††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Grain size in low loss superconducting Ta thin films on c-axis sapphire

Sarah Garcia Jones  Department of Electrical, Computer, and Energy
Engineering, University of Colorado Boulder, Boulder, Colorado 80309, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Nicholas Materise  Department of Physics, Colorado School of Mines, Golden,
Colorado 80401, USA  Ka Wun Leung  Department of Physics, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR,
China  Brian D. Isakov  Department of Electrical, Computer, and Energy
Engineering, University of Colorado Boulder, Boulder, Colorado 80309, USA  Xi
Chen  Department of Physics, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China  Jiangchang Zheng
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong SAR, China  András Gyenis  Department of
Electrical, Computer, and Energy Engineering, University of Colorado Boulder,
Boulder, Colorado 80309, USA  Berthold Jaeck<EMAIL_ADDRESS>Department of
Physics, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong SAR, China  HKUST IAS Center for Quantum Technologies, The
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong
Kong SAR, China  Corey Rae H. McRae<EMAIL_ADDRESS>Department of
Electrical, Computer, and Energy Engineering, University of Colorado Boulder,
Boulder, Colorado 80309, USA  Department of Physics, University of Colorado,
Boulder, Colorado 80309, USA  National Institute of Standards and Technology,
Boulder, Colorado 80305, USA

###### Abstract

In recent years, the implementation of thin-film Ta has led to improved
coherence times in superconducting circuits. Efforts to further optimize this
materials set have become a focus of the subfield of materials for
superconducting quantum computing. It has been previously hypothesized that
grain size could be correlated with device performance. In this work, we
perform a comparative grain size experiment with $\alpha$-Ta on $c$-axis
sapphire. Our evaluation methods include both room-temperature chemical and
structural characterization and cryogenic microwave measurements, and we
report no statistical difference in device performance between small- and
larger-grain-size devices with grain sizes of 924 nm2 and 1700 nm2,
respectively. These findings suggest that grain size is not correlated with
loss in the parameter regime of interest for Ta grown on c-axis sapphire,
narrowing the parameter space for optimization of this materials set.

###### pacs:

##  I Introduction

Superconducting qubits are a promising avenue for scalable quantum computing
devices due to their high-fidelity operation Kjaergaard  _et al._ (2020);
Zhang  _et al._ (2021); Hyyppä  _et al._ (2022); Howard  _et al._ (2022).
Recent advances in qubit design, packaging, and control have shrunken the gap
toward their practical use Preskill (2018); Arute  _et al._ (2019). Still,
dielectric losses due to bulk substrates, surface oxides, and amorphous or
defect-ridden material interfaces limit the coherence of superconducting
qubits and ancillary devices Place  _et al._ (2021); Crowley  _et al._ (2023);
Read  _et al._ (2022). Microscopically, materials loss is largely associated
with the excitation of two-level systems (TLS) that dominate microwave losses
in the technologically relevant range of low temperatures and single-photon
numbers Müller, Cole, and Lisenfeld (2019); McRae  _et al._ (2020). Materials
engineering has been identified as a leading route for improvement of
superconducting qubit coherence by reducing the effect of TLS Siddiqi (2021).

Recent works demonstrate improved qubit performance when
$\mathrm{\alpha}$-phase tantalum (Ta) replaces niobium (Nb) as the
superconducting thin film base layer for device fabrication Place  _et al._
(2021); Wang  _et al._ (2022). These findings are further supported by loss
measurements of superconducting microwave resonators Crowley  _et al._ (2023);
Alegria  _et al._ (2023); Shi  _et al._ (2022); Lozano  _et al._ (2022) and it
is believed that the loss reduction is afforded by the simple oxide structure
of the Ta film surface Place  _et al._ (2021). Further evidence for this is
suggested by recent work capping Nb films with Ta for improved qubit
performance Bal  _et al._ (2023).

A detailed materials study of Nb-based qubits links the bulk properties of the
polycrystalline films to qubit losses Premkumar  _et al._ (2021). Small
crystalline grain sizes were found to correlate with increased qubit losses,
which could arise from TLS present at the subsurface grain boundary oxides in
Nb films Premkumar  _et al._ (2021). Hence, the grain size of the
superconducting base layer has recently been debated as a promising process
parameter to further minimize microwave losses in Ta films. Moreover,
controlled A/B-testing studies would be desirable to firmly establish this
relation and it remains unknown whether grain size effects on microwave losses
extend to resonators based on Ta films, whose surface and subsurface oxide
structure differs from that of Nb films.

The goal of this work is to probe the relationship between grain size and
microwave losses for $\alpha$-Ta films grown on c-axis sapphire, a substrate
commonly used for Ta growth Place  _et al._ (2021); Alegria  _et al._ (2023).
To this end, we perform microwave loss measurements of coplanar waveguide
resonators made from magnetron-sputtered $\alpha$-Ta films with large and
small grain sizes. We compare the losses of both types of films across thirty
resonators from multiple chips and report no statistical difference between
the performance of films with small and larger grain sizes. In combination
with results from the chemical and crystallographic thin film
characterizations, our observations indicate that grain size does not play a
significant role in microwave losses for $\alpha$-Ta films across the tested
parameter regime.

##  II Ta Growth and Characterization

Figure 1: Structural and chemical characterization of Ta films. (a) X-ray
diffraction spectra of $2\Theta$-scans for measurements of the ‘SGS2’ and
‘LGS2’ samples. The inset displays the corresponding spectrum for the ‘LGS2’
sample over a larger angle ($\Theta$) range. The detected diffraction peaks
are labeled with the corresponding Miller indices of the $\alpha$-Ta phase.
(b) Atomic force microscopy topographies of the ‘SGS2’ (left) and ‘LGS2’
(right) sample surfaces. (d) Electron binding energy spectra of the Ta 4f core
level obtained from X-ray photoelectron spectroscopy measurements at the
surface of the ‘SGS2’ and ‘LGS2’ samples. The dominant Ta oxidation states are
indicated. (d) Least squares fit (open squares) to an XPS spectrum (solid
black line) recorded at the surface of the LGS2 sample. Contributions to the
spectrum by the Ta 4f5/2 (magenta color) and Ta 4f7/2 (green color) core
levels of Ta, (solid lines), Ta3+ (dotted lines), and Ta5+ (dashed lines) were
modeled by using Gaussian profiles. Figure 2: Resonator loss power curves with
small grain size (left) and larger grain size (right). Total loss $\delta$
minus high power loss $1/Q_{\mathrm{i,HP}}$ as a function of average number of
photons in the cavity $\langle n_{\mathrm{ph}}\rangle$ for all devices
measured in this work - SGS1 (light blue), SGS2 (blue), LGS1 (red), and LGS2
(orange) resonators. Lines denote best fits to the TLS model (Eq. (4)). 95$\%$
confidence intervals for Lorentzian fits to each data point are given, as well
as the prediction interval for each TLS-curve fit.

Ta films of nominal 200 nm thickness were deposited on $c$-axis sapphire
wafers (2" diameter, 550 $\mu$m thickness, from Hefei Keijing Materials
Technology) using dc magnetron sputtering. Prior to deposition, the as-
purchased substrates were cleaned via ultrasonication in acetone, isopropanol,
and deionized water for 5 min each and blown dry with nitrogen of purity 4N.
To deposit thin films with different grain sizes, two different substrate
temperatures $T=400\,^{\circ}$C (sample label ‘SGS’ or ‘small grain size’) and
$T=500\,^{\circ}$C (sample label ‘LGS’ or ‘larger grain size’) during the
deposition were chosen, while other deposition parameters (background pressure
$\leqslant 1\times 10^{-7}$ Torr, argon pressure 3 mTorr, deposition power 150
W, deposition rate 3.6 nm/min) were not changed. The deposition was carried
out without the use of a seed layer. A $T=600\,^{\circ}$C sample was also
grown, but no increase in grain size was detected, so this sample is not
included in the detailed film comparison.

The large-scale diffraction spectrum of the films (Fig. 1(a) inset) is
dominated by a set of two peaks, which can be associated with diffraction at
the [110] and [220] planes of the $\alpha$-Ta[220] phase. A comparably small
diffraction signal, which rises just above the background signal, is detected
at 2$\Theta\approx 33.7^{\circ}$ that can be associated with the
[002]-diffraction of the tantalum $\beta$-phase. Our observations indicate the
Ta films prepared for this study predominantly nucleate in the $\alpha$-Ta
phase. This finding is consistent with previous reports on 200 nm thick
$\alpha$-Ta films on $c$-axis sapphire, which were deposited under comparable
conditions Place  _et al._ (2021). The close-up view of the $\alpha$-Ta[110]
peaks for the ‘SGS2’ and ‘LGS2’ samples is shown in the main panel of Fig.
1(a). The diffraction peak of the ’LGS2’ sample ($\sigma=0.4^{\circ}$) has a
smaller full-width-half-maximum $\sigma$ compared to that of the ’SGS2’ sample
($\sigma=0.5^{\circ}$). While this observation is indicative of a larger
average grain size in the ’LGS2’ sample, we note that the Scherrer equation is
less suited to quantitatively analyze the grain size in this case, owing to
the grain shape anisotropy and significant grain size variations (see AFM
measurements below). We further observed a small deviation in the
[002]-diffraction angle both between the ’SGS2’ (2$\Theta=38.1^{\circ}$) and
’LGS2’ (2$\Theta=38.3^{\circ}$) sample, as well as with respect to the nominal
bulk value (2$\Theta=38.505^{\circ}$). This can be attributed to the presence
of strain in the thin film structure, which appears slightly more pronounced
in the ‘SGS2’ sample.

To characterize the crystalline grain size of the Ta films deposited at
different substrate temperatures, we carried out atomic force microscopy (AFM)
measurements (tapping mode). The resulting AFM topographies for samples ‘SGS2’
and ‘LGS2’ are shown in Fig. 1(b). Both topographies are characterized by
elongated crystalline grains oriented along the hexagonal basal plane of the
sapphire surface, consistent with previous reports Alegria  _et al._ (2023).
Moreover, the grains of ‘LGS2’ exhibit a visibly larger grain size area $G$
than those of ‘SGS2’, consistent with our expectations in light of the
substrate temperatures during deposition. To quantify these grain size
differences, we applied a watershed algorithm Rabbani and Ayatollahi (2015) to
determine $G$, which is an average across several 1 $\mu$m2 surface areas per
sample and several samples for each deposition condition. This approach was
previously applied to quantify grain sizes of Nb films Premkumar  _et al._
(2021). We obtain $G=924\pm 51\,\mathrm{nm}^{2}$ for the ‘SGS2’ and $G=1700\pm
29\,\mathrm{nm}^{2}$ for the ‘LGS2’ sample, respectively. Interestingly, the
average grain size $G=1732\pm 92\,\mathrm{nm}^{2}$ of samples deposited at a
substrate temperature $T=600\,^{\circ}$C is comparable to that of the
$T=500\,^{\circ}$C deposition SI .

