# Quantum Correlations in Jahn-Teller Molecular Systems Simulated with
Superconducting Circuits

Ali Pedram  Onur Pusuluk  and Ö E. Müstecaplıoğlu  Department of Physics, Koç
University, Sarıyer, Istanbul, 34450, Turkey
apedram19@ku.edu.tr,opusuluk<EMAIL_ADDRESS>
###### Abstract

We explore quantum correlations, in particular, quantum entanglement, among
vibrational phonon modes as well as between electronic and vibrational degrees
of freedom in molecular systems, described by Jahn-Teller mechanism.
Specifically, to isolate and simplify the phonon- electron interactions in a
complex molecular system, the basis of our discussions is taken to be the
proposal of simulating two-frequency Jahn- Teller systems using
superconducting circuit quantum electrodynamics systems (circuit QED) by Tekin
Dereli and co-workers in 2012. We evaluate the quantum correlations, in
particular entanglement between the vibrational phonon modes, and present
analytical explanations using a single privileged Jahn-Teller mode picture.
Furthermore, spin-orbit entanglement or quantum correlations between
electronic and vibrational degrees of freedom are examined, too. We conclude
by discussing experimental feasibility to detect such quantum correlations,
considering the dephasing and decoherence in state-of-the-art superconducting
two-level systems (qubits).

##  1 Introduction

Quantum entanglement in natural systems, in particular molecules, has been
attracted much attention for its fundamental and practical significance [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Due to their complex structure
and many interactions within, it is difficult to isolate the quantum
entanglement route in molecules. One such route is the case where electronic
and vibrational degrees of freedom are coupled via so-called Jahn-Teller
interaction [12, 9, 6, 8]. Dereli and coworkers have proposed that quantum
simulation of molecules using superconducting circuits can serve to capture
the essential physics of multi-mode Jahn-Teller effect [16]. Here, we take
advantage of the Dereli and co-workers’ methods and explore quantum
entanglement in two-mode Jahn-Teller molecules simulated by superconducting
circuits .

The contribution by Dereli and coworkers is not only serving for the purpose
of illuminating complex quantum processes in molecules but also provides an
unprecedented application direction for the ultrastrong coupling regime of
superconducting circuit QED. Jahn-Teller coupling can be comparable to
involved frequencies of the vibrational phonons and hence it requires an
artificial system where ultrastrong coupling can be realized [6, 17, 18].
Realization of such strong interaction regimes are challenging in the usual
cavity QED but it is possible with the advent of superconducting circuit QED
[19, 20, 21]. Flux qubits and transmission line resonators are ideal platforms
to realize such strong photon-qubit interactions, where photons can play the
role of vibrational phonons. Vibrational modes in Jahn-Teller molecules can be
optical and acoustic phonons and hence an ideal platform to simulate such a
system would need tunable and wide range of frequencies in addition to the
strong coupling. In the case of superconducting circuits even optomechanical-
like interactions can be simulated by using transmission line resonators [22,
23, 24, 25].

Typical focus on Jahn-Teller molecules in chemistry is limited to
energetically low-lying states and their spectral properties, such as optical
absorption. We explore quantum correlations and look for potential quantum
entanglement in ground state of the molecular system with two-mode Jahn-Teller
coupling. Due to routine methods of measuring quantum correlations in
superconducting circuit QED, our results can be significant for practical
entanglement generation in addition to illuminating the nature of entanglement
in natural molecules. In addition, our quantum simulation and effective
privileged mode approach can be further extended to explore the interplay of
quantum correlations, chaos, and phase transitions [26].

The rest of the paper is organized is as follows. We first give a short review
of the two-mode Jahn-Teller model and the method of privileged mode as
described by Dereli and coworkers in Sec. II. We present our quantum
entanglement measures and their evaluation results in Sec. III. Sec. IV is the
conclusion of our analysis of the quantum entanglement results in different
parameter regimes from the perspective of effective privileged mode approach.

