Critical behavior in ultra-strong-coupled oscillators
We investigate the strong coupling regime of a linear - coupled harmonic oscillator system, by performing a direct diagonalization of the hamiltonian. It is shown that the - coupled hamiltonian can be equivalently described by a Mach-Zehnder-type interferometer with a quadratic unitary operation in each of its arms. We show a sharp transition of the unitary operation from an elliptical phase rotator to an elliptical squeezer as the coupling gets stronger, which leads to the continuous generation of entanglement, even for a significantly thermal state, in the ultra-strong coupled regime. It is also shown that this critical regime cannot be achieved by a classical Hookian coupling. Finally, the effect of a finite-temperature environment is analyzed, showing that entanglement can still be generated from a thermal state in the ultra-strong coupled regime, but is destroyed rapidly.
pacs:42.50Ct, 42.50.St, 03.67.Bg
Current address: ]LPQM, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
Strong coupling, which is one of the key ingredients for the manipulation and control of quantum systems, makes quantum-mechanical predictions distinct from classical ones. In fact, the definition of strong coupling in quantum mechanics has evolved over the years. In atomic and optical physics, strong coupling was defined to show Autler-Townes splitting Autler and Townes (1955) where atom-field interaction is stronger than the atomic decay rate. The next age of strong coupling was defined in cavity quantum electrodynamics (QED), where an atom-field interaction strong in comparison to the cavity and atomic decay rates can be realized. A series of important experiments have been performed to show the nonclassical nature of the atom-field interaction for Rydberg atoms in microwave cavities Raimond et al. (2001); Haroche and Raimond (2006). This regime was recently also attained with a cavity optomechanical system Verhagen et al. (2012). But with a system consisting of superconducting Josephson junctions coupled to microwave fields in stripline resonators, the interaction strength can be made even stronger than the atom-field coupling realized previously in cavity QED Wallraff et al. (2004); Niemczyk et al. (2010). These so-called circuit QED systems can reach coupling strengths comparable to the transition frequency of the relevant qubit system. In these systems, we find new phenomena such as the breakdown of the rotating-wave approximation and the emergence of a unique deep strong coupling regime Casanova et al. (2010).
A system of coupled oscillators has been of interest in various contexts, as several physical systems are represented by harmonic oscillators such as nanomechanical oscillators, electromagnetic fields and a bunch of two-level systems. Hopfield studied the quantization of an electromagnetic field in a dispersive medium using a coupled harmonic oscillator model Hopfield (1958). Recently, such a system of coupled oscillators has been considered as a possible mechanism to generate entanglement in an array of nanoelectromechanical devices Eisert et al. (2004); Audenaert et al. (2002); Galve et al. (2010). In this paper, we study ultra-strong coupling between harmonic oscillators where the coupling rate is comparable to or larger than the natural frequencies of the harmonic oscillators. Strongly coupled oscillators, where the counter-rotating terms are not negligible have been studied in the context of twin-photon generation Ciuti et al. (2005). The study of such systems are supported by recent experimental progress wherein such a regime is becoming accessible, for instance in the interaction of multiple quantum well structures with terahertz electromagnetic fields Günter et al. (2009). In this paper, we go even further than the strong coupling regime, and find a critical point in the coupling strength at which the evolution of the oscillator states radically changes. An equivalent regime was studied for the interaction between an atom and a field Casanova et al. (2010); Bina et al. (2012), where photon number wavepackets were found to experience collapse and revivals across parity chains in Hilbert space. The model that we consider here, is quite different in the sense that we have a bosonic system living in an infinite dimensional Hilbert space, so that there is no such parity-defined dynamics. However, the model is exactly solvable and we find critical effects from the dynamics of the coupled oscillators, which get manifested in their entanglement behaviour.
We consider a system of two bosonic oscillators interacting via a quadratic interaction, whose hamiltonian is described by
where the dimensionless quadrature operators satisfy the canonical commutation relation, i.e. . For instance, the Dicke model of superradiance Dicke (1968) is approximately described by such a hamiltonian. In the context of cavity optomechanical systems, such a linearized hamiltonian describes the dynamics when the mechanics is being driven by a strong coherent field Mancini and Tombesi (1994). In fact, this linearized model has been investigated previously in the optomechanics context Akram et al. (2010) in the rotating wave approximation (RWA), which is possible when the system is relatively weakly coupled.
In this paper, we will show that in the ultra-strong coupling regime, the non-RWA terms turn out to be not only important but also to critically change the oscillator interaction. After identifying a unitary transformation exactly diagonalizing the hamiltonian in (1), we show its equivalence to a Mach-Zehnder-type setup with unitary operations in its arms, which explains the generation of entanglement in the model. At the critical point when the coupling becomes ultra-strong, we observe the transition of the unitary operation from an elliptic rotator to a squeezer. This transition is clearly reflected in the entanglement dynamics of the oscillators. It is interesting to note that the classical Hookian model for classical harmonic oscillators is not able to reach this critical point.
