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it is believed that the direct detection of gravitational waves ( gws ) will bring the era of gravitational wave astronomy .
the interferometer detectors are now under operation and awaiting the first signal of gws @xcite .
it is also known that pulsar timing arrays ( ptas ) can be used as a detector for gws @xcite .
these detectors are used to search for very low frequency ( @xmath0 ) gravitational waves , where the lower limit of the observable frequencies is determined by the inverse of total observation time @xmath1 .
indeed , the total observation time has a crucial role in ptas , because ptas are most sensitive near the lower edge of observable frequencies @xcite . taking into account its sensitivity ,
the first direct detection of the gravitational waves might be achieved by ptas .
the main target of ptas is the stochastic gravitational wave background ( sgwb ) generated by a large number of unresolved sources with the astrophysical origin or the cosmological origin in the early universe .
the promising sources are super massive black hole binaries @xcite , cosmic ( super)string @xcite , and inflation @xcite .
previous studies have assumed that the sgwb is isotropic and unpolarized @xcite .
these assumptions are reasonable for the primary detection of the sgwb , but the deviation from the isotropy and the polarizations should have rich information of sources of gravitational waves .
recently , the cross - correlation formalism has been generalized to deal with anisotropy in the sgwb @xcite .
result of this work enables us to consider arbitrary levels of anisotropy , and a bayesian approach was performed by using this formalism @xcite . on the other hand , for the anisotropy of the sgwb , the cross - correlation formalism has been also developed in the case of interferometer detectors @xcite .
as to the polarization , there are works including the ones motivated by the modified gravity @xcite
. we can envisage supermassive black hole binaries emit circularly polarized sgwb due to the chern - simons term @xcite .
there may also exist cosmological sgwb with circular polarization in the presence of parity violating term in gravity sector @xcite . in this paper
, we investigate the detectability of circular polarization in the sgwb by ptas .
we characterize sgwb by the so called stokes @xmath2 parameter @xcite and calculate generalized overlap reduction functions ( orfs ) so that we can probe the circular polarization of the sgwb .
we also discuss a method to separate the intensity ( @xmath3 mode ) and circular polarization ( @xmath2 mode ) of the sgwb .
the paper is organized as follows . in section [ sec :
stokes parameters for a plane gravitational wave ] , we introduce the stokes parameters for monochromatic plane gravitational waves , and clarify the physical meaning of the stokes parameters @xmath3 and @xmath2 . in section [ sec : formulation ] , we formulate the cross - correlation formalism for anisotropic circularly polarized sgwb with ptas .
the basic framework is essentially a combination of the formalism of @xcite , and the polarization decomposition formula of the sgwb derived in @xcite . in section [ sec : the generalized overlap reduction function for circular polarization ] , we calculate the generalized orfs for the @xmath2 mode .
the results for @xmath3 mode are consistent with the previous work @xcite . in section [ sec : separation method ] , we give a method for separation between the @xmath3 mode and @xmath2 mode of the sgwb .
the final section is devoted to the conclusion . in appendixes , we present analytic results for the generalized overlap reduction functions . in this paper
, we will use the gravitational units @xmath4 .
let us consider the stokes parameters for plane waves traveling in the direction @xmath5 , which can be described by @xmath6 \
, \\ & & h_{xy}(t , z)=h_{yx}(t , z)={\rm re}[b_{\times}\mathrm{e}^{-iw(t - z ) } ] \ .\end{aligned}\ ] ] for an idealized monochromatic plane wave , complex amplitudes @xmath7 and @xmath8 are constants .
polarization of the plane gws is characterized by the tensor , ( see @xcite and also electromagnetic case @xcite ) @xmath9 where @xmath10 take @xmath11 .
any @xmath12 hermitian matrix can be expanded by the pauli and the unit matrices with real coefficients .
hence , the @xmath13 hermitian matrix @xmath14 can be written as @xmath15 where @xmath16 by analogy with electromagnetic cases , @xmath17 and @xmath2 are called stokes parameters . comparing with , we can read off the stokes parameters as @xmath18= b_{+}^{\ast}b_{\times}+ b_{\times}^{\ast}b_{+},\\ v&=&-2{\rm i m } [ b_{+}^{\ast}b_{\times}]=i ( b_{+}^{\ast}b_{\times}- b_{\times}^{\ast}b_{+}).\label{stv}\end{aligned}\ ] ] apparently , the real parameter @xmath3 is the intensity of gws . in order to reveal the physical meaning of the real parameter @xmath2 , we define the circular polarization bases @xcite @xmath19 from the relation @xmath20 we see @xmath21
thus , we can rewrite the stokes parameters - as @xmath22 from the above expression , we see that the real parameter @xmath2 characterizes the asymmetry of circular polarization amplitudes .
the other parameters @xmath23 and @xmath24 have additional information about linear polarizations by analogy with the electromagnetic cases .
alternatively , we can also define the tensor @xmath25 in circular polarization bases @xmath26 where @xmath27 .
note that the stokes parameters satisfy a relation @xmath28 next , we consider the transformation of the stokes parameters under rotations around the @xmath5 axis . the rotation around the @xmath5 axis is given by @xmath29 where @xmath30 is the angle of the rotation .
the gws traveling in the direction @xmath5 @xmath31 transform as @xmath32 where we took the transverse traceless gauge @xmath33 after a short calculation , we obtain @xmath34 using and , the four stokes parameters ( [ sti])-([stv ] ) transform as @xmath35 as you can see , the parameters @xmath23 and @xmath24 depend on the rotation angle @xmath30 .
this reflects the fact that @xmath23 and @xmath24 parameters characterize linear polarizations .
note that this transformation is similar to the transformation of electromagnetic case except for the angle @xmath36 and can be rewritten as @xmath37
in this section , we study anisotropic distribution of sgwb and focus on the detectability of circular polarizations with pulsar timing arrays .
we combine the analysis of @xcite and that of @xcite . in sec.[subsec : the spectral ] , we derive the power spectral density for anisotropic circularly polarized sgwb @xmath38 .
then we also derive the dimensionless density parameter @xmath39 which is expressed by the frequency spectrum of intensity @xmath40 @xcite . in sec.[subsec : the signal ] , we extend the generalized orfs to cases with circular polarizations characterized by the parameter @xmath2 . for simplicity ,
we consider specific anisotropic patterns with @xmath41 expressed by the spherical harmonics @xmath42 . in the transverse traceless gauge , metric perturbations @xmath43 with a given propagation direction @xmath44
can be expanded as @xcite @xmath45 where the fourier amplitude satisfies @xmath46 as a consequence of the reality of @xmath43 , @xmath47 , @xmath48 is the frequency of the gws , @xmath49 are spatial indices , @xmath50 label polarizations .
note that the fourier amplitude @xmath51 satisfies the relation @xmath52 where @xmath53 was defined by .
the polarized tensors @xmath54 are defined by @xmath55 where @xmath56 and @xmath57 are unit orthogonal vectors perpendicular to @xmath58 .
the polarization tensors satisfy @xmath59 with polar coordinates , the direction @xmath44 can be represented by @xmath60 and the polarization basis vectors read @xmath61 we assume the fourier amplitudes @xmath62 are random variables , which is stationary and gaussian .
however , they are not isotropic and unpolarized .
the ensemble average of fourier amplitudes can be written as @xcite @xmath63 where @xmath64 here , the bracket @xmath65 represents an ensemble average , and @xmath66 is the dirac delta function on the two - sphere .
the gw power spectral density @xmath38 is a hermitian matrix , and satisfies @xmath67 because of the relation @xmath46 .
therefore , we have the relations @xmath68 note that the stokes parameters are not exactly the same as the expression of , but they have the relation and characterize the same polarization .
we further assume that the sgwbs satisfy @xmath69 we also assume the directional dependence of the sgwb is frequency independent @xcite .
this implies the gw power spectral density is factorized into two parts , one of which depends on the direction while the other depends on the frequency .
because of the transformations - , the parameters @xmath3 and @xmath2 have spin 0 and the parameters @xmath70 have spin @xmath71 @xcite . to analyze the sgwb on the sky , it is convenient to expand the stokes parameters by spherical harmonics @xmath72
. however , since @xmath70 parameters have spin @xmath71 , they have to be expanded by the spin - weighted harmonics @xmath73 @xcite .
thus , we obtain @xmath74 in this paper , we study specific anisotropic patterns with @xmath41 for simplicity .
therefore , we can neglect @xmath23 and @xmath24 from now on .
thus , the gw power spectral density becomes @xmath75 where @xmath76 so , we focus on the parameters @xmath3 and @xmath2 . in what follows , we will use the following shorthand notation @xmath77 next , we consider the dimensionless density parameter @xcite @xmath78 where @xmath79 is the critical density , @xmath80 is the present value of the hubble parameter , @xmath81 is the energy density of gravitational waves , and @xmath82 is the energy density in the frequency range @xmath48 to @xmath83 .
the bracket @xmath65 represents the ensemble average .
however , actually , we take a spatial average over the wave lengths @xmath84 of gws or a temporal average over the periods @xmath85 of gws . here
, we assumed the ergodicity , namely , the ensemble average can be replaced by the temporal average .
using , , , as well as @xmath46 and @xmath86 , we get @xmath87 then we define @xmath88 hence , the dimensionless quantity @xmath39 in is given by @xmath89 where the spherical harmonics are orthogonal and normalized as @xmath90 using @xmath91 , we obtain @xmath92 without loss of generality , we normalize the monopole moment as @xmath93 so , becomes @xmath94 the time of arrival of radio pulses from the pulsar is affected by gws .
consider a pulsar with frequency @xmath95 located in the direction @xmath96 . to detect the sgwb ,
let us consider the redshift of the pulse from a pulsar @xcite @xmath97 where @xmath98 is a frequency detected at the earth and @xmath96 is the direction to the pulsar .
the unit vector @xmath44 represents the direction of propagation of gravitational plane waves .
we also defined the difference between the metric perturbations at the pulsar @xmath99 and at the earth @xmath100 as @xmath101 the gravitational plane waves at each point is defined as @xmath102 for the sgwb , the redshift have to be integrated over the direction of propagation of the gravitational waves @xmath44 : @xmath103 we choose a coordinate system @xmath104 and assume that the amplitudes of the metric perturbation at the pulsar and the earth are the same .
then becomes @xmath105 and therefore , reads @xmath106 where we have defined the pattern functions for pulsars @xmath107 note that our convention for the fourier transformation is @xmath108 therefore , the fourier transformation of can be written as @xmath109 in the actual signals from a pulsar , there exist noises .
hence , we need to use the correlation analysis .
we consider the signals from two pulsars @xmath110 where @xmath111 labels the pulsar . here
, @xmath112 denotes the signal from the pulsar and @xmath113 denotes the noise intrinsic to the measurement .
we assume the noises are stationary , gaussian and are not correlated between the two pulsars .
to correlate the signals of two measurements , we define @xmath114 where @xmath1 is the total observation time and @xmath115 is a real filter function which should be optimal to maximize signal - to - noise ratio . in the case of interferometer
, the optimal filter function falls to zero for large @xmath116 compered to the travel time of the light between the detecters .
since the signals of two detectors are expected to correlate due to the same effect of the gravitational waves , the optimal filter function should behave this way .
then , typically one of the detectors is very close to the other compared to the total observation time @xmath1 .
therefore , the total observation time @xmath1 can be extended to @xmath117 @xcite .
in contrast , in the case of pta , it is invalid that @xmath1 is very large compered to the travel time of the light between the pulsars .
nevertheless , we can assume that one of the two @xmath1 can be expanded to @xmath117 , because in situations @xmath118 and @xmath119 it is known that we can ignore the effect of the distance @xmath120 of pulsars .
in this case , it is clear that any locations of the pulsars are optimal and optimal filter function should behave like as the interferometer case @xcite . using these assumptions @xmath118 and @xmath119 , we can rewrite as @xmath121 where @xmath122 note that @xmath123 satisfies @xmath124 , because @xmath125 is real .
moreover , to deal with the unphysical region @xmath126 we require @xmath127 .
thus , @xmath123 becomes real .
taking the ensemble average , using @xmath128 , @xmath118 , and assuming the noises in the two measurements are not correlated , we get @xmath129\ , \label{s2}\end{aligned}\ ] ] where we have defined @xmath130 the functions @xmath131 and @xmath132 are called the generalized orfs , which describe the angular sensitivity of the pulsars for the sgwb . note that , as we already mentioned , we consider the cases of @xmath41 for simplicity
. then we have assumed @xmath118 and @xmath119 , this assumption implies that approximately becomes @xmath133 due to the rapid oscillation of the phase factor .
therefore , the distance @xmath120 of the pulsars does not appear in the generalized orfs , and hence the generalized orfs do not depend on the frequency . as you can see from ,
the correlation of the two measurements involve both the total intensity and the circular polarization .
however , the degeneracy can be disentangled by using separation method , which will be discussed in the section [ sec : separation method ] .
in this section , we consider the generalized orfs for circular polarizations : @xmath134 where we defined @xmath135 in the above , we have used and the fact that the generalized orfs do not depend on frequency . for computation of the generalized orfs for circular polarizations ,
it is convenient to use the computational frame @xcite defined by @xmath136 where @xmath137 is the angular separation between the two pulsars . using - , , and
, one can easily show that @xmath138 we therefore get @xmath139 the explicit form of the spherical harmonics reads @xmath140 where @xmath141 is the normalization factor .
the associated legendre functions are given by @xmath142 and @xmath143 with the legendre functions @xmath144\ .\label{pl}\end{aligned}\ ] ] using the spherical harmonics , becomes @xmath145 where we have used the fact that the function of @xmath146 is odd parity in the case of @xmath147 and is even parity in the case of @xmath148 .
note that the generalized orfs for circular polarizations are real functions . in the case of @xmath149 and/or @xmath150 ,
the integrand in vanishes .
therefore , we can not detect circular polarizations for these cases .
this fact for @xmath151 implies that we do not need to consider auto - correlation for a single pulsar .
this is the reason why we neglected auto - correlation term in . integrating ,
we get the following form for @xmath152 : @xmath153 for @xmath154 , we have obtained @xmath155 \ , \\ \gamma^{v}_{1 - 1}&=&\gamma^{v}_{11 } \ , \end{aligned}\ ] ] recall that @xmath156 .
the derivation of this formula for @xmath154 can be found in appendix [ sec : angular integral of the generalized overlap reduction function for dipole circular polarization ] . for @xmath157 , we derived the following : @xmath158\ , \\ \gamma^{v}_{2 - 1}&=&\gamma^{v}_{21}\ , \\
\gamma^{v}_{22}&=&-\frac{\sqrt{30\pi}}{6}(1-\cos\xi)\left[2-\cos\xi+6\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{2 - 2}&=&-\gamma^{v}_{22}\ , \end{aligned}\ ] ] for @xmath159 , the results are @xmath160\ , \\ \gamma^{v}_{3 - 1}&=&\gamma^{v}_{31}\ , \\ \gamma^{v}_{32}&=&\frac{\sqrt{210\pi}}{24}(1-\cos\xi)\left[8 - 5\cos\xi-\cos^2\xi+24\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{3 - 2}&=&-\gamma^{v}_{3 - 2}\ , \\ \gamma^{v}_{33}&=&-\frac{\sqrt{35\pi}}{16}\sin\xi\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\left[11 - 6\cos\xi-\cos^2\xi+32\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{3 - 3}&=&\gamma^{v}_{33}\ .\end{aligned}\ ] ] in fig .
[ gv ] , we plotted these generalized orfs as a function of the angular separation between the two pulsars @xmath137 .
it is apparent that considering the @xmath2 mode does not make sense when we only consider the isotropic ( @xmath152 ) orf . on the other hand ,
when we consider anisotropic ( @xmath161 ) orfs , it is worth taking into account polarizations .
the polarizations of the sgwb would give us rich information both of super massive black hole binaries and of inflation in the early universe .
as a function of the angular separation between the two pulsars @xmath137 . in fig .
[ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig .
[ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig .
[ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig .
[ gv](d ) , the orfs for the octupole ( l=3 ) are plotted .
the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( a ) @xmath152 as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig .
[ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig .
[ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig .
[ gv](d ) , the orfs for the octupole ( l=3 ) are plotted .
the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( b ) @xmath154 as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig .
[ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig .
[ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig .
[ gv](d ) , the orfs for the octupole ( l=3 ) are plotted .
the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( c ) @xmath157 as a function of the angular separation between the two pulsars @xmath137 . in fig .
[ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig .
[ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig .
[ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig .
[ gv](d ) , the orfs for the octupole ( l=3 ) are plotted .
the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( d ) @xmath159 using the same procedure described in the above to derive the generalized orfs for circular polarizations , we can also derive the generalized orfs for the intensity @xmath166 where @xmath167 the angular integral in this case was performed in @xcite .
the results are summarized in appendix [ sec : the generalized overlap reduction function for intensity ] .
in this section , we separate the @xmath3 mode and @xmath2 mode of the sgwb with correlation analysis @xcite . to this aim ,
we use four pulsars ( actually we need at least three pulsars ) , and define correlations of @xmath168 @xmath169 where @xmath170 label the pulsars . comparing with , we obtain @xmath171 \ ,
\label{1c12}\\ & & c_{34}(f)=\sum_{lm}^{l=3}\left[c_{lm}^{i}i(f)\gamma_{lm,34}^{i}+c_{lm}^{v}v(f)\gamma_{lm,34}^{v}\right ] \ .\label{1c34}\end{aligned}\ ] ] if the @xmath3 mode and @xmath2 mode of the sgwb are dominated by a certain @xmath172 and @xmath173 , and become @xmath174 \ , \label{2c12 } \\ & & c_{34}(f)=\left[c _ { l m}^{i}i(f)\gamma _ { l m,34}^{i}+c _ { l ' m'}^{v}v(f)\gamma _ { l ' m',34}^{v}\right ] \ .\label{2c34}\end{aligned}\ ] ] to separate the intensity and the circular polarization , we take the following linear combinations @xmath175 where we defined coefficients @xmath176 as you can see , @xmath177 contains only @xmath40 , and @xmath178 contains only @xmath179 .
for the signal @xmath180 , the formulas corresponding to and are given by @xmath181 \ , \label{sp}\end{aligned}\ ] ] where @xmath182 denotes @xmath3 and @xmath2 .
we assume @xmath183 and that the noise in the four pulsars are not correlated .
we also assume that the ensemble average of fourier amplitudes of the noises @xmath184 is of the form @xmath185 where @xmath186 is the noise power spectral density .
the reality of @xmath187 gives rise to @xmath188 and therefore we obtain @xmath189 . without loss of generality
, we can assume @xmath190 then we obtain corresponding noises @xmath191 : @xmath192\ , \label{np}\end{aligned}\ ] ] where @xmath193^{1/2 } \label{sn12 } \ , \quad s_{n,34}(f ) \equiv [ s_{n,3}(f)s_{n,4}(f)]^{1/2 } \label{sn34 } \ .\end{aligned}\ ] ] using the inner product @xmath194 \ , \end{aligned}\ ] ] we can rewrite , as @xmath195 therefore , the optimal filter function can be chosen as @xmath196 using , we get optimal signal - to - noise ratio @xmath197^{1/2}\ .\label{snr}\end{aligned}\ ] ] plugging , , and into , we obtain @xmath198^{1/2}\ , \\
{ \rm snr}_{v}&=&\left[t\int_{-\infty}^{\infty}df\,\,\frac{\left(c^{v}_{{l}'{m}'}\right)^{2}v^{2}(f)\left(\gamma_{{l}'{m}',34}^{v}\gamma^{i}_{{l}{m},12}-\gamma_{{l}'{m}',12}^{v}\gamma^{i}_{{l}{m},34}\right)^2}{\left(\gamma^{i}_{{l}{m},12}\right)^2s^{2}_{n,34}(f)+\left(\gamma^{i}_{{l}{m},34}\right)^2s^{2}_{n,12}(f)}\right]^{1/2}\ .\end{aligned}\ ] ] if we assume all of the noise power spectral densities are the same , becomes @xmath199 thus , the compiled orfs can be defined as @xmath200^{1/2}}\ , \\ \gamma_{12:34}^{v}&\equiv&\frac{\gamma_{{l}'{m}',34}^{v}\gamma^{i}_{{l}{m},12}-\gamma_{{l}'{m}',12}^{v}\gamma^{i}_{{l}{m},34}}{\left[\left(\gamma^{i}_{{l}{m},12}\right)^2+\left(\gamma^{i}_{{l}{m},34}\right)^2\right]^{1/2}}\ .\end{aligned}\ ] ] this compiled orfs @xmath201 and @xmath202 describe the angular sensitivity of the four pulsars for the pure @xmath3 and @xmath2 mode of the sgwb , respectively .
note that , to do this separation , we must know a priori the coefficients @xmath203 and @xmath204 .
if we do not assume , the generalized orfs depend on the frequency . in this case
, it seems difficult to calculate these coefficients .
we next consider the case that @xmath3 mode and/or @xmath2 mode dominant in two or more @xmath205 . in this case , if we have a priori knowledge of the values of @xmath206 in each of @xmath205 for coefficients
@xmath203 and @xmath204 , we can separate @xmath3 mode and @xmath2 mode . for example , assume that @xmath3 mode is dominated by @xmath207 , while @xmath2 mode is dominated by @xmath208 , then and become @xmath209\ , \label{3c12}\\ & & c_{34}(f)=\left[c^{i}_{00}i(f)\left(\gamma_{00,34}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)+c_{11}^{v}v(f)\gamma_{11,34}^{v}\right]\ .\label{3c34}\end{aligned}\ ] ] thus , we can separate @xmath3 mode and @xmath2 mode by using linear combinations @xmath210\ , \\
d_{v}&\equiv&a_{v}c_{34}(f)+b_{v}c_{12}(f ) \nonumber\\ & = & c_{11}^{v}v(f)\left[\gamma_{11,34}^{v}\left(\gamma_{00,12}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)-\gamma_{11,12}^{v}\left(\gamma_{00,34}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)\right]\ , \end{aligned}\ ] ] where @xmath211 as in the previous calculations , we can get the compiled orfs @xmath212^{1/2}}\ , \label{gi1234}\\ \gamma_{12:34}^{v}&\equiv&\frac{\gamma_{11,34}^{v}\left(\gamma_{00,12}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)-\gamma_{11,12}^{v}\left(\gamma_{00,34}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)}{\left[\left(\gamma_{00,12}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)^2+\left(\gamma_{00,34}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)^2\right]^{1/2}}\ .\label{gv1234}\end{aligned}\ ] ] [ cols="^,^ " , ] in fig .