##  2 Privileged mode approach to two-mode Jahn-Teller model: A short review

###  2.1 two-mode Jahn-Teller model and circuit QED simulation

Jahn-Teller transition happens in some molecules that can end in a preferred
geometrical configuration among a set of degenerate ones due to the symmetry
breaking by vibrational phonon and localized electron interaction. Hamiltonian
describing this phenomenon is known as Jahn-Teller model [27]. Jahn-Teller
coupling belongs to the general class of spin-boson interactions. Vibrational
degrees of freedom of molecules are quantized in terms of phonons with boson
statistics. Optical and acoustic phonons have high and low frequencies and can
be described by an approximate two-mode model, a particular Jahn-Teller model,
also known as the Herberg-Teller model, which will be our focus here. Dereli
and coworkers have shown that E$\times(\beta_{1}+\beta_{2})$ type Herzberg-
Teller model can be simulated by using the superconducting circuit (sc)
Hamiltonian (we take $\hbar=1$)

$\displaystyle\hat{H}_{\text{sc}}=\hat{H}_{0}+\hat{H}_{\text{int}}$ (1)

where the non-interacting and interacting terms respectively are given by

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2}\hat{\sigma}_{z}+\sum_{i=1}^{2}\omega_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i},$
(2) $\displaystyle\hat{H}_{\text{int}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}g_{i}(\hat{a}_{i}+\hat{a}_{i}^{\dagger})\hat{\sigma}_{x}+J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1}).$
(3)

Here, $\omega_{q}$ and $\omega_{i}$ denote the transition frequency of the
qubit and the frequencies of the modes of two lumped element LC resonators
($i=1,2$). Creation ($\hat{a}_{i}^{\dagger}$) and annihilation ($\hat{a}_{i}$)
operators of the modes obey the Weyl-Heisenberg (bosonic) algebra. The two-
qubit states simulate the diabatic electronic states in actual molecules. Two
energy levels of the qubit, $|1\rangle,|2\rangle$ are represented by the Pauli
spin-$1/2$ operators $\hat{\sigma}_{z}=|2\rangle\langle 2|-|1\rangle\langle
1|,\hat{\sigma}_{x}=|1\rangle\langle 2|+|2\rangle\langle 1|$. The resonators
are coupled to a flux qubit with the interaction coefficients $g_{i}$. Their
difference can simulate the case of spatial anisotropy of molecules. We scale
the interaction coefficients with the corresponding mode frequencies and write
$g_{i}=\omega_{i}k_{i}$. The displacement-spin coupling form of the
interaction is the signature of the Jahn-Teller model. The last term, however,
makes the circuit QED model different than the original Jahn-Teller
Hamiltonian. It arises due to the inductive coupling of the two resonators.
This term can be made relatively smaller than the ultrastrong coupling
coefficient by optimizing the system geometry by extending the distance
between the LC resonators using a sufficiently large flux qubit. In the
privileged mode description, this term will not play a significant role.

###  2.2 Privileged Mode Approach to Two-Mode Jahn-Teller Model

Privileged mode is an energetically favored superposition of the two resonator
modes which can be expressed in terms of the annihilation operators as

$\displaystyle\hat{b}_{1}=\frac{1}{k_{p}}(k_{1}\hat{a}_{1}+k_{2}\hat{a}_{2}),$
(4)
$\displaystyle\hat{b}_{2}=\frac{1}{k_{p}}(k_{2}\hat{a}_{1}-k_{1}\hat{a}_{2}),$
(5)

where $k_{p}^{2}=k_{1}^{2}+k_{2}^{2}$. Next to the privileged mode, there is
also an energetically disadvantaged mode, whose annihilation operator is
denoted by $\hat{b}_{2}$, and the transformation from the resonator modes to
the privileged and disfavored modes is one-to-one. The orthogonal
transformation coefficients are determined by maximizing $k_{p}^{2}\omega_{p}$
and using the orthogonality condition of the transformation [28].

The terms of the transformed Hamiltonian for the privileged mode is the same
as the single mode Jahn-Teller model

$\displaystyle
H_{\text{JT}}=\frac{\omega_{q}}{2}\hat{\sigma}_{z}+\omega_{p}\hat{b}_{1}^{\dagger}\hat{b}_{1}+g_{p}(\hat{b}_{1}+\hat{b}_{1}^{\dagger})\hat{\sigma}_{x},$
(6)

where the privileged mode frequency and its coupling rate to the flux qubit
are denoted by
$\omega_{p}=(\omega_{1}k_{1}^{2}+\omega_{2}k_{2}^{2})/k_{p}^{2}$ and
$g_{p}=\omega_{p}k_{p}$, respectively. The rest of the total transformed
Hamiltonian terms include the hopping between the privileged and disadvantaged
modes. The full hamiltonian for the superconducting circuit after the
transformation becomes