Ii Diagonalization of the hamiltonian
One can codify the operator hamiltonian (1) in terms of a hamiltonian matrix,
expressed in the basis defined by the column vector of quadratures , so that where we also define the diagonal block matrices corresponding to positions and momenta.
The usual diagonalization of such a quadratic hamiltonian proceeds by defining normal mode operators () and their hermitian conjugates, which are linear combinations of the bare quadratures satisfying . Specifically, the normal mode energies are found to be , from which we already observe an interesting regime where is imaginary, for .
The transformation to normal modes is just a symplectomorphism McDuff and Salamon (1998). The hamiltonian matrix can be diagonalized by a symplectic matrix such that it is block diagonal of the form . The diagonal hamiltonian matrix is then . From the general conditions to be satisfied by symplectic transformations, we get that for , . The unitary representation of in the Hilbert space of the system is,
parameterized by real constants and . This diagonalizes to iff.
The diagonal hamiltonian is then given by,
where we denote and . Each normal mode is described by a quadratic hamiltonian, which looks very similar to a standard free hamiltonian for an oscillator. But using the condition (4), it is found that the last term in (5) is zero (negative) when (), so that these are standard harmonic oscillators only when . The first (second) mode in (5) is associated with (), which we call the ‘’ (‘’) mode. When , the ‘’ mode does not have a bound spectrum, but is rather a free particle, while the ‘’ mode is still a harmonic oscillator. Increasing the coupling strength further, for , the anomalous ‘’ mode is dynamically unstable since it is driven by a force derived from the inverted harmonic potential. So, the earlier observation of going imaginary is reflected by this qualitative change in the dynamical behaviour of the system, as the coupling strength crosses its critical value .
Iii Inequivalence to classical coupled oscillators
An obvious intuition one would have from thinking about the classical oscillating systems is that the model (1) must be the quantized version of the classical Hookian hamiltonian for coupled oscillators,
where is the common mass of the particle, the common natural frequency and the Hookian coupling rate. The oscillators are assumed to be identical for simplicity (a generalization is straightforward). Introducing dimensionless quadratures, with the renormalized frequency , the hamiltonian becomes,
where is the renormalized coupling strength. So, it is clear that in the quantum version of the classical Hookian problem, the ultra strong coupling regime (here, ) is not possible to achieve.
The ultra strong coupling transition of the form exhibited by the hamiltonian (1) seems to be a property of such quantum coupling models, and not something that is seen easily in the standard harmonic oscillator model. In particular though, for optomechanical problems, the linearization of the actual radiation pressure interaction (which leads to ) is valid only for a short interaction time. Otherwise, the displacement of the mechanics gets too large and the approximation leading to the interaction hamiltonian breaks down in the strong coupling regime Law (1995); Pace et al. (1993); Vanner et al. (2011).
Iv Entanglement dynamics
In the resonant case, i.e. and so , the transformation operator in Eq.(3) becomes exactly the same as the 50:50 beam splitter operator. Thus the evolution operator corresponding to the hamiltonian (1) can be factorized into , where . It is now obvious that in the resonant case, the dynamics is equivalently represented by a Mach-Zehnder interferometer, with 50:50 beam-splitters as shown in Fig. 1; but with some active operations in both arms Paternostro et al. (2005).
To determine the kind of active operations, consider the individual evolutions generated by which are of the quadratic form, (). The evolution of the quadrature operators induced by such a hamiltonian is
The transformation shows the behavior of an elliptical rotator where the product of the quadrature scaling factors is unity. It preserves the quadrature uncertainties, leading to at most, squeezing of the mode. Importantly, if either or is negative, the elliptical rotator becomes an elliptical squeezer – this is precisely the situation for the ‘’ mode when the coupling is ultra strong, i.e. . It is also worth noting that in the general non-resonant case, both the beam-splitting and two-mode squeezing terms appear in , meaning that instead of the passive beam-splitters at the entry and exit ports, one will have another active device in the equivalent picture.
From the equivalent Mach-Zehnder picture, the source of entanglement is obvious – squeezing in the input fields is a necessary condition for Gaussian entangled output from a beam splitter Kim et al. (2002a). The interaction leads to significantly more entanglement when , than the case where . To explicitly study dynamic entanglement, we consider each harmonic oscillator initially in its thermal equilibrium so that the total density operator , where with the canonical partition function and mean excitation number . at temperature with the Boltzmann constant .
Fig. 2(a) shows the dynamic entanglement, measured by the logarithmic negativity Vidal and Werner (2002); *LeeKim00; Plenio (2005), for a thermal state with a single average excitation (see the appendix A for explicit formulas of the log-negativity). We observe that for , the entanglement has an oscillatory behaviour, with typical periods related to the frequencies of the normal modes. When , the entanglement monotonically increases, disregarding the small oscillations due to the ‘’ normal mode. The same behaviour is observed for larger values of the coupling constant () in Fig. 2(b), where we plot the “seralian” Serafini et al. (2004) which, in the unitary case, is a sufficient entanglement monotone (see Appendix A for details). The entanglement keeps increasing with time (neglecting the small oscillations), and in particular at a fixed time, the larger is the value of the coupling constant , the larger is the entanglement achieved. This particular behaviour can be explained also by looking at the eigenvalues of the Hamiltonian matrix . For , the matrix is positive definite and, as proved in Genoni et al. (2012), the corresponding unitary operator will recur to the identity for a given time; for larger values of , the matrix is not positive definite anymore, the recurrence property of the evolution is lost and the entanglement increases with time.
