km, shifting the x-axis of all $\dot{E}$ plots to the right. Furthermore,
especially MSPs have a range of masses, of order only 60%, neglected in our
single $I$ value. Luminosity may also depend on parameters other than
$\dot{E}$, such as the inclination angles, which would contribute to the
scatter. The correlation is stronger at higher spindown powers, and is less
pronounced for the MSPs. It remains to be understood whether this is due to
high energy processes “shrinking” as $\dot{E}$ tends toward the “deathline”
near $10^{33}$ erg s-1, or is because e.g. of the compactness of the MSP
magnetosphere.

Highly polarized radio beams generally occur only for pulsars with
$\dot{E}>10^{34}$ erg$s^{-1}$ (Weltevrede & Johnston, 2008; Johnston & Kerr,
2018). Smith et al. (2019) suggested it is an unlikely coincidence that both
radio polarization and gamma-ray emission require the same minimum spin
parameters for independent reasons – that these two phenomena share the same
deathline may mean that the same electron populations produce both signals.

###  8.5 Towards 4PC and 400 gamma-ray pulsars

A small number ($\sim 20$) of our gamma-ray pulsars are below the 4FGL catalog
detection threshold. Thus we expect that the bulk of future pulsed discoveries
will correspond to the thousand or more currently unidentified LAT sources.
Requiring non-variability and a pulsar-like spectrum reduces this to a few
hundred. It thus appears unlikely that the LAT will ever more than double the
size of the current sample.

However, another hundred gamma-ray pulsars is possible, and dozens are likely.
Machine-learning methods applied to the 4FGL sources typically find $\sim 70$
pulsar-like unassociated objects with $|b|>10^{\circ}$ (Mirabal & Bonaca,
2021). Future high-sensitivity radio surveys with SKA are predicted to find
$27,000$ young pulsars and $3,000$ MSPs (Keane et al., 2015). Radio timing
solutions for those co-located with LAT source would allow gamma-ray folding,
at least for the years following the radio discovery. A radio position,
period, and orbital period for the binaries would improve the sensitivity and
decrease the computing cost for gamma-ray blind searches. Without waiting for
SKA, TRAPUM at MeerKAT and FAST are already finding new pulsars. Sustained
timing of the new discoveries will require thoughtful management of available
resources. The insights we have achieved in this work into the shapes of
pulsar spectra will help quantify how many of the unidentified LAT sources are
likely to be pulsars without pulsed detections.

We intend to maintain the online catalog (see the footnote at the end of
Section 1). We will add new gamma-ray pulsars as they accumulate. If new
analyses are performed, e.g. on-peak spectral analysis, we will add those
results.

Pulsar observations enable investigations over a spectacular range of topics,
from fundamental physics to variations of the interstellar medium, including
several aspects of stellar evolution, and more. Our gamma-ray pulsar sample is
so large that even sub-samples selected for specific studies have more members
than did the entire collection acquired by Compton Gamma-ray Observatory, over
twenty years ago. Discoveries enabled by exploiting the sample will expand for
many years to come, at the same time that Fermi LAT keeps finding more, and
better characterizes those already known.

##  Appendix A Radio Timing for This Work

PSR J0653+4706:  PSR J0653+4706 was discovered with LOFAR (van Haarlem et al.,
2013) at 135 MHz in a targeted pulsar survey towards unidentified Fermi LAT
gamma-ray sources (Pleunis et al., 2017; Bassa et al., 2017b). The survey used
semi-coherent dedispersion to minimize the effects of dispersive smearing at
these low frequencies while retaining sensitivity to MSPs (Bassa et al.,
2017a). The pulsar has a 4.76 ms spin period and a dispersion measure of
$\mathrm{DM}=25.74\pm 0.14$ pc cm-3, placing it at a distance of $\sim 900$ pc
with both the NE2001 and the YMW16 models. It is in a 5.84 day binary system
with a $M_{\mathrm{c}}\gtrsim 0.21$ M⊙ binary companion (Bassa et al., 2018).
The timing solution used in the catalog is obtained from LOFAR timing
observations covering a 2.8 yr timespan. Subsequently, a $5.2$ year solution
using Jodrell Bank Observatory L-band observations confirmed the model.

The LOFAR data indicate a flux density of $5\pm 2$ mJy at 150 MHz. An 1800 s
Green Bank Telescope observation at 820 MHz, with 200 MHz bandwidth, yielded a
$37.2\sigma$ detection. For 2 polarizations, gain $G=2$ K Jy -1, system
temperature T${}_{sys}=29$ K, and a 12% duty cycle seen in both the 150 and
820 MHz profiles, we obtain a flux density of $260\,\mu$Jy, calculated using
the PRESTO sum_profiles.py routine.

Analysis of 101 positive detections at 1400 MHz with the Lovell telescope at
Jodrell Bank Observatory yields flux densities of $89\pm 11$ and $10\pm
3\,\mu$Jy for the main peak and interpulse, respectively. The uncertainties
are statistical, and include variations due to interstellar scintillation. The
total is $S_{1400}=99\pm 23\,\mu$Jy, having added the 20% systematic
calibration uncertainty in quadrature. The spectral index obtained from these
three measurements agrees well with the average value of $1.7$ of the pulsar
population.

The JBO ROACH timing data show no significant pulsar proper motion. We found
PMDEC $=-1.5(2)$ and PMRA $=0.5(1.8)$. The Doppler correction is hence
$0.98<(1-\xi)<1$, with $(1-\xi)=\dot{E}^{int}/\dot{E}$ as in Section 4.3.

PSR J1023+0038:  PSR J1023$+$0038 was the first transitional millisecond
pulsar to be discovered (Archibald et al., 2009), and it underwent a
transition from a rotationally-powered MSP state to an accretion-powered state
in 2013 (Stappers et al., 2014). A faint source coincident with the pulsar was
detected by the LAT in the pre-transition data (Nolan et al., 2012), and the
flux from this source increased five-fold when accretion began (Stappers et
al., 2014).

Using the radio timing ephemeris of Archibald et al. (2013), we folded the
pre-transition data (from the beginning of the LAT data to 2013 June),
discovering significant pulsations, with $H=62$. This ephemeris contains a
single orbital period derivative to account for long-term trends in the
orbital period, but does not account for shorter-timescale orbital phase
variations seen in radio timing observations. We refined this solution using
an unbinned timing analysis on the LAT data, also using only one orbital
period derivative, resulting in an increased $H=92$.

We also folded the post-transition data using a timing model built from X-ray
timing by Jaodand et al. (2016) and X-ray and optical timing from Illiano et
al. (2023). We fit a polynomial function to the orbital phase deviations
measured by Illiano et al. (2023), and folded the LAT data using this orbital
model and the spin-down model of Jaodand et al. (2016). No pulsations were
detected. We also searched for pulsations with slightly different spin periods
and spin-down rates, and with slight shifts in orbital phase, using a sliding
window approach, but again did not detect any significant pulsations.