[ cg ] we show some compiled orfs @xmath213 ( left panels ) and @xmath214 ( right panels ) as a function of the two angular separations @xmath137 and @xmath215 for two pulsar pairs , respectively .
we used the expressions of @xmath2 mode and @xmath3 mode ( see appendix [ sec : the generalized overlap reduction function for intensity ] ) , and we assumed @xmath216 for simplicity . in fig .
[ cg](a ) and [ cg](b ) , the @xmath3 mode is dominated by @xmath217 and @xmath2 mode is dominated by @xmath218 . in fig .
[ cg](c ) and [ cg](d ) , the @xmath3 mode is dominated by @xmath219 and @xmath2 mode is dominated by @xmath218 . in fig .
[ cg](e ) and [ cg](f ) , the @xmath3 mode is dominated by @xmath207 and @xmath2 mode is dominated by @xmath218 . in fig .
[ cg](e ) and [ cg](f ) , the @xmath3 mode is dominated by @xmath220 and @xmath2 mode is dominated by @xmath218 . by definition , in the case of @xmath221 ,
the compiled orfs are zero .
we have studied the detectability of the stochastic gravitational waves with ptas . in most of the previous works ,
the isotropy of sgwb has been assumed for the analysis .
recently , however , a stochastic gravitational wave background with anisotropy have been considered .
the information of the anisotropic pattern of the distribution should contain important information of the sources such as supermassive black hole binaries and the sources in the early universe .
it is also intriguing to take into account the polarization of sgwb in the pta analysis .
therefore , we extended the correlation analysis to circularly polarized sgwb and calculated generalized overlap reduction functions for them .
it turned out that the circular polarization can not be detected for an isotropic background .
however , when the distribution has anisotropy , we have shown that there is a chance to observe circular polarizations in the sgwb .
we also discussed how to separate polarized modes from unpolarized modes of gravitational waves .
if we have a priori knowledge of the abundance ratio for each mode in each of @xmath205 , we can separate @xmath3 mode and @xmath2 mode in general .
this would be possible if we start from fundamental theory and calculate the spectrum of sgwb .
in particular , in the case that the signal of lowest @xmath222 is dominant , we performed the separation of @xmath3 mode and @xmath2 mode explicitly .
this work was supported by grants - in - aid for scientific research ( c ) no.25400251 and " mext grant - in - aid for scientific research on innovative areas no.26104708 and `` cosmic acceleration''(no.15h05895 ) .
in this appendix , we perform angular integration of the generalized orf for dipole ( @xmath154 ) circular polarization ( see @xcite ) : @xmath223 where we have defined @xmath224 .
it is obvious that in the case of @xmath225 , integrand of the generalized orf is zero , because of @xmath226 , then we obtain @xmath227 then , using - , we calculate @xmath228 and we find @xmath229 therefore we only have to consider the dipole generalized orf in the case of @xmath154 , @xmath230 : @xmath231 where @xmath232 first , to calculate @xmath233 , we use contour integral in the complex plane . defining @xmath234 and substituting @xmath235 into , we can rewrite @xmath233 as @xmath236 } \ , \end{aligned}\ ] ] where @xmath237 denotes a unit circle .
we can factorize the denominator of the integrand and get @xmath238 where @xmath239 hereafter , the upper sign applies when @xmath240 and the lower one applies when @xmath241 .
note that we only consider the region @xmath242 , so we have used the relation @xmath243 in above expression . in the region
@xmath244 , @xmath245 is inside the unit circle @xmath237 except for @xmath246 and @xmath247 is outside the unit circle @xmath237 .
now , we can perform the integral using the residue theorem @xmath248 where @xmath249 the residues inside the unit circle @xmath237 can be evaluated as @xmath250\right\ } = \frac{i(z_{+}+z_{-})}{2\sqrt{1-x^2}\sin\xi } \ , \end{aligned}\ ] ] @xmath251 thus , we obtain @xmath252 next , we consider @xmath253 defined in
. using , we can calculate @xmath253 as @xmath254 similarly , we can evaluate @xmath255 given in . to calculate @xmath255 in the complex plane , we again substitute into and obtain @xmath256 we use the residue theorem @xmath257 where @xmath258 the residues inside the unit circle @xmath237 can be calculated as @xmath259\right\ } = \frac{i(z_{+}^2+z_{-}^2)}{4\sqrt{1-x^2}\sin\xi } \ , \end{aligned}\ ] ] @xmath260 therefore , @xmath255 becomes @xmath261 substituting to , we can calculate @xmath262 : @xmath263 finally , substituting and into , we get the generalized orf for @xmath264 @xmath265\ .\end{aligned}\ ] ] as a function of the angular separation between the two pulsars @xmath137 .
[ gi](a ) shows monopole ( l=0 ) , fig .
[ gi](b ) shows dipole ( l=1 ) , fig .
[ gi](c ) shows quadrupole ( l=2 ) and fig .
[ gi](d ) shows octupole ( l=3 ) .
the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( a ) @xmath152 as a function of the angular separation between the two pulsars @xmath137 .
[ gi](a ) shows monopole ( l=0 ) , fig .
[ gi](b ) shows dipole ( l=1 ) , fig .
[ gi](c ) shows quadrupole ( l=2 ) and fig .
[ gi](d ) shows octupole ( l=3 ) .
the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( b ) @xmath154 as a function of the angular separation between the two pulsars @xmath137 . fig . [ gi](a ) shows monopole ( l=0 ) , fig .
[ gi](b ) shows dipole ( l=1 ) , fig .
[ gi](c ) shows quadrupole ( l=2 ) and fig .
[ gi](d ) shows octupole ( l=3 ) .
the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( c ) @xmath157 as a function of the angular separation between the two pulsars @xmath137 .
[ gi](a ) shows monopole ( l=0 ) , fig .
[ gi](b ) shows dipole ( l=1 ) , fig .
[ gi](c ) shows quadrupole ( l=2 ) and fig .
[ gi](d ) shows octupole ( l=3 ) . the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( d ) @xmath159
in this appendix , we show orfs for the intensity @xcite .
the following form for @xmath152 was derived in @xcite , and our expressions are identical to their expressions : @xmath271\ , \end{aligned}\ ] ] for , @xmath154 , we calculated as @xmath272\ , \\ \gamma^{i}_{11}&=&\frac{\sqrt{6\pi}}{12}\sin\xi\left[1 + 3(1-\cos\xi)\left\{1+\frac{4}{1+\cos\xi}\log\left(\sin\frac{\xi}{2}\right)\right\}\right]\ , \\ \gamma^{i}_{1 - 1}&=&-\gamma^{i}_{11}\ , \end{aligned}\ ] ] for @xmath157 , we obtain @xmath273\ , \\ \gamma^{i}_{21}&=&-\frac{\sqrt{30\pi}}{60}\sin\xi\left[21 - 15\cos\xi-5\cos^2\xi+60\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{2 - 1}&=&-\gamma^{i}_{2 - 1}\ , \\ \gamma^{i}_{22}&=&\frac{\sqrt{30\pi}}{24}(1-\cos\xi)\left[9 - 4\cos\xi-\cos^2\xi+24\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{2 - 2}&=&\gamma^{i}_{22}\ ,
\end{aligned}\ ] ] for @xmath159 , it is straightforward to reach the following @xmath274\ , \\ \gamma^{i}_{31}&=&\frac{\sqrt{21\pi}}{48}\sin\xi(1-\cos\xi)\left[34 + 15\cos\xi+5\cos^2\xi+\frac{96}{1+\cos\xi}\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 1}&=&-\gamma^{i}_{31}\ , \\ \gamma^{i}_{32}&=&-\frac{\sqrt{210\pi}}{48}(1-\cos\xi)\left[17 - 9\cos\xi-3\cos^2\xi-\cos^3\xi+48\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 2}&=&\gamma^{i}_{32}\ , \\ \gamma^{i}_{33}&=&\frac{\sqrt{35\pi}}{48}\frac{(1-\cos\xi)^2}{\sin\xi}\left[34 - 17\cos\xi-4\cos^2\xi-\cos^3\xi+96\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 3}&=&-\gamma^{i}_{33}\ .\end{aligned}\ ] ] these are plotted in fig .
the generalized orfs of total intensity are different from that of circular polarization in that the value for @xmath149 is non - trivial .
then the @xmath3 mode orfs for @xmath275 have value even in the case of @xmath151 .
this implies that we can consider auto - correlation for a single pulsar .
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rapid progress in the design and manufacture of optical fiber systems is a result of worldwide demand for ultra - high bit - rate optical communications .
this explains the growing interest of the soliton community in soliton - based optical fiber communication systems .
this area of research was considerably advanced in recent years @xcite .
the most remarkable results include the application of the concept of the dispersion management to _ temporal optical solitons _ and soliton - based optical transmission systems , and the discovery of the so - called _ dispersion managed soliton_. high - speed optical communications require effective components such as high - performance broadband computer networks that can be developed by employing the concept of the bit - parallel - wavelength ( bpw ) pulse transmission that offers many of the advantages of both parallel fiber ribbon cable and conventional wavelength - division - multiplexing ( wdm ) systems @xcite .
expanding development in the study of the soliton fiber systems has been observed in parallel with impressive research on their spatial counterparts , optical self - trapped beams or _
spatial optical solitons_. one of the key concepts in this field came from the theory of multi - frequency wave mixing and cascaded nonlinearities where a nonlinear phase shift is produced as a result of the parametric wave interaction @xcite .
in all such systems , the nonlinear interaction between the waves of two ( or more ) frequencies is the major physical effect that can support coupled - mode multi - frequency solitary waves .
the examples of temporal and spatial solitons mentioned above have one common feature : they involve the study of solitary waves in multi - component nonlinear models .
the main purpose of this paper is to overview several different physical examples of multi - mode and/or multi - frequency solitary waves that occur for the pulse or beam propagation in nonlinear optical fibers and waveguides . for these purposes , we select three different cases : multi - wavelength solitary waves in bit - parallel - wavelength optical fiber links , multi - colour spatial solitons due to multistep cascading in optical waveguides with quadratic nonlinearities , and quasiperiodic solitons in the fibonacci superlattices .
we believe these examples display both the diversity and richness of the multi - mode soliton systems , and they will allow further progress to be made in the study of nonlinear waves in multi - component nonintegrable physical models .
because the phenomenon of the long - distance propagation of _ temporal optical solitons _ in optical fibers @xcite is known to a much broader community of researchers in optics and nonlinear physics , first we emphasize _ the difference between temporal and spatial solitary waves_. indeed , for a long time stationary beam propagation in planar waveguides has been considered somewhat similar to the pulse propagation in fibers .
this approach is based on the so - called _ spatio - temporal analogy _ in wave propagation , meaning that the propagation coordinate @xmath0 is treated as the evolution variable and the spatial beam profile along the transverse direction in waveguides , is similar to the temporal pulse profile in fibers .
this analogy is based on a simple notion that both beam evolution and pulse propagation can be described by the cubic nonlinear schrdinger ( nls ) equation .
however , contrary to the widely accepted opinion , there is a crucial difference between temporal and spatial solitons . indeed , in the case of the nonstationary pulse propagation in fibers , the operation wavelength is usually selected near the zero point of the group - velocity dispersion .
this means that the absolute value of the fiber dispersion is small enough to be compensated by a weak nonlinearity such as that produced by the ( very weak ) kerr effect in optical fibers which leads to a relative nonlinearity - induced change in the refractive index .
therefore , nonlinearity in such systems is always weak and it should be well modeled by a cubic nls equation which is known to be integrable by means of the inverse - scattering technique . however , for very short ( e.g. , fs ) pulses the cubic nls equation describing the long - distance propagation of pulses should be corrected to include additional terms that would account for such effects as higher - order dispersion , raman scattering , etc .
all such corrections can be taken into account with the help of the perturbation theory @xcite .
thus , in fibers nonlinear effects are weak and they become important only when dispersion is small ( near the zero - dispersion point ) affecting the pulse propagation over large distances ( of order of hundreds of meters or even kilometers ) .
the situation changes dramatically when we consider the propagation of multi - wavelength pulses with almost equal group velocities .
the corresponding model is described by a nonintegrable and rather complicated system of coupled nls equations , which we briefly discuss below . in contrary to the pulse propagation in optical fibers ,
the physics underlying the stationary beam propagation in planar waveguides and bulk media is different . in this case
an optical beam is generated by a continuous wave ( cw ) source and it is time independent .
however , when the beam evolves with the propagation distance @xmath0 , it diffracts in the transverse spatial directions .
then , a nonlinear change in the refractive index should compensate for the beam spreading caused by diffraction _ which is not a small effect_. that is why to observe spatial solitons as self - trapped optical beams , much larger nonlinearities are usually required , and very often such nonlinearities are not of the kerr type ( e.g. they saturate at higher intensities ) .
this leads to the models of generalized nonlinearities with the properties of solitary waves different from those described by the integrable cubic nls equation .
propagation distances involved in the phenomenon of the beam self - focusing and spatial soliton propagation are of the order of millimeters or centimeters .
to achieve such large nonlinearities , one needs to use the optical materials with large nonlinearity - induced refractive index .
one of the possible way to overcome this difficulty is to use the so - called _ cascaded nonlinearities _ of noncentrosymmetric optical materials where nonlinear effects are accumulated due to parametric wave interaction under the condition of the wave phase matching .
such parametric wave - mixing effects generate novel classes of spatial optical solitons where resonant parametric coupling between the envelopes of two ( or more ) beams of different frequencies supports stable spatially localised waves even in a bulk medium ( see details in ref .
it is this kind of multi - component solitary waves that we discuss below .
a growing demand for high - speed computer communications requires an effective and inexpensive computer interconnection .
one attractive alternative to the conventional wdm systems is bpw single - mode fiber optics links for very high bandwidth computer communications @xcite .
they differ from the wdm schemes in that no parallel to serial conversion is necessary , and parallel pulses are launched simultaneously on different wavelengths .
when the pulses of different wavelengths are transmitted simultaneously , the cross - phase modulation can produce an interesting _ pulse shepherding effect _
@xcite , when a strong ( `` shepherd '' ) pulse enables the manipulation and control of pulses co - propagating on different wavelengths in a multi - channel optical fiber link . to describe the simultaneous transmission of @xmath1 different wavelengths in a nonlinear optical fiber , we follow the standard derivation @xcite and obtain a system of @xmath1 coupled nonlinear schrdinger ( nls ) equations @xmath2 ) : @xmath3 { \displaystyle \qquad + \chi_j \left ( |a_j|^2 + 2 \sum_{m \neq j } |a_m|^2
\right ) a_j = 0 , } \end{array}\ ] ] where , for the @xmath4th wave , @xmath5 is the slowly varying envelope , @xmath6 and @xmath7 are the group velocity and group - velocity dispersion , respectively , and the nonlinear coefficients @xmath8 characterize the kerr effect
. equations ( [ eq : nls_dim ] ) do not include the fiber loss , since the fiber lengths involved in bit - parallel links are only a small fraction of the attenuation length .
we measure the variables in the units of the central wavelength channel ( say , @xmath9 ) , and obtain the following normalized system of the @xmath1 coupled nls equations , @xmath10 { \displaystyle \qquad + \gamma_j \left(|u_j|^2 + 2 \sum_{m\neq j } |u_m|^2\right ) u_j = 0 , } \end{array}\ ] ] where @xmath11 , @xmath12 is the incident optical power in the central channel , @xmath13 , @xmath14 , so that @xmath15 . for the operating wavelengths spaced @xmath16 nm apart within the band @xmath17 nm , the coefficients @xmath18 and @xmath19 are different but close to @xmath20 . initially , in eq .
( [ eq : nls ] ) , we omit the mode walk - off effect described by the parameters @xmath21 ( so that @xmath22 ) .
this effect will be analysed later in this section . to analyze the nonlinear modes , i.e. localized states of the bpw model ( [ eq : nls ] )
, we look for stationary solutions in the form , @xmath23 and therefore obtain the system of equations for the normalized mode amplitudes , @xmath24 { \displaystyle \frac{1}{2 } \alpha_n \frac{d^2u_n}{dt^2 } + \gamma_n \left ( |u_n|^2 + 2\sum_{m \neq n } |u_m|^2 \right ) u_n = \lambda_n u_n , } \end{array}\ ] ] where @xmath25 , the amplitudes and time are measured in the units of @xmath26 and @xmath27 , respectively , and @xmath28 . system ( [ eq : nls_nn ] ) has _ exact analytical solutions _ for @xmath1 coupled components , the so - called _
bpw solitons_. indeed , looking for solutions in the form @xmath29 , @xmath30 , we obtain the constraint @xmath31 , and a system of @xmath1 coupled algebraic equations for the wave amplitudes , @xmath32 in a special symmetric case , we take @xmath33 , and the solution of those equations is simple @xcite : + @xmath34^{-1/2}$ ] .
analytical solutions can also be obtained in the _ linear limit _
, when the central frequency pulse ( at @xmath35 ) is large . then , linearizing eqs .
( [ eq : nls_nn ] ) for small @xmath36 , we obtain a decoupled nonlinear equation for @xmath37 and @xmath38 decoupled linear equations for @xmath39 . each of the latter possess a localized solution provided @xmath40 , where @xmath41 ^ 2 $ ] . in this limit
the central soliton pulse @xmath37 ( `` shepherd pulse '' ) can be considered as inducing an effective waveguide that supports a fundamental mode @xmath39 with the corresponding cutoff @xmath42 .
since , by definition , the parameters @xmath43 and @xmath44 are close to @xmath20 , we can verify that the soliton - induced waveguide supports maximum of two modes ( fundamental and the first excited one ) .
this is an important physical result that explains the effective robustness of the pulse guidance by the shepherding pulse . to demonstrate a number of unique properties of the multi - channel bpw solitons , we consider the case @xmath45 in more details .
a comprehensive discussion of the case @xmath46 can be found in the preprint @xcite .
we select the following set of the normalized parameters : @xmath47 , @xmath48 , and @xmath49 .
solitary waves of this four - wavelength bpw system can be found numerically as localized solutions of eqs .
( [ eq : nls_nn ] ) . figure [ fig : bpw1 ] presents the lowest - order families of such localized solutions . in general
, they are characterized by @xmath38 parameters , but we can capture the characteristic features by presenting power dependencies along the line @xmath50 in the parameter space @xmath51 .
the power of the central - wavelength component ( @xmath35 ) does not depend on @xmath52 ( straight line @xmath53 ) .
thin dashed , dotted , and dash - dotted curves correspond to the three separate single - mode solitons of the multi - channel bpw system , ( 1 ) , ( 2 ) , and ( 3 ) , respectively , shown with the corresponding branches of ( 0 + 1 ) , ( 0 + 2 ) , and ( 0 + 3 ) two - mode solitons .
the latter curves start off from the bifurcation points on the @xmath37 branch at @xmath54 , @xmath55 , and @xmath56 , respectively .
close separation of those curves is the result of closeness of the parameters @xmath43 and @xmath44 for @xmath57 .
thick solid curves in fig .
[ fig : bpw1 ] correspond to the two- ( 1 + 2 ) and three - mode ( 0 + 1 + 2 ) localized solutions .
the latter solutions bifurcate and give birth to four - wavelength solitons ( 0 + 1 + 2 + 3 ) ( branch a - b ) .
two examples of such four - wave composite solitons are shown in fig .
[ fig : bpw1 ] ( bottom row ) .
the solution b is close to an exact sech - type solution at @xmath58 ( described above ) for @xmath45 , whereas the solution a is close to that approximately described in the linear limit in the vicinity of a bifurcation point .
importantly , for different values of the parameters @xmath59 , the uppermost bifurcation point for this branch ( open circle in fig . [
fig : bpw1 ] ) is not predicted by a simple linear theory and , due to the nonlinear mode coupling , it gets shifted from the branch of the central - wavelength soliton ( straight line ) to a two - mode branch ( 0 + 1 + 2 ) ( thick solid curve ) . as a result , if we start on the right end of the horizontal branch and follow the lowest branches of the total power @xmath60 in fig .
[ fig : bpw1 ] , we pass the following sequence of the soliton families and bifurcation points : @xmath61 . if the modal parameters are selected closer to each other , the first two links of _ the bifurcation cascade _ disappear ( i.e. the corresponding bifurcation points merge ) , and the four - mode soliton bifurcates directly from the central - wavelength pulse , as predicted by the linear theory .
note however that the sequence and location of the bifurcation points is a function of the cross - section of the parameter space @xmath51 , and the results presented above correspond to the choice @xmath62 . the qualitative picture of the cascading bifurcations preserves for other values of @xmath1 .
in particular , near the bifurcation point a mixed - mode soliton corresponds to the localized modes guided by the central - wavelength soliton ( shepherd ) pulse .
the existence of such soliton solutions is a key concept of bpw transmission when the data are launched in parallel carrying a desirable set of bits of information , all guided by the shepherd pulse at a selected wavelength .
effects of the walk - off on the multi - channel bpw solitons seems to be most dangerous for the pulse alignment in the parallel links . for nearly equal soliton components ,
it was shown long time ago @xcite that nonlinearity can provide an effective trapping mechanism to keep the pulses together . for the shepherding effect , the corresponding numerical simulations are presented in figs .
