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Bistability in the frequency response of a driven SQUID ring-radio frequency resonator system
24 February 1997
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 226 ( 1997) 275-279
Bistability in the frequency response of a driven SQUID ring-radio frequency resonator system R. Whiteman I, J. Diggins, V. SchZjllmann, G. Buckling, T.D. Clark*+2, R.J. Prance, H, Prance, J.F. Ralph, A. Widom3 Physical electronics Group, School of Engineering. Universi~
of Sussex,Brighton, Sussex BNI 9QT, UK
Received 28 August 1996; revised manuscript received 3 December 1996; accepted for publication 12 December 1996 Communicated by A.P. Fordy
Abstract
We discuss the experimental observation of the frequency response of a parallel LC tank circuit inductively coupled to a SQUID ring. We show that when the rms rf flux that couples to the ring is a significant fraction of the superconducting flux quantum, (p, (= h/Ze), the resonance exhibits a cyclic fold bis~bility. PACS: 85.25.Dq; 47.2O.K~ Keywords: SQUID, Nonlinear dynamics; Bifurcations
In a series of theoretically based articles (for example Refs. [I] and [2]) we have reported on the dynamical behaviour of a radio frequency (rf) resonant “tank” circuit, inductively coupled to a single weak link SQUID ring, i.e. one Josephson weak link enclosed by a thick superconducting ring (for a comprehensive review of SQUID behaviour see Ref. [ 31). The dynamical behaviour discussed in these references is calculated from the nonlinear equation of motion of the coupled system, where the screening current response has the form of an almost discontinuous sawtooth with respect to an applied external flux,
* Corresponding author. ’ E-mail: r.r.whitem~~sussex.ac.uk. a E-mail: [email protected]. s Permanent address: Department of Physics, Northeastern University, Boston, MA 02115, USA.
cent improvements in electronic technique the situation has changed and we are now in a position to explore some of the experimental consequences of such nonlinear responses. It is convenient to describe the screening current in terms of its quantum mechanical expectation value, which is given by the derivative [4] (I,( Qi,)), = -dE,( @,)/&I&, where E,(@,) is the @,-dependent (and @,-periodic) energy of the SQUID ring (geometric inductance A and weak link capacitance C) in the eigenstate K (with K = 0, 1,2, . . .) , found by solving the time independent Schriidinger equation [ 51. The SQUID ring behaviour here is parameterized in terms of the matrix element, kv/2 (weak link critical current 2ev), for coherent Josephson pair tunnelling through the weak link and the characteristic ring oscillator frequency w, = l/&E. The ring magnetic susceptibility in state K is [ 4,151 xx(@x)
=
0375-9601/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PIf SO375-9601(96)00945-O
-~~2&(@x)/@:.
(1)
276
R. Whitenran et al. /Phgics
IinCt)
Fig. 1. Schematic of the coupled SQUID ring-tank circuit system including a separate static magnetic flux (c&r) bias coil, where A is the ring inductance. L, and Cr are, respectively, the tank circuit inductance and capacitance, R, is the tank circuit resistance on resonance, @, is the rf flux in the tank circuit coil, M is the mutual inductance between this coil and the SQUID ring, lin( t) is the rf drive signal. V,,, is the rf voltage response of the coupled system to Iin ( 1).
Experimentally, the ring magnetic susceptibility is the quantity of primary interest since, in the limit of weak coupling between the SQUID ring and the tank circuit (i.e. where K2x(Qx) < 1, with a coupling parameter K as defined below) the resonant frequency, jr, of this coupled system is related (to first order) to the magnetic susceptibility by the expression [ 61
fr(@x) =
fR
&+ K2xx(@x)’
Here, the coupling parameter K = (M*/L,A) ‘/2, fR is the resonant frequency of the bare tank circuit, Lt is the inductance of the tank circuit coil, and M is the mutual inductance between this coil and the SQUID ring. From Eq. (2)) it is clear that if we monitor fr (@,) we can infer xK (@,) and, in turn, from calculation infer (I,(@,)),. Since the dependence of (Zs)K on QX is generally very nonlinear, the precise behaviour of the ring-tank circuit system has to be found by solving the full nonlinear system equation of motion [ 1,2]
Here, C, is the tank circuit capacitance, Rt is the resistance of the tank circuit on resonance, @+is the rf flux
Letters A 226 (I 997) 275-279
in the tank circuit inductor (coil) as shown schematically in Fig. 1. The fraction of the rf flux coupled between the coil and the SQUID ring is p = M/L, and the total flux Cpy applied to the ring is the sum of the static bias flux @,,,,, and the rf flux @rf = &. The forcing term, 1i” (t) , is the time dependent drive exciting the tank circuit which may contain both coherent and noise contributions [ 71. In this paper we restrict ourselves to the ground state (K = 0) only where, typically, the ground state screening current (Is (a,))~ has the form of a slightly clipped sawtooth. It is, of course, highly relevant to discuss the ground state adiabaticity (or loss) of the SQUID ring in large amplitude (N @,) rf fields. Although not directly part of the work described in this paper, we have solved the time dependent Schrodinger equation for the quantum mechanical SQUID ring in a time dependent sinusoidal electromagnetic field [ 81. Given typical minimum ground to first excited state energy differences (in frequency units) of four to five hundred GHz, which can be inferred from the ground state frequency shift data, the effect of rf (20 MHz), at an amplitude of N QO, on transitions between these states is entirely negligible. In order to follow fr( @,) experimentally we needed to observe the frequency response of the ring-tank circuit system as a function of the static (i.e. incremented, point by point) flux exsbt, and plot the resonant frequency, fr, as a function of &rat. In practice, this was achieved by using a spectrum analyser; in our case, given our special requirement, one of our own design. This spectrum analyser, shown in block diagram form in Fig. 2, provided a constant current amplitude drive signal, which could be swept backwards and forwards over the frequency range of interest. This drive signal was used to excite the tank circuit. External program routines, written in National Instruments Labview software, allowed us to set QXstat, acquire data on the frequency response of the system for each value of this flux and find fr. The rf voltage response V,,, across the tank circuit due to the drive current was first boosted by a liquid helium cooled GaAsFET preamplifier which was followed by further amplification using a low noise room temperature receiver. The output from this amplifier chain was then fed into the spectrum analyser system for processing. In our experiments we used Zimmerman-type, niobium point contact SQUID rings [ 93. With in situ me-
R. Whiteman et al./Physics
Letters A 226 (1997) 275-279
277
’ Spectrum Analyser
’
-I
jRoom temperature;
i_...___ receiver
t
i
i
3
Fig. 4. The ground state screening current expectation value for the parameters rio,, = O.O43@i/A and fiv = 0.047@$/.4.