This non-detection implies that the flux increase due to accretion is not an
enhancement of the pulsed flux (seen in other non-accreting redbacks, e.g. An
et al. 2020; Clark et al. 2021) , but is instead an additional unpulsed
component. If the five-fold increase in gamma-ray flux is indeed due to
unpulsed emission from the accretion process, and the gamma-ray pulsations
remain unaffected by the accretion, then the pulsed fraction would be reduced
to $20\%$ of the pre-transition level, and this would leave the pulsations
undetectable to our analysis. Like PSR J1227$-$4853 (Johnson et al., 2015), it
therefore remains unclear whether or not this system emits gamma-ray
pulsations in the accretion-powered state.

PSR J1827$-$0849:  Applying the methods of Clark et al. (2017) and Section 3.4
to LAT data around the position of a steep-spectrum radio source coincident
with unidentified LAT source 3FGL J1827.6-0846, found by Frail et al. (2016),
we discovered gamma-ray pulsations with spin period $2.24$ ms in 2016. The
radio positional uncertainties of $\sim 2^{\prime\prime}$ have greatly reduced
the computing cost and allowed us to increase our sliding coherence window
length to $2^{23}$ s (or $\sim 97$ days) and still perform the search on the
ATLAS cluster212121https://www.aei.mpg.de/43564/atlas-computing-cluster in
about 1 day per source.

The associated radio source has flux density $229.9\pm 14.8$ mJy in the GMRT
150 MHz All-Sky Survey, and $777\pm 126$ mJy in VLSSr at 74 MHz. NVSS has no
detection, giving $S_{1400}<0.8$ mJy per beam. Table 8 lists an insignificant
proper motion estimate from LAT timing as per Section 3.5. C.J. Clark and co-
workers are studying other LAT unidentified sources co-located with steep-
spectrum radio sources and will detail their methods and results in future
work.

PSR J1833$-$3840:  This $1.87$ ms pulsar was discovered in 2015 by F. Camilo,
in observations of unidentified LAT sources using the Parkes radio telescope,
using the methods reported by Camilo et al. (2015). Later, Frail et al. (2018)
independently highlighted it in a search for steep radio spectrum pulsar
candidates in LAT unidentified sources using the GMRT 150 MHz all-sky survey
(TGSS ADR1). Fewer than 10% of known pulsars have such high spindown power,
$\dot{E}=1.1\times 10^{35}$ erg s-1, in the Milky Way or in globular clusters.
The orbital period is $0.9$ days. In 2020, then-undergraduate Camryn Phillips
phase-connected GBT timing observations during PINT code development she was
conducting. We phase-folded the gamma-ray data using her ephemeris and
obtained the pulsed detection reported here.

PSR J1852-1310:  The $P=4.31$ ms isolated pulsar PSR J1852$-$1310 was
discovered in a 1900 s pointing on MJD 57793 towards a LAT unidentified source
with a pulsar-like spectrum, at 800 MHz using the GUPPI backend on the Green
Bank radio telescope. It was initially named PSR J1852$-$13\. Routine timing
observations at 1400 MHz with the Lovell telescope at Jodrell Bank Observatory
began on MJD 57818. As of MJD 59927, 124 arrival times have been acquired
using ROACH, allowing a phase-connected rotation ephemeris, provided with the
online material. The spindown power is $\dot{E}=4.95\times 10^{33}$ erg s-1.
For DM $=44.95\pm 0.01$ pc cm-3 in the direction of this pulsar, the YMW16
model gives distance $1.27\pm 0.5$ kpc. The integral energy flux of co-located
4FGL J1852.2$-$1309 thus gives $L_{\gamma}=(5.8\pm 1.0)\times 10^{32}$ erg
s-1, for a gamma-ray efficiency of 11%.

GBT timing observations at 1400 MHz between MJD 58698 and 59050 yielded 49
times of arrival. Combining these with the JBO data slightly improves the
MSP’s proper motion measurement, and we obtain PMDEC $=-6.6\pm 1.4$ and PMRA
$=-1.84\pm 0.25$. The Doppler correction is hence $(1-\xi)=0.91$, as defined
in Section 4.3, such that $\dot{E}^{\mathrm{i}nt}=(4.6\pm 0.2)\times 10^{33}$
erg s-1.

Gamma-ray phase folding $14.5$ years of data using the JBO timing model
revealed pulsations with test statistic $H=122$ using the simple weights
parameter $\mu_{w}=4.3$. Model weights increase this to $H=133$. $H$ increases
steadily for the $8.5$ years before the start of the timing model, indicating
that the MSPs spindown rate is stable, well-described using only spin
frequency and its first derivative. The radio source is bright ($250\sigma$ in
the discovery observation) and the profile has sharp features, hence this MSP
may rate inclusion in gravitational wave searches.

PSR J1857+0943:  This $5.36$ ms pulsar has long been known for its stable
rotation rate (Segelstein et al., 1986). Gamma-ray phase folding using the
rotation ephemeris provided by Reardon et al. (2021) yielded H-test $>30$,
using simple weights. Located in a high background region, $(l,b)=(42.2,3.12)$
degrees, the gamma-ray profile is ill-defined but appears to be dominated by a
single narrow peak. The timing parallax distance from Table LABEL:tbl-doppler
and the integral energy flux of the co-located LAT source 4FGL J1857.2+0941
give $L_{\gamma}=(1.27\pm 0.31^{0.6}_{-0.9})\times 10^{33}$ erg s-1, where the
first uncertainty comes from the flux measurement and the second from the
distance, about 30% of the spindown power.

PSR J1901$-$0125:  The same procedure as described for PSR J1827$-$0849,
above, led to the 2016 discovery of $2.79$ ms gamma-ray pulsations. F. Camilo
then re-analyzed GBT radio data from previous observations of the LAT source
position using the LAT rotation ephemeris and discovered radio pulsations and
measured the DM. Table 3 lists $S_{1400}=2.50$ mJy, extrapolated from the
Frail et al. (2016) GMRT flux density of $362\pm 15$ mJy at 150 MHz and radio
spectral index $-2.23$. Table 8 lists a $2.4\sigma$ proper motion measurement
using the methods of Section 3.5. Clark and co-workers are studying other LAT
unidentified sources co-located with steep-spectrum radio sources and will
detail their methods and results in future work.