To detect the possible influence of the crystalline grain size on the surface
oxide structure, we performed X-ray photoelectron spectroscopy (XPS)
measurements (Kraxios Ultra DLD; X-ray source: Al K$\alpha$ line
$E=1486.6\,$eV) on the ‘SGS2’ and ‘LGS2’ samples. We note that these samples
did not undergo surface treatment to remove native surface oxides prior to XPS
measurements. The resulting XPS spectra in Fig. 1(c) show the photo-electron
count as a function of the electron binding energy for the Ta-4f core level.
The spectra are dominated by a four peak structure, which is predominantly
composed of the spin-orbit split Ta0 and Ta5+ doublets that can be assigned to
the metallic Ta bulk and the Ta2O5 at the film surface, respectively McGuire,
Schweitzer, and Carlson (1973); Himpsel  _et al._ (1984).

We quantify the relative contributions of the different Ta oxidation states to
the observed XPS spectra by applying a least-squares fit based on Gaussian
profiles. We find a three doublet structure composed of six Gaussians, as
shown in Fig. 1(d), can most accurately describe these spectra. The additional
third doublet exhibits a core level shift of $\approx 1.1\,$eV and can be
assigned to the Ta3+ oxidation state of the Ta2O3 suboxide Himpsel  _et al._
(1984). The resulting relative contributions of Ta, Ta3+, and Ta5+ obtained
from these fits are shown in Table 1 and reveal a near identical chemical
structure of the tantalum film surface for both samples. this is consistent
with their almost identical XPS spectra (cf. Fig. 1(c)). The relative spectral
weight of the Ta0 and Ta5+ peaks at the given incident X-ray energy is in
close agreement with that found in previous XPS studies of tantalum films and
indicates a surface oxide thickness of approximately 2 nm Place  _et al._
(2021).

Table 1: Relative atomic concentration of different tantalum oxidation states in the ’SGS2’ and ’LGS2’ samples as obtained from fits to the XPS spectra. | Small grain size | Larger grain size
---|---|---
Oxidation state | atomic % | atomic %
Ta0 | 18 | 20
Ta3+ | 17 | 18
Ta5+ | 65 | 62

##  III Device Design and Fabrication

All devices are coplanar waveguide resonators fabricated using the same
designs as reported by Kopas  _et al._ Kopas  _et al._ (2022). Nominally
identical designs and fabrication procedures were used for all samples. Prior
to etching, the samples were cleaned via ultrasonication in toluene, acetone,
methanol, and isopropanol, then patterned using optical lithography and
AZ-P4330-RS photoresist. The films were etched in a single 4 minute CF4/N2
Inductively Coupled Plasma – Reactive Ion Etch (Panasonic E640). Since the Ta
films were deposited on sapphire substrates, the etches did not produce any
trenching into the substrate. After etching, the resist was submerged in AZ
300 T stripper at 80 ∘C for 1 hour. After stripping, the samples were diced
and again cleaned ultrasonically in toluene, acetone, methanol, and
isopropanol before being wire bonded for cryogenic microwave measurement.
Optical images of the resonators are shown in Fig. 3.

Inverse coupling quality factors, $1/Q_{c}$, of the fabricated resonators are
presented in Supplementary Materials Table 1 and range from $1.18\times
10^{-6}$ to $6.61\times 10^{-6}$ across all devices. This is a larger spread
of values with a trend towards smaller coupling factors than the simulated
$1/Q_{c}$ values of these designs, which ranged from $1.95\times 10^{-6}$ to
$2.02\times 10^{-6}$. Kopas  _et al._ (2022) This variation is likely due to a
slight over etch of the devices during fabrication, which is congruent with a
thinner measured conductor width than the lithography designs used (design:
$6\mu\,\mathrm{m}$, measured: $5.5\mu\,\mathrm{m}$).

Figure 3: Optical microscope images of coplanar waveguide resonators. (a) Full
chip image of a representative chip. All circuits measured contain eight
resonators with identical couplers. (b) Close up of feedline and resonator
base. Conductor width is 5.5 $\mu m$ and gap is 3.8 $\mu m$.

##  IV Cryogenic Microwave Measurement

Figure 4: TLS loss in Ta on c-axis sapphire superconducting microwave
resonators with small and larger grain size. Left: Box and whiskers
comparison, indicating median values (red line) and 95$\%$ confidence interval
of median (red triangles), with outliers shown as crosses. Right: Histogram of
TLS loss for all devices in this experiment.

We perform transmission measurements on CPW resonators mounted to the mixing
chamber (MXC) plate of a FormFactor (formerly Janis) JDry 250 dilution
refrigerator (DR) at a mixing chamber temperature of $\sim$10 mK using a
Keysight PNA N5222B vector network analyzer (VNA). The input power varied over
ten orders of magnitude in estimated photon power to accurately extract the
dominant two level system (TLS) loss Gao (2008).

Nominally identical gold-plated oxygen-free high conductivity copper sample
boxes house the resonator chips and are mounted to a plate perpendicular to
the MXC plate with additional mu-metal shielding surrounding all samples. Two
Radiall R583 six-way microwave switches allow multiple samples to be measured
on the same pair of coaxial input and output lines.

The transmission data for each resonator ($S_{21}$ of the two-port
$S$-parameter matrix measured by the VNA) is first normalized with a circle
fit Probst  _et al._ (2015), and then fit to a diameter-corrected asymmetric
Lorentzian model of the form Khalil  _et al._ (2012),

$\displaystyle S_{21}(f)$ $\displaystyle=$ $\displaystyle 1-\frac{Q_{l}\ /\
Q_{c}\ e^{i\phi}}{1+2iQ\frac{f-f_{0}}{f_{0}}}$ (1) $\displaystyle Q_{i}^{-1}$
$\displaystyle=$ $\displaystyle
Q_{l}^{-1}-\mathrm{Re}\left\\{\hat{Q}^{-1}_{c}\right\\}$ (2)
$\displaystyle\hat{Q}_{c}^{-1}$ $\displaystyle=$ $\displaystyle
Q_{c}^{-1}e^{i\phi}$ (3)

where $f_{0}$ is the resonance frequency, $\phi$ is the asymmetry angle,
$Q_{c}$ is coupling quality factor, $Q_{l}$ is the loaded quality factor, and
$Q_{i}$ is the internal quality factor. These parameters are fit with their
corresponding 95% confidence intervals from a least squares fitting routine
BCQ (2023). A secondary fit of the loss $\delta=Q_{i}^{-1}$ as a function of
average number of photons $\left<n_{\mathrm{ph}}\right>$ and fixed temperature
$T$ follows from the sum of the TLS loss contribution $\delta_{\mathrm{TLS}}$
and an offset term $1/Q_{\mathrm{HP}}$ that accounts for power-independent
losses dominating at higher powers McRae  _et al._ (2020)

$\displaystyle\delta\left(\left<n_{\mathrm{ph}}\right>,T\right)$
$\displaystyle=$
$\displaystyle\delta_{\mathrm{TLS}}\left(\left<n_{\mathrm{ph}}\right>,T\right)+1/Q_{\mathrm{HP}}$
(4)
$\displaystyle\delta_{\mathrm{TLS}}\left(\left<n_{\mathrm{ph}}\right>,T\right)$
$\displaystyle=$ $\displaystyle
F\delta_{\mathrm{TLS}}^{0}\frac{\tanh\left(\frac{\hbar\omega_{0}}{k_{B}T}\right)}{\left(1+\frac{\left<n_{\mathrm{ph}}\right>}{n_{c}}\right)^{\beta}}$
(5)

where $n_{c}$ is the critical photon number at which TLS saturate at low
power, $\omega_{0}=2\pi f_{0}$ is the angular resonance frequency, $\beta$ is
an exponent interpolating between the non-interacting TLS model $\beta=1/2$
and interacting TLS model $\beta<1/2$ Gao (2008); Phillips (1972); De Graaf
_et al._ (2018), $\delta_{\mathrm{TLS}}^{0}$ is the intrinsic TLS loss, $F$ is
the geometry-dependent filling factor, $\hbar$ is the reduced Planck constant,
and $k_{B}$ is the Boltzmann constant.

In Fig. 2, we plot the loss power dependence for all large and small grain
size devices used in this work. The high power losses are subtracted to
emphasize the similar power dependence (line shape) and low power loss (TLS
saturation loss) between the two samples, without the confounding factor of
high-power losses which are known to be caused by a myriad of sources external
to the device materials McRae  _et al._ (2020). Figure 4 further highlights
this point, as the medians from the box and whisker plots of the intrinsic TLS
losses for the large and small grain size Ta films coincide with one another
and their respective histograms give similar variances.

Table 2: Mean parameter values in A/B grain size comparison. | Small grain size | Large grain size
---|---|---
Grain area ($\mathrm{nm^{2}}$) | $924\pm 51$ | $1700\pm 29$
$F\delta_{\mathrm{TLS}}^{0}$ ($\times 10^{-6}$) | 2.19 $\pm$ 0.07 | 2.17 $\pm$ 0.03

##  V Literature Comparison

Figure 5: State-of-the-art literature comparison of TLS
($F\delta_{\mathrm{TLS}}^{0}$) and low-power ($\delta_{\mathrm{LP}}$) loss
values in CPW resonators. Filling factor $F$ is estimated by plotting loss as
a function of CPW gap width $g$. Grey lines denote lines of constant interface
loss. Filled symbols denote TLS loss values, while empty symbols represent low
power loss (TLS loss values unavailable).

Mean parameter values for the two resonator populations are summarized in Tab.
2. Though our small-geometry resonators are very sensitive to TLS loss, we see
no statistical difference in TLS loss between the small-grain and large-grain
devices, despite the difference in grain size (with the large grain size
devices being almost twice as big in area). This difference in grain size is
similar to that seen in Nb films in Ref. Premkumar  _et al._ (2021), where a
difference in grain size of a factor of two was correlated with a difference
in qubit $T_{1}$ of almost two, and a difference in resonator TLS loss was
also detected.

We refer to recent studies, especially the work by Lozano  _et al._ (2022), to
estimate the number of devices required to adequately sample the device-to-
device variation in $F\delta_{\mathrm{TLS}}^{0}$. With more than ten devices
of each grain size, we exceed the number of devices for each variation (choice
of control parameter value) in Refs. Lozano  _et al._ (2022); Alegria  _et
al._ (2023); Crowley  _et al._ (2023); Jia  _et al._ (2023).

