The circuit quantum electrodynamical Josephson interferometer
© Jirschik and Hartmann; licensee Springer on behalf of EPJ. 2014
Received: 20 September 2013
Accepted: 5 December 2013
Published: 29 January 2014
Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensated by coherent input drives. Here we investigate a system consisting of three transmission line resonators, where the two outer ones are driven by coherent input sources and the central resonator interacts with a superconducting qubit. Whereas a low excitation number regime of such a device has been considered previously with a numerical integration, we here specifically address the high excitation density regime. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov or linear expansion in quantum fluctuations which can be understood as an approximation for weak nonlinearities. As the second method we introduce a combination of mean-field decoupling for the photon tunneling with an exact approach to a driven Kerr nonlinearity which can be understood as an approximation for low tunneling rates. In contrast to the low excitation regime we find that for high excitation numbers the anti-bunching of output photons from the central cavity does not monotonously disappear as the tunnel coupling between the resonators is increased.
In recent years, a new direction of research in cavity quantum electrodynamics (cavity-QED) has developed, in which multiple cavities that are coupled via the exchange of photons are considered. Such setups are particularly intriguing if the cavities are connected to form regular arrays and if the strong coupling regime is achieved in each cavity of the array. Such devices would give rise to novel structures where coherent light matter interactions exceed dissipative processes simultaneously in multiple locations of the array [1–6].
Whereas it is rather challenging to build mutually resonant high finesse cavities in the optical regime, it is for microwave photons perfectly feasible to engineer large arrays of mutually resonant cavities on one chip in an architecture known as circuit-QED [6, 7]. Here multiple superconducting transmission line resonators with virtually identical lengths in the centimeter range can be coupled via capacitors or inductive links . The individual transmission line resonators can feature strong optical nonlinearities by coupling them to superconducting qubits such as transmons [8, 9] or phase qubits .
Yet in all experiments that involve light-matter interactions, some photons will inevitably be lost from the structure due to imperfect light confinement or emitter relaxation. To compensate for such losses, cavity arrays are thus most naturally studied in a regime where an input drive continuously replaces the dissipated excitations. This mode of operation eventually gives rise to a driven dissipative regime, where the dynamical balance of loading and loss processes leads to the emergence of stationary states [10–13]. The properties of these stationary states may however vary significantly if one changes system parameters such as the photon tunneling rate between cavities, the light-matter interaction in a cavity or even the relative phase between a pair of coherent input drives [12, 14].
A device that is ideally suited for studying the effect of relative phases between input drives is the so called quantum optical Josephson interferometer , which consists of two coherently driven linear cavities connected through a central cavity with a single-photon nonlinearity. Here, the interplay between tunneling and interactions in the steady state of the system has been analyzed for regimes of weak input drives where the number of excitations in each cavity is rather small by Gerace et al. . For opposite phases of the input drives one finds a destructive interference which suppresses population of the central cavity, whereas for input drives in phase the central cavity is populated with anti-bunched excitations due to its strong nonlinearity.
For the considered regime of low excitation numbers the approach  employed a full numerical analysis relying on an excitation number truncation of the Hilbert spaces. For high input drives however, the theoretical description of the system poses a challenge as conventional numerical methods quickly become computationally infeasible. Hence, such a numerical approach cannot describe regimes with more intense input drives where excitation numbers begin to grow and is therefore unable to explore a possible transition from a quantum regime with anti-bunched output photons to a classical regime with uncorrelated output photons.
Here we present approaches that are capable of describing this transition. For a parameter regime with a weak nonlinearity in the central resonator we expand the intra cavity fields to linear order in the quantum fluctuations around their coherent parts. This Bogoliubov-type expansion provides a good approximation provided the density of quantum fluctuations around the coherent background is small compared to the excitation number in the latter. This regime is realized for weak nonlinearities in the central resonator.
The remainder of the paper is organized as follows, in Section 2 we describe the setup and model we consider and in Section 3 we revisit the numerical approach by Gerace et al.  to present results for time resolved correlation functions which so far have not been considered. In the next Section 4 we discuss the approach based on a Bogoliubov expansion and the results it yields. In Section 5 we then introduce our mean-field approach based on an exact solution for the central resonator. Finally Section 6 presents conclusions and a discussion of the parameter regimes covered by each of the discussed descriptions.
the dot denotes a time derivative and γ describes the excitation loss rate of an individual resonator site. Since the excitation losses are continuously compensated by a coherent input drive, a dynamical equilibrium leading to a steady state will be established. We are here interested in properties of this steady state, such as the mean excitation number and its correlation functions, mainly for the central resonator.
where . For vacuum input fields, we thus obtain an out photon flux for the central resonator.
The significance of the physical interpretation is to have a value for the probability to measure a second particle at time after a particle has been measured at time t and compare this to a coherent field where . Therefore, if it is less likely to measure a second particle. In this case we speak of an anti-bunched light field that is necessarily non-classical, whereas for we have bunched light meaning a higher probability than for a coherent field. We note that the -function for the output fields is identical to the -function of intra-cavity fields provided the input fields are in vacuum.