J2256$-$1024:  Crowter et al. (2020) found $\mu=9\pm 3$ mas yr-1 using GBT
timing data. NRT has also monitored this pulsar, but with a substantial gap
between the end of the GBT data and the beginning of the NRT observations.
This gap and limited frequency coverage cause a strong degeneracy between the
spindown parameters, time-varying dispersion measure, and orbital frequency
evolution. We therefore perform a hybrid analysis, using the methods of
Section 3.5, in which we use LAT data to determine the “long-term” parameters
(F0, F1, FB0, FB1, and FB2). We then fix these parameters and fit for the
astrometry using the radio data, finding a proper motion of PMRA $=2.80\pm
0.05$, PMDEC $=-5.30\pm 0.11$ from which $\mu=6.0\pm 0.1$ mas yr-1. A
systematic error comparable to or larger than the statistical uncertainty is
likely, due to the inhomogeneous data used in the analysis.

##  Appendix B Sample light curves, profile evolutions, and spectra.

Figures B-1 to B-4 show examples of LAT pulsar light curves in different
energy bands, with the radio pulse profiles overlaid when available. See
Section 5 for details about the construction of these light curves. All the
other pulse profiles are available in the supplementary online material,
detailed in Appendix D.

Figures B-5 to B-8 show examples of energy-resolved light curve fitting
analysis summaries, as described in Section 5. The plots for the remaining
pulsars are in the supplementary online material.

Figure B-1: Same as Figure 9, for PSR J0007+7307. Figure B-2: Same as Figure
9, for PSR J0633+1746 (the Geminga pulsar). Figure B-3: Same as Figure 9, for
PSR J1311$-$3430\. The $\star$ symbol indicates that we lacked information
needed for absolute phase alignment of the radio and gamma-ray profiles (see
Section 5.1). Figure B-4: Same as Figure 9, for PSR J2229+6114. The $\star$
symbol indicates that we lacked information needed for absolute phase
alignment of the radio and gamma-ray profiles (see Section 5.1). Figure B-5:
Same as Figure 12, for PSR J0007+7303. Here, the P3 (bridge) component was too
weak to fit in the different energy intervals, hence the points are not shown
(see Section 5.3).  Figure B-6: Same as Figure 12, for PSR J0633+1746 (the
Geminga pulsar). Figure B-7: Same as Figure 12, for PSR J1311$-$3430. Figure
B-8: Same as Figure 12, for PSR J2229+6114.

Figure B-9 shows sample spectral energy distributions (SEDs) provided as part
of 4FGL-
DR3222222https://fermi.gsfc.nasa.gov/ssc/data/access/lat/12yr_catalog/. The
supplementary online material for this pulsar catalog includes a compilation
of the SEDs of the gamma-ray pulsars, in PDF format. It also includes the
spectral parameters with their covariance matrices and the data points
necessary to reproduce the SEDs, and a python script that performs the task.

Figure B-9: Typical gamma-ray spectral energy distributions (SEDs). The plots
were made using the python script sed_plotter.py and data files in the Online
Material. The SEDs for all the gamma-ray pulsars detected as point sources (TS
$>25$) are in the Online Material as well. Similar to those provided with
4FGL-DR3, they use the refits done for some pulsars, described in Section 6.
When $b_{fr}$ exists, both $b_{23}$ and $b_{fr}$ fits are shown, with the
$b_{fr}$ uncertainty envelope.

##  Appendix C Additional Notes on Spectral Analysis

We study here the distribution of the model parameters and derived physical
quantities as they vary among the 116 pulsars which have both $b$-free and
$b=2/3$ fits available. These results are presented in Figure C-1. In brief,
they are

  * •

$\Gamma$, the spectral index at the reference energy $E_{0}$, is stable,
varying by $<$10% over the effective range in $b$.

  * •

The curvature at the reference energy, $d$, varies more than $\Gamma$, by up
to $\sim$50%. If considering $b<1$, the total range is only about $\pm$20%.

  * •

$\Gamma_{100}$ varies almost linearly with $b$ before saturating at $b=1.2$.
The shift is up to $1$ unit harder for pulsars with $b$ near 0, and $0.5$
units softer for (mostly) MSPs with $b\geq 1.2$. Thus, clearly, freeing $b$
profoundly alters the inferred low-energy spectral index.

  * •

The curvature at the peak, $d_{\mathrm{p}}$, is stable if the intrinsic value
of $b>0.5$. For pulsars with spectra more consistent with small $b$, the
$b$-free pulsars become sharper, with a difference in log-width ranging up to
$2\times$. It is not (yet) clear if this difference primarily “fixes” the true
width of the peak or delivers a better low-energy index.

  * •

The peak energy $E_{\mathrm{p}}$ varies by $<$10–15% in the “core” $b$ range
and $<25$% overall. The largest deviations are for the MSPs with fit values
$b>1$ which move to higher $E_{\mathrm{p}}$.

In summary, the physical properties measured at the peak, $d_{\mathrm{p}}$ and
$E_{\mathrm{p}}$, are robustly measured in both $b$-free and $b=2/3$ fits and
so we can make inferences on their distribution using the entire pulsar
sample. On the other hand, the low-energy spectrum is unreliable, and values
inferred from the $b=2/3$ model could be very far from the true values.

Figure C-1: Comparisons of base and derived parameters between the
$b=\frac{2}{3}$ and $b$-free populations. See main text for further
discussion.

##  Appendix D Description of the online catalog files

Online material is available through this Journal as well as at
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/3rd_PSR_catalog/. Here we
describe the files posted as of publication. Additional material may be added
in the future. The “xxx” in the file names is a multi-digit timestamp of the
file creation date. Contact the authors to request missing information.

3PC_Catalog+SEDs_xxx.fits, 3PC_Catalog_xxx.xls: The pulsar catalog “flat
file”, in FITS and spreadsheet formats. Each has 294 rows, one for each gamma-
ray pulsar. The 112 columns contain basic pulsar properties (name, position,
spin parameters, distance, radio flux density, et cetera) as well as the
spectral and profile parameters described in this article. The FITS version
includes a subtable (“HDU”) with the detailed spectral fit parameters and
covariance matrix needed to reproduce the Spectral Energy Distributions
(SEDs), provided in 3PC_SEDs.tgz (individual plots) and 3PC_AllSEDs.pdf (all
plots concatenated into a single file). 3PC_SEDPlotter.tgz contains a python
script that reads 3PC_Catalog+SEDs_xxx.fits to generate the SED plots. Most
Figures and Tables in this article were created from the FITS and spreadsheet
files: the python script Example_3PC_CatalogReader.py is a simple example.
README_3PC_Catalog+SEDs_xxx.txt provides more detail.

3PC_ProfileData_xxx.tgz: Contains two files for each pulsar, one in FITS
format and the other in simple text (ascii), containing the weighted gamma-ray
phase histograms and, when they exist, the fit to the histogram and/or the
radio profile. The python script Example_3PC_PlotProfileData.py illustrates
how to read the files to reproduce the plots provided in
3PC_ProfilePlotsPDF_xxx.tgz and 3PC_ProfilePlotsPNG_xxx.tgz. The plots of
gamma-ray peak evolution with energy, as in Figures 12 and B-5 to B-8, are
provided in 3PC_LC_Eres_pdf_xxx.tgz, 3PC_LC_Eres_png_xxx.tgz.
README_ProfileData.txt provides details.