$\displaystyle\hat{H}_{\text{sc}}=\frac{\omega_{q}}{2}\hat{\sigma}_{z}+(\omega_{p}+\frac{Jk_{1}k_{2}}{k_{p}^{2}})\hat{b}_{1}^{\dagger}\hat{b}_{1}+(\tilde{\omega}_{p}-\frac{Jk_{1}k_{2}}{k_{p}^{2}})\hat{b}_{2}^{\dagger}\hat{b}_{2}+[c+J\frac{k_{2}^{2}-k_{1}^{2}}{k_{p}^{2}}](\hat{b}_{1}^{\dagger}\hat{b}_{2}+\hat{b}_{2}^{\dagger}\hat{b}_{1})$
(7)
$\displaystyle+k_{p}[\omega_{p}(\hat{b}_{1}^{\dagger}+\hat{b}_{1})+c(\hat{b}_{2}^{\dagger}+\hat{b}_{2})]\hat{\sigma}_{x}.$

Here we have
$\tilde{\omega}_{p}=(\omega_{1}k_{2}^{2}+\omega_{2}k_{1}^{2})/k_{p}^{2}$. To
ensure the terms for the disadvantaged mode are negligible, perturbation
theory requires that the parameters $J$ and $c=\Delta k_{1}k_{2}/k_{p}^{2}$
must be sufficiently small where $\Delta=\omega_{1}-\omega_{2}$. The latter
condition can be realized if one resonator mode interacts with the flux qubit
in the ultrastrong coupling regime (without loss of generality, we label it as
$i=1$). The other mode, $i=2$, can be coupled strongly to the flux qubit, and
we demand the condition $g_{1}>g_{2}$ to be satisfied. The hopping rate $J$
between the modes must be smaller or comparable to $g_{2}$.

##  3 Results: Quantum Entanglement in the Jahn-Teller Ground State and
Privileged Mode Interpretation

We quantify the quantum entanglement in the ground state of the Jahn-Teller
model using the logarithmic negativity [29] as the measure of entanglement.
The logarithmic negativity, which is an upper bound to the distillable
entanglement, is defined as

$\displaystyle
E_{N}(A|\bar{A})=log_{2}\mathinner{\\!\left\lVert\rho_{A\bar{A}}^{T_{A}}\right\rVert},$
(8)

where $A$ is a subsystem of a total quantum system in which its complement is
$\bar{A}$, $\rho^{T_{A}}$ signifies partial transpose with respect to the
subsystem $A$, and $\mathinner{\\!\left\lVert\rho\right\rVert}$ is the trace
norm that equals to $\mathrm{tr}[\sqrt{\rho\rho^{\dagger}}]$. Here,
$E_{N}(A|\bar{A})$ quantifies the entanglement shared between the subsystems
$A$ and $\bar{A}$ when their joint state is described by the density matrix
$\rho_{A\bar{A}}$.

For the case of our model, correlations might arise between the flux qubit
($S$), the privileged mode ($B_{1}$), and the disadvantaged mode ($B_{2}$). To
capture an adequate description of the correlations in the tripartite ground
state $\rho_{SB_{1}B_{2}}$ due to the Jahn-Teller Hamiltonian, we consider the
bipartitions $S|B_{1}B_{2}$, $S|B_{1}$, $S|B_{2}$, and $B_{1}|B_{2}$ in what
follows. To measure the entanglement shared between two of the three
subsystems, we first compute the reduced density matrix of the subsystems
under consideration by tracing out the degrees of the freedom of the third
one.

According to our calculations of $E_{N}(S|B_{1}B_{2})$, $E_{N}(S|B_{1})$,
$E_{N}(S|B_{2})$, and $E_{N}(B_{1}|B_{2})$, it is sufficient to set the Fock
space dimension $N$ of the resonators to $10$ to analyze the behaviour of
ground-state quantum entanglement with respect to the Hamiltonian parameters.
We have done the same calculations for higher-dimensional Hilbert spaces up
until $N=20$, and they gave similar results for the strong and ultra-strong
coupling regimes.