[ fig : bpw2](a - d ) for the four - channel bpw system .
initially , we launch a composite four - mode soliton as an unperturbed solution a [ see fig . [
fig : bpw1 ] ] of eqs .
( [ eq : nls ] ) , without walk - off and centered at @xmath63 .
when this solution evolves along the propagation direction @xmath0 in the presence of small to moderate relative walk - off ( @xmath64 for @xmath65 ) , its components remain strongly localized and mutually trapped [ fig .
[ fig : bpw2](a , b ) ] , whereas it loses some energy into radiation for much larger values of the relative mode walk - off [ fig .
[ fig : bpw2](c , d ) ] .
recent progress in the study of cascading effects in optical materials with quadratic ( second - order or @xmath66 ) nonlinear response has offered new opportunities for all - optical processing , optical communications , and optical solitons @xcite .
most of the studies of cascading effects employ parametric wave mixing processes with a single phase - matching and , as a result , two - step cascading @xcite .
for example , the two - step cascading associated with type i second - harmonic generation ( shg ) includes the generation of the second harmonic ( @xmath67 ) followed by reconstruction of the fundamental wave through the down - conversion frequency mixing ( dfm ) process ( @xmath68 ) .
these two processes are governed by one phase - matched interaction and they differ only in the direction of power conversion . the idea to explore more than one simultaneous nearly phase - matched process , or _
double - phase - matched ( dpm ) wave interaction _ , became attractive only recently @xcite , for the purposes of all - optical transistors , enhanced nonlinearity - induced phase shifts , and polarization switching .
in particular , it was shown @xcite that multistep cascading can be achieved by two second - order nonlinear cascading processes , shg and sum - frequency mixing ( sfm ) , and these two processes can also support a novel class of multi - colour parametric solitons @xcite , briefly discussed below . to introduce the simplest model of multistep cascading ,
we consider the fundamental beam with frequency @xmath69 entering a noncentrosymmetric nonlinear medium with a quadratic response . as a first step ,
the second - harmonic wave with frequency @xmath70 is generated via the shg process .
as a second step , we expect the generation of higher order harmonics due to sfm , for example , a third harmonic ( @xmath71 ) or even fourth harmonic ( @xmath72 ) @xcite .
when both such processes are nearly phase matched , they can lead , via down - conversion , to a large nonlinear phase shift of the fundamental wave @xcite .
additionally , the multistep cascading can support _ a novel type of three - wave spatial solitary waves _ in a diffractive @xmath66 nonlinear medium , _ multistep cascading solitons_. we start our analysis with the reduced amplitude equations derived in the slowly varying envelope approximation with the assumption of zero absorption of all interacting waves ( see , e.g. , ref .
@xcite ) . introducing the effect of diffraction in a slab waveguide geometry
, we obtain @xmath73 { \displaystyle \qquad\qquad\qquad\qquad + \chi_{2}a_{2 } a_{1}^{\ast}e^{-i\delta k_{2}z } = 0 , } \\*[9pt ] { \displaystyle 4
i k_{1 } \frac{\partial a_{2}}{\partial z } + \frac{\partial^{2 } a_{2}}{\partial x^{2 } } + \chi_{4 } a_{3 } a_{1}^{\ast } e^{-i\delta k_{3}z } } \\*[9pt ] { \displaystyle \qquad\qquad\qquad\qquad + \chi_{5 } a_{1}^{2 } e^{i\delta k_{2}z } = 0 , } \\*[9pt ] { \displaystyle 6 i k_{1}\frac{\partial a_{3}}{\partial z } + \frac{\partial^{2 } a_{3}}{\partial x^{2 } } + \chi_{3}a_{2}a_{1}e^{i\delta k_{3}z } = 0 , } \end{array}\ ] ] where @xmath74 , @xmath75 , and @xmath76 , and the nonlinear coupling coefficients @xmath77 are proportional to the elements of the second - order susceptibility tensor which we assume to satisfy the following relations ( no dispersion ) , @xmath78 , @xmath79 , and @xmath80 . in eqs .
( [ physeqns ] ) , @xmath81,@xmath82 and @xmath83 are the complex electric field envelopes of the fundamental harmonic ( fh ) , second harmonic ( sh ) , and third harmonic ( th ) , respectively , @xmath84 is the wavevector mismatch for the shg process , and @xmath85 is the wavevector mismatch for the sfm process .
the subscripts ` 1 ' denote the fh wave , the subscripts ` 2 ' denote the sh wave , and the subscripts ` 3 ' , the th wave .
following the technique earlier employed in refs .
@xcite , we look for stationary solutions of eq .
( [ physeqns ] ) and introduce the normalised envelope @xmath86 , @xmath87 , and @xmath88 according to the relations , @xmath89 { \displaystyle a_{2 } = \frac{2 \beta k_{1}}{\chi_{2}}e^{2i\beta z + i\delta k_{2 } z } v , } \\*[9pt ] { \displaystyle a_{3 } = \frac{\sqrt{2\chi_{2}}\beta k_{1}}{\chi_{1}\sqrt{\chi_{5}}}e^{3i\beta z + i\delta k z } u , } \end{array}\ ] ] where @xmath90 . renormalising the variables as @xmath91 and @xmath92 , we finally obtain a system of coupled equations , @xmath93 { \displaystyle 2i \frac{\partial v}{\partial z } + \frac{\partial^{2 } v}{\partial x^{2 } } - \alpha v + \frac{1}{2 } w^{2 } + w^{\ast}u = 0 , } \\*[9pt ] { \displaystyle 3i\frac{\partial u}{\partial z } + \frac{\partial^{2 } u}{\partial x^{2 } } - \alpha_{1}u + \chi vw = 0 , } \\*[9pt ] \end{array}\ ] ] where @xmath94 and @xmath95 are two dimensionless parameters that characterise the nonlinear phase matching between the parametrically interacting waves .
dimensionless material parameter @xmath96 depends on the type of phase matching , and it can take different values of order of one .
for example , when both shg and sfm are due to quasi - phase matching ( qpm ) , we have @xmath97 $ ] , where @xmath98 .
then , for the first - order @xmath99 qpm processes ( see , e.g. , ref .
@xcite ) , we have @xmath100 , and therefore @xmath101 . when sfm is due to the third - order qpm process ( see , e.g. , ref .
@xcite ) , we should take @xmath102 , and therefore @xmath103 . at
last , when sfm is the fifth - order qpm process , we have @xmath104 and @xmath105 .
dimensionless equations ( [ normal ] ) present a fundamental model for three - wave multistep cascading solitons in the absence of walk - off . additionally to the type
i shg solitons ( see , e.g. , refs @xcite ) , the multistep cascading solitons involve the phase - matched sfm interaction ( @xmath106 ) that generates a third harmonic wave .
two - parameter family of localised solutions consists of three mutually coupled waves .
it is interesting to note that , similar to the case of nondegenerate three - wave mixing @xcite , eqs .
( [ normal ] ) possess an exact solution . to find it , we make a substitution @xmath107 , @xmath108 and @xmath109 , and obtain unknown parameters from the following algebraic equations @xmath110 valid for @xmath111 and @xmath112 .
equations ( [ exactsol ] ) have two solutions corresponding to _ positive _ and _ negative _ values of the amplitude ( @xmath113 ) .
this indicates a possibility of multi - valued solutions , even within the class of exact solutions . in general , three - wave solitons of eqs .
( [ normal ] ) can be found only numerically .
figures [ fig : tr1](a ) and [ fig : tr1](b ) present two examples of solitary waves for different sets of the mismatch parameters @xmath114 and @xmath115 .
when @xmath116 [ see fig .
[ fig : tr1](a ) ] , which corresponds to an unmatched sfm process , the amplitude of the third harmonic is small , and it vanishes for @xmath117 . to summarise different types of three - wave solitary waves , in fig .
[ fig : tr2 ] we plot the dependence of the total soliton power defined as @xmath118 .
it is clearly seen that for some values of @xmath119 ( including the exact solution at @xmath120 shown by two filled circles ) , there exist _ two different branches _ of three - wave solitary waves , and only one of those branches approaches , for large values of @xmath119 , a family of two - wave solitons of the cascading limit ( fig . [ fig : tr2 ] , dashed ) . the slope of the branches changes from negative ( for small @xmath119 ) to positive ( for large @xmath119 ) , indicating a possible change of the soliton stability .
however , the detailed analysis of the soliton stability is beyond the scope of this paper ( see , e.g. , refs .
@xcite ) .
another type of multistep cascading parametric processes which involve only two frequencies , i.e. _ two - colour multistep cascading _
, can occur due to the vectorial interaction of waves with different polarization .
we denote two orthogonal polarization components of the fundamental harmonic ( fh ) wave ( @xmath121 ) as a and b , and two orthogonal polarizations of the second harmonic ( sh ) wave ( @xmath122 ) , as s and t. then , a simple multistep cascading process consists of the following steps .
first , the fh wave a generates the sh wave s via type i shg process .
then , by down - conversion
sa - b , the orthogonal fh wave b is generated .
at last , the initial fh wave a is reconstructed by the processes sb - a or ab - s , sa - a .
two principal second - order processes aa - s and ab - s correspond to _ two different components _ of the @xmath66 susceptibility tensor , thus introducing additional degrees of freedom into the parametric interaction .
different cases of such type of multistep cascading processes are summarized in table [ tab : dpm ] . to demonstrate some of the unique properties of the multistep cascading
, we discuss here how it can be employed for soliton - induced waveguiding effects in quadratic media . for this purpose
, we consider a model of two - frequency multistep cascading described by the principal dpm process ( c ) ( see table [ tab : dpm ] above ) in the planar slab - waveguide geometry .
using the slowly varying envelope approximation with the assumption of zero absorption of all interacting waves , we obtain @xmath123 { \displaystyle 2 i k_{1}\frac{\partial b}{\partial z } + \frac{\partial^{2 } b } { \partial x^{2 } } + \chi_2 s b^{\ast}e^{-i\delta k_2 z } = 0 , } \\*[9pt ] { \displaystyle 4 i k_{1 } \frac{\partial s}{\partial z } + \frac{\partial^{2 } s}{\partial x^{2 } } + 2 \chi_1 a^2 e^{i\delta k_1 z } + 2 \chi_2 b^2 e^{i\delta k_2 z } = 0 , } \end{array}\end{aligned}\ ] ] where @xmath74 , the nonlinear coupling coefficients @xmath77 are proportional to the elements of the second - order susceptibility tensor , and @xmath124 and @xmath125 are the corresponding wave - vector mismatch parameters . to simplify the system ( [ eq_1 ] ) , we look for its stationary solutions and introduce the normalized envelopes @xmath126 , @xmath127 , and @xmath128 according to the following relations , @xmath129 , @xmath130 , and @xmath131 , where @xmath132 , @xmath133 , and @xmath134 , and the longitudinal and transverse coordinates are measured in the units of @xmath135 and @xmath136 , respectively .
then , we obtain a system of normalized equations , @xmath137 { \displaystyle i \frac{\partial v}{\partial z } + \frac{\partial^{2 } v}{\partial x^{2 } } - \alpha_1 v + \chi v^{\ast}w= 0 , } \\*[9pt ] { \displaystyle 2i \frac{\partial w}{\partial z } + \frac{\partial^{2}w}{\partial x^{2 } } - \alpha w+\frac{1}{2}(u^2+v^2)= 0 , } \end{array}\ ] ] where @xmath138 , @xmath139 , and @xmath140 .
first of all , we notice that for @xmath141 ( or , similarly , @xmath142 ) , the dimensionless eqs .
( [ eq_n ] ) reduce to the corresponding model for the two - step cascading due to type i shg discussed earlier @xcite , and its stationary solutions are defined by the equations for real @xmath126 and @xmath128 , @xmath143 { \displaystyle \frac{d^2 w}{d x^{2 } } - \alpha w + \frac{1}{2 } u^2 = 0 , } \end{array}\ ] ] that possess a one - parameter family of two - wave localized solutions @xmath144 found earlier numerically for any @xmath145 , and also known analytically for @xmath146 , @xmath147 ( see ref .
@xcite ) .
then , in the small - amplitude approximation , the equation for real orthogonally polarized fh wave @xmath127 can be treated as an eigenvalue problem for an effective waveguide created by the sh field @xmath148 , @xmath149 v = 0.\ ] ] therefore , an additional parametric process allows to propagate a probe beam of one polarization in _ an effective waveguide _ created by a two - wave spatial soliton in a quadratic medium with fh component of another polarization .
however , this type of waveguide is different from what has been studied for kerr - like solitons because it is _ coupled parametrically _ to the guided modes and , as a result , the physical picture of the guided modes is valid , rigorously speaking , only in the case of stationary phase - matched beams . as a result , the stability of the corresponding waveguide and localized modes of the orthogonal polarization it guides is a key issue . in particular , the waveguide itself ( i.e. _ two - wave parametric soliton _ ) becomes unstable for @xmath150 @xcite . in order to find the guided modes of the parametric waveguide created by a two - wave quadratic soliton
, we have to solve eq .
( [ eq_eigen ] ) where the exact solution @xmath148 is to be found numerically .
then , to address this problem analytically , approximate solutions can be used , such as those found with the help of the variational method @xcite .
however , the different types of the variational ansatz used do not provide a very good approximation for the soliton profile at all @xmath114 .
for our eigenvalue problem ( [ eq_eigen ] ) , the function @xmath148 defines parameters of the guided modes and , in order to obtain accurate results , it should be calculated as close as possible to the exact solutions found numerically . to resolve this difficulty , below we suggest a novel `` almost exact '' solution that _ would allow to solve analytically many of the problems involving quadratic solitons _
, including the eigenvalue problem ( [ eq_eigen ] ) .
first , we notice that from the exact solution at @xmath146 and the asymptotic result for large @xmath114 , @xmath151 , it follows that the sh component @xmath148 of eqs .
( [ eq_2 ] ) remains almost self - similar for @xmath152 .
thus , we look for the sh field in the form @xmath153 , where @xmath154 and @xmath155 are to be defined .
the solution for @xmath156 should be consistent with this choice of the shape for sh , and it is defined by the first ( linear for @xmath126 ) equation of the system ( [ eq_2 ] ) .
therefore , we can take @xmath126 in the form of the lowest guided mode , @xmath157 , that corresponds to an effective waveguide @xmath148 . by matching the asymptotics of these trial functions with those defined directly from eqs .
( [ eq_2 ] ) at small and large @xmath158 , we obtain the following solution , @xmath159 @xmath160 here , the third relation allows us to find @xmath154 for arbitrary @xmath114 as a solution of a cubic equation , and then to find all other parameters as functions of @xmath114 . for mismatches in the interval @xmath161
, the parameter values change monotonically in the regions : @xmath162 , @xmath163 , and @xmath164 .
it is really amazing that the analytical solution ( [ eq_s]),([eq_p ] ) provides _ an excellent approximation _ for the profiles of the two - wave parametric solitons found numerically , with the relative errors not exceeding 1%3% for stable solitons
( e.g. when @xmath165 ) . as a matter of fact
, we can treat eqs .
( [ eq_s ] ) and ( [ eq_p ] ) as an _
approximate scaling transformation _ of the family of two - wave bright solitons .
moreover , this solution allows us to capture some remarkable internal similarities and distinctions between the solitons existing in different types of nonlinear media .
in particular , as follows from eqs .
( [ eq_s ] ) and ( [ eq_p ] ) , the fh component and the self - consistent effective waveguide ( created by the sh field ) have approximately the same stationary transverse profiles as for one - component solitons in a kerr - like medium with power - law nonlinear response @xcite . for @xmath166 ( @xmath167 ) and @xmath168 ( @xmath169 )
our general expressions reduce to the known analytical solutions , and the fh profile is exactly the same as that for solitons in quadratic and cubic kerr media , respectively .
on the other hand , the strength of self - action for quadratic solitons depends on the normalized phase mismatch @xmath114 and , in general , the beam dynamics for parametric wave mixing can be very different from that observed in kerr - type media .
now , the eigenvalue problem ( [ eq_eigen ] ) can be readily solved analytically .
the eigenmode cutoff values are defined by the parameter @xmath119 that takes one of the discrete values , @xmath170 , where @xmath171^{1/2}$ ] .
number @xmath172 stands for the mode order @xmath173 , and the localized solutions are possible provided @xmath174 .
the profiles of the corresponding guided modes are @xmath175 where @xmath176 , @xmath177 is the hypergeometric function , and @xmath178 is the mode s amplitude which can not be determined within the framework of the linear analysis . according to these results , a two - wave parametric soliton creates , a multi - mode waveguide and larger number of the guided modes can be observed for smaller @xmath114 .
figures [ fig : al1](a , b ) show the dependence of the mode cutoff values @xmath179 for a fixed @xmath180 , and @xmath181 for a fixed @xmath114 , respectively . for the case
@xmath103 , the dependence has a simple form : @xmath182 ^ 2 $ ] .
because a two - wave soliton creates an induced waveguide parametrically coupled to its guided modes of the orthogonal polarization , the dynamics of the guided modes _ may differ drastically _ from that of conventional waveguides based on the kerr - type nonlinearities .
figures show two examples of the evolution of guided modes . in the first example
[ see fig .
[ fig : wave_w](a - c ) ] , a weak fundamental mode is amplified via parametric interaction with a soliton waveguide , and the mode experiences a strong power exchange with the orthogonally polarized fh component through the sh field .
this process is accompanied by only a weak deformation of the induced waveguide [ see fig . [
fig : wave_w](a ) dotted curve ] .
the resulting effect can be interpreted as a power exchange between two guided modes of orthogonal polarizations in a waveguide created by the sh field . in the second example , the propagation is stable [ see fig . [
fig : wave_w](d ) ] . when all the fields in eq .
( [ eq_n ] ) are not small , i.e. the small - amplitude approximation is no longer valid , the profiles of the three - component solitons should be found numerically . however , some of the lowest - order states can be calculated approximately using the approach of the `` almost exact '' solution ( [ eq_s]),([eq_p ] ) described above , which is presented in detail elsewhere @xcite .
moreover , a number of the solutions and their families can be obtained in _
an explicit analytical form_. for example , for @xmath183 , there exist two _ families of three - component solitary waves _ for any @xmath152 , that describe soliton branches starting at the bifurcation points @xmath184 at : ( i ) the soliton with a zero - order guided mode for @xmath185 : @xmath186 , @xmath187 , @xmath188 , and ( ii ) the soliton with a first - order guided mode for @xmath103 : @xmath189 , @xmath190 , @xmath188 , where @xmath191 and @xmath192 . for a practical realization of
the dpm processes and the soliton waveguiding effects described above , we can suggest two general methods .
the first method is based on the use of _ two commensurable periods _ of the quasi - phase - matched ( qpm ) periodic grating . indeed , to achieve dpm
, we can employ the first - order qpm for one parametric process , and the third - order qpm , for the other parametric process .
taking , as an example , the parameters for linbo@xmath193 and aa - s @xmath194 and bb - s @xmath195 processes @xcite , we find two points for dpm at about 0.89 @xmath196 m and 1.25 @xmath196 m .
this means that a single qpm grating can provide simultaneous phase - matching for two parametric processes .
for such a configuration , we obtain @xmath197 or , interchanging the polarization components , @xmath198 . the second method to achieve the conditions of dpm processes
is based on the idea of _ quasi - periodic qpm grating _
specifically , fibonacci optical superlattices provide an effective way to achieve phase - matching at _ several incommensurable periods _ allowing multi - frequency harmonic generation in a single structure .
we describe the properties of such structures in the next section .
for many years , solitary waves have been considered as _ coherent localized modes _ of nonlinear systems , with particle - like dynamics quite dissimilar to the irregular and stochastic behavior observed for chaotic systems @xcite . however , about 20 years ago akira hasegawa , while developing a statistical description of the dynamics of an ensemble of plane waves in nonlinear strongly dispersive plasmas , suggested the concept of a localized envelope of random phase waves @xcite . because of the relatively high powers required for generating self - localized random waves , this notion remained a theoretical curiosity until recently , when the possibility to generate spatial optical solitons by a partially incoherent source was discovered in a photorefractive medium @xcite .
the concept of incoherent solitons can be compared with a different problem : the propagation of a soliton through a spatially disordered medium . indeed , due to random scattering on defects , the phases of the individual components forming a soliton experience random fluctuations , and the soliton itself becomes _ partially incoherent _ in space and time . for a low - amplitude wave ( linear regime )
spatial incoherence is known to lead to a fast decay . as a result
, the transmission coefficient vanishes exponentially with the length of the system , the phenomenon known as anderson localization @xcite .
however , for large amplitudes ( nonlinear regime ) , when the nonlinearity length is much smaller than the anderson localization length , a soliton can propagate almost unchanged through a disordered medium as predicted theoretically in 1990 @xcite and recently verified experimentally @xcite . these two important physical concepts , spatial self - trapping of light generated by an incoherent source in a homogeneous medium , and suppression of anderson localization for large - amplitude waves in spatially disordered media , both result from the effect of strong nonlinearity .
when the nonlinearity is sufficiently strong it acts as _ an effective phase - locking mechanism _ by producing a large frequency shift of the different random - phase components , and thereby introducing _ an effective order _ into an incoherent wave packet , thus enabling the formation of localized structures . in other words ,
both phenomena correspond to the limit when the ratio of the nonlinearity length to the characteristic length of ( spatial or temporal ) fluctuations is small . in the opposite limit , when this ratio is large , the wave propagation is practically linear . below we show that , at least for aperiodic inhomogeneous structures , solitary waves can exist in the intermediate regime in the form of _ quasiperiodic nonlinear localized modes_. as an example , we consider shg and nonlinear beam propagation in _ fibonacci optical superlattices _ , and demonstrate numerically the possibility of spatial self - trapping of quasiperiodic waves whose envelope amplitude varies quasiperiodically , while still maintaining a stable , well - defined spatially localized structure , _ a quasiperiodic envelope soliton_. we consider the interaction of a fundamental wave with the frequency @xmath69 ( fh ) and its sh in a slab waveguide with quadratic ( or @xmath66 ) nonlinearity . assuming the @xmath66 susceptibility to be modulated and the nonlinearity to be of the same order as diffraction , we write the dynamical equations in the form @xmath199 { \displaystyle i\frac{\partial w}{\partial z } + \frac{1}{4 } \frac{\partial^2 w}{\partial x^2 } + d(z ) u^2 e^{i\beta z } = 0 , } \end{array}\ ] ] where @xmath200 and @xmath201 are the slowly varying envelopes of the fh and sh , respectively .
the parameter @xmath202 is proportional to the phase mismatch @xmath203 , @xmath204 and @xmath205 being the wave numbers at the two frequencies .
the transverse coordinate @xmath158 is measured in units of the input beam width @xmath206 , and the propagation distance @xmath0 in units of the diffraction length @xmath207 .
the spatial modulation of the @xmath66 susceptibility is described by the quasi - phase - matching ( qpm ) grating function @xmath208 . in the context of shg
, the qpm technique is an effective way to achieve phase matching , and it has been studied intensively @xcite . here
we consider a qpm grating produced by a quasiperiodic nonlinear optical superlattice .
quasiperiodic optical superlattices , one - dimensional analogs of quasicrystals @xcite , are usually designed to study the effect of anderson localization in the linear regime of light propagation .
for example , gellermann _ et al .