The median values of the losses for the untreated Ta resonators on silicon in
Ref. Lozano  _et al._ (2022) are comparable to the large- and small-grain size
median losses reported in Fig. 4(a).

Figure. 5 shows that the difference between the large- and small-grain size
losses in this study is imperceptible on the same scale as other A/B
comparisons, e.g. Refs. Alegria  _et al._ (2023); Lozano  _et al._ (2022)

##  VI Discussion

An enduring hypothesis in the superconducting qubit community has been that
larger grain size in superconducting thin films is an indication of improved
device performance. The simple and stable oxide structure of Ta differs from
that of Nb, where Premkumar  _et al._ (2021) reported that smaller grain size
films exhibited higher concentrations of suboxides in interface regions,
resulting in measurably higher losses in their Nb resonators. In this study,
we show that smaller grain size does not induce significant low power loss in
Ta thin films grown between 400 and 500 °C on c-axis sapphire.

Microwave measurements of low power loss suggest that there is no
statistically significant difference between the intrinsic TLS losses of the
two grain size Ta thin films. Chemical and structural analysis support this
interpretation, as the surface chemistry obtained by XPS is nearly identical
for the two films. This distinguishes densely packed Ta films with their
simple Ta2O5 surface oxide structure Place  _et al._ (2021) from Nb films
Premkumar  _et al._ (2021) for which subsurface grain boundary oxides
contribute a grain size dependent TLS channel. Following this train of
thought, we expect qubits and resonators fabricated from Ta films to exhibit
more uniform microwave losses than those fabricated from Nb. At the same time,
non-negligible concentrations of Ta3+ species found in our XPS measurements
indicates the presence of Ta2O3 suboxides at the Ta metal-Ta2O5 interface
consistent with a recent report McLellan  _et al._ (2023). Interestingly, we
also detect practical limits within which to tune the grain size of
[110]-oriented $\alpha$-Ta films deposited on $c$-axis sapphire: Deposition at
substrate temperatures below 400 °C favors the formation of the unwanted
$\beta$-phase Knepper, Stevens, and Baker (2006); Myers  _et al._ (2013)
whereas grain size does not respond to an increase of substrate temperature in
excess of 500 °C in our study. Thus, it would be interesting to explore other
substrates or sapphire surface orientations to promote larger grain sizes up
to the formation of single-crystalline Ta films. On the other hand, our study
suggests more sophisticated materials engineering efforts that focus on the
reduction of TLS losses at the immediate metal-air surface rather than on the
optimization of bulk properties, such as grain size, are required to further
reduce microwave losses below those reported in this and other recent studies
Lozano  _et al._ (2022); Alegria  _et al._ (2023). These efforts will benefit
from targeted A/B testing studies, such as is presented here, to address the
vast materials and processing parameter space in order to maximize state-of-
the-art superconducting qubit performance.

##  VII Conclusion

We performed millikelvin microwave transmission measurements of $\alpha$-phase
Ta microwave resonators with both large- and small-grain size sputter-
deposited on c-axis sapphire at two different growth temperatures. Structural
and chemical analysis reveal that the films differ only in their grain size
and not in their surface oxide types and concentrations, and crystal
structure. The extracted intrinsic TLS losses show no statistical difference
between the two film types, suggesting that, in this materials regime, grain
size does not significantly affect millikelvin, ultralow power dieletric loss.
We encourage future A/B experimentation to continue to reduce the fabrication
parameter space and to identify correlations between other room-temperature
materials characterization parameters and low-power, low-temperature microwave
performance of devices.

###### Acknowledgements.

The authors would like to thank Qiming Shao and Jiacheng Liu for their
assistance with the sputter deposition, and Carlos Torres Castanedo, David
Garcia, Tony McFadden, and Dominic Goronzy for helpful discussions and
feedback on the manuscript. N.M. acknowledges funding from the Graduate
Fellowships for STEM Diversity. B.J. acknowledges funding from the Croucher
Foundation.

## Author Declarations

### Conflicts of Interest

The authors have no conflicts of interest.

### Commercial Products Disclaimer

Certain commercial instruments are identified to specify the experimental
study adequately. This does not imply endorsement by NIST or that the
instruments are the best available for the purpose.

### Author Contributions

Sarah Garcia Jones: Resonator sample fabrication (lead), Writing - review &
editing; Nicholas Materise: Writing - original draft (lead), Writing - review
& editing (supporting), Data acquisition (equal), Software (equal); Ka Wun
Leung: Materials fabrication (lead) ; Brian D. Isakov: Writing - review &
editing; Chen Xi: Materials characterization (lead); András Gyenis: Writing -
review & editing, Resonator sample fabrication; Bertold Jaeck:
Conceptualization (equal), Writing - review & editing (lead), Visualization
(equal), Formal analysis (equal), Methodology (lead); Corey Rae H. McRae:
Conceptualization (equal), Writing - review & editing (lead), Visualization
(equal), Formal analysis (equal), Methodology (lead).

## Data Availability

The data that support the findings of this study are openly available at
https://zenodo.org/record/8161535, reference number 10.5281/zenodo.8161535.

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††thanks: Corresponding author

# Entanglement in phase-space distribution for an anisotropic harmonic
oscillator in noncommutative space

Pinaki Patra<EMAIL_ADDRESS>Department of Physics, Brahmananda Keshab
Chandra College, Kolkata, India-700108

###### Abstract

The bi-partite Gaussian state, corresponding to an anisotropic harmonic
oscillator in a noncommutative space (NCS), is investigated with the help of
Simon’s separability condition (generalized Peres-Horodecki criterion). It
turns out that, to exhibit the entanglement between the noncommutative
coordinates, the parameters (mass and frequency) have to satisfy a unique
constraint equation.
We have considered the most general form of an anisotropic oscillator in NCS,
with both spatial and momentum non-commutativity. The system is transformed to
the usual commutative space (with usual Heisenberg algebra) by a well-known
Bopp shift. The system is transformed into an equivalent simple system by a
unitary transformation, keeping the intrinsic symplectic structure
($Sp(4,\mathbb{R})$) intact. Wigner quasiprobability distribution is
constructed for the bipartite Gaussian state with the help of a Fourier
transformation of the characteristic function. It is shown that the
identification of the entangled degrees of freedom is possible by studying the
Wigner quasiprobability distribution in phase space. We have shown that the
coordinates are entangled only with the conjugate momentum corresponding to
other coordinates.

Anisotropic oscillator; Peres-Horodecki criterion; Wigner distribution;
Noncommutative space; Coherent state; Symplectic diagonalization

##  I Introduction

There is a consensus that most of the theories of quantum gravity appear to
predict departures from classical gravity only at energy scales on the order
of $10^{19}$ GeV qg1 ; qg2  (By way of comparison, the LHC was designed to run
at a maximum collision energy of $14$ TeV lhc ). At that energy scale, the
space-time structure is believed to be deformed in such a manner that the
usual notion of commutative space will be ceased csseased1 ; csseased2 . It is
almost a Gospel that the fundamental concept of space-time is mostly
compatible with quantum theory in noncommutative space (NCS) Witten ; Juan .
Formalisms of Quantum theory in NCS have been successfully applied to various
physical situations ncsho ; ncsosc ; ncstopics ; ncsfluid . For the reviews on
NCS, one can see ncsreview1 ; ncsreview2 .
Perhaps, due to the present-day technological limitation of attainable energy,
the unified quantum theory of gravity lacks any direct experimental evidence.
Moreover, the huge gap between the required ($10^{19}$ GeV) and the currently
achievable energy scales suggests that we have to wait a long to successfully
perform quantum gravity (QG) experiments in the regime of present-day
colliders. However, instead of building a larger collider, there are various
suggestions for the passive experiments regarding the QG, with the help of
interferometry gravitationalwave1 ; catstatedeformed1 ; catstatedeformed2 ;
catstatedeformed3 , in particular with the utilization of Coherent states
(CS), for which the maximum contrast in the interference pattern is achieved
interfero1 . In the context of CS, the bipartite Gaussian states are
particularly important for their frequent appearance in systems with small
oscillations bipartitegaussian1 ; bipartitegaussian2 . For example, the
vibration of the test mass (corresponding to the masses on which the
interferometric mirrors are placed) under the effects of the tidal forces of
gravitational waves, can be demonstrated by an anisotropic oscillator
gravitationalwave1 ; gravitationalwave2 ; gravitationalwave3 ;
gravitationalwave4 , which exhibits a bipartite Gaussian state, upon
consideration in a two-dimensional noncommutative space entangledgaussian1 ;
entangledgaussian2 ; entangledgaussian3 ; entangledgaussian4 .
The entanglement properties of the bipartite state, corresponding to the
noncommutative spatial degrees of freedom, are the central theme of the
present article. A general separability criterion for a bipartite Gaussian
state was proposed in separability1 , which is a generalization for the
continuous variable states, over the discrete qubits separability2 ;
separability3 .
At first, using a Bopp’s shift in co-ordinates boppsshift1 ; boppsshift2 ;
boppsshift3 ; boppsshift4 , we have transformed the Hamiltonian
($\hat{H}_{nc}$) of the anisotropic harmonic oscillator in noncommutative
space, to the equivalent Hamiltonian ($\hat{H}_{c}$) in commutative space. The
structure of the $\hat{H}_{c}$ is similar to the Hamiltonians corresponding to
a charged particle moving in presence of an inhomogeneous magnetic field. One
of the major computational difficulties in the construction of the CS
structure of an anisotropic oscillator in a magnetic field lies in the
diagonalization of the quadratic system, keeping the intrinsic symplectic
structure intact. In this article, to construct a linear canonical
transformation (the group $Sp(4,\mathbb{R})$) to reduce our Hamiltonian to a
simplified representative one, we have revisited the formalisms stated in
separability4 ; separability5 ; separability6 ; separability7 ; separability8
. The ground state of the anisotropic harmonic oscillator (AHO) turns out to
be a bipartite Gaussian state. Utilizing Simon’s separability criterion
separability1 , we have obtained the restrictions on the masses
($m_{1},m_{2}$) and the parameters ($\theta,\eta$) (minimal length) of the
noncommutative coordinates and momentums, for which the CS is separable.
For an entangled state, the identification of the entangled degrees of freedom
is not straightforward. With the help of the well-known Wigner
quasiprobability distribution (WQD) wigner1 ; wigner2 ; wigner3 ; wigner4 ;
wigner5 ; wigner6 , we have identified the coordinates, which are entangled.
Since WQD is the Fourier transform of the expectation value of the
characteristic function of the random variables, it offers a phase-space
description of a quantum system. It is believed that one can conceptually
identify where the quantum corrections enter a problem by comparing it with
the classical version. However, philosophical debates in this regard are
inevitable, which we shall avoid in our present article. Rather, we shall
indulge ourselves in the computation of WQD for our problem, keeping in mind
that, being entirely real, WQD simplifies both the calculation and the
interpretation, of the results. In our case, the random variables are the co-
ordinate $(\hat{x}_{1},\hat{x}_{2})$ and momentum $(\hat{p}_{1},\hat{p}_{2})$
operators in commutative space. The study of the WQD suggests that for the
system under consideration (AHO in NCS) coordinate degrees of freedom are
entangled with the conjugate momentum corresponding to another coordinate. In
particular, $\hat{x}_{1}$ is entangled with $\hat{p}_{2}$ and $\hat{x}_{2}$ is
entangled with $\hat{p}_{1}$.
The organization of the article is the following. First, we have briefly
discussed the system under consideration. Then, we diagonalized the
Hamiltonian, with the help of a $Sp(4,\mathbb{R})$ canonical transformation.
Coherent states are formed from the annihilation operators. After that,
Simon’s separability criterion is utilized to identify the constraints on the
parameters, which can be used as an entanglement criterion for our system. The
identification of the entangled degrees of freedom was performed with the help
of the study of Wigner quasiprobability phase space distribution.