Figure 1: Ground state entanglement in Jahn-Teller model in terms of the
logarithmic negativity $E_{N}$. $\Delta=\omega_{1}-\omega_{2}$ is the
frequency difference between the vibration modes. The parameters are fixed to
explore the strong coupling regime such that $J=0$ and
$k_{1}=k_{2}=k=0.1/\sqrt{2}$. Figure 2: Ground state entanglement in Jahn-
Teller model in terms of the logarithmic negativity $E_{N}$.
$\Delta=\omega_{1}-\omega_{2}$ is the frequency difference between the
vibration modes. The parameters are fixed to explore the ultra-strong coupling
regime such that $J=0$ and $k_{1}=k_{2}=k=1/\sqrt{2}$. Figure 3: Ground state
entanglement in Jahn-Teller model in terms of the logarithmic negativity
$E_{N}$. $\kappa=k_{2}-k_{1}$ is the difference between the factors that
rescale the coupling constants $g_{i}$ with respect to frequencies
$\omega_{i}$. The parameters are fixed such that $J=0$ and
$\omega_{1}=2\omega_{2}=0.1$. Figure 4: Ground state entanglement in Jahn-
Teller model in terms of the logarithmic negativity $E_{N}$.
$\kappa=k_{2}-k_{1}$ is the difference between the factors that rescale the
coupling constants $g_{i}$ with respect to frequencies $\omega_{i}$. The
parameters are fixed such that $J=0$ and $\omega_{1}=2\omega_{2}=1$. Figure 5:
Ground state entanglement in Jahn-Teller model in terms of the logarithmic
negativity $E_{N}$ when $J=0$ and $\omega_{1}-\omega_{2}=\Delta=k_{1}=k_{2}$.
Figure 6: Ground state entanglement in Jahn-Teller model in terms of the
logarithmic negativity $E_{N}$ when $J\neq 0$. Here,
$k_{1}=k_{2}=k=1/\sqrt{2}$ and $\omega_{1}=2\omega_{2}=0.2$.

We first examine the effect of the frequency difference between the vibration
modes on the entanglement shared in the Jahn-Teller ground state. To this end,
and for the sake of simplicity, we take $k_{1}=k_{2}=k$, $J=0$ and
$\omega_{1/2}=1\pm\Delta/2$ by rescaling all the parameters with respect to
the qubit transition frequency $\omega_{q}$. It is clear from Eq. 7 that as
$\Delta$ is increased, both the coupling between the two resonators and the
coupling of the flux qubit to the disadvantaged mode becomes stronger.
Therefore, the flux qubit and the disadvantaged mode correlations are expected
to constitute a significant portion of the total correlations as we increase
$\Delta$. Fig. 1 justifies this expectation in the strong coupling regime
where $k=0.1/\sqrt{2}$. The effective single privileged mode description of
the system is valid for $|\Delta|<0.1$. The entanglement between the flux
qubit and the disadvantaged mode remains negligible until this limit. In the
two-mode regime, $E_{N}(S|B_{2})$ rises slowly with further increases of
$\Delta$. When $\Delta=2$, the flux qubit becomes coupled to both modes with
the same strength, as a result of which $E_{N}(S|B_{1})$ and $E_{N}(S|B_{2})$
take the same value. Although the hopping constant $J$ is set to zero,
$E_{N}(B_{1}|B_{2})$ takes a small but nonzero value at this point.

For the case of ultra-strong coupling ($k=1/\sqrt{2}$) shown in Fig. 2, the
general trend of ground-state entanglement does not appear different from the
strong coupling case. However, the stronger coupling produces stronger
entanglement between the flux qubit and the two resonators in this case. The
observation of the steeper increase in the entanglement between the flux qubit
and the disadvantaged mode is delayed until $|\Delta|=1$, only after which the
privileged mode approximation fails.

We also explore the ground state entanglement by changing the difference
between the scaling factors appearing in the Jahn-Teller couplings
$g_{i}=k_{i}\omega_{i}$. To do so, we set $J=0$, $\omega_{1}=2\omega_{2}$, and
$k_{2/1}=(1\pm\kappa)/2$. We use $k_{2}-k_{1}=\kappa$ as our control
parameter. The results for $\omega_{1}=0.1$ and $\omega_{1}=1$ are
respectively given in Figs. 3 and 4. When the flux qubit and the privileged
mode are resonant, i.e., $\omega_{q}=\omega_{1}=1$, we end up with higher
values for $E_{N}(S|B_{1})$. The coupling of the privileged mode to the flux
qubit is one order of magnitude larger than both the coupling of the
disadvantaged mode and the hopping between the modes for $\kappa\geq 2/3$ in
both cases. So, $E_{N}(S|B_{1})$ approximately equals $E_{N}(S|B_{1}B_{2})$ in
both cases as long as the privileged mode approximation works well.