_ measured the optical transmission properties of quasiperiodic dielectric multilayer stacks of sio@xmath209 and tio@xmath209 thin films and observed a strong suppression of the transmission @xcite . for qpm gratings , a nonlinear quasiperiodic superlattice of litao@xmath193 , in which two antiparallel ferro - electric domains are arranged in a fibonacci sequence ,
was recently fabricated by zhu _
et al . _
@xcite , who measured multi - colour shg with energy conversion efficiencies of @xmath210 .
this quasiperiodic optical superlattice in litao@xmath193 can also be used for efficient direct third harmonic generation @xcite .
the quasiperiodic qpm gratings have two building blocks a and b of the length @xmath211 and @xmath212 , respectively , which are ordered in a fibonacci sequence [ fig .
[ fig : d_z](a ) ] .
each block has a domain of length @xmath213=l ( @xmath214=l ) with @xmath215=@xmath216 ( shaded ) and a domain of length @xmath217=@xmath218 [ @xmath219=@xmath220 with @xmath215=@xmath221 ( white ) . in the case of @xmath66
nonlinear qpm superlattices this corresponds to positive and negative ferro - electric domains , respectively .
the specific details of this type of fibonacci optical superlattices can be found elsewhere @xcite . for our simulations presented below
we have chosen @xmath222= @xmath223= 0.34 , where @xmath224= @xmath225 is the so - called _
golden ratio_. this means that the ratio of length scales is also the golden ratio , @xmath226= @xmath224 .
furthermore , we have chosen @xmath227=0.1 .
the grating function @xmath208 , which varies between @xmath216 and @xmath221 according to the fibonacci sequence , can be expanded in a fourier series @xmath228 where @xmath229=@xmath230=0.52 for the chosen parameter values .
hence the spectrum is composed of sums and differences of the basic wavenumbers @xmath231=@xmath232 and @xmath233=@xmath234 .
these components fill the whole fourier space densely , since @xmath231 and @xmath233 are incommensurate .
figure [ fig : d_z](b ) shows the numerically calculated fourier spectrum @xmath235 .
the lowest - order `` fibonacci modes '' are clearly the most intense . to analyze the beam propagation and shg in a quasiperiodic qpm grating one
could simply average eqs .
( [ dynam ] ) .
to lowest order this approach always yields a system of equations with constant mean - value coefficients , which does not allow to describe oscillations of the beam amplitude and phase .
however , here we wish to go beyond the averaged equations and consider the rapid large - amplitude variations of the envelope functions .
this can be done analytically for periodic qpm gratings @xcite . however , for the quasiperiodic gratings we have to resolve to numerical simulations .
thus we have solved eqs .
( [ dynam ] ) numerically with a second - order split - step routine . at the input of the crystal
we excite the fundamental beam ( corresponding to unseeded shg ) with a gaussian profile , @xmath236 we consider the quasiperiodic qpm grating with matching to the peak at @xmath237 , i.e. , @xmath238=@xmath237=82.25 .
first , we study the small - amplitude limit when a weak fh is injected with a low amplitude .
figures [ fig : soliton](a , b ) show an example of the evolution of fh and sh in this effectively linear regime . as is clearly seem from fig .
[ fig : soliton](b ) the sh wave is excited , but both beams eventually diffract .
when the amplitude of the input beam exceeds a certain threshold , self - focusing and localization should be observed for both harmonics .
figures [ fig : soliton](c , d ) show an example of the evolution of a strong input fh beam , and its corresponding sh . again
the sh is generated , but now the nonlinearity is so strong that it leads to self - focusing and mutual self - trapping of the two fields , resulting in a spatially localized two - component soliton , despite the continuous scattering of the quasiperiodic qpm grating .
it is important to notice that the two - component localized beam created due to the self - trapping effect is quasiperiodic by itself . as a matter of fact , after an initial transient its amplitude oscillates in phase with the quasiperiodic qpm modulation @xmath208 .
this is illustrated in fig .
[ fig : oscillations ] , where we show in more detail the peak intensities in the asymptotic regime of the evolution . the oscillations shown in fig . [
fig : oscillations ] are in phase with the oscillations of the qpm grating @xmath208 , and we indeed found that their spectra are similar .
our numerical results show that the quasiperiodic envelope solitons can be generated for a broad range of the phase - mismatch @xmath238 .
the amplitude and width of the solitons depend on the effective mismatch , which is the separation between @xmath238 and the nearest strong peak @xmath235 in the fibonacci qpm grating spectrum [ see fig . [
fig : d_z](b ) ] .
thus , low - amplitude broad solitons are excited for @xmath238-values in between peaks , whereas high - amplitude narrow solitons are excited when @xmath238 is close to a strong peak , as shown in fig .
[ fig : soliton](c , d ) .
to analyse in more detail the transition between the linear ( diffraction ) and nonlinear ( self - trapping ) regimes , we have made a series of careful numerical simulations @xcite . in fig .
[ fig : transmission ] we show the transmission coefficients and the beam widths at the output of the crystal versus the intensity of the fh input beam , for a variety of @xmath238-values .
these dependencies clearly illustrate the universality of the generation of localised modes for varying strength of nonlinearity , i.e. a quasiperiodic soliton is generated only for sufficiently high amplitudes .
this is of course a general phenomenon also observed in many nonlinear isotropic media .
however , here the self - trapping occurs for quasiperiodic waves , with the quasiperiodicity being preserved in the variation of the amplitude of both components of the soliton .
we have overviewed several important physical examples of the multi - component solitary waves which appear due to multi - mode and/or multi - frequency coupling in nonlinear optical fibers and waveguides .
we have described several types of such multi - component solitary waves , including : ( i ) multi - wavelength solitary waves in multi - channel bit - parallel - wavelength fiber transmission systems , ( ii ) multi - colour parametric spatial solitary waves due to multistep cascading in quadratic materials , and ( iii ) quasiperiodic envelope solitons in fibonacci optical superlattices .
these examples reveal some general features and properties of multi - component solitary waves in nonintegrable nonlinear models , also serving as a stepping stone for approaching other problems of the multi - mode soliton coupling and interaction .
the work was supported by the australian photonics cooperative research centre and by a collaborative australia - denmark grant of the department of industry , science , and tourism ( australia ) . for an overview of quadratic spatial solitons ,
see l. torner , in : _
beam shaping and control with nonlinear optics _ , f. kajzer and r. reinisch , eds .
( plenum , new york , 1998 ) , p. 229 ; yu . s. kivshar , in : _ advanced photonics with second - order optically nonlinear processes _ , a. d. boardman , l. pavlov , and s. tanev , eds .
( kluwer , dordretch , 1998 ) , p. 451 |
a cluster category is a certain 2-calabi - yau orbit category of the derived category of a hereditary abelian category .
cluster categories were introduced in @xcite in order to give a categorical model for the combinatorics of fomin - zelevinsky cluster algebras @xcite .
they are triangulated @xcite and admit ( cluster-)tilting objects , which model the clusters of a corresponding ( acyclic ) cluster algebra @xcite . each cluster in a fixed cluster algebra comes together with a finite quiver , and in the categorical model this quiver is in fact the gabriel quiver of the corresponding tilting object @xcite .
a principal ingredient in the construction of a cluster algebra is quiver mutation .
it controls the exchange procedure which gives a rule for producing a new cluster variable and hence a new cluster from a given cluster .
exchange is modeled by cluster categories in the acyclic case @xcite in terms of a mutation rule for tilting objects , i.e. a rule for replacing an indecomposable direct summand in a tilting object with another indecomposable rigid object , to get a new tilting object .
quiver mutation describes the relation between the gabriel quivers of the corresponding tilting objects .
analogously to the definition of the cluster category , for a positive integer @xmath0 , it is natural to define a certain @xmath1-calabi - yau orbit category of the derived category of a hereditary abelian category .
this is called the _
@xmath0-cluster category_. implicitly , @xmath0-cluster categories was first studied in @xcite , and their ( cluster-)tilting objects have been studied in @xcite .
combinatorial descriptions of @xmath0-cluster categories in dynkin type @xmath2 and @xmath3 are given in @xcite . in cluster categories
the mutation rule for tilting objects is described in terms of certain triangles called _ exchange triangles_. by @xcite the existence of exchange triangles generalizes to @xmath0-cluster categories .
it was shown in @xcite that there are exactly @xmath1 non - isomorphic complements to an almost complete tilting object , and that they are determined by the @xmath1 exchange triangles defined in @xcite .
the aim of this paper is to give a combinatorial description of mutation in @xmath0-cluster categories . _ a priori _
, one might expect to be able to do this by keeping track of the gabriel quivers of the tilting objects .
however , it is easy to see that the gabriel quivers do not contain enough information . we proceed to associate to a tilting object a quiver each of whose arrows has an associated colour @xmath4 . the arrows with colour 0 form the gabriel quiver of the tilting object .
we then define a mutation operation on coloured quivers and show that it is compatible with mutation of tilting objects .
a consequence is that the effect of an arbitrary sequence of mutations on a tilting object in an @xmath0-cluster category can be calculated by a purely combinatorial procedure .
our definition of a coloured quiver associated to a tilting object makes sense in any @xmath1-calabi - yau category , such as for example those studied in @xcite .
we hope that our constructions may shed some light on mutation of tilting objects in this more general setting . in section 1 , we review some elementary facts about higher cluster categories . in section 2 , we explain how to define the coloured quiver of a tilting object , we define coloured quiver mutation , and we state our main theorem . in sections 3 and 4 , we state some further lemmas about higher cluster categories , and we prove certain properties of the coloured quivers of tilting objects .
we prove our main result in sections 5 and 6 . in sections 7 and 8
we point out some applications . in section 9
we interpret our construction in terms of @xmath0-cluster complexes . in section 10
, we give an alternative algorithm for computing coloured quiver mutation .
section 11 discusses the example of @xmath0-cluster categories of dynkin type @xmath2 , using the model developed by baur and marsh @xcite .
we would like to thank idun reiten , in conversation with whom the initial idea of this paper took shape .
let @xmath5 be an algebraically closed field , and let @xmath6 be a finite acyclic quiver with @xmath7 vertices .
then the path algebra @xmath8 is a hereditary finite dimensional basic @xmath5-algebra let @xmath9 be the category of finite dimensional left @xmath10-modules .
let @xmath11 be the bounded derived category of @xmath10 , and let @xmath12 $ ] be the @xmath13th shift functor on @xmath14 .
we let @xmath15 denote the auslander - reiten translate , which is an autoequivalence on @xmath14 such that we have a bifunctorial isomorphism in @xmath14 @xmath16 ) \simeq
d{\operatorname{hom}\nolimits}(b,\tau a).\ ] ] in other words @xmath17 \tau$ ] is a serre functor .
let @xmath18 $ ] .
the @xmath0-cluster category is the orbit category @xmath19 $ ] .
the objects in @xmath20 are the objects in @xmath14 , and two objects @xmath21 are isomorphic in @xmath20 if and only if @xmath22 in @xmath14 .
the maps are given by @xmath23 .
by @xcite , the category @xmath20 is triangulated and the canonical functor @xmath24 is a triangle functor .
we denote therefore by @xmath25 $ ] the suspension in @xmath20 .
the @xmath0-cluster category is also krull - schmidt and has an ar - translate @xmath15 inherited from @xmath14 , such that the formula ( [ ar ] ) still holds in @xmath20 . if follows that @xmath17 \tau$ ] is a serre functor for @xmath20 and that @xmath20 is @xmath1-calabi - yau , since @xmath26 $ ] .
the indecomposable objects in @xmath14 are of the form @xmath27 $ ] , where @xmath28 is an indecomposable @xmath10-module and @xmath29 .
we can choose a fundamental domain for the action of @xmath18 $ ] on @xmath14 , consisting of the indecomposable objects @xmath27 $ ] with @xmath30 , together with the objects @xmath31 $ ] with @xmath28 an indecomposable projective @xmath10-module .
then each indecomposable object in @xmath20 is isomorphic to exactly one of the indecomposables in this fundamental domain .
we say that @xmath32 $ ] has degree @xmath33 , denoted @xmath34 ) = d$ ] .
furthermore , for an arbitrary object @xmath35 in @xmath36 , we let @xmath37 $ ] be the @xmath10-module which is the ( shifted ) direct sum of all summands @xmath38 of @xmath39 with @xmath40 . in the following theorem
the equivalence between ( i ) and ( ii ) is shown in @xcite and the equivalence between ( i ) and ( iii ) is shown in @xcite .
let @xmath41 be an object in @xmath20 satisfying @xmath42 ) = 0 $ ] for @xmath43 .
then the following are equivalent * if @xmath44 ) = 0 $ ] for @xmath43 then @xmath45 is in @xmath46 . * if @xmath47 ) = 0 $ ] for @xmath43 then @xmath45 is in @xmath46 .
* @xmath41 has @xmath48 indecomposable direct summands , up to isomorphism .
here @xmath46 denotes the additive closure of @xmath41 . a ( cluster-)tilting object @xmath41 in an @xmath0-cluster is an object satisfying the conditions of the above theorem . for a tilting object @xmath49 , with each @xmath50 indecomposable , and @xmath51 an indecomposable direct summand , we call @xmath52 an almost complete tilting object .
we let @xmath53 denote the @xmath5-space of irreducible maps @xmath54 in a krull - schmidt @xmath5-category @xmath55 .
the following crucial result is proved in @xcite and @xcite .
[ p : number ] there are , up to isomorphism , @xmath1 complements of an almost complete tilting object .
let @xmath51 be an indecomposable direct summand in an @xmath0-cluster tilting object @xmath56 .
the complements of @xmath57 are denoted @xmath58 for @xmath59 , where @xmath60 . by @xcite
, there are @xmath1 exchange triangles @xmath61 here the @xmath62 are in @xmath63 and the maps @xmath64 ( resp .
@xmath65 ) are minimal left ( resp .
right ) @xmath63-approximations , and hence not split mono or split epi . note that by minimality , the maps @xmath64 and @xmath65 have no proper zero summands .
we first recall the definition of quiver mutation , formulated in @xcite in terms of skew - symmetric matrices .
let @xmath66 be a quiver with vertices @xmath67 and with no loops or oriented two - cycles , where @xmath68 denotes the number of arrows from @xmath13 to @xmath69 .
let @xmath70 be a vertex in @xmath71 .
then , a new quiver @xmath72 is defined by the following data @xmath73 it is easily verified that this definition is equivalent to the one of fomin - zelevinsky .
now we consider coloured quivers .
let @xmath0 be a positive integer .
an @xmath0-coloured ( multi-)quiver @xmath71 consists of vertices @xmath67 and coloured arrows @xmath74 , where @xmath75 .
let @xmath76 denote the number of arrows from @xmath13 to @xmath70 of colour @xmath77 .
we will consider coloured quivers with the following additional conditions . *
no loops : @xmath78 for all @xmath79 .
* monochromaticity : if @xmath80 , then @xmath81 for @xmath82 * skew - symmetry : @xmath83 .
we will define an operation on a coloured quiver @xmath71 satisfying the above conditions .
let @xmath70 be a vertex in @xmath71 and let @xmath84 be the coloured quiver defined by @xmath85 in an @xmath0-cluster category @xmath20 , for every tilting object @xmath86 , with the @xmath50 indecomposable , we will define a corresponding @xmath0-coloured quiver @xmath87 , as follows .
let @xmath88 be two non - isomorphic indecomposable direct summands of the @xmath0-cluster tilting object @xmath41 and let @xmath89 denote the multiplicity of @xmath90 in @xmath91 .
we define the @xmath0-coloured quiver @xmath87 of @xmath41 to have vertices @xmath13 corresponding to indecomposable direct summands @xmath50 , and @xmath92 .
note , in particular , that the @xmath93-coloured arrows are the arrows from the gabriel quiver for the endomorphism ring of @xmath41 . by definition
, @xmath87 satisfies condition ( i ) .
we show in section [ s : higher ] that ( ii ) is satisfied ( this also follows from @xcite ) , and in section [ s : symmetry ] that ( iii ) is also satisfied . the aim of this paper is to prove the following theorem , which is a generalization of the main result of @xcite .
[ t : main ] let @xmath86 and @xmath94 be @xmath0-tilting objects , where there is an exchange triangle @xmath95
. then @xmath96 . in the case
@xmath97 the coloured quiver of a tilting object @xmath41 is given by @xmath98 and @xmath99 where @xmath100 denotes the number of arrows in the gabriel quiver of @xmath41
. then coloured mutation of the coloured quiver corresponds to fz - mutation of the gabriel quiver .
let @xmath6 be @xmath101 with linear orientation , i.e. the quiver @xmath102 .
the ar - quiver of the 2-cluster category of @xmath103 is @xmath104 & & { i_3 } \ar[dr ] & & * + + [ o][f-]{p_3[1 ] } \ar[dr ] & & i_1[1 ] \ar[dr ] & & p_1[2]\ar[dr ] \\ p_2[2 ] \ar[ur ] \ar[dr ] & & { p_2 } \ar[ur ] \ar[dr ] & & * + [ o][f-]{i_2 } \ar[ur ] \ar[dr ] & & p_2[1 ] \ar[ur ] \ar[dr ] & & i_2[1]\ar[ur]\ar[dr ] & & p_2[2 ] \\ & { p_3 } \ar[ur ] & & * + [ o][f-]{i_1 } \ar[ur ] & & p_1[1 ] \ar[ur ] & & i_3[1]\ar[ur ] & & p_3[2 ] \ar[ur ] & & } \ ] ] the direct sum @xmath105 $ ] of the encircled indecomposable objects gives a tilting object .
its coloured quiver is @xmath106 & i_2 \ar@<0.6ex>^{(0)}[r ] \ar@<0.6ex>^{(2)}[l ] & p_3[1 ] \ar@<0.6ex>^{(2)}[l ] } \ ] ] now consider the exchange triangle @xmath107 \to i_3[1 ] \to\ ] ] and the new tilting object @xmath108 \amalg p_3[1]$ ] .
the coloured quiver of @xmath109 is @xmath110 \ar@<0.6ex>^{(1)}[r ] & i_3[1 ] \ar@<0.6ex>^{(2)}[r ] \ar@<0.6ex>^{(1)}[l ] & p_3[1 ] \ar@<0.6ex>^{(0)}[l ] \ar@<0.6ex>^{(2)}@/^3.5pc/[ll ] } \ ] ]
in this section we summarize some further known results about @xmath0-cluster categories .
most of these are from @xcite and @xcite .
we include some proofs for the convenience of the reader
. tilting objects in @xmath111 give rise to partial tilting modules in @xmath9 , where a _
partial tilting module _ @xmath28 in @xmath9 , is a module with @xmath112 .
[ l : partial ] * when @xmath41 is a tilting object in @xmath36 , then each @xmath113 is a partial tilting module in @xmath9 . * the endomorphism ring of a partial tilting module has no oriented cycles in its ordinary quiver .
\(a ) is obvious from the definition .
see ( * ? ? ?
4.2 ) for ( b ) . in the following note that degrees of objects are always considered with a fixed choice of fundamental domain , and sums and differences of degrees are always computed modulo @xmath1 . [
l : div ] assume @xmath114 .
* @xmath115 for any indecomposable exceptional object @xmath39 .
* we have that @xmath116 * the distribution of degrees of complements is one of the following * * there is exactly one complement of each degree , or * * there is no complement of degree @xmath0 , two complements in one degree @xmath117 , and exactly one complement in all degrees @xmath118 . * if @xmath119 , then @xmath120 . * for @xmath121 we have @xmath122 ) = \begin{cases }
k & \text { if $ c'-c+t = 0 ( { \operatorname{mod}\nolimits}m+1)$ } \\ 0 & \text { else } \end{cases}\ ] ] \(a ) follows from the fact that @xmath123 for exceptional objects and the definition of maps in a @xmath0-cluster category .