##  II Anisotropic harmonic oscillator in noncommutative space

We have studied the phenomenology of a two-dimensional anisotropic harmonic
oscillator in noncommutative (NC) space, with both the configuration and
momentum space non-commutativity. We shall consider the following Bopp’s shift
boppsshift1 ; boppsshift2 ; boppsshift3 ; boppsshift4 , which connects the NC-
space co-ordinates ($\hat{\tilde{X}}_{1},\hat{\tilde{X}}_{2}$) and momentum
($\hat{\tilde{P}}_{1},\hat{\tilde{P}}_{2}$) to the commutative space co-
ordinates ($\hat{x}_{1},\hat{x}_{2}$) and momentum
($\hat{p}_{1},\hat{p}_{2}$).

$\displaystyle\begin{array}[]{cc}\hat{\tilde{X}}_{1}=l_{0}(\hat{x}_{1}-\frac{\theta}{2\hbar}\hat{p}_{2}),&\hat{\tilde{P}}_{1}=l_{0}(\hat{p}_{1}+\frac{\eta}{2\hbar}\hat{x}_{2}),\\\
\hat{\tilde{X}}_{2}=l_{0}(\hat{x}_{2}+\frac{\theta}{2\hbar}\hat{p}_{1}),&\hat{\tilde{P}}_{2}=l_{0}(\hat{p}_{2}-\frac{\eta}{2\hbar}\hat{x}_{1}).\end{array}$
(3)

In boppsshift3 , it was shown that the noncommutative algebra is invariant
under scale factor $l_{0}$, so that all values of $l_{0}$ describe the same
physical model. For simplicity, we shall consider $l_{0}=1$ in our present
study. With this choice, the NC extension of quantum mechanics in two
dimensions is realized by the following commutation relations.

$\displaystyle\begin{array}[]{ccc}[\hat{\tilde{X}}_{1},\hat{\tilde{X}}_{2}]=i\theta,&[\hat{\tilde{P}}_{1},\hat{\tilde{P}}_{2}]=i\eta,&[\hat{\tilde{X}}_{i},\hat{\tilde{P}}_{j}]=i\hbar_{e}\delta_{ij},\end{array}$
(5)
$\displaystyle\mbox{with}\;\hbar_{e}=(1+\frac{\theta\eta}{4\hbar^{2}})\hbar.$
(6)

Whereas, the usual Heisenberg algebra for the commutative space reads

$\displaystyle[\hat{x}_{1},\hat{x}_{2}]=[\hat{p}_{1},\hat{p}_{2}]=0,\;[\hat{x}_{i},\hat{p}_{j}]=i\hbar\delta_{ij}.$
(7)

Where the Kronecker delta ($\delta_{ij}$) is defined as

$\displaystyle\delta_{ij}=\left\\{\begin{array}[]{cc}1,&i=j\\\ 0,&i\neq
j\end{array}\right.\;\;\;;i,j=1,2.$ (10)

We would like to mention that the noncommutative parameters $\theta$, $\eta$,
and the effective Planck constant
$\hbar_{e}=(1+\frac{\theta\eta}{4\hbar^{2}})\hbar$ are experimentally
measurable quantities for the scale factor $l_{0}=1$ boppsshift3 ; boppsshift4
.
In this NC space (NCS), let us consider, a two dimensional anisotropic
harmonic oscillator (AHO), which is described by the Hamiltonian

$\hat{H}_{nc}=\frac{1}{2m_{1}}\hat{\tilde{P}}_{1}^{2}+\frac{1}{2m_{2}}\hat{\tilde{P}}_{2}^{2}+\frac{1}{2}m_{1}\tilde{\omega}_{1}^{2}\hat{\tilde{X}}_{1}^{2}+\frac{1}{2}m_{2}\tilde{\omega}_{2}^{2}\hat{\tilde{X}}_{2}^{2}.$
(11)

Without loss of generality, we shall consider the mass ($m_{1},m_{2}$) and the
frequencies ( $\tilde{\omega}_{1},\tilde{\omega}_{2}$) are constants and can
have only positive values. We would like to mention that the Hamiltonian (11)
has different masses in different directions as well as different harmonic
potentials. Thus, it is anisotropic in two ways.
Although the linear transformation (3) changes the symplectic structure (thus,
not unitary), it enables us to convert the Hamiltonian in the NCS into a
modified Hamiltonian in the commutative space having an explicit dependence on
the deformation parameters $\theta,\eta$. Then the states of the system are
functions on the ordinary Hilbert space. The dynamics of the system are now
governed by the Schrödinger equation with the NC parameters $\theta$ and
$\eta$ dependent Hamiltonian, which reads

$\hat{H}=\frac{1}{2\mu_{1}}\hat{p}_{1}^{2}+\frac{1}{2\mu_{2}}\hat{p}_{2}^{2}+\frac{1}{2}\mu_{1}\omega_{1}^{2}\hat{x}_{1}^{2}+\frac{1}{2}\mu_{2}\omega_{2}^{2}\hat{x}_{2}^{2}+\hat{L}_{d},$
(12)

with

$\hat{L}_{d}=\nu_{1}\left\\{\hat{x}_{2},\hat{p}_{1}\right\\}-\nu_{2}\left\\{\hat{x}_{1},\hat{p}_{2}\right\\}.$
(13)

New parameters ($\mu_{1},\mu_{2},\omega_{1},\omega_{2},\nu_{1},\nu_{2}$ are
connected with the previous
($m_{1},m_{2},\tilde{\omega}_{1},\tilde{\omega}_{2}$) one by

$\displaystyle\frac{1}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{1}}+\frac{1}{4\hbar^{2}}m_{2}\tilde{\omega}_{2}^{2}\theta^{2},$
(14) $\displaystyle\frac{1}{\mu_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{2}}+\frac{1}{4\hbar^{2}}m_{1}\tilde{\omega}_{1}^{2}\theta^{2},$
(15) $\displaystyle\mu_{1}\omega_{1}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}\tilde{\omega}_{1}^{2}+\frac{\eta^{2}}{4\hbar^{2}m_{2}},$ (16)
$\displaystyle\mu_{2}\omega_{2}^{2}$ $\displaystyle=$ $\displaystyle
m_{2}\tilde{\omega}_{2}^{2}+\frac{\eta^{2}}{4\hbar^{2}m_{1}},$ (17)
$\displaystyle\nu_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4m_{1}\hbar}(\eta+m_{1}m_{2}\tilde{\omega}_{2}^{2}\theta),$
(18) $\displaystyle\nu_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4m_{2}\hbar}(\eta+m_{1}m_{2}\tilde{\omega}_{1}^{2}\theta).$
(19)

It is easy to see that the deformed angular momentum operator $\hat{L}_{d}$
will be reduced to the usual angular momentum operator for an isotropic
oscillator (IHO).
On the way of comparison of our system with an anisotropic oscillator in a
magnetic field, we note that an anisotropic oscillator (potential
$V_{an}(x_{1},x_{2})=k_{1}x_{1}^{2}/2+k_{2}x_{2}^{2}/2$) with charge $e$ and
effective mass $M=(\mu_{1},\mu_{2})$, under the magnetic vector potential

$\vec{\mathcal{A}}=(-\alpha_{01}x_{2},\alpha_{02}x_{1}),$ (20)

corresponds to the Hamiltonian

$\hat{H}_{mag}=\frac{1}{2\mu_{1}}\hat{p}_{1}^{2}+\frac{1}{2\mu_{2}}\hat{p}_{2}^{2}+\frac{1}{2}\alpha_{1}\hat{x}_{1}^{2}+\frac{1}{2}\alpha_{2}\hat{x}_{2}^{2}+\hat{L}_{d}.$
(21)

Since in $2$-dimension curl produces a pseudo-scalar, the vector potential
(20) corresponds to the homogeneous magnetic field

$B_{h}=\alpha_{01}+\alpha_{02}.$ (22)

One can identify that our system is equivalent to the anisotropic charged
oscillator in the magnetic field $B_{h}$ with the parameter values

$\displaystyle\begin{array}[]{ccc}\alpha_{1}=k_{1}+\frac{e^{2}\alpha_{02}^{2}}{\mu_{2}c^{2}},&\alpha_{2}=k_{2}+\frac{e^{2}\alpha_{01}^{2}}{\mu_{1}c^{2}},&\\\
\nu_{1}=\frac{e\alpha_{01}}{2\mu_{1}c},&\nu_{2}=\frac{e\alpha_{02}}{2\mu_{2}c},&B_{h}=\frac{2c}{e}(\mu_{1}\nu_{1}+\mu_{2}\nu_{2}),\end{array}$
(25)

$c$ being the speed of light in free space.
Moreover, one can see that these types of systems are also equivalent to the
two-dimensional Maxwell-Chern-Simon’s model in long wavelength limit, which
has a potential application in the theory of anyons Dunne .
For an IHO, the usual angular momentum operator ($\hat{L}$) commutes with the
Hamiltonian. Thus the problem can be reduced to the two noninteracting
harmonic oscillators. However, for AHO ($\nu_{1}\neq\nu_{2}$), the process of
diagonalization is not straightforward. In the next section, the
diagonalization of (12) is illustrated.

