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NOISE DOWN-CONVERSION IN A PUMPED
JOSEPHSON JUNCTION
Y. Taur
To cite this version:
JOURNAL DE PHYSIQUE
Colloque
C6, supplgment au
no8, Tome 39, aotit 1978, page
C6-575
NOISE DOWN-CONVERSION IN
A
PUMPED JOSEPHSON JUNCTIONNASA Goddard I n s t i t u t e for Space Studies, Goddard Space F l i g h t Center, New York, N.Y. 10025, USA.
R6sumd.- Nous avons d t u d i d numdriquement l a c o n v e r s i o n du b r u i t v e r s l e s b a s s e s frdquence's d a n s une j o n c t i o n Josephson pompde e x t d r i e u r e m e n t . I1 appara'it que l ' e x c s s d e b r u i t b a s s e - f r d q u e n c e dans un mdlangeur Josephson p e u t Z t r e e x p l i q u d p a r l e s d i v e r s e s c o n t r i b u t i o n s provenant d e l a c o n v e r s i o n du b r u i t p r d s e n t
P
l a f r d q u e n c e Josephson e t aux f r d q u e n c e s h a r - moniques d e l a f r d q u e n c e d e pompe.A b s t r a c t . - Downward f r e q u e n c y c o n v e r s i o n i n an e x t e r n a l l y pumped Josephson j u n c t i o n i s s t u - d i e d n u m e r i c a l l y . C a l c u l a t i o n shows t h a t t h e e x c e s s low-frequency n o i s e i n a Josephson m i x e r can be accounted f o r by t h e c o n t r i b u t i o n s of down-converted n o i s e from t h e Josephson f r e q u e n c y and harmonics of t h e pump f r e q u e n c y .
INTRODUCTION.- The s e n s i t i v i t y of a low-noise Josephson d e v i c e i s u l t i m a t e l y l i m i t e d by t h e i n - t r i n s i c low-frequency f l u c t u a t i o n s / I / . Noise much g r e a t e r than thermal h a s been observed i n mm- wave p o i n t - c o n t a c t m i x e r s 121. The e x c e s s n o i s e i s mainly c o n t r i b u t e d by components a t harmonics of
t h e pump and Josephson f r e q u e n c i e s which a r e down- converted by t h e j u n c t i o n . However, no g e n e r a l s o l u t i o n e x i s t s t o t h i s h i g h l y n o n l i n e a r problem, e x c e p t i n t h e c a s e of no e x t e r n a l pumping / 3 / . Assuming t h e r m a l n o i s e d r i v i n g t h e j u n c t i o n , an a n a l o g s i m u l a t o r has been used t o compute t h e low- frequency n o i s e i n a pumped Josephson j u n c t i o n / 4 / . Although t h e s i m u l a t o r r e s u l t a g r e e s r e a s o n a b l y w e l l w i t h e x p e r i m e n t a l d a t a , i t does n o t a c c o u n t a d e q u a t e l y f o r n o i s e c o n t r i b u t i o n s from v a r i o u s frequency-conversion p r o c e s s e s . I n t h i s p a p e r , we d e s c r i b e a n u m e r i c a l method u s i n g a d i g i t a l compu- t e r t o d e r i v e a l l t h e down-conversion c o e f f i c i e n t s from v a r i a t i o n s i n t h e s t a t i c I-V c u r v e . I n a d d i t i o n t o a q u a n t i t a t i v e u n d e r s t a n d i n g of t h e problem, t h e s e c o e f f i c i e n t s a l s o e n a b l e u s t o c a l c u l a t e t h e low-frequency n o i s e s p e c t r a l d e n s i t y when t h e junc- t i o n i s d r i v e n by non-thermal n o i s e i n t h e quantum l i m i t . m i c r o b r i d g e s /5,6/. The e q u a t i o n f o r a c u r r e n t - b i a s e d ( v a l i d f o r most low-impedance j u n c t i o n s ) j u n c t i o n w i t h a r e s i s t i v e s h u n t R i s d $ / d ~
+
s i n 4 = idc+
i s i n Q T.
