserves an special mention. In this case, it is obvious that the number of linearly-independent Hermitian operators which are necessary to retrieve the information about the density matrix is infinite. For instance, a harmonic oscillator can be described through all its moments and , which are linearly independent barnett (). Indeed, in case of a total quantum synchronization, the cardinality is , which opens several theoretical challenges. A prototypical model is a network of van der Pol (vdP) oscillators. The quantum version of this model has recently attracted much attention, and several proposals for engineering it in the oscillating dynamics of trapped ions or nanomechanical oscillators has been put forward Lee2013 (); Walter2014 (); Lee2014 (). We will study below an example of degree of quantumness with quantum vdP oscillators.
We illustrate now our definition in a setup consisting of two coupled cavities with a qubit in each of them, and investigate the synchronization between the two-level systems, in which the natural observables are Pauli operators. In our work, we are interested not only in one of the components of the spin operator Giorgi2013 (); Ameri2015 (), but in all of them. In this case, the maximum degree of quantumness happens when the expectation values of the three Pauli operators, , and , are synchronized with their counterparts, , and , respectively. The two cavity-qubit systems, depicted in Fig. 1, are coupled through a hopping term in the degrees of freedom of the cavities. A weak driving field on one of the two-level systems is counterbalanced by the dissipation in both cavities with a decay rate . The dynamics of such systems can be described by the master equation
where the Hamiltonian is given by
in which () stands for the annihilation (creation) operator of the cavity modes, while () stands for the lowering (rising) operator of the qubits ( is the ground state while is the excited state). Here, is the frequency of the cavity modes, is the frequency of the qubits, is the cavity-qubit coupling, is the hopping strength between the cavity modes, and and are the amplitude and the frequency of the driving field, respectively (see Fig. 1). The driving field acts only on one of the qubits, and supplies energy into the system to compensate the losses due to dissipation.
We can distinguish two parts in the system, namely, the master system, corresponding to the driven qubit and its respective cavity, and the slave system, which is the other cavity-qubit subsystem. It is useful to rewrite Eq. (Degree of Quantumness in Quantum Synchronization) in terms of the creation and annihilation operators of the normal modes of the coupled cavity-cavity system by diagonalizing the quadratic form (considering ),
The diagonalization leads to the new creation and annihilation operators and . We also define the collective spin operators and . This yields
Considering the case of equal qubit frequencies, we set the frequency of the external driving field quasi-resonant to the atomic transition and to the high-frequency normal mode (). In the interaction picture with respect to , we have
If we assume , then the interaction part regarding the low-frequency normal mode in Eq. (Degree of Quantumness in Quantum Synchronization) oscillates very rapidly with respect to the other parts, so it is negligible due to the rotating-wave approximation. Also, assuming the bad-cavity limit (), the field variables can be adiabatically eliminated zoller (), so that the reduced dynamics for the qubits, in the interaction picture, is given by
The approximation allows us to identify that the high-frequency normal mode effectively act as a common reservoir that couples the qubits. Such effective coupling will be responsible for the onset of synchronization. If the qubits have small detunings, , we can replace the Hamiltonian in Eq. (7) by .
A numerical simulation of the evolution of the expectation values of Pauli operators according to Eq. (1) for three sets of parameters is depicted in Fig. 2, as well as the mutual information mutual () between the qubits. In Fig. 2(a), we observe that the external driving field induces a total quantum synchronization of the qubits (maximum degree of quantumness ), since all Pauli operators get almost immediately synchronized, even in amplitude. If the driving field is off, see Fig. 2(b), the qubits spontaneously synchronize. However, it is a partial quantum synchronization since just and get synchronized (), while for both qubits exhibits a pure exponential decay. In Fig. 2(c), we consider a case in which the external driving field can only induce (anti-)synchronization in , i.e., a classical synchronization (). In Ref. Ameri2015 (), the authors propose the use of quantum mutual information as a steady-state order parameter for signaling the presence or the absence of quantum synchronization, claiming that it is well defined for every bipartite quantum state and does not depend on the particular details of the system. They suggest that synchronized systems should converge to a steady state having large mutual information. The intuition behind this proposal is that, under a quantum dynamics, quantum correlations emerge tend to emerge. Surprisingly, every case shown in Fig. 2 exhibits synchronization without any kind of correlations between the qubits in the steady state. Therefore, as already noticed for entanglement Zhirov2009 (); Lee2014 (); Manzano2013 (); Mari2013 (); Ameri2015 (), there is not a one-to-one correspondence between correlations in the steady state and synchronization, and this relation strongly depends on the specific details of the system. From our point of view, quantum mutual information is a signal of quantum synchronization, but the opposite is not true, and a quantum dynamics can yield quantum synchronization without generating a high mutual information.
Distinct resonator frequencies can be created by means of different resonator lengths, and the interaction between different cavity modes, known as mode mixing, occurs via tunneling of photons. The corresponding hopping term which connects both resonators in the Hamiltonian, can be implemented using a superconducting quantum interference device (SQUID) made of a superconducting loop interrupted by two Josephson junctions (JJs), provided that Felicetti2014 (). Each of the resonators, on the other hand, contains a superconducting qubit, which can be coupled to cavities well beyond the value that we require, as it has been reported very recently by Bosman et al. Bosman2016 (). Additionally, they can reach coherence times as high as s Koch2007 (); Paik2011 (); Rigetti2012 (), while the plotted amount of time in Fig. 2, on the other hand, corresponds to the order of s, which means that the observation of hundreds of oscillations is available. Finally the driving field on one of the qubits is implemented via a coherent microwave source Mallet2009 ().
We finally focus on a model made of two coupled quantum vdP oscillators. We introduce an in-phase synchronizing Hamiltonian
The master equation is given by Lee2013 ()
where are energy insertion parameters, and the ones corresponding to quadratic dissipation.
The lowest moments for this model are depicted in Fig. 3, showing a transient oscillatory behavior in which synchronization is observed in all of them, except for . The quantum synchronization figure of merit , introduced by A. Mari et al. Mari2013 (), is also plotted, where and , with and the canonical variables of each oscillator. These simulations suggest that almost full degree of quantum synchronization is attained. However, the existence of infinite moments gives rise to the theoretical challenge of proving that higher moments are actually synchronized, since the numerical approach is limited. This, therefore, means that there are still open questions in the case of continuous variables which should be addressed in future research.
Conclusions.— We have proposed a quantifier of the quantumness of a quantum synchronization process based on quantum information techniques. Indeed, we define the degree of quantumness in terms of the number of synchronized non-commuting observables. This approach is different to the previous works, since we are not proposing another measure of synchronization, but of the quantumness of the generated quantum synchronization. We study in detail the case of finite-dimensional systems, illustrating it with two cavity-qubit systems, in which we show that all possible degrees of quantumness may be reached. Additionally, we show that this setup is feasible in superconducting circuits with current technology. Finally, we analyze the case for continuous variables, illustrating it with two quantum van der Pol oscillators, where we show that there are still open questions which should be addressed in the future.
We thank L. García-Álvarez, I. L. Egusquiza, R. Zambrini and F. Deppe for fruitful discussions. This work was supported by São Paulo Research Foundation (FAPESP) Grants No. 2013/23512-7 and 2014/24576-1, Spanish MINECO/FEDER FIS2015-69983-P, and Basque Government IT986-16.
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