##  III diagonalization of the system

The Hamiltonian (12) can be expressed in the following quadratic form.

$\hat{H}=\frac{1}{2}X^{T}\hat{\mathcal{H}}X,$ (26)

where

$\displaystyle
X=\left(\hat{X}_{1},\hat{X}_{2},\hat{X}_{3},\hat{X}_{4}\right)=\left(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2}\right)^{T},\;\hat{\mathcal{H}}=\left(\begin{array}[]{cc}\hat{C}&\hat{A}^{T}\\\
\hat{A}&\hat{B}\end{array}\right),$ (29)
$\displaystyle\mbox{with}\;\;\hat{B}=\left(\begin{array}[]{cc}\mu_{2}\omega_{2}^{2}&0\\\
0&\frac{1}{\mu_{2}}\end{array}\right),\;\hat{C}=\left(\begin{array}[]{cc}\mu_{1}\omega_{1}^{2}&0\\\
0&\frac{1}{\mu_{1}}\end{array}\right),\;\hat{A}=\left(\begin{array}[]{cc}0&2\nu_{1}\\\
-2\nu_{2}&0\end{array}\right).$ (36)

$X^{T}$ denotes the matrix transposition of $X$. One can identify that

$\displaystyle\hat{\mathcal{S}}\hat{\mathcal{J}}_{4}+\hat{\mathcal{J}}_{4}\hat{\mathcal{S}}^{T}=0,$
(37)

where

$\displaystyle\hat{\mathcal{S}}=\hat{\mathcal{J}}_{4}\hat{\mathcal{H}},\;\;\mbox{with}\;\;\hat{\mathcal{J}}_{2n}=\left(\begin{array}[]{cc}0&\hat{\mathbb{I}}_{n}\\\
-\hat{\mathbb{I}}_{n}&0\end{array}\right),$ (40)

$\hat{\mathbb{I}}_{n}$ being the $n\times n$ identity matrix. From the
equation (37), we can conclude that $\mathcal{S}$ is a symplectic matrix
($\mathcal{S}\in Sp(4,\mathbb{R})$). Since our co-ordinates are the Cartesian
type, the natural range (spectrum) for each of them is the entire real line.
Moreover, the commutation relations

$[\hat{X}_{\alpha},\hat{X}_{\beta}]=-(\Sigma_{y})_{\alpha\beta};\;(\alpha,\beta=1,2,3,4),$
(41)

generate an intrinsic symplectic matrix

$\displaystyle\hat{\Sigma}_{y}=\mbox{diag}(\sigma_{y},\sigma_{y}),$ (42)

where the Pauli matrices are given by

$\displaystyle\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\;\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\;\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (49)

Here we have considered $\hbar=1$, which will be followed throughout our
present discussion.
The present section aims to diagonalize $\hat{H}$ keeping the symplectic
structure (41) intact.
The structure constants of the closed quasi-algebra of the co-ordinates
($\hat{X}_{i},i=1,..,4.$) with respect to the Hamiltonian ($\hat{H}$), induce
the following commutation relation.

$\left[i\hat{H},X\right]=\hat{\Omega}X,$ (50)

where

$\hat{\Omega}=i\hat{\Sigma}_{y}\hat{\mathcal{H}}.$ (51)

The next step is to diagonalize the matrix $\hat{\Omega}$. We would like to
relegate the detailed calculation of the diagonalization of $\hat{\Omega}$ to
Appendix A.
The diagonal Hamiltonian is given by

$\hat{H}_{D}=\frac{1}{2}\zeta^{\dagger}\hat{\Sigma}\zeta=\lambda_{1}\left(\hat{\zeta}_{1}^{\dagger}\hat{\zeta}_{1}+\frac{1}{2}\right)+\lambda_{2}\left(\hat{\zeta}_{2}^{\dagger}\hat{\zeta}_{2}+\frac{1}{2}\right),$
(52)

where
$\hat{\Sigma}=\mbox{diag}(\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2})$.
The explicit forms of $\zeta_{i},\;i=1,2$ and $\zeta_{i}^{\dagger},\;i=1,2$,
along with their algebra are given in subsections C and D of Appendix-A. The
detailed calculations of the four distinct purely imaginary eigenvalues ($\mp
i\lambda_{1},\mp i\lambda_{2}$) of $\hat{\Omega}$ are given in the
subsection-B of the Appendix-A.
Next section deals with the separability of the bipartite state of our system.

##  IV Separability of the bipartite state

The factorized form (52) of $\hat{H}_{D}$ suggests that the states of the
system may be expressed as

$\displaystyle|n_{1},n_{2}\rangle=\frac{1}{\sqrt{n_{1}!n_{2}!}}(\hat{\zeta}_{1}^{\dagger})^{n_{1}}(\hat{\zeta}_{2}^{\dagger})^{n_{2}}|0,0\rangle;n_{1},n_{2}=0,1,2,3,.....$
(53)

The corresponding energy eigenvalues are

$E_{n_{1},n_{2}}=\left(n_{1}+\frac{1}{2}\right)\lambda_{1}+\left(n_{2}+\frac{1}{2}\right)\lambda_{2}.$
(54)

For our purpose, we shall only need the ground state, which is given by

$\hat{\zeta}_{1}|0,0\rangle=\hat{\zeta}_{2}|0,0\rangle=0.$ (55)

A $2$-mode pure Gaussian state ($|\alpha,z\rangle$) is obtained from the
$2$-mode vacuum state $|0,0\rangle$ by gaussian1 ; gaussian2

$|\alpha,z\rangle=\hat{S}_{2}(z)\hat{D}_{2}(\alpha)|0,0\rangle,$ (56)

where the displacement operator
$\hat{D}_{2}(\alpha)=e^{\alpha^{T}\zeta_{a_{\dagger}}-\alpha^{\dagger}\zeta_{a}}$
and the squeezed operator
$\hat{S}_{2}(z)=e^{\frac{1}{2}(\zeta_{a}^{\dagger}z\zeta_{a_{\dagger}}-\zeta_{a}^{T}z^{\dagger}\zeta_{a})}$
are given through $\zeta_{a}=(\hat{\zeta}_{1},\hat{\zeta}_{2})^{T}$,
$\zeta_{a_{\dagger}}=(\hat{\zeta}_{1}^{\dagger},\hat{\zeta}_{2}^{\dagger})^{T}$,
along with the displacement vector $\alpha$ and a complex symmetric squeeze
matrix $z$. It is evident that $|0,0\rangle$ is a two mode Gaussian state with
$\alpha^{T}=(0,0),z=\hat{0}_{2\times 2}$. The advantage to use Gaussian states
is that they are entirely characterized by the covariance matrix. It means
that typical issues of continuous variables quantum information theory, which
are generally difficult to handle in an infinite Hilbert space, can be faced
up with the help of finite matrix theory gaussian3 .
The explicit form of $|0,0\rangle$ in position $\\{|x\rangle\\}$
representation reads

$\psi_{0}(x_{1},x_{2})=\langle
x_{1},x_{2}|0,0\rangle=A_{0}e^{-\Lambda_{11}x_{1}^{2}-\Lambda_{22}x_{2}^{2}-\Lambda_{12}x_{1}x_{2}}.$
(57)

Where

$\displaystyle\Lambda_{11}$ $\displaystyle=$
$\displaystyle\frac{\mu_{1}\mu_{2}\tau_{\lambda}\sqrt{\Delta_{\lambda}}}{4\delta_{\lambda}},$
(58) $\displaystyle\Lambda_{22}$ $\displaystyle=$
$\displaystyle\frac{\mu_{2}\tau_{\lambda}(\mu_{2}\omega_{2}^{2}-4\mu_{1}\nu_{1}^{2})}{4\delta_{\lambda}},$
(59) $\displaystyle\Lambda_{12}$ $\displaystyle=$
$\displaystyle\frac{2i\mu_{2}}{\delta_{\lambda}}(4\mu_{1}\nu_{1}^{2}\nu_{2}-\mu_{2}\nu_{2}\omega_{2}^{2}+\mu_{1}\nu_{1}\sqrt{\Delta_{\lambda}}),$
(60)

with

$\delta_{\lambda}=\mu_{2}(\omega_{2}^{2}+\sqrt{\Delta_{\lambda}})-4\mu_{1}\nu_{1}^{2}.$
(61)

$\tau_{\lambda}$ and $\Delta_{\lambda}$ are the trace and determinant of
$\hat{\Sigma}$, respectively. The normalization constant is given by
$|A_{0}|=\sqrt[4]{4\Lambda_{11}\Lambda_{22}/\pi^{2}}$.
It is worth noting that $\Lambda_{11}$ and $\Lambda_{22}$ are real, whereas
$\Lambda_{12}$ is purely imaginary, which makes the computation of expectation
values very simple. In particular,

$\displaystyle\begin{array}[]{cccc}\langle\hat{x}_{1}\rangle=0,&\langle\hat{x}_{1}^{2}\rangle=\frac{1}{4\Lambda_{11}},&\langle\hat{x}_{2}\rangle=0,&\langle\hat{x}_{2}^{2}\rangle=\frac{1}{4\Lambda_{22}},\\\
\langle\hat{p}_{1}\rangle=0,&\langle\hat{p}_{1}^{2}\rangle=\frac{d}{4\Lambda_{22}},&\langle\hat{p}_{2}\rangle=0,&\langle\hat{p}_{2}^{2}\rangle=\frac{d}{4\Lambda_{11}},\\\
\langle\hat{x}_{1}\hat{x}_{2}\rangle=0,&\langle\hat{p}_{1}\hat{p}_{2}\rangle=0,&\langle\hat{x}_{1}\hat{p}_{2}\rangle=\frac{i\Lambda_{12}}{4\Lambda_{11}},&\\\
\langle\left\\{\hat{x}_{1},\hat{p}_{1}\right\\}\rangle=0,&\langle\left\\{\hat{x}_{2},\hat{p}_{2}\right\\}\rangle=0,&\langle\hat{p}_{1}\hat{x}_{2}\rangle=\frac{i\Lambda_{12}}{4\Lambda_{22}}.&\end{array}\;;d=4\Lambda_{11}\Lambda_{22}-\Lambda_{12}^{2}.$
(66)

Since $\Lambda_{12}$ is purely imaginary, all the expectation values in (66)
are real. Moreover, the variances of the observables give