3PC_TimingModels_xxx.tgz: A compressed archive of the 294 rotation ephemerides
(“.par files”) used to calculate the gamma-ray rotational phases using
software such as Tempo2 or PINT. README_3PC_TimingModels.txt provides details.

Phased, weighted FT1 FITS files: available at
https://heasarc.gsfc.nasa.gov/FTP/fermi/data/lat/catalogs/3PC/photon/. Two
subdirectories called 3deg_50MeV and 15deg_20MeV each contain one file for
each gamma-ray pulsar, in the standard Fermi LAT data format, with additional
columns PHASE and MODEL_WEIGHT. The 277 files in 3deg_50MeV include photons
with energies $>50$ MeV, within $3^{\circ}$ of the pulsar position, and were
used to create the gamma-ray phase histograms. They occupy roughly 30 Gb of
disk space, with a large variation in single file size depending on the
background intensity (generally decreasing away from the Galactic plane) and
pulsar intensity. The files in 15deg_20MeV are on average $15\times$ larger,
including photons with energies $>20$ MeV within $15^{\circ}$ of the pulsar
position. They are suitable for spectral analyses (e.g. using the gtlike Fermi
tool) or to explore the lowest LAT photon energies. README_3PC_PhotonData_.txt
describes the FT1 files in more detail.

3PC_sensitivity_xxx.fits : The all-sky Fermi LAT pulsar sensitivity, in FITS
format, useful for e.g. population syntheses.
Example_3PC_SensitivityMap_xxx.py is a python script that reads the FITS file
and generates a plot as in Figure 25. README_3PC_SensitivityMap_xxx.txt
provides details.

Finally, we provide HTML files for each pulsar, to allow a visual summary of
the information for each gamma-ray pulsar, named 3PC/J0007+7303_LAT.html and
so on.

Acknowledgments: The Fermi-LAT Collaboration acknowledges generous ongoing
support from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat à l’Energie
Atomique and the Centre National de la Recherche Scientifique / Institut
National de Physique Nucléaire et de Physique des Particules in France, the
Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in
Italy, the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), High Energy Accelerator Research Organization (KEK) and Japan
Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg
Foundation, the Swedish Research Council and the Swedish National Space Board
in Sweden.
Additional support for science analysis during the operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy
and the Centre National d’Études Spatiales in France. This work performed in
part under DOE Contract DE-AC02-76SF00515.
The Parkes radio telescope is part of the Australia Telescope which is funded
by the Commonwealth Government for operation as a National Facility managed by
CSIRO. The Green Bank Telescope is operated by the National Radio Astronomy
Observatory, a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. The Arecibo Observatory
is part of the National Astronomy and Ionosphere Center (NAIC), a national
research center operated by Cornell University under a cooperative agreement
with the National Science Foundation. The Nançay Radio Observatory is operated
by the Paris Observatory, associated with the French Centre National de la
Recherche Scientifique (CNRS). The Lovell Telescope is owned and operated by
the University of Manchester as part of the Jodrell Bank Centre for
Astrophysics with support from the Science and Technology Facilities Council
of the United Kingdom. The International LOFAR Telescope and the Westerbork
Synthesis Radio Telescope are operated by ASTRON, the Netherlands Institute
for Radio Astronomy. This work made extensive use of the ATNF pulsar catalog
(Manchester et al., 2005). IG, AKH, and DAS thank the ISSI (Bern, Switzerland)
for financial support of Team 459 meetings that helped to improve the present
work.

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# Landau-Zener transition between two quantum dots coupled by resonant
tunneling

M. E. Raikh  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112

###### Abstract

We consider the transition of electron between two quantum dots in which the
discrete levels are swept past each other with a constant velocity. If a
direct tunneling between the dot levels was allowed, an electron will be
transferred between the dots when the levels cross. This transfer is described
in terms of the conventional Landau-Zener theory. We assume that direct
tunneling between the dots is forbidden. Rather, the transfer is due to the
resonant tunneling via a discrete impurity level separating the dots. Then the
description of the electron transfer reduces to a three-state (two dots plus
impurity) Landau-Zener transition. Transition probability depends on the
relative positions of the resonant level and the energy at which the levels
cross. It also depends on the left-right asymmetry of tunneling between the
impurity and the left(right) dots. We calculate the transition probability in
different limits of the horizontal (in space) and vertical (in energy)
impurity level positions.

##  I Introduction

For several decades after it was introduced, the Landau-Zener physicsLandau ;
Zener ; Majorana ; Stukelberg  was invoked in relation to the atomic
collisions, see e.g. Refs. atomic0, ; atomic1, . In the past two decades the
relevance of the Landau-Zener-based description was demonstrated for qubits
based on superconducting circuitsdot1 ; dot2  and quantum dot
interferometers,dot3 ; dot4 ; dot5  see also the reviews review1, ; review2, .

On the theory side, the studies of the transitions taking place upon drive-
induced crossing of two levels have expanded to include multistate Landau-
Zener transitions, which take place when several levels cross upon application
of a linear drive. In the course of analysis of the multistate transitions a
number of models which allow an exact solution had been identified.Demkov1968
; Carroll ; Carroll1 ; Bow1 ; Bow2 ; Bow3 ; Bow5 ; Sinitsyn2004 ; Shytov ;
Sinitsyn1 ; Sinitsyn2 ; Volkov ; Sinitsyn3

One of realizations of a two-state Landau-Zener model is a solid state qubit
representing a double quantum dot, where two dots are coupled by tunneling and
are driven by external voltage. In this paper we consider the simplest
realization of the three-state model within the resonant-tunneling setup.
Namely, we assume that a direct tunneling between the dots is forbidden.
Rather, the electron transfer between the dots is due to the resonant
tunneling via a discrete impurity level separating them. While the
corresponding three-state model is not solvable, we apply the perturbative and
semiclassical descriptions to calculate the tunneling probability between the
driven dot levels.

Figure 1: (Color online) Illustration of the resonant tunneling configuration.
Impurity level is coupled to the level in the left dot with a coupling
coefficient $V_{1}$ and to the level in the right dot with a coupling
coefficient $V_{2}$.