\(b ) follows from the fact that @xmath124 ) \neq 0 $ ] , since in the exchange triangles , the @xmath125 are not split mono and ( c ) follows from ( b ) .
considering the two different possible distributions of complements , we obtain from ( c ) that if @xmath126 and @xmath127 and @xmath128 , then @xmath129 .
consider the case @xmath130 .
we can assume @xmath131 , since else the statement is void .
hence we can clearly assume that @xmath132 .
there is an exchange triangle induced from an exact sequence in @xmath9 , @xmath133.\ ] ] it is clear that @xmath134 , t_i^{(c-1 ) } ) = 0 $ ] , since @xmath131 .
we claim that also @xmath135 .
this holds since @xmath136 is a partial tilting object in @xmath10 , and so there are no cycles in the endomorphism ring , by lemma [ l : partial ] .
hence also @xmath137 follows , and this finishes the proof for ( d ) .
for ( e ) we first apply @xmath138 to the exchange triangle @xmath139 and consider the corresponding long - exact sequence , to obtain that @xmath140 ) = \begin{cases } k & \text { if $ t = 1 $ } \\ 0 & \text { if $ t=0 $ or $ t \in \{2 , \dots , m \}$ } \end{cases}.\ ] ] now consider @xmath141)$ ] . when @xmath142 , we have that @xmath143 ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c+u+1 ) } , t_i^{(c)}[v+1 ] ) \simeq \\ { \operatorname{hom}\nolimits}(t_i^{(c-1 ) } , t_i^{(c)}[v+m - u ] ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_i^{(c-1)}[1+u - v ] ) .
\end{gathered}\ ] ] when @xmath144 , we have that @xmath145 ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c+u-1 ) } , t_i^{(c)}[v-1 ] ) \simeq \\ { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_i^{(c)}[v - u]).\ ] ] combining these facts , ( e ) follows .
[ l : div2 ] the following statements are equivalent * @xmath146 ) = 0 $ ] * @xmath90 is not a direct summand in @xmath147 * @xmath50 is not a direct summand in @xmath148 furthermore , @xmath149 ) = 0 $ ] for @xmath150 .
note that @xmath151 , so ( b ) and ( c ) are equivalent .
consider the exact sequence @xmath152 ) \to { \operatorname{hom}\nolimits}(t_i^{(c ) } , b_j^{(0)}[1 ] ) \to \\ { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_j^{(1)}[1 ] ) \to { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_j^{(0)}[2 ] ) \to \end{gathered}\ ] ] coming from applying @xmath153 to the exchange triangle @xmath154 the first and fourth terms are always zero . using [ l : div](e ) we get that the second term ( and hence the third ) is non - zero if and only if @xmath155 and @xmath50 is a direct summand in @xmath148 .
( @xcite)[l : composing ] for @xmath156 , the composition @xmath157 \circ h_k^{(v-2)}[2 ] \circ \cdots \circ h_k^{(v - l+1)}[l-1 ] \colon t_k^{(v ) } \to t_k^{(v - l)}[l]\ ] ] is non - zero and a basis for @xmath158)$ ] . for @xmath97 ,
see @xcite .
assume @xmath159 .
for the first claim see @xcite , while the second claim then follows from lemma [ l : div](e ) .
we include an independent proof of the following crucial property .
@xcite [ l : disjoint ] @xmath160 and @xmath161 has no common non - zero direct summands whenever @xmath162 . when @xmath97 , this is proved in @xcite .
assume @xmath163 .
we consider two cases , @xmath164 or @xmath165 .
consider first the case @xmath166 . without loss of generality
we can assume @xmath167 and @xmath168 , and that @xmath169 .
assume that there exists a ( non - zero ) indecomposable @xmath170 , which is a direct summand in @xmath171 and in @xmath172 .
we have that @xmath173 by lemma [ l : div](b ) .
assume first @xmath174 .
then the exchange triangle @xmath175 is induced from the degree 0 part of the derived category , and hence from an exact sequence in @xmath9 .
then the endomorphism ring of the partial tilting module @xmath176 has a cycle , which is a contradiction to lemma [ l : partial ] .
assume now that @xmath177 .
then @xmath178 , where 0 can only occur if @xmath179 . if @xmath180 , then clearly @xmath181 , and hence the partial tilting module @xmath182 contains a cycle , which is a contradiction .
assume that @xmath183 ( and hence @xmath179 ) . then @xmath184 . if @xmath181 , we get a contradiction as in the previous case .
if @xmath185 , consider the exchange triangle @xmath186 which is induced from an exact sequence in @xmath9 .
hence there is a _ non - zero _
map @xmath187 obtained by composing @xmath188 with the monomorphism @xmath189 , and thus there are cycles in the endomorphism ring of the partial tilting module @xmath190 , a contradiction .
this finishes the case with @xmath191 .
assume now that @xmath192 .
then we have @xmath193 .
since @xmath194 and @xmath195 , we have by lemma [ l : div](c ) that @xmath196 .
so without loss of generality we can assume @xmath197 .
assume that @xmath198 . then @xmath199 using lemma [ l : div](c ) and the fact that @xmath194
. then also @xmath200 .
but @xmath201 , so @xmath202 , contradicting the fact that @xmath195 .
@xmath87 satisfies condition ( ii ) .
let @xmath203 be a tilting object . in this section we show that the coloured quiver @xmath87 satisfies condition ( iii ) .
[ p : symmetry ] with the notation of the previous section , we have @xmath204 . by lemma [ l : div2 ]
we only need to consider the case @xmath205 .
it is enough to show that @xmath206 .
we first prove [ l : non - van ] let @xmath207 be irreducible in @xmath208 . then the composition @xmath209
\circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] is non - zero .
we have already assumed @xmath210 .
assume @xmath211 \circ h_i^{(0)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(0)}[-c ] \to t_i^{(m)}[-c+1]\ ] ] is zero .
this means that @xmath212 must factor through @xmath213 .
since @xmath50 is by assumption a summand in @xmath214 , we have that @xmath50 is not a summand in @xmath148 by proposition [ l : disjoint ] .
since @xmath215 , we have that @xmath90 is not a direct summand in @xmath147 .
this means that @xmath216 is not irreducible in @xmath208 , a contradiction .
so @xmath209 \circ h_i^{(0)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m)}[-c+1]$ ] is non - zero .
assume @xmath217 .
if the composition @xmath209 \circ h_i^{(0)}[-c ] \circ h_i^{(m)}[-c+1]$ ] is zero , then @xmath209 \circ h_i^{(0)}[-c]$ ] factors through @xmath218 \to t_i^{(m)}[-c+1].\ ] ] we claim that @xmath219 , b_i^{(m-1)}[-c+1 ] ) \simeq { \operatorname{hom}\nolimits}(t_j^{(c ) } , b_i^{(m-1)}[1 ] ) = 0 $ ] .
this clearly holds if @xmath90 is not a summand of @xmath220 .
in addition we have that @xmath221 ) = 0 $ ] since @xmath217 , using lemma [ l : div](e ) .
this is a contradiction , and this argument can clearly be iterated to see that @xmath209 \circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] is non - zero , using lemma [ l : div](e ) . we now show that any irreducible map @xmath207 gives rise to an irreducible map @xmath222
. consider the composition @xmath223 \overset{g_j^{(c)}[-c]}{\longrightarrow } t_j^{(c)}[-c ] \longrightarrow t_i^{(m - c+1)}.\ ] ] since @xmath50 is a summand in @xmath214 by assumption , it is not a summand in @xmath224 .
thus , @xmath224 is in @xmath225 .
since @xmath226)= 0 $ ] for any @xmath39 in @xmath227 , the composition vanishes . using the exchange triangle @xmath223 \overset{g_j^{(c)}[-c]}{\longrightarrow } t_j^{(c)}[-c ] \overset{h_j^{(c)}[-c]}{\longrightarrow } t_j^{(c-1)}[-c+1],\ ] ] we see that @xmath209 \circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] factors through the map @xmath228 \overset{h_j^{(c)}[-c]}{\longrightarrow } t_j^{(c-1)}[-c+1]$ ] , i.e. there is a commutative diagram @xmath229 \ar^{g_j^{(c)}[-c]}[r ] & t_j^{(c)}[-c ] \ar^{h_j^{(c)}[-c]}[r ] \ar[d ] & t_j^{(c-1)}[-c+1 ] \ar[r ] \ar^{\phi_1}[dl ] & \\ & t_i^{(m - c+1 ) } & & } \ ] ] similarly , using the exchange triangle @xmath230 \overset{g_j^{(c-1)}[-c+1]}{\longrightarrow } t_j^{(c-1)}[-c+1 ] \overset{h_j^{(c-1)}[-c+1]}{\longrightarrow } t_j^{(c-2)}[-c+2]\ ] ] we obtain a map @xmath231 \to t_i^{(m - c+1)}$ ] repeating this argument @xmath79 times we obtain a map @xmath232 , such that @xmath233 \circ \phi_c = \alpha[-c ] \circ \gamma_i^{(0,c)}$ ]
. @xmath234 \ar_{h_j^{(c)}[-c]}[d ] \ar[r ] & t_i^{(m - c+1 ) } \\ t_j^{(c-1)}[-c+1 ] \ar_{h_j^{(c-1)}[-c+1]}[d ] \ar^{\phi_1}[ur ] & \\ t_j^{(c-2)}[-c+2 ] \ar_{h_j^{(c-2)}[-c+2]}[d ] \ar^{\phi_2}[uur ] & \\
\vdots \ar[d ] & \\
t_j \ar^{\phi_c}[uuuur ] & \\ & } \ ] ] we claim that [ l : irred ] there is a map @xmath235 , such that @xmath233 \circ \beta = \alpha[-c ] \circ \gamma_i^{(0,c)}$ ] , and such that @xmath236 is irreducible in @xmath237
. let @xmath238 be a minimal left @xmath239-approximation , with @xmath240 in @xmath241 and @xmath242 in @xmath243 .
let @xmath244 be as above , and factor it as @xmath245 since @xmath246 factors through @xmath247 $ ] , we have that @xmath233 \psi '' = 0 $ ] , so we have @xmath248(\psi ' \epsilon ' + \psi '' \epsilon'')= \gamma_j^{(c , c)}[-c ] \psi ' \epsilon'.\ ] ] hence , let we let @xmath249 and since the summands in @xmath250 are isomorphisms , it is clear that @xmath236 is irreducible .
next , assume @xmath251 is a basis for the space of irreducible maps from @xmath252 to @xmath50 .
then , by lemma [ l : non - van ] the set @xmath253 is also linearly independent . for each @xmath254 , consider the corresponding map @xmath255 , such that @xmath233 \circ \beta_t = \alpha_t[-c ] \circ \gamma_i^{(0,c)}$ ] , and which we by lemma [ l : irred ] can assume is irreducible . assume a non - trivial linear combination @xmath256 is zero
. then also @xmath257 \circ \beta_t ) = \sum k_t \alpha_t \circ \gamma_i^{(0,c)}=0 $ ] . but
this contradicts lemma [ l : non - van ] since @xmath258 is irreducible .
hence it follows that @xmath259 is also linearly independent .
hence , in the exchange triangle @xmath260 , we have that @xmath90 appears with multiplicity at least @xmath261 in @xmath262 .
so , we have that @xmath206 , and the proof of the proposition is complete .
in this section we show how mutation in the vertex @xmath70 affects the complements of the almost complete tilting object @xmath263 . as
before , let @xmath264 be an @xmath0-tilting object , and let @xmath94 . we need to consider @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] for all possible values of @xmath266 . however , we have the following restriction on the colour of arrows . [
p : limits ] assume @xmath267 and @xmath268
. then @xmath269 .
consider the exchange triangle @xmath270 .
note that @xmath90 is a direct summand in the middle term @xmath271 by the assumption that @xmath272 .
consider also the exchange triangle @xmath273 .
pick an arbitrary non - zero map @xmath274 , and consider the map @xmath275 .
it suffices to show that whenever @xmath276 , then @xmath277 is not irreducible in @xmath46 .
so assume that @xmath276 .
we claim that there is a commutative diagram @xmath278 \ar[d ] & t_j \amalg x ' \ar[r ] \ar^{\left ( \begin{smallmatrix } h & 0 \\ 0 & 0 \end{smallmatrix } \right)}[d ] & t_i^{(e+1 ) } \ar[r ] \ar[d ] & \\
t_i^{(c ) } \ar[r ] & t_k \amalg z \ar[r ] & t_i^{(c+1 ) } \ar[r ] & } \ ] ] where the rows are the exchange triangles .
the composition @xmath279 is zero since * if @xmath280 @xmath281 by using @xmath276 and lemma [ l : div](e ) * if @xmath282 , there is no non - zero composition @xmath283 hence the leftmost vertical map exists , and then the rightmost map exists , using that @xmath20 is a triangulated category .
then , since @xmath284 , t_i^{(c ) } ) = 0 $ ] by lemma [ l : div](e ) , there is a map @xmath285 , such that @xmath286 .
hence there is map @xmath287 such that @xmath288 . by restriction
we get @xmath289 under the assumption @xmath290 we have that @xmath291 can not be irreducible in @xmath292 .
hence @xmath293 , where @xmath51 is not summand in @xmath294 .
also , by proposition [ l : disjoint ] we have that @xmath90 is not a summand in @xmath294 .
if @xmath295 was irreducible in @xmath296 , then there would be an irreducible map @xmath297 in @xmath298 , and since @xmath299 , this does not hold , by proposition [ l : disjoint ] .
hence , @xmath300 , where @xmath90 is not a direct summand of @xmath301 . also by proposition [ l : disjoint ] we have that @xmath51 is not a summand of @xmath301 . by ( [ factor ] ) , this shows that @xmath274 is not irreducible in @xmath46 .
let @xmath302 .
for @xmath303 , let @xmath304 denote the complements of @xmath305 , where there are exchange triangles @xmath306 we first want to compare @xmath304 with @xmath307 . [ l : samecomp ] assume that @xmath308 for @xmath309 and that @xmath310 .
* for @xmath311 , the minimal left @xmath312-approximation @xmath313 is also an @xmath314-approximation . * for @xmath315
, we have @xmath316 . by assumption @xmath90
is not a direct summand in any of the @xmath317 .
assume there is a map @xmath318 and consider the diagram @xmath319 \ar[r ] & t_i^{(u ) } \ar[r ] \ar [ d ] & b_i^{(u ) } \ar [ r ] & \\ & t_j^{(1 ) } & & } \ ] ] since @xmath320)= 0 $ ] by lemma [ l : div2 ] , we see that the map @xmath318 factors through @xmath313 .
hence the minimal left @xmath312-approximation @xmath313 is also an @xmath314-approximation , so we have proved ( a ) .
then ( b ) follows directly .
[ l : comp ] assume that @xmath321 and there are exchange triangles @xmath322 and @xmath323 where @xmath324 and @xmath325 , i.e. @xmath326 and @xmath327 , where @xmath51 is not isomorphic to any direct summand in @xmath328 . * the composition @xmath329 is a left @xmath314-approximation .
* there is a triangle @xmath330 with @xmath331 in @xmath332 and @xmath333 .
* there is a triangle @xmath334 .
consider an arbitrary map @xmath335 with @xmath45 in @xmath314 .
we have that @xmath336 ) = 0 $ ] , by lemma [ l : div2 ] .
hence , by applying @xmath337 to the triangle ( [ i - tri ] ) we get that @xmath338 factors through @xmath339 . by applying @xmath337 to the triangle ( [ j - tri ] ) , and using that @xmath340 ) = 0 $ ] , we get that @xmath338 factors through @xmath341 .
this proves ( a ) . for ( b ) and ( c )
we use the exchange triangles ( [ i - tri ] ) and ( [ j - tri ] ) and the octahedral axiom to obtain the commutative diagram of triangles @xmath342 \ar@{=}[d ] & ( t_j)^p \amalg x \ar[r ] \ar[d ] & t_i^{(e+1 ) } \ar[d ] \ar[r ] & \\
t_i^{(e ) } \ar[r ] & ( t_k)^{pq } \amalg y^p \amalg x \ar[r ] \ar[d ] & c \ar[r ] \ar[d ] & \\ & ( t_j^{(1)})^p \ar@{=}[r ] & ( t_j^{(1)})^p \ar[r ] & } \ ] ] by ( a ) the map @xmath343 is a left @xmath314-approximation , and by lemma [ l : samecomp ] we have that @xmath344 .
hence @xmath345 , where @xmath331 is in @xmath346 , and with no copies isomorphic to @xmath51 in @xmath328 .
note that the induced @xmath314-approximation is in general not minimal .
[ l : modtri ] assume @xmath321 and @xmath347 .
* then there is a triangle @xmath348 where @xmath216 is a minimal left @xmath314-approximation , and @xmath331 is as in lemma [ l : comp ] .
* there is an induced exchange triangle @xmath349 where @xmath350 . *
@xmath351 .
consider the exchange triangle @xmath352 \to t_i^{(e+1 ) } \to b_i^{(e+1 ) } \to\ ] ] and the triangle from lemma [ l : comp ] ( b ) @xmath353 apply the octahedral axiom , to obtain the commutative diagram of triangles @xmath354 \ar[r ] \ar@{=}[d ] & t_i^{(e+1 ) } \ar[r ] \ar[d ] & b_i^{(e+1 ) } \ar[d ] \ar[r ] & \\
t_i^{(e+2)}[-1 ] \ar[r ] & ( t_i^{(e+1 ) } ) ' \amalg c ' \ar[r ] \ar[d ] & g \ar[r ] \ar[d ] & \\ & ( t_j^{(1)})^p \ar@{=}[r ] & ( t_j^{(1)})^p \ar[r ] & } \ ] ] since @xmath90 does not occur as a summand in @xmath294 by proposition [ l : disjoint ] , we have that @xmath355 ) = 0 $ ] . hence the rightmost
triangle splits , so we have a triangle @xmath356 \to ( t_i^{(e+1 ) } ) ' \amalg c ' \to b_i^{(e+1 ) } \amalg ( t_j^{(1)})^p \to\ ] ] by lemma [ l : div2 ] we have that @xmath357)= 0 $ ] . by lemma [ l : div](e )
we get that @xmath358 ) = 0 $ ] , and clearly @xmath359 ) = 0 $ ] , for @xmath360 .
we hence get that all maps @xmath361 , with @xmath45 in @xmath362 , factor through @xmath363 .
minimality is clear from the triangle ( [ octa - tri ] ) .
this proves ( a ) , and ( b ) follows from the fact that @xmath331 contains no copies of @xmath90 , and hence splits off .
( c ) is a direct consequence of ( b ) .
[ p : summarize ] * if @xmath308 for @xmath364 , then @xmath365 for all @xmath366 . * if @xmath321 and @xmath347 , then @xmath365 for @xmath367 .
\(a ) is a direct consequence of [ l : samecomp ] . for
( b ) note that by lemmas [ l : samecomp ] and [ l : modtri ] we have @xmath365 for @xmath368 and @xmath369 . for @xmath370
consider the exchange triangles @xmath371 since @xmath372 ) = 0 $ ] by lemma [ l : div2 ] and @xmath373 , it is clear that the map @xmath374 is a left @xmath375-approximation .
hence ( b ) follows .
this section contains the proof of the main result , theorem [ t : main ] . as before , let @xmath264 be an @xmath0-tilting object , and let @xmath94 .
we will compare the numbers of @xmath77-coloured arrows from @xmath13 to @xmath69 , in the coloured quivers of @xmath41 and @xmath109 , i.e. we will compare @xmath376 and @xmath377 .
we need to consider an arbitrary @xmath41 whose coloured quiver locally looks like @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] for any possible value of @xmath266 . our aim is to show that the formula @xmath378 holds .
the case where @xmath379 is directly from the definition .
the case where @xmath380 follows by condition ( ii ) for @xmath381 . for the rest of the proof
we assume @xmath382 .
we will divide the proof into four cases , where @xmath383 denotes the number of arrows from @xmath13 to @xmath70 , and @xmath384 . * @xmath385 * @xmath386 , @xmath321 and @xmath387 * @xmath386 , @xmath321 and @xmath388 . * @xmath386 and @xmath389 note that in the three first cases , the formula reduces to @xmath390 and in the first two cases it further reduces to @xmath391 case i.
we first consider the situation where there is no coloured arrow @xmath392 , i.e. @xmath308 for all @xmath393 .
that is , we assume @xmath87 locally looks like this @xmath265 & t_j \ar^{(d)}[r ] & t_k } \ ] ] with @xmath394 arbitrary .
it is a direct consequence of proposition [ p : summarize ] that @xmath395 for all @xmath393 which shows that the formula holds .
+ + case ii .
we consider the setting where we assume @xmath87 locally looks like this @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] with @xmath321 and @xmath396 .
we then claim that we have the following , which shows that the formula holds . in the above setting @xmath395 for all @xmath393 .
it follows directly from proposition [ p : summarize ] that @xmath395 for @xmath397 .
we claim that @xmath398 . by lemma [ l : comp ]
we have the ( not necessarily minimal ) left @xmath314-approximation @xmath399 first , assume that @xmath51 does not appear as a summand in @xmath326 , then the same holds for @xmath400 , and hence for @xmath401 which is a direct summand in @xmath400
. next , assume @xmath51 appears as a summand in @xmath271 , and hence in @xmath39 .
then @xmath51 is by proposition [ l : disjoint ] not a summand in @xmath294 , and by lemma [ l : modtri ] we have that @xmath51 is also not a summand in @xmath331 .
therefore @xmath51 appears with the same multiplicity in @xmath271 as in @xmath401 , also in this case .
we now show that @xmath395 for @xmath402 . if @xmath403 , then @xmath404 for @xmath402 and we are finished .
so assume @xmath405 , i.e. @xmath51 does not appear as a direct summand of @xmath39 .
consider the map @xmath406 we have that @xmath407 . by assumption , @xmath51 is not a direct summand in @xmath408 , and thus not in @xmath331 .
hence it follows that @xmath409 .
since , by proposition [ p : summarize ] we have for @xmath410 , that @xmath316 and the left @xmath411-approximation coincide with the left @xmath412-approximations of @xmath413 , it now follows that @xmath395 for all @xmath393 .