$\Delta\hat{x}_{1}\Delta\hat{p}_{1}=\Delta\hat{x}_{2}\Delta\hat{p}_{2}=\frac{1}{2}\sqrt{1+\frac{|\Lambda_{12}|^{2}}{4\Lambda_{11}\Lambda_{22}}},$
(67)

which means the state $\psi_{0}$ is a squeezed state for nonzero deformation
parameters. However, for the usual commutative space ($\Lambda_{12}=0$) the
squeezing goes off and $\psi_{0}$ is reduced to a coherent state.
To study the separability criterion, let us consider the variance matrix

$\displaystyle\hat{V}=\left(\begin{array}[]{cc}V_{11}&V_{12}\\\
V_{12}^{T}&V_{22}\end{array}\right),$ (70)

with

$\displaystyle V_{11}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\langle\hat{x}_{1}^{2}\rangle&\langle\\{\hat{x}_{1},\hat{p}_{1}\\}\rangle\\\
\langle\\{\hat{x}_{1},\hat{p}_{1}\\}\rangle&\langle\hat{p}_{1}^{2}\rangle\end{array}\right),$
(73) $\displaystyle V_{22}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\langle\hat{x}_{2}^{2}\rangle&\langle\\{\hat{x}_{2},\hat{p}_{2}\\}\rangle\\\
\langle\\{\hat{x}_{2},\hat{p}_{2}\\}\rangle&\langle\hat{p}_{2}^{2}\rangle\end{array}\right),$
(76) $\displaystyle V_{12}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\langle\hat{x}_{1}\hat{x}_{2}\rangle&\langle\hat{x}_{1}\hat{p}_{2}\rangle\\\
\langle\hat{p}_{1}\hat{x}_{2}\rangle&\langle\hat{p}_{1}\hat{p}_{2}\rangle\end{array}\right).$
(79)

The necessary and sufficient condition for the separability of a bipartite
Gaussian state reads separability1

$\displaystyle\Delta_{1}\Delta_{2}+\left(\frac{1}{4}-|\Delta_{12}|\right)^{2}-\mbox{Trace}(V_{11}\tilde{V}_{12}V_{22}\tilde{V}_{21})\geq(\Delta_{1}+\Delta_{2}).$
(80)

Where $\Delta_{1},\;\Delta_{2},\;\Delta_{12}$ are the determinant of
$V_{11},\;V_{22},V_{12}$ respectively, and

$\displaystyle\tilde{V}_{ij}=J_{2}V_{ij}J_{2},\;\;V_{21}=V_{12}^{T}.$ (81)

The symplectic matrix $J_{2}$ is given by (40).
Using the explicit forms (66) in (80), we can see that the bipartite state is
separable only for

$\Lambda_{12}^{2}\geq 0.$ (82)

However, $\Lambda_{12}$ is purely imaginary. Therefore, (82) holds only for
the case

$\Lambda_{12}=0.$ (83)

(83) is satisfied with one of the following conditions.

  * •

$\theta=\eta=0$: Corresponding to a 2-D anisotropic harmonic oscillator in
commutative-space.

  * •

$\tilde{\omega}_{1}=\tilde{\omega}_{2}$: The anisotropy due to the mass and
the noncommutative parameters.

  * •

If $m_{1},m_{2}$ and $\tilde{\omega}_{1},\tilde{\omega}_{2}$ satisfy

$\theta m_{1}\tilde{\omega}_{1}=\frac{\eta}{m_{2}\tilde{\omega}_{2}}.$ (84)

In other words, for nonzero values of the noncommutative parameters (nonzero
$\theta,\eta$) the bipartite states are entangled except for the parameter
values (84). In particular, this entanglement behavior solely depends on the
noncommutative parameters, which is in agreement with the result of
entangledgaussian1 . However, we would like to mention that, in
entangledgaussian1 , only the position-position noncommutativity was
considered, whereas, we have considered both position-position and momentum-
momentum non-commutativity. That means, our result is a generalization of the
previously reported results. In the following subsection, we have shown that
this type of entanglement is generated due to the noncommutative geometry.

###  IV.1 The entanglement is induced due to the noncommutative geometry

If we consider the limit $\theta\to 0,\eta\to 0$, the eigenvalues of
$\hat{\Omega}$ is reduecd to

$\lambda=\\{\pm i\omega_{1},\pm i\omega_{2}\\},$ (85)

which means

$\displaystyle\begin{array}[]{ccc}\tau_{\lambda}=2(\omega_{1}+\omega_{2}),&\Delta_{\lambda}=\omega_{1}^{2}\omega_{2}^{2},&\delta_{\lambda}=\mu_{2}\omega_{2}(\omega_{1}+\omega_{2}),\\\
\Lambda_{11}=\frac{1}{2}\mu_{1}\omega_{1},&\Lambda_{22}=\frac{1}{2}\mu_{2}\omega_{2},&\Lambda_{12}=0.\end{array}$
(88)

The expectation values then reduced to

$\displaystyle\begin{array}[]{cccc}\langle\hat{x}_{1}\rangle=0,&\langle\hat{x}_{1}^{2}\rangle=\frac{1}{2\mu_{1}\omega_{1}},&\langle\hat{x}_{2}\rangle=0,&\langle\hat{x}_{2}^{2}\rangle=\frac{1}{2\mu_{2}\omega_{2}},\\\
\langle\hat{p}_{1}\rangle=0,&\langle\hat{p}_{1}^{2}\rangle=\frac{1}{2}\mu_{1}\omega_{1},&\langle\hat{p}_{2}\rangle=0,&\langle\hat{p}_{2}^{2}\rangle=\frac{1}{2}\mu_{2}\omega_{2},\\\
\langle\hat{x}_{1}\hat{x}_{2}\rangle=0,&\langle\hat{p}_{1}\hat{p}_{2}\rangle=0,&\langle\hat{x}_{1}\hat{p}_{2}\rangle=0,&\\\
\langle\left\\{\hat{x}_{1},\hat{p}_{1}\right\\}\rangle=0,&\langle\left\\{\hat{x}_{2},\hat{p}_{2}\right\\}\rangle=0,&\langle\hat{p}_{1}\hat{x}_{2}\rangle=0.&\end{array}$
(93)

That means, $\hat{x}_{1}$ and $\hat{p}_{2}$, as well as $\hat{x}_{2}$ and
$\hat{p}_{1}$ are not correlated for usual commutative space. However, from
(66), we can see that the expectation values of $\hat{x}_{1}\hat{p}_{2}$ and
$\hat{p}_{1}\hat{x}_{2}$ are nonzero for nonzero parameters of noncommutative
geometry ($\theta,\eta$). This indicates the correlations between
$\hat{x}_{1}$ and $\hat{p}_{2}$, as well as $\hat{x}_{2}$ and $\hat{p}_{1}$.
This type of entanglement is due to the noncommutative geometry.
In the next section, we have illustrated this fact with the help of the Wigner
quasiprobability distribution.

##  V Wigner Distribution

The characteristic function ($\hat{M}(\vec{\tau},\vec{\theta})$) of the random
variable $X=(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2}),$ is defined by

$\hat{M}(\vec{\tau},\vec{\theta})=e^{i\mbox{Trace}(t^{T}X)}.$ (94)

with the parameter

$t=(\tau_{1},\theta_{1},\tau_{2},\theta_{2}).$ (95)

The Fourier transformation of the expectation value of $\hat{M}$ is known as
the Wigner quasiprobability distribution (WQD). For example, the WQD for a
quantum state $\psi(x_{1},x_{2})$ may be given by

$\displaystyle
W(x_{1},p_{1},x_{2},p_{2})=\frac{2}{\pi\hbar^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}[\psi^{*}(x_{1}-t_{1},x_{2}-t_{2})e^{-\frac{2i}{\hbar}(t_{1}p_{1}+t_{2}p_{2})}$
$\displaystyle\psi(x_{1}+t_{1},x_{2}+t_{2})]dt_{1}dt_{2}.$ (96)

Using (57) in (V) and considering (58), (59), (60) and (66), the direct
computation produces the WQD for the anisotropic oscillator in NCS. In
particular,

$\displaystyle
W(x_{1},p_{1},x_{2},p_{2})=\frac{2}{\pi\hbar^{2}}\exp(-2\langle\hat{p}_{1}^{2}\rangle
x_{1}^{2}-2\langle\hat{p}_{2}^{2}\rangle
x_{2}^{2}-2\langle\hat{x}_{1}^{2}\rangle
p_{1}^{2}-2\langle\hat{x}_{2}^{2}\rangle p_{2}^{2}$
$\displaystyle+4\langle\hat{x}_{1}\hat{p}_{2}\rangle
x_{2}p_{1}+4\langle\hat{p}_{1}\hat{x}_{2}\rangle x_{1}p_{2}).$ (97)

projection on $(x_{1},p_{1})=(1,1)$ plane.

projection on $(x_{2},p_{2})=(1,1)$ plane.

projection on $(p_{1},p_{2})=(1,1)$ plane.

projection on $(x_{1},x_{2})=(1,1)$ plane.

Figure 1: Wigner distribution $W(1,1,x_{2},p_{2})$, $W(x_{1},p_{1},1,1)$,
$W(x_{1},1,x_{2},1)$ and $W(1,p_{1},1,p_{2})$ shows no entanglement effect.

Projection on $(p_{1},x_{2})=(1,1)$ plane

Projection on $(x_{1},p_{2})=(1,1)$ plane

Figure 2: Wigner distribution $W(x_{1},1,1,p_{2})$ and $W(1,x_{2},p_{1},1)$
are exhibiting the entanglement effects. They are not appearing as a separated
Gaussian distribution.

For the illustration purpose, let us choose

$\displaystyle\langle\hat{x}_{1}^{2}\rangle=\langle\hat{x}_{2}^{2}\rangle=\langle\hat{p}_{1}^{2}\rangle=\langle\hat{p}_{2}^{2}\rangle=\frac{1}{2},$
$\displaystyle\langle\hat{x}_{1}\hat{p}_{2}\rangle=\langle\hat{p}_{1}\hat{x}_{2}\rangle=-\frac{1}{2}.$
(98)

Since our $W$ is on four-dimensional phase space, we have to project it onto
two-dimensional space to visualize it graphically. The FIG. 1 represents the
projection of WQD onto $(x_{1},p_{1})=(1,1)$, $(x_{2},p_{2})=(1,1)$,
$(p_{1},p_{2})=(1,1)$ and $(x_{1},x_{2})=(1,1)$ planes respectively. They
exhibit no entanglement effect. Being projected onto the two-dimensional
phase-space, they appear to be an individual Gaussian distribution. However,
from FIG. 2, one can envisage the entanglement of the co-ordinates
$(x_{1},p_{2})$ and $(x_{2},p_{1})$. In particular, $x_{1}$ is entangled with
$p_{2}$ and $x_{2}$ is entangled with $p_{1}$. This confirms the indications
we had to get from the correlations in the previous section. In particular,
the figures are consistent with the correlations matrix (variance matrix) of
the last section.