P P (1)Here i s t h e s u p e r c o n d u c t i n g phase d i f f e r e n c e , idc and i a r e t h e d c and pump c u r r e n t s normalized t o
P
t h e c r i t i c a l c u r r e n t I of t h e j u n c t i o n , T = 2eRI / h ) t i s t h e normalized t i m e , and 52 = (M/2eRIc)
P w i s t h e d i m e n s i o n l e s s pumpfrequency. For g i v e n P v a l u e s of 52 and i e q u a t i o n ( 1 ) can be s o l v e d P P' n u m e r i c a l l y on a d i g i t a l computer t o o b t a i n t h e s t a t i c I-V c u r v e , i . e . t h e normalized d c v o l t a g e vdc = l i m $ ( T ) / T a s a f u n c t i o n of idc. To evalua- T- t e t h e c o e f f i c i e n t of down-conversion from a s i g n a l f r e q u e n c y Q s , we add a s m a l l c u r r e n t 6 i s i n (Qs' + 8) t o e q u a t i o n ( I ) and look f o r f i r s t - o r d e r v a r i a t i o n s i n t h e I-V c u r v e . L i n e a r r e s p o n s e i s found i n two c a s e s : ( i ) Qs = nQ and ( i i ) Q s =
P n52
+
Q where n i s an i n t e g e r and 52 = vdc i s t h e P j ' j normalized Josephson f r e q u e n c y . I n t h e f i r s t c a s e , a s m a l l c u r r e n t a t nQ P m o d i f i e s t h e s h e of a l l t h e pump-induced Josephsons t e p s , a s shown i n t h e example of F i g u r e ] ( a ) . For t h e c a s e of i n t e r e s t where 52 5 1 , t h i s r e s u l t s i n
P
a change i n t h e d c c u r r e n t by a s l i g h t amount 6 i o which i s n e a r l y independent of v o l t a g e between NUMERICAL CALCULATION.- I n t h e a n a l y s i s , we assume
s u c c e s s i v e s t e p s . Because of t h e e x a c t harmonic t h e R e s i s t i v e l y Shunted J u n c t i o n (RSJ) model, which
r e l a t i o n s h i p between Cis and 52 i n t h e c a l c u l a t i o n , h a s been shown t o d e s c r i b e s u c c e s s f u l l y t h e I - V P
t h e s i g n a l phase f o r maximum c u r r e n t r e s p o n s e i s c u r v e s of RF-biased p o i n t c o n t a c t s and s h o r t
€I = 0' f o r odd n and 8 = 90" f o r even n. Therefore,
3
Also w i t h Columbia ~ a d i a t i o n L a b o r a t o r y , Columbia the coefficients of current conversion, a i d c / a i n = U n i v e r s i t y , New York, N.Y. 10027. U.S.A.
1
6io/6is1
max* can b e found f o r n = 1 , 2 , 3 , e t c .NORMAL12 ED DC VOLTAGE
Fig.
1 :Response of static I-V curves to a small
signal for (a) n
:Rs/Rp
=2
(8 =go0), and (b)
ns/n
=0.4.
P
In the second case when a small current at nR
f 61P j
is applied, the pump-induced steps remain unchanged
while new Josephson steps appear symmetrically at
voltages vdc
=kR
f$2. (k is an integer), as shown
P
3in Figure I(b).
The step size 26i depends linearly
j
on 6is, but is independent of the signal phase
8.Therefore, one can also evaluate the current con-
version coefficients, ai
dc
fain +
.=16i. ./6isl
,
for
J
R
=nR
f Q =(n
tr)R
at a bias voltage vdf-=Qj.
s
p
j
P
There is no first-order down-conversion from hlgher
order
( > I )harmonics of the Josephson frequency 131.
Although there is no dc effect when we
consider the small signal to consist only of noise
components at the same frequencies, similar conver-
sion processes still occur. If we assume the junc-
tion to be driven by thermal noise due to the re-
sistance R, the low-frequency spectral density of
current fluctuations (normalized by 4kT/R) is
m
fi2
=1
+2
g
(aidc/ainI2
+2
,o(aidc/ain+,P
at a bias voltage vdc
=rQD. Here we include the
(2)
direct noise component and double sideband conver-
sions of high-frequency noise. A larger value of f i 2
is obtained for small R since there are significant
P
contributions from more harmonic frequencies. Of
particular interest is the degenerate case in which
r
=k
+
112, i.e.
a bias midway between pump-indu-
ced steps, where the second summation consists of
only half as many terms and
$ 2is a minimum. We have
calculated all the significant conversion coeffi-
cients for
i2 =0.16 and 0.50, with r
=112 and a
P
pump current which suppresses the zeroth step to
I 12 as summarized in Table I It follows from
Equation (2) that
B2 =6.11 for R
=0.16 and
B2 =P
3.25 for R
=0.50, in excellent agreement with the
P
low-frequency noise observed in the analog simula-
tion of a Josephson mixer 141. These conversion co-
efficients are insensitive to noise rounding, in
contrast to the equivalent noise temperature which
depends on the dynamic resistance and is strongly
influenced by the quadratic effect of noise/2/.
-- HARMONIC
MIXING
4laf
I2
3
4
5
JOSEPHSON
ns/ap,
MIXING