##  II Resonant tunneling current

Resonant-tunneling configuration is depicted in Fig. 1. Let $a_{1}$ and
$a_{2}$ be the amplitudes to find electron in the left and right dots,
respectively, while $c_{0}$ is the amplitude to find an electron on the
impurity. These amplitudes are related by the system of equations

$\displaystyle i{\dot{a}}_{1}=\frac{vt}{2}a_{1}+V_{1}c_{0},$ (1)
$\displaystyle i{\dot{a}}_{2}=-\frac{vt}{2}a_{2}+V_{2}c_{0},$ (2)
$\displaystyle i{\dot{c}}_{0}={\varepsilon}_{\scriptscriptstyle
0}c_{0}+V_{1}a_{1}+V_{2}a_{2}.$ (3)

Here $v$ is the velocity with which the levels in the dots are swept past each
other. The impurity level, $\varepsilon_{\scriptscriptstyle 0}$, is coupled to
the dots with coupling coefficients $V_{1}$ and $V_{2}$. As the levels are
swept, they cross pairwise at time moments

$t_{1}=\frac{2\varepsilon_{\scriptscriptstyle
0}}{v},~{}~{}~{}~{}t_{2}=-\frac{2\varepsilon_{\scriptscriptstyle
0}}{v},~{}~{}~{}~{}t_{12}=0.$ (4)

Individual Landau-Zener transitions take place at $t=t_{1}$ and at $t=t_{2}$
(see Fig. 2), while at $t=t_{12}$ the transition does not occur, since the dot
levels are decoupled.

If individual transitions take place independently, their respective
probabilities are given by the Landau-Zener formula

${\cal P}_{1}=1-\exp\Bigg{(}-\frac{2\pi
V_{1}^{2}}{\frac{1}{4}v}\Bigg{)},~{}~{}~{}~{}{\cal
P}_{2}=1-\exp\Bigg{(}-\frac{2\pi V_{2}^{2}}{\frac{1}{4}v}\Bigg{)}.$ (5)

Velocity $\frac{v}{4}$ in the denominator of the exponents is the arithmetic
average of the velocity $\frac{v}{2}$ of the dot level and the velocity of the
impurity level, which is zero.

Another consequence of independence of the partial transitions is that the net
probability, ${\cal P}$, of tunneling between the dots is equal to the product
${\cal P}_{1}{\cal P}_{2}$. Our goal is to calculate ${\cal P}$ in the domains
when the above decoupling is not permitted. Then the resonant current is
expressed via ${\cal P}$ as

$I=\frac{e}{\tau}{\cal P},$ (6)

where $\tau$ is the period of the voltage sweep.

Figure 2: (Color online) Levels in the left and the right dot are swept by the
gate voltage as $\frac{vt}{2}$ and $-\frac{vt}{2}$, respectively. At time
moment $t_{1}=\frac{2\varepsilon_{\scriptscriptstyle 0}}{v}$ resonant level
crosses the level in the left dot, while at time moment
$t_{2}=-\frac{2\varepsilon_{\scriptscriptstyle 0}}{v}$ it crosses the level in
the right dot. As a result of two Landau-Zener transitions the electron is
transferred between the dots.

##  III Semiclassical treatment

Assuming that $a_{1}$, $a_{2}$ and $c_{0}$ are proportional to the factor
$\exp\left[i\int\limits_{\cal C}^{t}dt^{\prime}{\cal L}(t^{\prime})\right]$ ,
while the prefactors are the slow functions of time. Substituting this form
into the system Eq. 1 we obtain

$\displaystyle a_{1}=-\frac{V_{1}c_{\scriptscriptstyle 0}}{{\cal
L}+\frac{vt}{2}},~{}~{}a_{2}=-\frac{V_{2}c_{\scriptscriptstyle 0}}{{\cal
L}-\frac{vt}{2}},$ (7) $\displaystyle c_{\scriptscriptstyle
0}=-\frac{V_{1}a_{1}+V_{2}a_{2}}{{\cal L}+\varepsilon_{\scriptscriptstyle
0}}.$ (8)

Combining Eqs. 7 and 8 leads to the following cubic equation for ${\cal L}$

$\displaystyle\left[{\cal
L}^{2}-\left(\frac{vt}{2}\right)^{2}\right]\left({\cal
L}+\varepsilon_{\scriptscriptstyle 0}\right)$
$\displaystyle=\Big{(}V_{1}^{2}+V_{2}^{2}\Big{)}{\cal
L}+\Big{(}V_{2}^{2}-V_{1}^{2}\Big{)}\frac{vt}{2}.$ (9)

We will analyze how the solutions of this equation evolve upon gradual
decrease of $\varepsilon_{\scriptscriptstyle 0}$.

###  III.1 $\large{\varepsilon}_{\scriptscriptstyle 0}$

In the lowest order in couplings $V_{1}$ and $V_{2}$ the roots of Eq. III are
${\cal L}_{\scriptscriptstyle 0}=-\varepsilon_{\scriptscriptstyle 0}$ and
${\cal L}_{\pm}=\pm\Big{(}\frac{vt}{2}\Big{)}$. The first root corresponds to
the resonant tunneling of electron via the impurity, while the other two roots
describe the conventional tunneling. To find the coupling-dependent correction
to the first root we set ${\cal L}_{\scriptscriptstyle
0}=-\varepsilon_{\scriptscriptstyle 0}$ in the square brackets and in the
right-hand side of Eq. III. This yields

${\cal L}_{\scriptscriptstyle 0}=-\varepsilon_{\scriptscriptstyle
0}+\frac{-\varepsilon_{\scriptscriptstyle
0}(V_{1}^{2}+V_{2}^{2})+\frac{vt}{2}(V_{2}^{2}-V_{1}^{2})}{\varepsilon_{\scriptscriptstyle
0}^{2}-\left(\frac{vt}{2}\right)^{2}}.$ (10)

The assumption that the second term in Eq. 10 is a correction to
$\varepsilon_{\scriptscriptstyle 0}$ amounts to the condition
$(V_{1}^{2}+V_{2}^{2})\ll\varepsilon_{\scriptscriptstyle 0}^{2}$, i. e. the
condition that the impurity level is high enough.

The other two roots of Eq. III are obtained by neglecting ${\cal L}$ in the
bracket $\big{(}{\cal L}+\varepsilon_{\scriptscriptstyle 0}\big{)}$. Then the
equation Eq. III becomes quadratic and can be cast in the form

$\left[{\cal
L}_{\pm}-\frac{V_{2}^{2}+V_{1}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right]^{2}=\left[\frac{vt}{2}+\frac{V_{2}^{2}-V_{1}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right]^{2}+\left(\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}\right)^{2}.$ (11)

Upon an appropriate shifts of ${\cal L}$ and $t$, Eq. (11) can be reduced to a
standard form,
${\cal{\tilde{L}}}^{2}=\Big{(}\frac{v{\tilde{t}}}{2}\Big{)}^{2}+J^{2}$,
corresponding to the gap $J=\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}$. The probability of the electron transfer is expressed via this gap in
the form conventional for the Landau-Zener transition

$\displaystyle{\cal P}=1-\exp\left(-\frac{2\pi J^{2}}{v}\right)$
$\displaystyle=1-\exp\left[-\frac{2\pi}{v}\Bigg{(}\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}\Bigg{)}^{2}\right].$ (12)