+ case iii .
we now consider the setting with @xmath414 non - zero , @xmath388 and @xmath321 .
that is , we assume @xmath87 locally looks like this @xmath265 \ar^{(e)}[r ] & t_j \ar^{(0)}[r ] & t_k } \ ] ] where @xmath415 by proposition [ p : limits ] , and where there are @xmath416 arrows from @xmath50 to @xmath51 .
[ l : formulas ] in the above setting , we have that @xmath381 is given by @xmath417 @xmath418 and @xmath419 we first deal with the case where @xmath420 and @xmath421 . by assumption @xmath39 in the triangle ( [ i - tri ] )
has @xmath422 copies of @xmath51 , so @xmath423 has @xmath424 copies of @xmath51 . hence to show ( [ form1 ] ) it is sufficient to show that @xmath331 in the triangle @xmath425 has no copies of @xmath51 .
this follows directly from the lemma [ l : modtri ] and the fact that @xmath51 ( by the assumption that @xmath421 and proposition [ l : disjoint ] ) is not a summand in @xmath294 . in this case
( [ form2 ] ) and ( [ form3 ] ) follow directly from proposition [ l : disjoint ] .
consider the case with @xmath426 and @xmath427 .
we have that @xmath39 in the triangle ( [ i - tri ] ) does not have @xmath51 as a direct summand .
assume @xmath51 appears as a direct summand of @xmath331 with multiplicity @xmath428 .
we claim that @xmath429 .
assume first @xmath430 , then on one hand @xmath51 appears with multiplicity @xmath431 in @xmath432 . on
the other hand @xmath51 appears with multiplicity @xmath433 in @xmath401 .
this contradicts proposition [ l : disjoint ] .
hence @xmath429 . therefore @xmath401 has @xmath434 copies of @xmath51 and ( [ form1 ] ) and ( [ form2 ] ) hold . if @xmath435 , then ( [ form3 ] ) follows directly from the above and proposition [ l : disjoint ] . in the case
@xmath436 , we also need to show that @xmath51 does not appear as a summand in @xmath437 for @xmath438 .
since @xmath439 , we have @xmath440 , and the result follows from proposition [ p : summarize ] .
now assume @xmath426 and @xmath441 .
assume @xmath442 , where @xmath51 is not a summand in @xmath443 .
now since @xmath444 with @xmath51 not a summand in @xmath328 , is a minimal left @xmath362-approximation , we have that @xmath445 and @xmath51 appears with multiplicity @xmath446 in the minimal left @xmath447-approximation of @xmath448 , hence @xmath51 can not appear as a summand in the minimal left @xmath375-approximation of @xmath449 .
hence @xmath450 , and we have completed the proof of ( [ form1 ] ) and ( [ form2 ] ) in this case .
the case ( [ form3 ] ) , i.e. @xmath451 follows from proposition [ l : disjoint ] .
+ case iv .
we now consider the case with @xmath452 .
assume first there are no arrows from @xmath70 to @xmath69 .
then we can use the symmetry proved in proposition [ p : symmetry ] and reduce to case i. the formula is easily verified in this case .
assume @xmath453 , again we can use the symmetry , this time to reduce to case iii .
it is straightforward to verify that the formula holds also in this case .
assume now that @xmath454 , i.e. we need to consider the following case @xmath455 \ar@<0.6ex>^{(m)}[r ] & t_j \ar@<0.6ex>^{(0)}[r ] \ar@<0.6ex>^{(0)}[l ] & t_k \ar@<0.6ex>^{(m)}[l ] \ar@<0.6ex>^{(m - c)}@/^3.5pc/[ll ] } \ ] ] now by proposition [ p : limits ] we have that @xmath79 is in @xmath456 .
assume there are @xmath457 @xmath77-coloured arrows the coloured quiver of @xmath109 is of the form @xmath458 \ar@<0.6ex>^{(0)}[r ] & t_j^{(1 ) } \ar@<0.6ex>^{(m)}[r ] \ar@<0.6ex>^{(m)}[l ] & t_k \ar@<0.6ex>^{(0)}[l ] \ar@<0.6ex>^{(m - c')}@/^3.5pc/[ll ] } \ ] ] and applying the symmetry of proposition [ p : symmetry ] we have that if @xmath421 , then @xmath459 by proposition [ p : limits ] .
hence for all @xmath460 we have that @xmath461 .
therefore it suffices to show that @xmath462 , for @xmath463 .
this is a direct consequence of the following .
assume we are in the above setting .
a map @xmath464 or @xmath465 is irreducible in @xmath46 if and only if it is irreducible in @xmath362 .
assume @xmath464 is not irreducible in @xmath362 , and that @xmath466 for some @xmath467 , with @xmath468 the indecomposable direct summands of @xmath45 .
note that by lemma [ l : div](a ) , we can assume that all @xmath469 and all @xmath470 are non - isomorphisms . if there is some index @xmath471 such that @xmath472 , the map @xmath470 factors through some @xmath473 in @xmath474 , since there are no @xmath475-coloured arrows @xmath476 or @xmath477 in the coloured quiver of @xmath41 .
this shows that @xmath464 is not irreducible in @xmath46 .
assume @xmath464 is not irreducible in @xmath46 , and that @xmath478 for some @xmath479 , with @xmath480 the indecomposable direct summands of @xmath481 . if there is some index @xmath471 such that @xmath482 , the map @xmath483 factors through @xmath484 , which is in @xmath485 , since there are no @xmath93-coloured arrows @xmath392 or @xmath486 in the coloured quiver of @xmath41 .
this shows that @xmath464 is not irreducible in @xmath362 . by symmetry
, the same property holds for maps @xmath465 .
thus we have proven that the formula holds in all four cases , and this finishes the proof of theorem [ t : main ] .
an @xmath0-cluster - tilted algebra is an algebra given as @xmath487 for some tilting object @xmath41 in an @xmath0-cluster category @xmath488 . obviously , the subquiver of the coloured quiver of @xmath41 given by the @xmath93-coloured maps is the gabriel quiver of @xmath487 .
an application of our main theorem is that the quivers of the @xmath0-cluster - tilted algebras can be combinatorially determined via repeated ( coloured ) mutation . for this one
needs transitivity in the tilting graph of @xmath0-tilting objects .
more precisely , we need the following , which is also pointed out in @xcite . any @xmath0-tilting object can be reached from any other @xmath0-tilting object via iterated mutation .
we sketch a proof for the convenience of the reader .
let @xmath109 be a tilting object in an @xmath0-cluster category @xmath20 of the hereditary algebra @xmath489 , and let @xmath490 be the @xmath491-cluster category of @xmath10 .
by @xcite , there is a tilting object @xmath41 of degree 0 , i.e. all direct summands in @xmath41 have degree 0 , such that @xmath41 can be reached from @xmath109 via mutation .
it is sufficient to show that the canonical tilting object @xmath10 can be reached from @xmath41 via mutation .
since @xmath41 is of degree 0 , it is induced from a @xmath10-tilting module .
especially @xmath41 is a tilting object in @xmath490 . since @xmath41 and @xmath10 are tilting objects in @xmath490 , by @xcite there are @xmath490-tilting objects @xmath492 , such that @xmath50 mutates to @xmath493 ( in @xmath490 ) for @xmath494 .
now each @xmath50 is induced by a tilting module for some @xmath495 where all @xmath496 are derived equivalent to @xmath497 . hence , each @xmath50 is easily seen to be an @xmath0-cluster tilting object . since @xmath493 differs from @xmath50 in only one summand the mutations in @xmath490 are also mutations in @xmath20 .
this concludes the proof .
a direct consequence of the transitivity is the following .
for an @xmath0-cluster category @xmath111 of the acyclic quiver @xmath71 , all quivers of @xmath0-cluster - tilted algebras are given by repeated coloured mutation of @xmath71 .
in this section , we discuss concrete computation with tilting objects in an @xmath0-cluster tilting category .
an exceptional indecomposable object in @xmath9 is uniquely determined by its image @xmath498 $ ] in the grothendieck group @xmath499 .
there is a map from @xmath500 to @xmath499 which , for @xmath501 , takes @xmath502 $ ] to @xmath503 $ ] .
an exceptional indecomposable in @xmath500 can be uniquely specified by its class in @xmath499 together with its degree .
the map from @xmath500 to @xmath499 does not descend to @xmath504 .
however , if we fix our usual choice of fundamental domain in @xmath500 , then we can identify the indecomposable objects in it as above .
let us define the combinatorial data corresponding to a tilting object @xmath41 to be @xmath87 together with @xmath505 , \deg t_i)$ ] for @xmath506 .
given the combinatorial data for a tilting object @xmath41 in @xmath504 , it is possible to determine , by a purely combinatorial procedure , the combinatorial data for the tilting object which results from an arbitrary sequence of mutations applied to @xmath41 . clearly , it suffices to show that , for any @xmath13 , we can determine the class and degree for @xmath507 .
if we can do that then , by the coloured mutation procedure , we can determine the coloured quiver for @xmath508 , and by applying this procedure repeatedly , we can calculate the result of an arbitrary sequence of mutations . since we are given @xmath87 , we know @xmath509 , and we can calculate @xmath510 $ ] .
now we have the following lemma : [ one ] @xmath511=[b_i^{(0)}]-[t^{(0)}_i]$ ] , and @xmath512 or @xmath513 , whichever is consistent with the sign of the class of @xmath511 $ ] , unless this yields a non - projective indecomposable object in degree @xmath0 , or an indecomposable of degree @xmath1 .
the proof is immediate from the exchange triangle @xmath514 . applying this lemma , and supposing that we are not in the case where its procedure fails
, we can determine the class and degree @xmath515 . by the coloured mutation procedure
, we can also determine the coloured quiver for @xmath516 .
we therefore have all the necessary data to apply lemma [ one ] again . repeatedly applying the lemma
, there is some @xmath69 such that we can calculate the class and degree of @xmath507 for @xmath517 , and the procedure described in the lemma fails to calculate @xmath518 .
we also have the following lemma : [ two ] @xmath519=[b_i^{(m)}]-[t^{(0)}_i]$ ] , and @xmath520 or @xmath521 , whichever is consistent with the sign of @xmath519 $ ] , unless this yields an indecomposable in degree @xmath522 .
applying this lemma , starting again with @xmath41 , we can obtain the degree and class for @xmath523 .
we can then determine the coloured quiver for @xmath524 , and we are now in a position to apply lemma [ two ] again . the last complement which lemma [ two ] will successfully determine is @xmath525 .
it follows that we can determine the degree and class of any complement to @xmath263 .
in this section , we discuss the application of our results to the study of the @xmath0-cluster complex , a simplicial complex defined in @xcite for a finite root system @xmath526 .
we shall begin by stating our results for the @xmath0-cluster complex in purely combinatorial language , and then briefly describe how they follow from the representation - theoretic perspective in the rest of the paper . for simplicity ,
we restrict to the case where @xmath526 is simply laced .
number the vertices of the dynkin diagram for @xmath526 from 1 to @xmath48 .
the @xmath0-coloured almost positive roots , @xmath527 , consist of @xmath0 copies of the positive roots , numbered @xmath491 to @xmath0 , together with a single copy of the negative simple roots .
we refer to an element of the @xmath13-th copy of @xmath528 as having colour @xmath13 , and we write such an element as @xmath529 .
since the dynkin diagram for @xmath526 is a tree , it is bipartite ; we fix a bipartition @xmath530 .
the @xmath0-cluster complex , @xmath531 , is a simplicial complex on the ground set @xmath527 .
its maximal faces are called @xmath0-clusters .
the definition of @xmath531 is combinatorial ; we refer the reader to @xcite .
the @xmath0-clusters each consist of @xmath48 elements of @xmath527 ( * ? ? ? * theorem 2.9 ) .
every codimension 1 face of @xmath531 is contained in exactly @xmath1 maximal faces ( * ? ? ?
* proposition 2.10 ) .
there is a certain combinatorially - defined bijection @xmath532 , which takes faces of @xmath531 to faces of @xmath531 ( * ? ? ?
* theorem 2.4 )
. it will be convenient to consider _ ordered @xmath0-clusters_. an ordered @xmath0-cluster is just a @xmath48-tuple from @xmath527 , the set of whose elements forms an @xmath0-cluster .
write @xmath533 for the set of ordered @xmath0-clusters . for each ordered @xmath0-cluster @xmath534
, we will define a coloured quiver @xmath535 .
we will also define an operation @xmath536 , which takes ordered @xmath0-clusters to ordered @xmath0-clusters , changing only the @xmath70-th element .
we will define both operations inductively .
the set @xmath537 of negative simple roots forms an @xmath0-cluster .
its associated quiver is defined by drawing , for each edge @xmath538 in the dynkin diagram , a pair of arrows .
suppose @xmath539 and @xmath540 .
then we draw an arrow from @xmath13 to @xmath70 with colour @xmath541 , and an arrow from @xmath70 to @xmath13 with colour @xmath0 .
suppose now that we have some ordered @xmath0-cluster @xmath542 , together with its quiver @xmath535 .
we will now proceed to define @xmath543 . write @xmath544 for the number of arrows in @xmath535 of colour @xmath541 from @xmath70 to @xmath69 .
define : @xmath545 let @xmath79 be the colour of @xmath546 .
we define @xmath543 by replacing @xmath546 by some other element of @xmath527 , according to the following rules : * if @xmath546 is positive and @xmath236 is positive , replace @xmath546 by @xmath547 . *
if @xmath546 is positive and @xmath236 is negative , replace @xmath546 by @xmath548 . *
if @xmath546 is negative simple @xmath549 , define @xmath550 by @xmath551 , and then replace @xmath546 by @xmath552 , with colour zero .
define the quiver for the @xmath0-cluster @xmath543 by the coloured quiver mutation rule from section 2 .
since any @xmath0-cluster can be obtained from @xmath537 by a sequence of mutations , the above suffices to define @xmath543 and @xmath535 for any ordered @xmath0-cluster @xmath542 .
the operation @xmath553 defined above takes @xmath0-clusters to @xmath0-clusters , and the @xmath0-clusters @xmath554 for @xmath555 are exactly those containing all the @xmath556 for @xmath557 .
the connection between the combinatorics discussed here and the representation theory in the rest of the paper is as follows .
@xmath527 corresponds to the indecomposable objects of ( a fundamental domain for ) @xmath36 .
the cluster tilting objects in @xmath36 correspond to the @xmath0-clusters .
the operation @xmath558 corresponds to @xmath25 $ ] . for further details on the translation ,
the reader is referred to @xcite .
the above proposition then follows from the approach taken in section [ sec : cc ] .
here we give an alternative description of coloured quiver mutation at vertex @xmath70 . 1 . for each pair of arrows @xmath559 & j\ar^{(0)}[r ] & k } \ ] ] with @xmath560 , the arrow from @xmath13 to @xmath70 of arbitrary colour @xmath79 , and the arrow from @xmath70 to @xmath69 of colour @xmath541 , add a pair of arrows : an arrow from @xmath13 to @xmath69 of colour @xmath79 , and one from @xmath69 to @xmath13 of colour @xmath561 .
2 . if the graph violates property ii , because for some pair of vertices @xmath13 and @xmath69 there are arrows from @xmath13 to @xmath69 which have two different colours , cancel the same number of arrows of each colour , until property ii is satisfied .
3 . add one to the colour of any arrow going into @xmath70 and subtract one from the colour of any arrow going out of @xmath70 .
the above algorithm is well - defined and correctly calculates coloured quiver mutation as previously defined .
fix a quiver @xmath71 and a vertex @xmath70 at which the mutation is being carried out . to prove that the algorithm is well - defined
, we must show that at step 2 , there are only two colours of arrows running from @xmath13 to @xmath69 for any pair of vertices @xmath13 , @xmath69 .
( otherwise there would be more than one way to carry out the cancellation procedure of step 2 . ) since in the original quiver @xmath71 , there was only one colour of arrows from @xmath13 to @xmath69 , in order for this problem to arise , we must have added two different colours of arrows from @xmath13 to @xmath69 at step 1 .
two colours of arrows will only be added from @xmath13 to @xmath69 if , in @xmath71 , there are both @xmath93-coloured arrows from @xmath70 to @xmath69 and from @xmath70 to @xmath13 . in this case , by property iii , there are @xmath562-coloured arrows from @xmath13 to @xmath70 and from @xmath69 to @xmath70 .
it follows that in step 1 , we will add both @xmath93-coloured and @xmath562-coloured arrows . applying proposition 5.1 , we see that any arrows from @xmath13 to @xmath69 in @xmath71 are of colour 0 or @xmath0 . thus , as desired , after step 1 , there are only two colours of arrows in the quiver , so step 2 is well - defined .
we now prove correctness .
let @xmath563 .
write @xmath76 for the number of @xmath79-coloured arrows from @xmath13 to @xmath70 in @xmath71 , and similarly @xmath564 for @xmath565 .
write @xmath566 and @xmath567 for the result of applying the above algorithm .
it is clear that only the final step of the algorithm is relevant for @xmath568 where one of @xmath13 or @xmath69 coincides with @xmath70 , and therefore that in this case @xmath569 as desired .
suppose now that neither @xmath13 nor @xmath69 coincides with @xmath70 .
suppose further that in @xmath71 there are no @xmath93-coloured arrows from either @xmath13 or @xmath69 to @xmath70 , and therefore also no @xmath0-coloured arrows from @xmath69 to @xmath13 or @xmath70 .
in this case , @xmath570 . in the algorithm , no arrows will be added between @xmath13 and @xmath69 in step 1 , and therefore no further changes will be made in step 2 .
thus @xmath571 , as desired .
suppose now that there are @xmath93-coloured arrows from @xmath70 to both @xmath13 and @xmath69 . in this case ,
@xmath572 . in this case , as discussed in the proof of well - definedness , an equal number of @xmath93-coloured and @xmath562-coloured arrows will be introduced at step 1 .
they will therefore be cancelled at step 2 .
thus @xmath573 as desired .
suppose now that there is a @xmath93-coloured arrow from @xmath70 to @xmath69 , but not from @xmath70 to @xmath13 .
let the arrows from @xmath13 to @xmath70 , if any , be of colour @xmath79 .
at step 1 of the algorithm , we will add @xmath574 arrows of colour @xmath79 to @xmath71 . by proposition 5.1 ,
the arrows in @xmath71 from @xmath13 to @xmath69 are of colour @xmath79 or @xmath575 .
one verifies that the algorithm yields the same result as coloured quiver mutation , in the three cases that the arrows from @xmath13 to @xmath69 in @xmath71 are of colour @xmath79 , that they are of colour @xmath575 but there are fewer than @xmath574 , and that they are of colour @xmath575 and there are at least as many as @xmath574 . the final case , that there is a @xmath93-coloured arrow from @xmath70 to @xmath13 but not from @xmath70 to @xmath69 , is similar to the previous one .
in @xcite , a certain category @xmath576 is constructed , which is shown to be equivalent to the @xmath0-cluster category of dynkin type @xmath2 .
the description of @xmath576 is as follows .
take an @xmath577-gon @xmath578 , with vertices labelled clockwise from 1 to @xmath577 .
consider the set @xmath39 of diagonals @xmath550 of @xmath578 with the property that @xmath550 divides @xmath578 into two polygons each having a number of sides congruent to 2 modulo @xmath0 . for each @xmath579 , there is an object @xmath580 in @xmath576 .
these objects @xmath580 form the indecomposables of the additive category @xmath576 .
we shall not recall the exact definition of the morphisms , other than to note that they are generated by the morphisms @xmath581 which exist provided that @xmath538 and @xmath582 are both diagonals in @xmath39 , and that , starting at @xmath70 and moving clockwise around @xmath578 , one reaches @xmath69 before @xmath13 .
a collection of diagonals in @xmath39 is called non - crossing if its elements intersect pairwise only on the boundary of the polygon
. an inclusion - maximal such collection of diagonals divides @xmath578 into @xmath583-gons ; we therefore refer to such a collection of diagonals as an @xmath583-angulation .
if we remove one diagonal @xmath550 from an @xmath583-angulation @xmath584 , then the two @xmath583-gons on either side of @xmath550 become a single @xmath585-gon .
we say that @xmath550 is a _ diameter _ of this @xmath585-gon , since it connects vertices which are diametrically opposite ( with respect to the @xmath585-gon ) .
if @xmath586 is another diameter of this @xmath585-gon , then @xmath587 is another maximal noncrossing collection of diagonals from @xmath39 .