Even when the initial state is significantly thermal, if the system is critically coupled, entanglement can be generated, as depicted in Fig. 3. In fact, we also observe that the purity of one oscillator is an important factor in the generation of entanglement. When one oscillator is pure, the maximum degree of entanglement achieved remains about the same regardless of the temperature of the other oscillator. The importance of single-system purity was also observed in some other interaction models Bose et al. (2001); Kim et al. (2002b)
iv.1 Dissipative dynamics
Let us consider now the case where the system interacts with a noisy environment. Because of the environment interaction we observe that also super-critical regime can stop to be dynamically unstable. Assuming that each bare oscillator interacts with its respective thermal environment with mean excitation , under the Born-Markov approximation, the evolution of the system density matrix is governed by the Kossakowski-Lindblad master equation,
The logarithmic negativity can be evaluated, and the results are plotted in Fig. 4 for various values of the coupling strenght. We observe that, as expected, also in the super-critical regime , the entanglement reaches a maximum value and then starts to decrease and eventually reaches zero for a given time . Not surprisingly, larger values of the coupling constant and lower values of the noisy parameters , and , correspond to larger maximum values of entanglement and to larger values of the zero-entanglement time .
We have investigated the ultra-strong-coupled regime of a linear coupled quantum harmonic oscillator system. Three different regimes of qualitatively different dynamical behaviour can be identified from the normal modes of the system. In particular, in the super-critical regime , the ‘’ mode has an anomalous accelerating motion, which can be understood as a consequence of a lack of translation symmetry in the hamiltonian (1), so that momentum conservation is not explicitly guaranteed. In the classical Hookian hamiltonian (6) such a symmetry is manifest, so that no dynamical anomaly is allowed in the strong coupling regime.
We further find that these three regimes characterized by their dynamical behaviour, is also uniquely characterized by the dynamics of entanglement; in particular, that the super-critical regime is one where entanglement can be unboundedly generated from even a highly thermal state. A similar behavior was studied in the context of the superradiant phase transition of the Dicke model, whose effective hamiltonian is of the form (1) after a Holstein-Primakoff transformation of a set of SU(2) operators into a bosonic operator.
Finally, we also note that the super-critical regime exhibits a finite entanglement even though the system maybe coupled to a decohering Markovian environment at a non-zero temperature.
Acknowledgements.MSK thanks Wonmin Son for discussions. This work was supported by the Qatar National Research Fund (NPRP 4-554-1-084). JL acknowledges support from a National Research Foundation of Korea (NRF) grant (No. 2010-0015059). MGG acknolwedges a fellowship support from UK EPSRC (EP/I026436/1) and thanks Alessio Serafini and Matteo Binazzi for interesting and useful discussions.
Appendix A Log-negativity for Gaussian states
Given a two-mode Gaussian state, its entanglement properties are fully characterized by its covariance matrix , which can be written in terms of matrices as
The log-negativity Vidal and Werner (2002) of the two-mode state can be evaluated as , where is the lowest symplectic eigenvalue of the corresponding partial transposed state. In formula we have
where the “seralian” is defined as
and where denotes the determinant of the matrix . It is known that the logarithmic negativity is an entanglement monotone Plenio (2005). One should also note that is directly related to the purity of the state, since . As a consequence, unitary evolution leaves invariant, so that the log-negativity is monotonous with . For this reason, when we consider the entanglement dynamics due to non-dissipative evolution, we may take the seralian to be an entanglement monotone.
Appendix B Solving the master equation
The master equation (7) can be cast into a c-number partial differential equation (PDE) by introducing the symmetric ordered characteristic function
where . We choose to express the PDE in terms of the quadratures parameters defined via , so that we get the Fokker-Planck equation,
where we define the vector , the associated gradient operator , the following matrices:
and where here is a permutation on the hamiltonian matrix (2) corresponding to the difference in the order chosen for the quadrature variables. The arrows on the gradient operators denote the direction in which the differential operators act.
As shown above, the system Hamiltonian only affects elliptic rotations or elliptic squeezing in the course of the dynamics, so that an initial Gaussian state localized at the origin in phase space remains so always; and since the bath is in thermal equilibrium, it too does not affect any finite displacements in phase space, but only scale and rotation changes. Thus, for our relevant case, without loss of generality, we choose the ansatz,
where now the time dependence is carried in the covariance matrix. Substituting this into (11), and then identifying the coefficients of the various bilinear products, we get the equation of motion for the covariance matrix viz.,
This has the solution,
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