Naturally, if any of two couplings is zero, there is no electron transfer. It
is less trivial that, the higher is the impurity level, the smaller is the
transfer probability. Note that, within the factor of the density of states
square, the effective tunnel parameter $J^{2}$ in the exponent can be
identified with the cotunneling conductance. Recall that the result Eq. III.1
was obtained in the semiclassical approximation, i.e. under the condition that
the exponent in Eq. III.1 is big. It is known that, for conventional Landau-
Zener transition, the exact result coincides with semiclassical. Concerning
the result Eq. III.1, it emerges without semiclassics if we neglect
${\dot{c}}$ in Eq. 1 and introduce instead of $a_{1}$ and $a_{1}$ the new
variables

$\displaystyle{\tilde{a}}_{1}=a_{1}\exp\left[{-i\left(\frac{V_{1}^{2}+V_{2}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right)t}\right],$
$\displaystyle{\tilde{a}}_{2}=a_{2}\exp\left[-{i\left(\frac{V_{1}^{2}+V_{2}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right)t}\right].$ (13)

With these new variables, the system 1 takes the form

$\displaystyle
i{\dot{\tilde{a}}}_{1}=\left(\frac{vt}{2}-\frac{V_{1}^{2}-V_{2}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right){\tilde{a}}_{1}-\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}{\tilde{a}}_{2},$ (14) $\displaystyle
i{\dot{\tilde{a}}}_{2}=-\left(\frac{vt}{2}-\frac{V_{1}^{2}-V_{2}^{2}}{2\varepsilon_{\scriptscriptstyle
0}}\right){\tilde{a}}_{2}-\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}{\tilde{a}}_{1},$ (15)

for which the result Eq. III.1 is exact. The condition that the gap is much
smaller than $\varepsilon_{\scriptscriptstyle 0}$ ensures that ${\dot{c}}_{0}$
in Eq. 1 can be neglected.

It is instructive to compare the probabilities of transfer in the cotunneling
regime and in the independent crossing regime when these probabilities are
small

${\cal
P}\Big{|}_{\text{cotunnel}}\approx\frac{2\pi}{v}\left(\frac{V_{1}V_{2}}{\varepsilon_{\scriptscriptstyle
0}}\right)^{2},~{}{\cal P}\Big{|}_{\text{independent}}\approx\left(\frac{8\pi
V_{1}V_{2}}{v}\right)^{2}.$ (16)

It follows from Eq. 16 that when the couplings $V_{1}$, $V_{2}$ are small the
criterion for cotunneling regime to apply is $\varepsilon_{\scriptscriptstyle
0}^{2}\gg v$.

Figure 3: (Color online) Graphic illustration of the solution of Eq. 18. In
the vicinity of the minimum at ${\cal L}_{1}=\tilde{\cal L}_{1}$ two roots of
the cubic equation are close to each other. Then the transport between the
dots is described by the conventional Landau-Zener theory.

###  III.2 Upon decreasing ${\varepsilon}_{\scriptscriptstyle 0}$

To analyze the cubic equation Eq. III in this limit we eliminate the quadratic
term by shifting the variable ${\cal L}$ as follows

${\cal L}=-\frac{\varepsilon_{\scriptscriptstyle 0}}{3}+{\cal L}_{1}.$ (17)

Then the exact equation for ${\cal L}_{1}$ takes the form

${\cal L}_{1}^{3}-{\cal L}_{1}F=G,$ (18)

where the functions $F$ and $G$ are defined as

$\displaystyle F=\frac{\varepsilon_{\scriptscriptstyle
0}^{2}}{9}+\left(\frac{vt}{2}\right)^{2}+V_{1}^{2}+V_{2}^{2},$ $\displaystyle
G=-\frac{\varepsilon_{\scriptscriptstyle
0}}{3}\left[\frac{2\varepsilon_{\scriptscriptstyle
0}^{2}}{9}-2\left(\frac{vt}{2}\right)^{2}+V_{1}^{2}+V_{2}^{2}\right]+\frac{vt}{2}\Big{(}V_{2}^{2}-V_{1}^{2}\Big{)}.$
(19)

The left-hand side of Eq. 18 has a minimum at ${\cal L}_{1}={\tilde{\cal
L}_{1}}=\left(\frac{F}{3}\right)^{1/2}$, see Fig. 3. The minimal value is
equal to $-2\left(\frac{F}{3}\right)^{3/2}$. Consider a particular case
$V_{2}=V_{1}=V$ and $t=0$. Then Eq. 18 takes the form

${}^{\prime}\left[1+\frac{1}{18}\left(\frac{\varepsilon_{\scriptscriptstyle
0}}{V}\right)^{2}\right]^{3/2}=\left(\frac{3}{8}\right)^{1/2}\left(\frac{\varepsilon_{\scriptscriptstyle
0}}{V}\right)\left[1+\frac{1}{9}\left(\frac{\varepsilon_{\scriptscriptstyle
0}}{V}\right)^{2}\right].$ (20)

The solution of the cubic equation is $\varepsilon_{\scriptscriptstyle
0}=\tilde{\varepsilon_{\scriptscriptstyle 0}}=1.55V$.

To find the time dependence of ${\cal L}_{1}$, we restore the time dependence
in the functions $F$ and $G$ and expand ${\cal L}_{1}^{3}-{\cal L}_{1}F$ in
Eq. 18 near the minimum. Then the time-dependent correction to the solution of
Eq. 20 reads

$\left[{\cal
L}_{1}-\left(\frac{F}{3}\right)^{1/2}\right]^{2}=\frac{G+2\left(\frac{F}{3}\right)^{3/2}}{3\left(\frac{F}{3}\right)^{1/2}}.$
(21)

At $t=0$ the right-hand side of Eq. 21 turns to zero. Upon expanding $G$ and
$F$ around $\tilde{\varepsilon}_{\scriptscriptstyle 0}$ and restoring
$\left(\frac{vt}{2}\right)^{2}$ in $G$ and $F$, Eq. 21 takes the form

$\left[{\cal
L}_{1}-\left(\frac{F}{3}\right)^{1/2}\right]^{2}=\alpha\left(\frac{vt}{2}\right)^{2}+\beta\Big{(}\varepsilon_{\scriptscriptstyle
0}-\tilde{\varepsilon_{\scriptscriptstyle 0}}\Big{)},$ (22)

where the coefficients of expansion $\alpha$ and $\beta$ are of the order of
$1$. We see that, for $\varepsilon_{\scriptscriptstyle
0}>\tilde{\varepsilon}_{\scriptscriptstyle 0}$, two of the three solutions of
the cubic equation Eq. III satisfy the quadratic equation having the Landau-
Zener form Eq. 11.