So far, a significant amount of entanglement has not appeared between the
privileged and disadvantaged modes in the two-mode Jahn-Teller regime. This
led us to change both $\omega_{i}$ and $k_{i}$ at the same time. For the sake
of simplicity, we set $J=0$, $\omega_{1/2}=1\pm\Delta/2$, and
$k_{1}=k_{2}=\Delta$ in Fig. 5. Here, the privileged mode approximation fails
when $\Delta\geq 0.2$. However, $E_{N}(S|B_{1}B_{2})$ and $E_{N}(S|B_{1})$
seem to overlap until $\Delta$ reaches 1. $E_{N}(B_{1}|B_{2})$ rises rapidly
after this point with further increases of $\Delta$. As of $\Delta=1.5$,
$E_{N}(S|B_{1})$ shows a sharp decline, whereas $E_{N}(S|B_{1}B_{2})$ remains
fixed at unit negativity. When the flux qubit becomes coupled to both modes
with the same strength at $\Delta=2$, a significant amount of entanglement
arises between the flux qubit and the two-mode subsystem and between the two
modes themselves. However, neither the privileged nor the disadvantaged are
strongly entangled with the flux qubit in this limit.

Finally, we consider the role of hopping constant $J$ on the ground state
entanglement in Fig. 6. Here, the remaining parameters are fixed such that
$k_{1}=k_{2}=k=1/\sqrt{2}$ and $\omega_{1}=2\omega_{2}=0.2$, for which the
privileged mode approximation works well for every value of $J$, i.e., the
privileged mode is coupled to the flux qubit with a strength one order of
magnitude larger than both its coupling strength to the disadvantaged mode and
the flux qubit-disadvantaged mode coupling strength. However,
$E_{N}(B_{1}|B_{2})$ and $E_{N}(S|B_{2})$ are always comparable to
$E_{N}(S|B_{1}B_{2})$ and $E_{N}(S|B_{1})$. If we doubled $k$,
$E_{N}(S|B_{1}B_{2})$ would also double but $E_{N}(B_{1}|B_{2})$ would
increase more than six times in this figure. Thus, a non-vanishing hopping
between the two modes results in nontrivial changes in the distribution of
entanglement between different subsystems in the Jahn-Teller model.

##  4 Conclusion

We examined the quantum entanglement in the two-mode Jahn-Teller ground state
simulated by a superconducting circuit using privileged mode description
following the approach by Dereli and co-workers [16].

We find that the two-mode Jahn-Teller ground state exhibits bipartite
entanglement between the qubit and both modes in the strong coupling regime of
the superconducting circuit quantum simulator. This entanglement is weakly
changed with the frequency difference (detuning) between the modes. The
disadvantaged mode is approximately unentangled from the qubit at slight
frequency differences, and it is unentangled from the privileged mode at all
detunings. As the detuning increases towards the regime where the privileged
mode approximation fails, both modes have comparable entanglement to the
qubit. Similar behavior with stronger entanglement is found in the ultrastrong
regime, too. The entanglement between the privileged mode and the qubit
increases significantly, while the disadvantaged mode entanglement to the
qubit is slowly decreasing, with the difference of the Jahn-Teller coupling
constants, simulated by the flux qubit and the transmission line resonator
interaction. We identified a parameter regime to entangle the privileged mode
to the disadvantaged mode where both modes are equally strongly coupled to the
qubit, and their detuning is significant so that the privileged mode
approximation fails. In this case, the qubit is disentangled from the modes,
and we only have entanglement between the bosonic modes. These conclusions are
made when the modes are not directly coupled. We have also investigated the
effect of direct coupling between the modes, which can be possible technically
in superconducting circuits. Direct interaction of the modes can further
increase the qubit-mode and the mode-mode entanglements. The enhancement is,
however, more pronounced for mode-mode entanglement.

Our results can be significant to simulate molecular systems with strong and
ultrastrong coupled superconducting circuits to illuminate their quantum
correlations and utilize them as large-scale quantum resources for quantum
technologies.

###  4.1 Acknowledgments

O. P. acknowledges support by the Scientific and Technological Research
Council of Turkey (TÜBİTAK), Grant No. (120F089).

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