##  VI Experimental connections

###  VI.1 Szilard engine cycle

One of the possibilities of results of the present study is to use it for the
Szilard engine cycle szilard ; szilard1 ; szilard2 ; szilard3 ; szilard4 ;
szilard5  for a bipartite system. Gaussian measurement performed by the second
party (corresponding to $x_{2}$) corresponds to the variance matrix
measurementgaussian

$\hat{\gamma}^{\pi_{2}}=\hat{R}(\theta_{\pi_{2}})\mbox{diag}(\mu/2,\mu^{-1}/2)\hat{R}^{T}(\theta_{\pi_{2}}),\;\;\mu\in[0,\infty).$
(99)

Where the rotation operator is given through the Pauli matrix by
$\hat{R}(\theta_{\pi_{2}})=\cos\theta_{\pi_{2}}\hat{\mathbb{I}}-i\sin\theta_{\pi_{2}}\hat{\sigma}_{y}$.
The measurement parameter $\mu=0$ and $\mu=1$ corresponds to homodyne and
heterodyne measurement respectively. The conditional state covariance matrix
is given by

$\hat{V}_{11}^{\pi_{2}}=\hat{V}_{11}-\hat{V}_{12}(\hat{V}_{22}+\hat{\gamma}^{\pi_{2}})^{-1}\hat{V}_{12}^{T}.$
(100)

Then the extractable work ($W_{se}=K_{B}T[S(V_{11})-S(V_{11}^{\pi_{2}})]$) due
to the measurement backreaction is given by

$W_{se}=\frac{1}{2}k_{B}T\ln\left(\frac{det(\hat{V}_{11})}{det(\hat{V}_{11}^{\pi_{2}})}\right),$
(101)

where we have used the Rényi entropy of order $2$,
$S_{2}(\rho)=-\ln(Tr(\rho^{2}))$, which in case of Gaussian state becomes
$S_{2}(V_{12})=\frac{1}{2}\ln(det(V_{12}))$.
In the concerned case of this present paper, for a heterodyne measurement
($\mu=1$), the extractable work is reduecd to

$W_{se}=-\frac{1}{2}k_{B}T\ln\left[\left(1+\frac{\Lambda_{12}^{2}}{(d+2\Lambda_{11})}\right)\left(1+\frac{\Lambda_{12}^{2}}{d(1+2\Lambda_{22})}\right)\right]\neq
0.$ (102)

In particular, the non-commutativity of the space opens up a possibility of
extractable work protocol.

###  VI.2 Determination of the signature of noncommutative of space: an Opto-
mechanical scheme

The optomechanical scheme is realized through an interaction between an
optical pulse and a mechanical oscillator expt1 ; expt2 . The setup consists
of a half-silvered mirror, two Fabry-Perot cavities, a laser source, and a
detector (for instance see expt1 ; expt2 ; expt3 ), on which a sequence of
four radiation pressure interactions form an optical cavity (resonator)
yielding the interaction operator

$\hat{\zeta}_{lm}=\hat{U}_{lm,p}\hat{U}_{lm,x}^{\dagger}\hat{U}_{lm,p}^{\dagger}\hat{U}_{lm,x},\;\mbox{with}\;\hat{U}_{lm,p}=e^{i\hat{n}_{l}\sum_{j=1}^{2}\lambda_{jl}\hat{\tilde{P}}_{j}^{m}},\;\hat{U}_{lm,x}=e^{i\hat{n}_{l}\sum_{j=1}^{2}\lambda_{jl}\hat{\tilde{X}}_{j}^{m}},$
(103)

where the index $m$ and $l$ stand for the identification of the mechanical
quantities and the optical degrees of freedom, respectively, and
$\lambda_{l}=(\lambda_{1l},\lambda_{2l})$ is the optical field interaction
length. $\hat{\zeta}_{lm}$ displaces the mechanical state around complete loop
in phase space, thus creating an additional phase, which is measurable in a
tabletop experiment, in principle. In particular, the mean of the optical
field operator

$\langle\hat{a}\rangle=\langle\alpha|\hat{\zeta}_{lm}^{\dagger}\hat{a}\hat{\zeta}_{lm}|\alpha\rangle=\langle\hat{a}\rangle_{QM}e^{-i\Theta}$
(104)

provides the additional phase ($\Theta$) on the light arising due to the
presence of the noncommutative oscillator. Whereas, the mean of the
annihilation operator $\hat{a}$ of the optical field $|\alpha\rangle$ (the
input optical coherent state with mean photon number $n_{p}$) using the
standard quantum mechanics (with commutative space) reads

$\langle\hat{a}\rangle_{QM}=\alpha
e^{-i|\lambda_{l}|^{2}-n_{p}(1-e^{-2i|\lambda_{l}|^{2}})}.$ (105)

The additional phase factor $\Theta$ depends on the product of the
noncommutative parameters ($\theta\eta$). That means the nonzero $\Theta$
implies both position-position and momentum-momentum noncommutativity.

##  VII Conclusions

In this paper, we have studied in detail the entanglement property of two
dimensional anisotropic (a fairly general form of anisotropy) harmonic
oscillator (AHO, where the entanglement is induced by both the spatial and
momentum non-commutativity. It turns out that the bipartite Gaussian state
(BGS) is almost always entangled. Simon’s separability criterion shows that
BGS is separable for a set of specific values of the masses and frequencies of
the oscillators. The explicit form of the constraint equation is determined in
the present paper. An additional interaction term, dependent upon the
noncommutativity parameter $\theta$ and $\eta$, is generated by Bopp’s shift,
which is the main source for the entanglement between the coordinates and
momentum degrees of freedom.
Wigner quasiprobability distribution (WQD) for the ground state of our system
is computed. Projecting the four-dimensional phase space WQD onto two
dimensions, we have graphically identified the exact degrees of freedom, which
are entangled. This is also demonstrated by calculating the correlations
(variances) between the observables. This opens up the possibility of direct
experimental observation through interferometry. In particular, the obtained
coherent state structure can be utilized in an optomechanical setup that helps
us to transfer the information of a noncommutative anisotropic oscillator to
the high-intensity optical pulse in terms of a sequence of opto-mechanical
interaction inside an optical resonator expt1 ; expt2 ; expt3 ; expt4 ; expt5
. Consequently, the optical phase shift is easily measurable with very high
accuracy through an interferometric system as stated in expt1 . This makes the
whole procedure much easier to collect the pieces of information of
noncommutative structures through optical systems already available to us. In
particular, by comparing the phase difference of the input signal and output
signal after interaction of the input signal with the mechanical anisotropic
oscillator inside the cavity, and comparing the phase difference with the
prediction of usual quantum mechanics, we can determine the signature of
quantum gravity (if any). Thus, it may bypass all the difficulties of probing
high energy scales through scattering experiments. However, we would like to
mention that, within this type of experimental set-up only the product
$\theta\eta$ are determined, not the individual $\theta$ and $\eta$. Moreover,
the typical size ( distance between mirrors) of the resonant cavity for a
Fabry-Perot Interferometer should be an order of a few kilometers to have any
detectable effect in this scenario expt1 .
Moreover, the amount of extractable work for a heterodyne measurement through
Szilard engine cycle for our bipartite system is outlined in this paper. This
shows the applicability of our results in thermodyanmic process, i.e, in
engineering applications.
In terms of constructor-theoretic principles marletto1 ; bose1 , if we observe
the entanglement effects in the measurement of the properties of two quantum
masses that interact with each other through gravity only, then we can
conclude that the mediator (gravity) got to have some quantum features
marletto1 ; bose1 . It doesn’t matter in what way gravity is quantum - whether
it’s loop quantum gravity or string theory or something else - but it has to
be a quantum theory. We have seen in the present paper that the entanglement
in co-ordinate degrees of freedom is solely depends on the noncommutative
parameters. Thus any experimental signature of entanglement for the present
scenario will indicate the signature of noncommutativity of space.

##  VIII Acknowledgement

I am grateful to the anonymous referees for their valuable suggestions.
This research was supported in part by the International Centre for
Theoretical Sciences (ICTS) for the online program - Non-Hermitian Physics
(code: $ICTS/nhp2021/3$).

##  IX Conflict of interest

The author declares that there is no conflict of interest for the present
article.

##  X Data availability statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  Appendix A Diagonalization of $\hat{\Omega}$

$\hat{\Omega}$ is a normal matrix. So, $\hat{\Omega}$ is diagonalizable
through a similarity transformation

$\hat{\Omega}_{D}=\hat{Q}^{-1}\hat{\Omega}\hat{Q}.$ (106)

$\hat{Q}^{-1}$ and $\hat{Q}$ are obtained by arranging the left and right
eigenvectors column-wise, respectively. Since, $\hat{\Omega}$ is not a
symmetric matrix ($\hat{\Omega}^{T}\neq\hat{\Omega}$), the left and right
eigenvectors of $\hat{\Omega}$ are not same. However, the left and right
eigenvalues are the same. The characteristic polynomial
($p(\lambda)=\det(\lambda\mathbb{\hat{I}}-\hat{\Omega}),$) of $\hat{\Omega}$
is given by

$p(\lambda)=\lambda^{4}+b\lambda^{2}+c.$ (107)

Where

$\displaystyle c$ $\displaystyle=$
$\displaystyle\omega_{x}^{2}\omega_{y}^{2},$ (108) $\displaystyle b$
$\displaystyle=$
$\displaystyle\alpha_{0}\omega_{x}^{2}+\frac{1}{\alpha_{0}}\omega_{y}^{2}+4\nu_{1}\nu_{2}(\sqrt{\alpha_{0}}+\frac{1}{\sqrt{\alpha_{0}}})^{2}.$
(109)

With

$\displaystyle\omega_{x}^{2}$ $\displaystyle=$ $\displaystyle
4\nu_{1}\nu_{2}(\frac{\mu_{1}\omega_{1}^{2}}{4\mu_{2}\nu_{2}^{2}}-1),$ (110)
$\displaystyle\omega_{y}^{2}$ $\displaystyle=$ $\displaystyle
4\nu_{1}\nu_{2}(\frac{\mu_{2}\omega_{2}^{2}}{4\mu_{1}\nu_{1}^{2}}-1),$ (111)
$\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle\frac{\mu_{2}\nu_{2}}{\mu_{1}\nu_{1}}.$ (112)

using the explicit forms of $\mu_{i},\nu_{i},\omega_{i},\;i=1,2$, one can show
that $c\geq 0,\;b\geq 0$ as follows.