Comparing Eq. 22 to Eq. III.1 leads immediately to the expression for the
transition probability

${\cal
P}=1-\exp\left[-\frac{2\pi}{v}\left(\frac{\beta}{\alpha^{1/2}}\right)\Big{(}\varepsilon_{\scriptscriptstyle
0}-\tilde{\varepsilon_{\scriptscriptstyle 0}}\Big{)}\right].$ (23)

To summarize our semiclassical analysis of the transmission probability,
${\cal P}$, for a symmetric position of the resonant level, $V_{1}=V_{2}=V$,
it depends on the dimensionless ratio,
$\Big{|}\frac{V}{\varepsilon_{\scriptscriptstyle 0}}\Big{|}$. When this ratio
is small, the transport between the dots is due to cotunneling, and ${\cal P}$
is small. As $\varepsilon_{\scriptscriptstyle 0}$ is gradually shifted down,
${\cal P}$ increases and passes through a maximum at ${\varepsilon}_{0}\sim
V$. All three states, left dot, impurity level, and the right dot are
represented in the wave-function approximately equally. As the level reaches a
critical value $\varepsilon_{\scriptscriptstyle
0}=\tilde{\varepsilon}_{\scriptscriptstyle 0}=1.55V$ from the above, the
transmission probability is small again, as follows from Eq. 23. Upon further
decrease of $\varepsilon_{\scriptscriptstyle 0}$, the transmission probability
keeps dropping down to the value $\left({\cal P}\right)^{2}$, predicted by the
independent crossing approximation.

###  III.3 Perturbative treatment

Perturbative treatment applies when the couplings of the impurity to the dots
differ strongly. Then the picture of the electron transfer between the dots
gets strongly simplified. Assume e.g. that $V_{2}\ll V_{1}$. This relation
implies that beatings of electron between the left dot and impurity is much
faster than the beatings between the impurity and the right dot. In this
limiting case, the Landau-Zener transition between the left dot and impurity
gets completed before electron accomplishes a single virtual tunneling from
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estimated as $\frac{1}{V_{2}}$. Then the condition that the second time is
longer reads $V_{1}V_{2}\ll v$. Under this condition, the probability of the
transfer of the electron from the impurity to the second dot can be calculated
perturbatively.

At time $t=t_{1}=2\frac{\varepsilon_{\scriptscriptstyle 0}}{v}$ the dot level
crosses the impurity level. To capture the vicinity of this crossing we
introduce a shifted time

$T=t-t_{1}$ (24)

and the new variables

$\displaystyle a_{1}=A_{1}(T)\exp\left(-i\varepsilon_{\scriptscriptstyle
0}T-\frac{vT^{2}}{8}\right),$ $\displaystyle
a_{2}=A_{2}(T)\exp\left(-i\varepsilon_{\scriptscriptstyle
0}T-\frac{vT^{2}}{8}\right),$ $\displaystyle
c_{0}=C_{0}(T)\exp\left(-i\varepsilon_{\scriptscriptstyle
0}T-\frac{vT^{2}}{8}\right).$ (25)

With these variables the system Eq. 1 acquires the form

$\displaystyle i{\dot{A}}_{1}=\frac{vT}{4}A_{1}+V_{1}C_{0},$ (26)
$\displaystyle
i{\dot{A}}_{2}=-\frac{v}{2}\left(\frac{3T}{2}+\frac{4\varepsilon_{\scriptscriptstyle
0}}{v}\right)A_{2}+V_{2}C_{0},$ (27) $\displaystyle
i{\dot{C}}_{0}=-\frac{vT}{4}C_{0}+V_{1}A_{1}+V_{2}A_{2}.$ (28)

If we neglect $V_{2}$, the system gets decoupled. Then the first and the third
equations reduce to the system describing a conventional Landau-Zener
transition with matrix element $V_{1}$ and velocity $\frac{v}{4}$. A textbook
expression for $A_{1}$ and $C_{0}$, which satisfies the initial condition
$C_{0}(-\infty)=0$, reads

$A_{1}(T)=D_{\nu}\Big{(}{\sqrt{\frac{v}{4}}}e^{\frac{\pi
i}{4}}T\Big{)},~{}C_{0}(T)=-i{\sqrt{\nu}}D_{\nu-1}\Big{(}{\sqrt{\frac{v}{4}}}e^{\frac{\pi
i}{4}}T\Big{)},$ (29)

where $D_{\nu}(z)$ and $D_{\nu-1}(z)$ are the parabolic cylinder functions,
whereas the parameter $\nu$ is defined as

$\nu=-i\frac{V_{1}^{2}}{\Big{(}\frac{v}{4}\Big{)}}.$ (30)

Perturbative incorporation of $V_{2}$ reduces to solving the second equation
of the system for $A_{2}$ while assuming that $C_{0}(T)$ is fixed. Upon
imposing a natural initial condition $A_{2}(-\infty)=0$, we get

$\displaystyle{\Big{|}}A_{2}(T){\Big{|}}^{2}$
$\displaystyle=V_{2}^{2}{\Bigg{|}}\int\limits_{-\infty}^{T}dT^{\prime}C_{0}(T^{\prime})\exp\left[-\frac{iv}{2}\Bigg{(}\frac{3T^{\prime
2}}{4}+\frac{4\varepsilon_{\scriptscriptstyle
0}T^{\prime}}{v}\Bigg{)}\right]{\Bigg{|}}^{2}.$ (31)

Substituting $C_{0}(T)$ from Eq. 29 and introducing a new variable
$z=T\sqrt{\frac{v}{4}}\exp\Big{(}\frac{\pi i}{4}\Big{)}$, we find the
following expression for $|A_{2}(\infty)|^{2}$

$\displaystyle|A_{2}(\infty)|^{2}$
$\displaystyle=\frac{4|\nu|V_{2}^{2}}{v}\Bigg{|}\int\limits_{-\infty}^{\infty}dzD_{\nu-1}(z)\exp\Big{[}-\frac{3z^{2}}{2}-\kappa
z\Big{]}\Bigg{|}^{2},$ (32)

where the parameter $\kappa$ is defined as

$\kappa=\frac{4\varepsilon_{\scriptscriptstyle
0}}{v^{1/2}}\exp\Big{(}\frac{\pi i}{4}\Big{)}.$ (33)

To evaluate the integral in Eq. 32 we use the following
representationRepresentation  of the parabolic cylinder function

$D_{\nu-1}(z)=-\Bigl{(}\frac{2}{\pi}\Bigr{)}^{1/2}e^{\frac{z^{2}}{4}}\int\limits_{0}^{\infty}du~{}e^{-\frac{u^{2}}{2}}u^{\nu}\sin\left(zu-\frac{\pi\nu}{2}\right).$
(34)

Upon substituting this representation into Eq. 32, the integral over $z$
becomes gaussian leading to

$\displaystyle|A_{2}(\infty)|^{2}$
$\displaystyle=\frac{16|\nu|V_{2}^{2}}{3v}\Bigg{|}\int\limits_{0}^{\infty}duu^{\nu}\exp\Big{(}-\frac{2}{3}u^{2}\Big{)}\sin\Big{(}\frac{\kappa
u}{3}+\frac{\pi\nu}{2}\Big{)}\Bigg{|}^{2}.$ (35)

Comparing Eqs. 34 and III.3, we conclude that the integral $du$ can also be
expressed via the function $D_{\nu-1}$

$|A_{2}(\infty)|^{2}=\frac{2\pi|\nu|V_{2}^{2}}{v}\Bigg{|}D_{\nu-1}\Big{(}-\frac{\kappa}{2\sqrt{3}}\Big{)}\Bigg{|}^{2}.$
(36)

Note now, that the combination $\frac{|\nu|V_{2}^{2}}{v}$ in the prefactor
with definition Eq. (30) of the parameter $\nu$ assumes the form
$\Big{(}\frac{V_{1}V_{2}}{v}\Big{)}^{2}$, which is the square of the parameter
established above from the qualitative consideration. Smallness of this
parameter ensures the applicability of the perturbative treatment.