( in particular , @xmath588 . ) for @xmath584 an @xmath583-angulation , let @xmath589
. then we have that @xmath590 is a basic ( @xmath0-cluster-)tilting object for @xmath576 , and all basic tilting objects of @xmath576 arise in this way .
it follows from the previous discussion that if @xmath591 is a basic tilting object , and @xmath592 , then the complements to @xmath593 will consist of the objects @xmath594 where @xmath586 is a diameter of the @xmath585-gon obtained by removing @xmath550 from the @xmath583-angulation determined by @xmath584 .
in fact , we can be more precise . define @xmath595 to be the diameter of the @xmath585-gon obtained by rotating the vertices of @xmath550 by @xmath13 steps counterclockwise ( within the @xmath585-gon )
. then @xmath596 . the coloured quiver @xmath598 of @xmath591 has an arrow from @xmath550 to @xmath586 if and only if @xmath550 and @xmath586 both lie on some @xmath583-gon in the @xmath583-angulation defined by @xmath584 . in this case
, the colour of the arrow is the number of edges forming the segment of the boundary of the @xmath583-gon which lies between @xmath550 and @xmath586 , counterclockwise from @xmath550 and clockwise from @xmath586 .
we return to the example from section 2 .
the quadrangulation of a decagon corresponding to the tilting object @xmath41 is on the left .
the quadrangulation corresponding to @xmath109 is on the right .
passing from the figure on the left to the figure on the right , the diagonal 27 ( which corresponds to the summand @xmath599 ) has been rotated one step counterclockwise within the hexagon with vertices 1,2,3,4,7,10 . |
the control and manipulation of single electrons in mesoscopic systems constitutes one of the key ingredients in nanoelectronics .
the study of single - electron sources@xcite in the high - frequency regime has attracted a great interest due to their potential application in quantum electron optics experiments , in metrology , and in quantum information processing based on fermionic systems.@xcite in this work we study the time evolution of a quantum dot ( qd ) tunnel coupled to a single electronic reservoir , as depicted schematically in fig . [ fig_scheme](a ) . in the presence of some time - dependent voltage modulations ,
this system defines the building block of the typical single - electron source , namely the mesoscopic capacitor.@xcite in the linear - response regime , the relaxation behavior of such a mesoscopic capacitor has been extensively studied theoretically@xcite and experimentally,@xcite revealing the quantization of the charge relaxation resistance.@xcite on the other hand , the application of _ nonlinear _ periodic potentials to the mesoscopic capacitor yields the controlled emission and absorption of electrons at giga - hertz frequencies.@xcite from these experiments the average charge as well as current correlations@xcite after each cycle of the potential applied have been extracted .
these results demonstrate the importance of investigating the dynamics of this kind of single - electron sources . in some of the recent realizations@xcite
the coulomb interaction is weak ; however , in small - sized qds the coulomb blockade is , in general , strong and it is , therefore , desirable to include it in the theoretical analysis @xcite since it may even dominate time - dependent phenomena , see e.g. ref . .
the time - evolution of interacting quantum dots after the coupling to the leads has been switched on , has , e.g. , been studied in refs . and references therein . , coupled to a normal lead with a tunneling strength @xmath0 .
dot occupations can be measured via the current passing through a nearby quantum point contact ( qpc ) capacitively coupled to the dot .
b ) qd attached to an additional superconducting contact .
c ) qd coupled to a ferromagnetic lead . ] here we investigate the exponential relaxation of a qd towards its equilibrium state after its has been brought out of equilibrium by applying , e.g. , a voltage step pulse .
we consider a voltage pulse that affects the occupation of only a single orbital energy level .
the level can be spin split due to coulomb interaction . in an earlier work,@xcite some of the present authors investigated the decay of charge and spin of such a single level qd .
it was found that the relaxation of charge and spin are given by rates which differ from each other due to coulomb repulsion . since the reduced density matrix of a qd with a single orbital level with spin is four dimensional , there are thus three rates which govern the relaxation of the diagonal elements of the density matrix towards equilibrium ( plus one which is always zero and corresponds to the stable stationary state ) .
in addition to the rates that govern charge and spin there is a third rate that appears in the relaxation of a single level qd with spin and with interaction .
this additional rate is the subject of this paper .
interestingly , this additional time scale is independent of the interaction and of the dot s level position .
it is shown to be related to two - particle effects and appears , e.g. , in the time - evolution of the mean squared deviations of the charge from its equilibrium value .
we study in detail the perturbations leading to a relaxation of the system with the additional decay rate only , and find that it is indeed related to two - particle correlations .
we also propose a procedure to separately read out the different relaxation rates occurring in the dynamics of the qd exploiting the sensitivity of a nearby quantum point contact to the occupation of the qd , see fig .
[ fig_scheme ] ( a ) . in order to further clarify the properties of the additional time scale , we extend our study to two other setups :
a qd proximized by an extra , superconducting electrode and tunnel coupled to a normal lead ; and a qd tunnel coupled to a ferromagnetic lead , see fig .
[ fig_scheme ] ( b ) and ( c ) .
we consider a quantum dot coupled to an electronic reservoir .
we assume that the single - particle level spacing in the dot is larger than all other energy scales , so that only one , spin - degenerate level of the qd spectrum is accessible . at a certain time
@xmath1 the system is brought out of equilibrium , e.g. by applying a gate potential , and afterwards relaxes to an equilibrium dictated by the hamiltonian @xmath2 .
the hamiltonian @xmath3 of the decoupled dot @xmath4 contains the spin - degenerate level @xmath5 and the on - site coulomb energy @xmath6 for double occupation of the dot .
the creation ( annihilation ) operator of an electron with spin @xmath7 on the dot is denoted by @xmath8 and @xmath9 is the corresponding number operator .
the reservoir is modeled by the hamiltonian @xmath10 , in which @xmath11 creates ( annihilates ) an electron with spin @xmath12 and momentum @xmath13 in the lead .
the coupling between the dot and the reservoir is described by the tunneling hamiltonian @xmath14 , where @xmath15 is a tunneling amplitude , which we assume to be independent of momentum and spin . by considering a constant density of states @xmath16 in the reservoir ,
the tunnel coupling strength @xmath0 is defined as @xmath17 . in the remainder of this paper , we focus on the relaxation behavior of the quantum dot to its equilibrium state and in particular on how this relaxation manifests itself in measurable quantities .
we are not interested in the dynamics of the reservoir , thus the trace over its degrees of freedom is performed to obtain the reduced density matrix of the qd .
the hilbert space is spanned by the four eigenstates of the decoupled dot hamiltonian , @xmath18 , where @xmath19 represents the unoccupied dot , the dot is in the state @xmath20 when being singly occupied with spin @xmath7 , and @xmath21 is the state of double occupation .
the energies related to these states are @xmath22 and @xmath23 , where we set the electrochemical potential of the reservoir to zero .
as we consider spin - conserving tunneling events , the off - diagonal elements of the reduced density matrix evolve independently of the diagonal ones ( which are the occupation probabilities ) .
we can , therefore , consider these probabilities alone , which arranged in a vector are given by @xmath24 and fulfill the condition @xmath25 .
the time evolution of the occupation probabilities is governed by the generalized master equation @xmath26 where the matrix elements @xmath27 of the kernel @xmath28 describe transitions from the state @xmath29 at time @xmath30 to a state @xmath31 at time @xmath32 .
we consider now the dynamics of the system after being brought out of equilibrium at time @xmath1 .
since for @xmath33 the total hamiltonian is time independent , the transition matrix elements depend only on the time difference @xmath34 , i.e. @xmath35 .
furthermore , we are interested in the exponential decay towards equilibrium . to be more specific , we will therefore consider only the leading , time - independent , prefactor of the exponential functions .
time - dependent corrections to the pre - exponential functions , that generally may appear,@xcite are disregarded .
furthermore , when focussing on times @xmath32 distant from the switching time @xmath1 , such that the difference @xmath36 is hence much larger than the decay time of the kernel @xmath37 , we can replace the lower limit of the integral in eq .
( [ eq_master ] ) by @xmath38 . expanding the probability vector @xmath39 in eq .
( [ eq_master ] ) around the measuring time @xmath32 we find@xcite @xmath40 here we introduced the laplace transform of the kernel @xmath41 , with @xmath42 and the @xmath43-th derivative of the kernel with respect to the laplace variable @xmath44_{z=0}$ ] .
the formal solution of eq .
( [ eq_masterexpand ] ) is given by @xmath45 which depends on the initial probability vector @xmath46 at @xmath47 , where the initial values for the system parameters are given by the ones just after the switching time @xmath1 .
the matrix @xmath48 includes markovian and non - markovian processes.@xcite in the following , we consider the limit of weak coupling between quantum dot and reservoir and limit ourselves to a perturbation expansion up to second order in @xmath0 , which is valid for the regime where the tunnel coupling @xmath0 is much smaller than the energy scale set by the temperature @xmath49 .
the perturbative expansion of @xmath48 is @xmath50 with @xmath51 and @xmath52 , where the number in the superscript represents the power of @xmath0 included in the transition matrix @xmath53 .
notice that the first non - markovian correction , i.e. the term @xmath54 is present in second - order in the tunnel coupling .
the evaluation of the kernel within a perturbative expansion can be performed using a real - time diagrammatic technique,@xcite which has been used in ref . in order to extract the exponential decay of spin and charge in the system studied here .
considering eq .
( [ eq_exp ] ) , we see that the rates defining the decay of the state into equilibrium are found from the eigenvalues of the matrix @xmath48 , which turn out to be real and non - positive .
the matrix @xmath48 is not hermitian , as expected since we deal with a dissipative system , and hence has different left and right eigenvectors , @xmath55 and @xmath56 .
the time - dependent probability vector , @xmath57 , can be expressed in terms of the right eigenvectors of @xmath48 , each being related to a decay with a different rate .
the left eigenvectors determine the observable that decay with a single time scale only , see also the appendix .
+ in the following we discuss the exponential relaxation towards equilibrium of the vector of occupation probabilities , in first order in the tunneling strength @xmath0 .
we start by briefly discussing the simplest case of a single spinless particle .
this limit is obtained , when a magnetic field much larger than the temperature is applied , @xmath58 .
the hilbert space of the system is two dimensional and spanned by the states @xmath19 and @xmath59 for the empty and singly - occupied dot respectively , whose occupation probabilities are arranged in the vector @xmath60 .
the decay to the stationary state is governed by matrix @xmath61 ( defined equivalently to @xmath62 but for the two - dimensional hilbert space for the problem at hand ) which contains a single relaxation rate , namely the tunnel coupling @xmath0 , as intuitively expected .
we now include the spin degree of freedom but disregard interactions .
the system is described by two independent hilbert spaces spanned by the states @xmath63 and @xmath64 with @xmath7 .
the probability vector for each spin @xmath12 can be written in terms of the eigenvalues and eigenvectors of the matrix @xmath61 ( for the two - dimensional hilbert space ) as @xmath65 \label{eq_psigma}\ ] ] where the right eigenvector corresponding to the eigenvalue zero of @xmath61 defines the occupation probabilities for the equilibrium state , @xmath66 , with the fermi function @xmath67^{-1}$ ] and the inverse temperature @xmath68 .
furthermore , @xmath69 is the vector representation of the number operator for dot electrons with spin @xmath12 , whose initial / equilibrium expectation value is obtained by multiplying it from the left into the initial / equilibrium probability vector , @xmath70 . the rate @xmath71 is obtained as the negative of the non - zero eigenvalue of @xmath61 , with the corresponding left eigenvector being @xmath72 .
the time evolution of the occupation of each spin state is governed by a single decay rate @xmath0 , @xmath73 this equation can be obtained making use of the fact that the time evolution of the expectation value of any operator , which describes an observable of the qd , is given by projecting its vector representation from the left onto eq .
( [ eq_psigma ] ) .
the time evolution of the total charge of the dot , @xmath74 , is also determined by a single relaxation rate @xmath71 .
this means that both charge and spin , which are quantities related with single - particle processes , do not evolve independently from each other and the corresponding decay is given by the same rate .
a similar non - interacting problem has been studied _ non - pertubatively _ in refs . and . as a next step we consider the squared deviation of the charge from its equilibrium value , @xmath75 ^ 2 $ ] .
its time evolution is obtained from eq .
( [ eq_psigma ] ) as @xmath76 ^ 2\rangle(t)-[\langle\hat{n}\rangle^{\text{eq}}]^2\\ & & = \sum_{\sigma=\uparrow,\downarrow}[1+\langle\hat{n}_{\sigma}\rangle^{\text{eq}}]\langle\hat{n}_{\sigma}\rangle(t)+2\langle\hat{n}_{\uparrow}\hat{n}_{\downarrow}\rangle(t)\nonumber\end{aligned}\ ] ] the last , two - particle term of this expression exhibits a decay rate given by @xmath77 .
this is in contrast to the spinless case , where such a term does not appear since double occupation is not possible .
such an additional exponential decay with the rate @xmath78 appears directly in the time evolution of the probability vector , when considering the full two - particle hilbert space spanned by the basis @xmath79 . in this basis , eq .
( [ eq_exp ] ) for the non - interacting regime can be written as : @xmath80\\ \frac{1}{2}\left[1 - 2f(\epsilon)\right]\\ \frac{1}{2}\left[1 - 2f(\epsilon)\right]\\ f(\epsilon ) \end{array}\right ) e^{-\gamma t}\left(\langle\hat{n}\rangle^\mathrm{in}-\langle\hat{n}\rangle^\mathrm{eq}\right)\nonumber \\
+ \left(\begin{array}{c } 0\\ \frac{1}{2}\\ -\frac{1}{2}\\ 0 \end{array}\right ) e^{-\gamma t}\langle\hat{s}\rangle^\mathrm{in } + \left ( \begin{array}{c } -1\\1\\1\\-1 \end{array } \right ) e^{-2\gamma t}\left ( \langle\hat{m}\rangle^\mathrm{in}-\langle\hat{m}\rangle^\mathrm{eq } \right ) \nonumber\\\end{aligned}\ ] ] where as before , @xmath81 defines the state at equilibrium . the decaying part of the probability vector can be divided into three contributions which appear depending on how the initial state at @xmath1 differs from the equilibrium state .
deviations of charge and spin from their equilibrium value relax with the same rate @xmath0 .
the corresponding expectation values are calculated by multiplying the probability vector eq .
( [ ptot ] ) from the left with the vector representation of the operators @xmath82 and @xmath83 which represent the charge and spin , respectively , in this two - particle basis .
the two left eigenvectors of the matrix @xmath84 with the same eigenvalue @xmath85 , are given by @xmath86 and @xmath87 .
the third contribution to the decay of the system into the equilibrium comes from the relaxation rate @xmath78 , which enters the probability vector in connection with a quantity @xmath88 , defined by the operator in vector notation @xmath89 the left eigenvector of @xmath62 with the eigenvalue @xmath90 is given by @xmath91 .
in contrast to charge and spin , the quantity represented by @xmath88 does not have a straightforward intuitive interpretation , since it depends on the quantum dot parameters at @xmath33 and on the temperature and chemical potential of the reservoir via the fermi functions . from now on we assume a finite on - site coulomb repulsion @xmath6 on the dot .
analogously to the noninteracting case discussed before , from eq .
( [ eq_exp ] ) we can write the time - dependent probability vector in terms of contributions exhibiting different decay times @xmath92\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ f(\epsilon+u ) \end{array}\right)e^{-\gamma_n t}\left(\langle\hat{n}\rangle^\mathrm{in}-\langle\hat{n}\rangle^\mathrm{eq}\right ) \nonumber \\ & & + \left(\begin{array}{c } 0\\ \frac{1}{2}\\ -\frac{1}{2}\\ 0 \end{array}\right ) e^{-\gamma_s t}\langle\hat{s}\rangle^\mathrm{in}+\left ( \begin{array}{c } -1\\1\\1\\-1 \end{array } \right ) e^{-\gamma_m t}\left ( \langle\hat{m}\rangle^\mathrm{in}-\langle\hat{m}\rangle^\mathrm{eq } \right)\ .\end{aligned}\ ] ] again , @xmath93 is the eigenvector of @xmath51 with the zero eigenvalue and represents the equilibrium state in lowest order in the tunnel coupling ( the explicit form of the four - dimensional matrix @xmath62 , together with its entire set of eigenvalues and eigenvectors , is given in the appendix ) .
in the two - particle basis , again @xmath94 represents the charge operator , and @xmath95 represents the spin operator .
the form of the operator @xmath88 is modified by the presence of finite coulomb interaction ; the explicit form will be discussed later in this sub - section ( see eq .
( [ eq_def_m ] ) below ) .
the initial and equilibrium expectation values for these operators , entering in the above eq .
( [ eq_solution ] ) , are obtained as @xmath96 , with @xmath97 .
explicit expressions for @xmath98 and @xmath99 are shown below .
the negative of the other three eigenvalues of @xmath62 directly determine the decay of charge , spin,@xcite and the quantity denoted by @xmath88 .
these decay rates read @xmath100\label{eq_lcharge1}\\ \gamma_s&=&\gamma\left[1-f(\epsilon)+f(\epsilon+u)\right]\label{eq_lspin1}\\ \gamma_m&=&2\gamma . \label{eq_lmal1}\end{aligned}\ ] ] notice that due to interaction , the relaxation rates for charge and spin ( @xmath101 and @xmath102 respectively ) differ from each other and depend on the level position @xmath5 , in contrast to the non - interacting case .
their dependence on the level position is shown in fig .
[ fig_decay ] . in the region for @xmath103 , @xmath101 is enhanced as the charge decays into the twofold degenerate state of single - occupation , whereas the spin relaxation in first order in @xmath0 is suppressed , since spin - flip processes are not possible .
however , the third decay rate , @xmath104 , remains fully energy independent as in the case with @xmath105 .
( blue , dashed line ) , @xmath101 ( red , dash - dotted line ) and @xmath102 ( green , solid line ) in units of @xmath0 as a function of the dot level position @xmath5 .
the temperature is @xmath106 and the interaction energy is @xmath107 . ]
the right eigenvectors occurring in eq .
( [ eq_solution ] ) each represent a change to the steady state density matrix that decays exponentially with rate @xmath108 ( @xmath109 ) .
therefore , a system being brought out of equilibrium by a symmetric deviation between @xmath110 and @xmath111 only , is decaying with a rate @xmath102 .
a deviation from equilibrium in which the occupation of the even sector , @xmath112 is symmetrically shifted from the odd sector , @xmath113 , is governed solely by the relaxation rate @xmath104 .
this right eigenvector is found to play an important role also in the low - temperature renormalization of this model .
@xcite an energy - dependent change in the occupation probabilities as prescribed by the second vector in eq .
( [ eq_solution ] ) yields a decay of the total charge of the system with the rate @xmath101 .
the conditions under which specific deviations from the equilibrium state should be performed in order to obtain a specific decay rate , are discussed in the following section .
the attribution of these relaxation rates to the charge , spin , and @xmath88 arises from the independent decay of these quantities , due to the explicit form of the _ left _ eigenvectors of @xmath62 .
the spin operator coincides with the left eigenvector associated to the eigenvalue @xmath114 and since it has a vanishing equilibrium value , the time evolution of its expectation value is given by @xmath115 equivalently , the left eigenvector corresponding to the eigenvalue @xmath116 , is @xmath117 .
it contains the charge operator @xmath118 and its equilibrium value @xmath119 $ ] .
hence , for the time evolution of the charge we find @xmath120 as a function of the dot level position @xmath5 .
the other parameters are : @xmath106 and @xmath107 . ] the quantity decaying with the rate @xmath104 alone is related to the left eigenvector @xmath91 , where the operator @xmath88 is given by @xmath121 its expectation value follows a time evolution equivalent to the one for the charge in eq .
( [ eq_nrelax ] ) : @xmath122 .
its equilibrium value @xmath123 $ ] , plotted in fig .
[ fig_meh ] , is - in contrast to spin and charge - not sensitive to the regime of single occupation on the quantum dot .
instead , it exhibits a feature close to the electron - hole symmetric point of the anderson model , indicating that @xmath88 represents a quantity which is affected by two - particle effects and it decays with a rate that is not modified by the coulomb interaction @xmath6 . already for the noninteracting case , we found that the rate @xmath78 appears as a consequence of introducing two particles in the system , and we considered the deviations from equilibrium charge as a quantity involving two - particle processes leading to such a decay rate .
also in the case for finite coulomb interaction , the time - dependent mean squared deviations @xmath124
^ 2\rangle(t)$ ] are suitable to reveal the relaxation rate @xmath125 .
their time evolution is obtained by means of eq .
( [ eq_solution ] ) and reads @xmath126 ^ 2\rangle(t ) - [ \langle \hat{n}\rangle^\mathrm{eq}]^2 & = & c\cdot\langle \hat{n}\rangle(t)-2\cdot\langle \hat{m}\rangle(t)\nonumber\\ \label{eq_variance}\end{aligned}\ ] ] ( red , solid line ) and the coefficient @xmath127 ( blue , dashed line ) as a function of the dot level position @xmath5 .
the other parameters are : @xmath106 and @xmath107 . ] where in front of the time - dependent charge @xmath128 the following coefficient appears : @xmath129 with @xmath130 the quantity @xmath131/\left[1+f(\epsilon)-f(\epsilon+u)\right]$ ] is the difference between the probability of doubly occupied and empty dot in equilibrium , which can also be related with the occupation of electrons and holes , @xmath132 . the behavior of @xmath133 is shown in fig . [ fig_coeff ] . for @xmath134 ,
when the dot is doubly occupied , @xmath135 ; for @xmath103 , when one electron and one hole are present in the system ( singly occupied dot ) , @xmath136 ; and for @xmath137 , when the system is completely `` filled with holes '' ( empty dot ) , @xmath138 .
the quantity @xmath127 is also shown in fig . [ fig_coeff ] ( blue dashed line ) , exhibiting a sign change around @xmath139 , the point at which the anderson model is electron - hole symmetric . by replacing @xmath140 , we go from the electron - like to the hole - like behavior , finding an inversion in the sign of @xmath127 , @xmath141 .
the function @xmath127 therefore indicates whether the spectrum of the quantum dot is electron - like or hole - like .
the mean squared deviations of the charge from its value at equilibrium is an example for a physical quantities showing a decay with @xmath104 ; it also includes the charge relaxation rate @xmath101 , which is found independently from the time evolution of the charge .
equivalently also the time - resolved charge variance , @xmath142 ^ 2\rangle(t)$ ] , or the time - resolved spin variance,@xcite @xmath143 , contain a contribution decaying with @xmath104 .
we now consider in detail which external perturbations are necessary in order to induce a decay of the _ full _ occupation probability vector with one certain relaxation rate only , in a controlled way .
furthermore , we address the conditions under which a single decay rate can be extracted more easily from the occupation of a single state by a measurement with a nearby quantum point contact ( qpc ) .
we first address the case of an infinitesimal perturbation ( linear response ) .
a small variation of the gate potential leads to a decay of the charge governed by the charge relaxation rate @xmath101 .
similarly , the infinitesimal variation of the zeeman splitting in the dot yields a decay with the spin relaxation rate @xmath102 . in order to obtain a decay of the state with the rate @xmath104 only , it is not sufficient to modulate the gate voltage , also the two - particle term in the hamiltonian , @xmath144 , needs to be varied .
the on - site repulsion @xmath6 could be changed , for example , by tuning the carrier density in a nearby two - dimensional electron gas , thereby controlling the screening of the electron - electron interaction in the dot . from eq .