###  A.1 Proof of $b\geq 0$, $c\geq 0$

We have

$\displaystyle b$ $\displaystyle=$
$\displaystyle\omega_{1}^{2}+\omega_{2}^{2}+6\nu_{1}\nu_{2},$ (113)
$\displaystyle c$ $\displaystyle=$
$\displaystyle\omega_{1}^{2}\omega_{2}^{2}+16\nu_{1}^{2}\nu_{2}^{2}-4\nu_{1}^{2}\omega_{1}^{2}\mu_{1}/\mu_{2}-4\nu_{2}^{2}\omega_{2}^{2}\mu_{2}/\mu_{1}.$
(114)

Clearly, $b>0$ for nonzero positive values of
$\omega_{1},\omega_{2},\nu_{1},\nu_{2}$.
First we observe that, $c$ can be factorized as

$c=16\nu_{1}^{2}\nu_{2}^{2}\left(\frac{\mu_{1}\omega_{1}^{2}}{4\mu_{2}\nu_{2}^{2}}-1\right)\left(\frac{\mu_{2}\omega_{2}^{2}}{4\mu_{1}\nu_{1}^{2}}-1\right).$
(115)

If we use the explicit forms of $\omega_{1},\omega_{2},\nu_{1},\nu_{2}$, we
have

$\displaystyle\frac{\mu_{1}\omega_{1}^{2}}{4\mu_{2}\nu_{2}^{2}}$
$\displaystyle=$
$\displaystyle\frac{4m_{2}^{2}\hbar^{2}}{(\eta+m_{1}m_{2}\theta\tilde{\omega}_{1}^{2})^{2}}\left(m_{1}\tilde{\omega}_{1}^{2}+\frac{\eta^{2}}{4\hbar^{2}m_{2}}\right)\left(\frac{1}{m_{2}}+\frac{1}{4\hbar^{2}}m_{1}\tilde{\omega}_{1}^{2}\theta^{2}\right)$
(116) $\displaystyle=$
$\displaystyle\left(1+\frac{4\hbar^{2}}{\eta^{2}}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)\left(1+\frac{\theta^{2}}{4\hbar^{2}}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)\left(1+\frac{\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)^{-2}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1+\frac{\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)^{2}}\left(\left(1+\frac{\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)^{2}+\frac{2\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{1}^{2}(\frac{2\hbar^{2}}{\eta\theta}+\frac{\eta\theta}{8\hbar^{2}}-1)\right)$
$\displaystyle=$ $\displaystyle 1+\frac{2\theta
m_{1}m_{2}\tilde{\omega}_{1}^{2}}{\eta\left(1+\frac{\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{1}^{2}\right)^{2}}\left(\frac{\sqrt{2}\hbar}{\sqrt{\eta\theta}}-\frac{\sqrt{\eta\theta}}{2\sqrt{2}\hbar}\right)^{2}\geq
1.$

Similarly, we see that

$\frac{\mu_{2}\omega_{2}^{2}}{4\mu_{1}\nu_{1}^{2}}=1+\frac{2\theta
m_{1}m_{2}\tilde{\omega}_{2}^{2}}{\eta\left(1+\frac{\theta}{\eta}m_{1}m_{2}\tilde{\omega}_{2}^{2}\right)^{2}}\left(\frac{\sqrt{2}\hbar}{\sqrt{\eta\theta}}-\frac{\sqrt{\eta\theta}}{2\sqrt{2}\hbar}\right)^{2}\geq
1.$ (117)

Therefore, we have

$c\geq 0.$ (118)

Moreover, the discriminant of the characteristic polynomial

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle b^{2}-4c$ (119)
$\displaystyle=$
$\displaystyle\left(\alpha_{0}\omega_{x}^{2}-\frac{1}{\alpha_{0}}\omega_{y}^{2}\right)^{2}+8\nu_{1}\nu_{2}(1+\alpha_{0})^{2}(\omega_{x}^{2}+\omega_{y}^{2})+\frac{4\nu_{1}^{2}\nu_{2}^{2}}{\alpha_{0}^{2}}(1+\alpha_{0})^{4}$

is positive definite. It can be easily verified that

$\lambda^{2}=\frac{1}{2}(-b\pm\sqrt{\Delta})\leq 0.$ (120)

###  A.2 Eigenvalues and eigenvectors of $\hat{\Omega}$

From (120), we can see that $\hat{\Omega}$ has four distinct imaginary eigen-
values

$\displaystyle\lambda=\left\\{-i\lambda_{1},i\lambda_{1},-i\lambda_{2},i\lambda_{2}\right\\}.$
(121)

Where

$\lambda_{1}=\frac{1}{\sqrt{2}}\sqrt{b+\sqrt{\Delta}},\;\lambda_{2}=\frac{1}{\sqrt{2}}\sqrt{b-\sqrt{\Delta}}.$
(122)

Let us consider the eigenvalue equations of $\hat{\Omega}$ as follows.

$\displaystyle u_{j}\hat{\Omega}$ $\displaystyle=$
$\displaystyle-i\lambda_{j}u_{j},\;j=1,2.$ (123) $\displaystyle
u_{j}^{*}\hat{\Omega}$ $\displaystyle=$ $\displaystyle
i\lambda_{j}u_{j}^{*}\;,j=1,2.$ (124) $\displaystyle\hat{\Omega}v_{j}$
$\displaystyle=$ $\displaystyle-i\lambda_{j}v_{j},\;j=1,2.$ (125)
$\displaystyle\hat{\Omega}v_{j}^{*}$ $\displaystyle=$ $\displaystyle
i\lambda_{j}v_{j}^{*},\;j=1,2.$ (126)

Then the following identities hold quite generally.

$u_{i}^{*}v_{j}=u_{i}v_{j}^{*},\;u_{i}v_{j}=u_{i}^{*}v_{j}^{*}=\delta_{ij},\;\forall
i,j=1,2.$ (127)

(127) suggests the following conditions between the left and right eigen-
vectors.

$v_{i}=-\Sigma_{y}u_{i}^{\dagger},\;(i=1,2).$ (128)

In our case, the explicit form of the left eigenvectors $u_{i},\;(i=1,2)$ are
given by

$\displaystyle
u_{i}=\frac{1}{k_{i}}\left(\begin{array}[]{c}-i\lambda_{i}\mu_{1}\mu_{2}(\lambda_{i}^{2}-\omega_{2}^{2}-4\nu_{1}\nu_{2})\\\
\mu_{2}(\lambda_{i}^{2}-\omega_{2}^{2})+4\mu_{1}\nu_{1}^{2}\\\
2\nu_{1}\mu_{1}\mu_{2}(\lambda_{i}^{2}-4\nu_{1}\nu_{2})+2\nu_{2}\mu_{2}^{2}\omega_{2}^{2}\\\
2i\lambda_{i}(\mu_{1}\nu_{1}+\mu_{2}\nu_{2})\end{array}\right)^{T}.$ (133)

The right eigenvectors can be constructed from (128) in a straighforward
manner.

###  A.3 Similarity transformation

Now the similarity transformation $\hat{Q}$ (as well as $\hat{Q}^{-1}$) can be
constructed by arranging the eigen-vectors columnwise. In particular,

$\displaystyle\hat{Q}$ $\displaystyle=$
$\displaystyle(v_{1},v_{1}^{*},v_{2},v_{2}^{*}),$ (134)
$\displaystyle\hat{Q}^{-1}$ $\displaystyle=$
$\displaystyle(u_{1}^{T},(u_{1}^{*})^{T},u_{2}^{T},(u_{2}^{*})^{T}).$ (135)

One can verify that

$\hat{Q}^{\dagger}=-\hat{\Sigma}_{z}\hat{Q}^{-1}\hat{\Sigma}_{y},$ (136)

with

$\hat{\Sigma}_{z}=\mbox{diag}(\sigma_{z},\sigma_{z}).$ (137)

Now, using (106), we have the diagonalization representation
$\hat{\Omega}_{D}$ of $\hat{\Omega}$ as

$\hat{\Omega}_{D}=\mbox{diag}(-i\lambda_{1},i\lambda_{1},-i\lambda_{2},i\lambda_{2}).$
(138)

The similarity transformation suggests a new co-ordinate vector $\hat{\zeta}$,
which is connected with the original co-ordinate vector $X$ of (29) through

$\hat{\zeta}=\hat{Q}^{-1}X.$ (139)

Then, using (136), we can write

$\displaystyle X^{\dagger}$ $\displaystyle=$
$\displaystyle\hat{\zeta}^{\dagger}(-\hat{\Sigma}_{z}\hat{Q}^{-1}\hat{\Sigma}_{y}).$
(140)

###  A.4 Equivalent dynamics in diagonal representation

The time-dependent Schrödinger equation

$\hat{H}\psi=i\hbar\frac{\partial\psi}{\partial t},$ (141)

is transformed with the similarity transformation as follows.

$\displaystyle\hat{Q}^{-1}\frac{1}{2}X^{\dagger}\hat{\mathcal{H}}X\psi\hat{Q}$
$\displaystyle=$ $\displaystyle i\hbar\hat{Q}^{-1}\frac{\partial\psi}{\partial
t}\hat{Q}.$ (142)
$\displaystyle\therefore\frac{1}{2}\hat{\zeta}^{\dagger}\Sigma\hat{\zeta}\tilde{\psi}_{Q}$
$\displaystyle=$ $\displaystyle i\hbar\frac{\partial\tilde{\psi}_{Q}}{\partial
t}.$ (143)

Where

$\displaystyle\hat{\Sigma}=\mbox{diag}(\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2}),$
(144) $\displaystyle\tilde{\psi}_{Q}=\hat{Q}^{-1}\psi\hat{Q}.$ (145)

That means, the new diagonal Hamiltonian

$\hat{H}_{D}=\frac{1}{2}\zeta^{\dagger}\hat{\Sigma}\zeta$ (146)

obeys the same dynamics as that of $\hat{H}$, with the new co-ordinate vector
$\zeta$.
If we compute the commutators of the components of $\hat{\zeta}$, then we can
identify that they forms a set of independent annihilation and creation
operators. In particular,

$\left[\hat{\zeta}_{k},\hat{\zeta}_{l}^{\dagger}\right]=\delta_{kl};\;k,l=1,2.$
(147)

Where

$\hat{\zeta}=(\hat{\zeta}_{1},\hat{\zeta}_{1}^{\dagger},\hat{\zeta}_{2},\hat{\zeta}_{2}^{\dagger})^{T}.$
(148)

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