Let us finally stress that although we refer to an implementation in circuit electrodynamics here, our calculations do in most parts not make use of details of this technology and are therefore applicable to other realizations as well. Yet we prefer to discuss them in the context of circuit QED as this technology offers excellent perspectives for realizing a coupling between resonators as we envision it here.
3 The numerical approach in the weak driving regime
In the regime where the laser intensity is rather low, we are able to numerically solve the master equation directly. Here, an entire regime of rather high nonlinearities , which lessen the probability of high intra-cavity excitations, can be covered for arbitrary values of the tunneling rate J. We introduce a cut-off in the excitation number to give a matrix expression for the excitation ladder operators in a Fock basis of bosonic number states. This reduces the infinitely high dimensional Hilbert space to a small number of dimensions and hence allows for the numerical calculation of all operator expectation-values. To corroborate our results we test their convergence as we increase the excitation number cut-off.
For the steady states we consider, as given in equation (6) is independent of the time t and only depends on the time delay τ. In the weak driving regime the results for no delay have already been covered in  and we therefore focus for this regime on the time-resolved second order correlation function for .
With the used approach of directly solving the master-equation numerically in a truncated Hilbert-space one can cover a regime for arbitrary values of J if the nonlinearity fulfills in the weak driving regime of . Nevertheless, this method is unable to show a clear crossover between anti-bunched and uncorrelated excitations because the weak drive and high nonlinearity restrict it to low numbers of excitations, implying anti-bunched statistics. Therefore we will present a method to extend the investigation of this set-up to the strongly driven regime, motivated by the possibility of high intra-resonator excitations.
4 The quantum fluctuations approach in the strong driving regime
where denotes the n th order of quantum fluctuations in the Liouvillian ℒ.
4.1 Steady state coherent background
4.2 Steady state quantum fluctuations
To analyze the validity of the solutions we find, we thus check whether they fulfill this property.
Considering the fact that we neglected the terms of third and fourth order in quantum fluctuations, which contain additional pre-factors of the nonlinearity U, we come to the conclusion that our approach can also be considered an expansion in U. As a consequence, we can deduce the results represent a good approximation in a regime of weak nonlinearities U. Moreover, the approach becomes much better for higher laser-input intensities because the coherent background excitation number in Eq. (11) scales much stronger with the laser input amplitude Ω than the number of quantum fluctuations in Eq. (19).
4.3 Limitations of the Bogoliubov approach
As a conclusion, we can safely assume that our approximation works well in a regime for strong input drives and low nonlinearities for arbitrary values of the tunneling rate. For larger drives, such as the range of validity extends to larger . Thus we are motivated to find access to the regime of larger nonlinearities because, as is evident from Figure 8, the assumption breaks down in that regime, primarily in the second resonator. We present an approach to this parameter regime in the next section.
5 P-Function mean-field approach to the low tunnelling regime
in the case. Our approach can thus be understood as a mean-field decoupling of the three cavities. With the consistency condition (22), the Hamiltonian (21) becomes a single site model which can be solved exactly using a P-function based method introduced by Drummond and Walls .
The result for can be put into Eq. (22) allowing for an expression of the ‘new’ drive . Based on this solution the correlation values for all higher normal-ordered momenta in Eq. (23) can be subsequently derived in a self-consistent manner.
5.1 Comparison to the numerical solution
6 Conclusion and discussion of accessible regimes
In summary we have complemented the discussion of a quantum optical Josephson interferometer with two methods. One is a linearization or Bogoliubov expansion of the intra-cavity fields around their coherent parts. The second method, which we have introduced here, combines an exact solution for the central resonator with a mean-field decoupling of the tunneling terms to the outer resonators.
With the proposed methods the circuit QED Josephson interferometer can be described for a significant range of all experimentally adjustable parameters, such as the input drive Ω, the nonlinearity U and the tunneling strength J for a given dissipation rate γ. If one is faced with low laser intensities and thus low intra-cavity excitations, the solution can be obtained by solving the master-equation numerically via an excitation cut-off in the matrix representation of the ladder operators as has been shown in Section 3. In a strong driving regime with intermediate to high tunneling rates but low nonlinearities a good approximation to the excitation output statistics can be given by a Bogoliubov expansion in the quantum fluctuations around the coherent background parts of the intra-cavity fields, which we presented in Section 4. However, for a strong input drive, low tunneling rates and intermediate to high values of the nonlinearity the Bogoliubov expansion breaks down. Here we derived a method in Section 5, which we termed P-function mean-field approach, that works extremely well in said regime. The only regime for which a good approximation could not be gained was one where the drive is strong enough to produce high intra-cavity excitations, but the tunneling rate and nonlinearity are still relatively high. As a result they are responsible for non-neglectible quantum effects in comparison to the drive or to each other and therefore neither an expansion in U nor in J would be valid. A summary of the validity ranges for each of the presented methods is sketched in Figure 1.
The authors thank Peter Degenfeld-Schonburg for fruitful discussions. This work is part of the Emmy Noether project HA 5593/1-1 and the CRC 631, both funded by the German Research Foundation, DFG.
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