( [ eq_solution ] ) we know that a dynamics given only by @xmath104 is obtained if the occupation of the even states are changed in the same direction , opposite to that of the single occupied states ; this condition is fulfilled if infinitesimal variations of the gate , @xmath145 , and of the interaction , @xmath146 , obey the relation : @xmath147)}{1+\exp(\beta\epsilon)}d\epsilon . \label{dvarm}\ ] ] this expression is represented in terms of field lines in fig . [ fig_field ] . an infinitesimal change tangential to the field line passing through the point corresponding to the initial values of @xmath5 and @xmath6 leads to a pure decay with @xmath104 .
for parameter variations that are not infinitesimal ( beyond linear response ) , a change only of the gate voltage results in a decay of the state with both rates @xmath101 and @xmath104 . from eq .
( [ eq_solution ] ) we find that a finite variation of the energy level and the interaction from an initial condition @xmath148 to @xmath149 resulting in a relaxation containing solely @xmath101 , satisfies the equation @xmath150 . \label{varn}\ ] ] a relaxation given _ only _ by the rate @xmath104 is found when the relation : @xmath151 \label{varm}\ ] ] is fulfilled . for different values of @xmath152 and @xmath153 , eq .
( [ varm ] ) produces again the field lines shown in fig .
[ fig_field ] . therefore ,
finite variations of the parameters between two points lying on _ the same _ field line yield a dynamics for the entire occupation probabilities vector @xmath154 governed only by @xmath104 . obviously , a generic variation in both @xmath5 and @xmath6 which does not fulfill the conditions specified by eqs .
( [ varn ] ) or ( [ varm ] ) exhibits a dynamics of the probabilities with two time scales : @xmath101 and @xmath104 . in fig .
[ fig_field ] it is observed that in the region @xmath155 the field lines are approximately horizontal , i.e , only the interaction @xmath6 needs to be varied while keeping the level position constant in order to see a dynamics of the probability governed by @xmath104 only .
in fact , in this regime the qd is predominantly empty and variations of the interaction strength @xmath6 do not affect the occupation of the dot .
this is the reason why this variation yields a dynamics in which the rate @xmath101 does not contribute . on the other hand , in the region for @xmath156 in order to avoid that the number of particles on the dot changes , which would lead to a relaxation with rate @xmath101 , a variation in @xmath6 needs to be accompanied by an opposite variation in @xmath5 , that is @xmath157 .
the crossover between the two regimes appears around the symmetry point of the anderson model , @xmath139 .
importantly , it is also possible to read out either the rate @xmath101 or the rate @xmath104 by varying the gate voltage only ( and , thus , not fulfilling eqs .
( [ dvarm ] ) and ( [ varm ] ) ) , which is easier to realize in an experiment . this can be done by measuring an observable that is sensitive to only one occupation probability , for instance the probability of the quantum dot being empty .
such a time - resolved read - out of the probability can be achieved by considering a qpc located nearby the system and tuned such that it conducts only if the qd is empty .
@xcite in the simplest model of the qpc , which assumes a very fast response , the operator corresponding to the current in the qpc is given by @xmath158 where @xmath159 is a constant current , given by the characteristics of the qpc potential .
the expectation value of the qpc current is simply @xmath160 . in this way
, the qpc effectively measures the dynamics of the occupation probability @xmath161 . according to eq .
( [ eq_solution ] ) , a modulation of the gate in which the initial value @xmath162 equals the equilibrium value @xmath99 leads to a pure decay with @xmath101 . instead , for a decay given by @xmath104 either the factor @xmath163 or the factor @xmath164/\left[1-f(\epsilon)+f(\epsilon+u)\right]$ ] in eq .
( [ eq_solution ] ) has to vanish . .
the on - site coulomb repulsion @xmath6 is constant and takes the value @xmath165 .
dashed blue line : @xmath5 changes from @xmath166 to @xmath167 , its slope yields the relaxation rate @xmath104 .
red dot - dashed line : in this case @xmath168 to @xmath169 , and the slope leads to @xmath101 . the black line is obtained if @xmath5 changes from @xmath170 to @xmath169 , in which both rates @xmath104 and @xmath101 are present . in all cases
we have subtracted the corresponding value for the current in the long - time limit . ]
results for the qpc current for different variations of the level position @xmath5 while @xmath6 is kept constant , are shown in the logarithmic plot in fig .
[ lines ] .
for clarity , we also subtracted the corresponding current in the long time limit , @xmath171 .
in particular , for a fixed value of @xmath6 equal to @xmath172 , we find that if the level position is changed from @xmath166 to @xmath167 , the time evolution of @xmath161 is governed entirely by the rate @xmath104 , giving rise to the straight , blue - dashed line in fig .
[ lines ] .
its slope is given by @xmath104 , making it possible to extract this relaxation rate from measurements of the current in the qpc .
however we can obtain a dynamics of @xmath161 given mainly by the rate @xmath101 by performing a variation in @xmath5 from @xmath166 to @xmath169 which results in the red dot - dashed straight line in fig .
[ lines ] ; again , the slope yields the corresponding relaxation rate which takes the value @xmath173 . finally , we show an example in which variations from @xmath170 to @xmath169 ( solid black line ) produce a dynamics of @xmath161 which includes two exponential decays with rates @xmath104 and @xmath101 . as a result ,
the curve exhibits a change in the slope , showing that a single rate will not be obtained by arbitrary variations of the parameters . in the previous sections we investigated the relaxation rates in first order in the tunnel coupling strength @xmath0 .
however , corrections due to higher order tunneling processes appear when the tunnel coupling gets stronger . besides quantitative corrections , this reveals an interesting new aspect . in second order in the tunnel coupling ,
the matrix @xmath174 included in the exponential decay takes the form @xmath175 .
the second - order corrections to the relaxation rates for charge and spin are given by:@xcite @xmath176w_{\mathrm{d}0 } } { 1-f(\epsilon)+f(\epsilon+u ) } \label{eq_charge_corr}\\ \gamma_\mathrm{s}^{(2 ) } & = & \sigma(\epsilon,\gamma , u ) \frac{\partial}{\partial\epsilon}\gamma_\mathrm{s}+\sigma_\gamma(\epsilon,\gamma , u)\gamma_\mathrm{s}+2w_\mathrm{sf}\ .
\label{eq_spin_corr}\end{aligned}\ ] ] these corrections contain renormalization terms as well as real cotunneling contributions .
on one hand , the renormalization terms contain an effect due to the level renormalization @xmath177 , with @xmath178 $ ] , @xmath179 and @xmath180 is the digamma function . on the other hand , the renormalization of the tunnel coupling appears , @xmath181 $ ] , with @xmath182 $ ] and where @xmath127 was defined in eq .
( [ eq_coeff ] ) .
real cotunneling contributions are manifest in terms of spin flips , @xmath183 , and coherent transitions changing the particle number on the dot by @xmath184 , @xmath185 and @xmath186 .
these cotunneling terms read @xmath187\\ w_{\mathrm{d}0 } & = & - \frac{2 \gamma } { e^{\beta(2\epsilon+u)}-1 } \left[\gamma\phi'(\epsilon)+\gamma\phi'(\epsilon+u)-\frac{2}{u}\sigma(\epsilon , u)\right],\nonumber\\\end{aligned}\ ] ] and @xmath188 w_{\mathrm{d}0}$ ] .
the way in which the cotunneling contributions enter in the respective charge and spin relaxation rates is related to the deviation of the state of the qd from equilibrium , given by eq .
( [ eq_solution ] ) in first order in @xmath0 . as an example we discuss the correction to the charge decay rate , second line of eq .
( [ eq_charge_corr ] ) . there
the factor @xmath184 appears due to the change in the charge by @xmath189 in a process bringing the dot from zero to double occupation and vice versa .
@xcite the fraction with which the transition from zero to double occupation , @xmath186 , enters the correction to the charge relaxation rate , @xmath190 , is given by the deviation from equilibrium of @xmath57 in the direction of @xmath161 , of the contribution which actually decays with @xmath101 only .
this is the first component of the second vector in eq .
( [ eq_solution ] ) .
equivalently , the transition from double to zero occupation , @xmath185 , enters with the fraction given by the fourth component of the same vector , namely by the deviation from equilibrium of @xmath57 in the direction of @xmath191 .
strikingly , in contrast to the charge and spin relaxation rates , @xmath104 does not get renormalized at all by second order tunneling processes : @xmath192 the reason for this is that the contribution due to @xmath0 renormalization and those due to coherent processes between empty and doubly occupied dot , cancel each other .
the lack of second order corrections , confirms that this relaxation rate is related to a quantity which is not sensitive to the coulomb interaction .
the fact that corrections are missing , is also found using a renormalization - group approach .
another important aspect of this missing second - order correction is that it is due to an exact cancelation of the contribution due to virtual second order processes , namely the @xmath0-renormalization , with real cotunneling contributions .
this is in contrast to , e.g. the conductance , where only the real cotunneling processes contribute far from resonances , while renormalization terms are limited to the resonant regions . until now , we considered the quantum dot to be coupled to a normal conducting lead .
however , the vicinity of a superconducting or a ferromagnetic reservoir induces correlations between electrons and holes or between charge and spin , respectively . in the following we study , in first order in the tunnel coupling strength @xmath0 , the influence of induced correlations on the relaxation rates of the dot .
the charge response of a noninteracting mesoscopic scattering region coupled to both normal and superconducting leads has been studied in refs . .
in the previous sections we have seen that the rate @xmath104 , which together with the time decay of charge and spin determines the relaxation of the qd to the equilibrium state , is independent of the level position and the coulomb interaction and that it enters in the time evolution of quantities sensitive to two - particle effects . it is therefore expected that the rate @xmath104 will directly influence the relaxation of the charge towards the equilibrium in a setup that naturally mixes the empty and doubly occupied states of the dot .
this situation is obtained if the qd is not only coupled to a normal lead ( with tunnel coupling strength @xmath0 ) but also to an _ additional _ superconducting contact ( with tunnel coupling strength @xmath193 ) , as shown in fig .
[ fig_scheme ] ( b ) .
we consider only the case when the superconductor is kept at the same chemical potential as the normal lead and we set both chemical potentials to zero .
the only purpose of the extra lead is here to induce superconducting correlations on the dot via the proximity effect .
to the original hamiltonian , @xmath194 , we now add the hamiltonian for the superconducting contact and its tunnel coupling to the qd , @xmath195 where @xmath196 is the the annihilation ( creation ) operator of electrons in the lead . in the limit of a large superconducting gap @xmath197
the effect of the additional contact can be cast in an effective hamiltonian of the dot which includes a coupling between electrons and holes in the qd , @xmath198 .
the eigenstates of the proximized dot are the states of single occupation @xmath20 and other two states which are superpositions of the empty and double occupied states of the dot ( due to andreev reflection ) : @xmath199 with energies given by @xmath200 , where the level detuning between @xmath19 and @xmath21 is @xmath201 and @xmath202 is the energy splitting between the @xmath203 and @xmath204 states.@xcite in the new basis @xmath205 , the vector representing the charge operator is expressed as @xmath206 and we expect that the effect of the mixing of electrons and holes will be visible in its time evolution . in first order in the tunnel - coupling strength to the normal reservoir @xmath0 and assuming @xmath207 , we find the relaxation rates @xmath208\\ \gamma_{s , s}&= & \gamma\left[1-f(\epsilon - e_-)+f(e_+-\epsilon)\right]\\ \gamma_{s,2}&=&2\gamma.\end{aligned}\ ] ] remarkably the eigenvalue @xmath209 remains unaffected , i.e. @xmath210 is not modified by the presence of the additional superconducting lead .
the spin on the dot , which is determined by the occupation probabilities of singly occupied states , still decays with a single relaxation rate given by @xmath211 , i.e. @xmath212 is an eigenvector of the kernel @xmath62 ( in the proximized basis ) .
in contrast , the decay of the charge to its equilibrium value is given by @xmath213 \left(e^{-\gamma_{s,2 } t}+e^{-\gamma_{s,1}t }
\right ) + \langle \hat{n}\rangle^{\mathrm{eq}}\nonumber\\ & & + a_{sc } \frac{1}{2}\left[\langle x\rangle^{\mathrm{in}}-\langle x\rangle^{\mathrm{eq}}\right ] \left(e^{-\gamma_{s,2 } t}-e^{-\gamma_{s,1}t}\right ) \nonumber \\ & & + \frac{1}{2}\left[\langle y\rangle^{\mathrm{in}}-\langle y\rangle^{\mathrm{eq}}\right ] \left(e^{-\gamma_{s,2 } t}-e^{-\gamma_{s,1}t}\right ) \label{eq_nsct}\end{aligned}\ ] ] with @xmath214 and where we defined the difference in the occupation of the @xmath215 states , @xmath216 and the quantity @xmath217 , with @xmath218 .
the charge evolves with two different time scales , @xmath219 and @xmath220 , instead of only one as in the normal case .
this is a direct consequence of the mixing of the states @xmath19 and @xmath21 induced by the superconducting contact .
this effect opens the possibility to extract this rate by measuring the time evolution of the charge in the proximized dot . even though the presence of a superconducting lead couples electrons and holes , the relaxation rate @xmath104 has not been modified .
since we associate this rate with processes involving two particles each with spin @xmath12 , it is expected that if the spin symmetry is broken by introducing a ferromagnetic contact , the rate @xmath104 will now be the sum of the tunneling rates for spin up and spin down electrons . in order to verify this
, we consider the hamiltonian used for the normal case and assume a spin - dependent density of states in the only reservoir attached to the quantum dot , see fig .
[ fig_scheme ] ( c ) .
this leads to spin - dependent tunnel couplings , @xmath221 and @xmath222 , which are included in the corresponding transition matrix @xmath62 .
diagonalization of @xmath62 yields the three relaxation rates : @xmath223 ^ 2 } \\
\gamma_{f,2 } & = & \gamma-\frac{1}{2}\sqrt{\left(\delta\gamma\right)^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } \\
\gamma_{f , m}&= & 2 \gamma\end{aligned}\ ] ] with @xmath224 and @xmath225 . as in the normal case
, there is an eigenvalue which does not depend on the level position nor on the interaction but on the sum of the different tunneling rates : @xmath226 .
the appearance of such a combination of the spin - dependent tunneling strengths in the relaxation rate , confirms the statement that two - particle processes involving electrons with both spin polarizations are at the basis of the decay rate @xmath104 . due to the ferromagnetic lead ,
the dynamics of spin and charge are now mixed .
the corresponding time evolution in first order in the tunnel coupling takes the form : @xmath227(e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t } ) \nonumber \\
\langle\hat{n}\rangle_f(t ) & = & \frac{1}{2}\left[\langle \hat{n}\rangle^{\mathrm{in}}-\langle \hat{n}\rangle^{\mathrm{eq}}\right](e^{-\gamma_{f,1}t}+e^{-\gamma_{f,2}t } ) + \langle \hat{n}\rangle^{\mathrm{eq}}\nonumber \\ & & + a_c\left[\langle \hat{n}\rangle^{\mathrm{in}}-\langle \hat{n}\rangle^{\mathrm{eq}}\right](e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t})\nonumber \\ & & + b_c\langle \hat{s}\rangle^{\mathrm{in}}(e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t})\label{eq_decay_ferro}\end{aligned}\ ] ] where we introduced the abbreviations : @xmath228}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } } \nonumber \\ b_s & = & \frac{\delta\gamma[1+f(\epsilon)-f(\epsilon+u)]}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } } \nonumber \\ a_c & = & a_s\nonumber \\ b_c & = & \frac{\delta\gamma[1-f(\epsilon)+f(\epsilon+u)]}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2}}. \nonumber\end{aligned}\ ] ] the last term in eq .
( [ spin_ferro ] ) shows that at finite time @xmath32 the initial charge influences the time evolution of the spin ; similarly , the initial spin enters explicitly in the dynamics of the charge , eq .
( [ eq_decay_ferro ] ) .
these terms persist in the non - interacting limit , revealing that the coupled evolution of charge and spin including two relaxation rates ( which for the non - interacting case take the form @xmath229 and @xmath230 ) is a direct consequence of the presence of the ferromagnetic contact .
in contrast , the factor @xmath231 vanishes for @xmath105 implying that it stems from the combined effect of the coulomb interaction and the breaking of the spin symmetry . as expected the independent evolution of charge and spin
is recovered in the limit @xmath232 .
the mixing of the dynamics of both , charge and spin , induced here by a ferromagnetic lead was found in ref .
for the case of lifted spin - degeneracy in the dot due to a finite zeeman splitting . note that for the hybrid as well as for the normal system , the sum of the energy - dependent relaxation rates equals @xmath78 , as long as the tunnel coupling @xmath0 is treated in first order , only .
we have studied the different time scales present in the evolution of the reduced density matrix of a single - level qd with coulomb interaction and tunnel coupled to a single reservoir , after being brought out of equilibrium . besides the relaxation rates for charge and spin
, we find an additional rate @xmath125 , which is independent of the energy level of the dot as well as of the interaction strength .
this relaxation is related to the presence of two particles in the dot and is found to be not sensitive to the coulomb interaction .
the time evolution of the square deviations of the charge from its equilibrium value is proposed as a physical quantity related with processes involving two - particles leading to the rate @xmath78 . in order to further elucidate the properties of this decay
, we analyzed the response of the system to specific variations of both , the interaction strength @xmath6 and the level position @xmath5 , finding that @xmath104 can be extracted from time - resolved measurements of the current passing through a nearby quantum point contact .
additionally , we analyzed two other setups : a dot proximized by a superconductor and coupled to a normal reservoir , and a dot coupled to a ferromagnetic lead .
in the hybrid normal - superconducting systems , we found that the time - resolved read - out of the charge represents another possibility to get access to the rate @xmath104 .
we thank michael moskalets , roman riwar and maarten wegewijs for fruitful discussion .
financial support by the ministry of innovation , nrw , the dfg via spp 1285 and ko 1987/5 , the european community s seventh framework programme under grant agreement no .
238345 ( geomdiss ) , as well as the swiss national science foundation , the swiss centers of excellence manep and qsit and the european marie curie itn , nanoctm is acknowledged .
the transition matrix for the normal case in the eigenbasis of the isolated qd @xmath233 , in first order in the tunneling strength @xmath0 , is calculated by means of fermi s golden rule and is given by : @xmath234 & 0 & 1-f(\epsilon+u))\\ f(\epsilon ) & 0 & -\left[1-f(\epsilon)+f(\epsilon+u)\right ] & 1-f(\epsilon+u))\\ 0 & f(\epsilon+u ) & f(\epsilon+u ) & -2\left[1-f(\epsilon+u)\right ] \end{array } \right)\end{aligned}\ ] ] as @xmath62 is non - hermitian it has different right and left eigenvectors , @xmath56 and @xmath55 . for a system with a well - defined steady state ( as the one we are considering here ) there must be at least a zero eigenvalue , @xmath238 .
@xcite the other eigenvalues are found to be the negative of @xmath100 \nonumber \\
\gamma_s&=&\gamma\left[1-f(\epsilon)+f(\epsilon+u)\right ] \\
\nonumber \label{evalues}\end{aligned}\ ] ] the right eigenvector corresponding with the zero eigenvalue , @xmath239 , determines the stationary density matrix ( which we also label as @xmath81 ) , whereas each one of the rest of the right eigenvectors represents a deviation out of the equilibrium density matrix which decays exponentially with a rate given by the negative of the corresponding eigenvalue : @xmath240[1-f(\epsilon+u)]\\ f(\epsilon)[1-f(\epsilon+u)]\\ f(\epsilon)[1-f(\epsilon+u)]\\ f(\epsilon)f(\epsilon+u)\\ \end{array}\right ) , \
\mathbf{r}_s=\frac{1}{2}\left(\begin{array}{c}0\\1 \\-1 \\
0\end{array}\right),\\ \mathbf{r}_n = \frac{1}{1-f(\epsilon)+f(\epsilon+u)}\left(\begin{array}{c } -[1-f(\epsilon)]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ f(\epsilon+u ) \end{array}\right ) , \ \mathbf{r}_m
= \left(\begin{array}{c } -1\\1\\1\\-1\end{array}\right)\ .\end{aligned}\ ] ] these left eigenvectors contain the operators for spin , charge and @xmath88 in vector representation , which can be understood in the following manner .
while in general the expectation value of an operator @xmath247 is found from @xmath248 , with the full density matrix @xmath249 , this can be considerably simplified in the situation considered here , where only diagonal elements of the reduced density matrix of the quantum dot , collected in the vector @xmath154 , play a role .
the expectation value of a quantum dot operator is then obtained by multiplying its vector representation from the left hand side onto the vector @xmath154 . to show an example the expectation value of the spin on the dot
is obtained by multiplying @xmath154 from left by the vector @xmath250 , yielding @xmath251 .
similarly , all other operators for quantum dot observables can be expressed in such a vector representation . |
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