The factor
$\Big{|}D_{\nu-1}\Big{(}-\frac{\kappa}{2\sqrt{3}}\Big{)}\Big{|}^{2}$ describes
, via the argument $\kappa$, the dependence of the tunnel conductance on the
level position $\varepsilon_{\scriptscriptstyle 0}$. At resonance,
$\varepsilon_{\scriptscriptstyle 0}=0$, using the representation Eq. 34, one
finds
$\Big{|}D_{\nu-1}\big{(}0\big{)}\Big{|}^{2}=\frac{\sinh^{2}\frac{\pi|\nu|}{2}}{\cosh\frac{\pi|\nu|}{2}}$.
Correction to the conductance at finite $\varepsilon_{0}$ follows from the
expansion

$\displaystyle\Bigg{|}\frac{D_{\nu-1}\Big{(}-\frac{\kappa}{2\sqrt{3}}\Big{)}}{D_{\nu-1}(0)}\Bigg{|}^{2}$
$\displaystyle\approx
1-\left(\frac{32}{3}\right)^{1/2}\frac{\varepsilon_{\scriptscriptstyle
0}}{v^{1/2}}\text{Re}\Bigg{\\{}\frac{\Gamma(\frac{\nu}{2}+1)}{\Gamma(\frac{\nu}{2}+\frac{1}{2})}e^{\frac{\pi
i}{4}}\Bigg{\\}}.$ (37)

As was discussed above, see Eq. 16, it is the parameter
$\frac{\varepsilon_{\scriptscriptstyle 0}^{2}}{v}$ that should be large for
cotunneling regime to unfold.

Note, that there is a formal correspondence between the standard Landau-Zener
result Eq. 29 for $C_{0}(T)$ and our perturbative result 36 for
$A_{2}(\infty)$. It is well-known, see e.g. Refs. review1, ; review2, , that
the Landau-Zener transition is accompanied by the oscillations of the level
occupation with time. Then we conclude that $|A_{2}(\infty)|^{2}$ contains an
oscillating term as a function of the resonant level position. Building up on
the formal correspondence between $|C_{0}(T)|^{2}$ and $|A_{2}(\infty)|^{2}$,
we can readily write down the expression for this oscillating term

$\cos\Bigg{[}\frac{2\varepsilon_{\scriptscriptstyle
0}}{v^{1/2}}\left(|\nu|+\frac{4\varepsilon_{\scriptscriptstyle
0}^{2}}{v}\right)^{1/2}-\Psi\Bigg{]},$ (38)

where the phase $\Psi$ is given by

$\Psi=|\nu|\ln\Bigg{\\{}\frac{\left(|\nu|+\frac{4\varepsilon_{\scriptscriptstyle
0}^{2}}{v}\right)^{1/2}-\frac{2\varepsilon_{\scriptscriptstyle
0}}{v^{1/2}}}{\left(|\nu|+\frac{4\varepsilon_{\scriptscriptstyle
0}^{2}}{v}\right)^{1/2}+\frac{2\varepsilon_{\scriptscriptstyle
0}}{v^{1/2}}}\Bigg{\\}}.$ (39)

While the term Eq. 38 is a small correction to the monotonic term, the
oscillations of the resonant-tunneling current with the level position are
highly nontrivial.

##  IV Comparison to three-level bow-tie model

The paper Ref. Carroll1,  is entirely devoted to the transfer probability in
the three-level model in the limit $\varepsilon_{\scriptscriptstyle
0}\rightarrow 0$. Quantitatively, this limit corresponds to
$|\varepsilon_{\scriptscriptstyle 0}|\ll V_{1},V_{2}$. Similar to the present
paper, it was assumed in Ref. Carroll1,  that the first and the third levels
are completely decoupled. Since in the limit $|\varepsilon_{\scriptscriptstyle
0}|\ll V_{1},V_{2}$, the position ${\varepsilon}_{\scriptscriptstyle 0}$
should drop out from the transfer probability, it can depend only on two
dimensionless combinations: $\frac{V_{1}^{2}}{v}$ and $\frac{V_{2}^{2}}{v}$.
Using the contour integral representation,Demkov1968  it was shown in Ref.
Carroll1, , see also Ref. Carroll, , that the result ${\cal P}={\cal
P}_{1}{\cal P}_{2}$, corresponding to the independent crossing, is valid.
Thus, upon increasing of $\varepsilon_{\scriptscriptstyle 0}$ , the
probability, ${\cal P}$, crosses over from independent crossing to the
cotunneling regime Eq. III.1, which applies for
$\varepsilon_{\scriptscriptstyle 0}^{2}\gg(V_{1}^{2}+V_{2}^{2}).$ The result
Eq. III.1 seems to be the only result predicting the analytical dependence of
${\cal P}$ on the level position, $\varepsilon_{\scriptscriptstyle 0}$. This
is certainly in agreement with conclusion of Ref. dot1, , supported
numerically, that the independent crossing approximation fails upon increasing
of $\varepsilon_{\scriptscriptstyle 0}$. Equally, the result Eq. 36 of the
perturbative treatment indicates the explicit dependence of the tunnel
probability on $\varepsilon_{\scriptscriptstyle 0}$.

General approach in the papers Carroll ; Carroll1 ; Bow1 ; Bow2 ; Bow3 ; Bow5
is based on the representation of the solution of the system Eq. 1 in the form
of the contour integral. The approach adopted in the present paper is based on
exploiting of the small parameter in the problem. In the three-state model,
the semiclassical equation for the quasienergy, ${\cal L}$, has three
solutions. Small parameter emerges when one of the solutions is much bigger
than the other two. Neglecting the fast solution and keepinng only two slow
solutionsRajesh0 ; Rajesh1  allows to reduce the problem to the effective
Landau-Zener transition. Other small parameter emerges when the coupling of
impurity level to one dot is much bigger than the coupling to the other dot.

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