Frequency modulated RFdrive
T=4?Ki Fig. 2. Block diagram of the spectrum SQUID-tank circuit arrangement.
21
analyser
and experimental
o-
;;
$
cc!
20.9.
20.8 a
xstat
Fig. 3. Experimental pattern, fr versus Gut, for a niobium point contact SQUID ring at T = 4.2 K with K2 = 0.0054,Qi = 1200, ,fa = 20.988 MHz, and a drive signal CL@,of && rms.
adjustment of the point contact at liquid helium temperatures, we found that we were routinely able to make point contacts that exhibited a sharp downward shift in fr at half integer values of the static bias flux, i.e. Qxstat = (n + l/2)@,, (where n is integer) . Such frequency shift behaviour is characteristic of SQUID rings with an almost sawtooth screening current response. To illustrate this experimentally, we chanical
show in Fig. 3 a sharp fr( @jXxstat) pattern for a niobium point contact SQUID ring-rf tank circuit system taken at 4.2 K. Our computational model of the system indicates that such frequency responses are consistent with an rms flux ,!A+ at the ring of N &QO [ 71. Here, fa = 20.988 MHz, K* = 0.0054 and the quality factor Qt of the coupled system at axstat = n@, is 1200. We note that with this value of K* ,and with the maximum ~(experimental) 5 10 at &,,,, = (n + l/2)@,, the condition for weak coupling, K*,y(@y) < 1, is satisfied. In Fig. 4 we show a calculated screening current pattern, consistent with the data of Fig. 3, where we have used the physically reasonable SQUID ring parameters fiiv = O.O47@z/A and b, = O.O43@z/A [ 61. Clearly, for such levels of @r compared with Qp, we might expect a Lorentzian frequency response for any value of bias flux n@, < Gxstat < (n + 1/2)Qp,. This is borne out by the integer and half integer bias state frequency response curves corresponding to the data of Fig. 3. These frequency responses are shown in Fig. 5 with the driving signal being swept in the conventional manner from low to high frequency. It is apparent that the half integer response is markedly reduced in peak amplitude compared with that for the integer bias. This amplitude reduction is entirely consistent with numerical simulations, as calculated from the nonlinear equation of motion for the coupled system (3)) and is discussed in Ref. [ 71. The inherent nonlinear nature of the screening current (Fig. 4) suggests that we might also expect decidedly nonlinear features to appear in the frequency response at larger levels of drive current. Indeed, experimentally we find that this is the case and the system does in fact behave like a classic, strongly driven
278
R. Whitman
et al. /Physics
Metiers A 226 (1997) 27.5-279
CD xstat= (n+l@o
2075
20.80
20.85
20.90
20.95
11M
21.05
:
5
Driving Frequency (MHz)
20.80
zJ.85
2o.vo
2095
t1.00
11.05
:
Driving Frequency (MHz)
Fig. 5. Experimental output voltage (amplitude) versus frequency for the coupled SQUID ring-tank circuit system of Fig. 3 at T = 4.2 K for integer and half integer static flux bias, measured at small levels of driving signal (@t = &&, rms).
20.7
Driving Frequency (MHz)
21.3
20.6
Driving Frequency (MHz)
21.2
Fig. 6. Experimental strongly driven SQUID ring-tank circuit output voltage (amplitude) versus frequency responses, for the system of Figs. 3 and 5, at the integer and half integer static flux bias states, with @, = 2&,/3 rms and T = 4.2 K.
nonlinear oscillator. As an example, again for the experimental run shown in Fig. 3, we present in Fig. 6 the integer (PI@,) and half integer (n + 1/2)Q0bias flux frequency responses, where the amplitude of the drive signal flux @+ is 40 times greater than that of Fig. 5. The solid triangles in Fig. 6 correspond to sweeping from low to high frequency, whilst the dashed triangles correspond to sweeping from high to low frequency. It is apparent that what are essentially discontinuous amplitude jumps (cuts) have appeared in the frequency responses. These cuts reflect a gross distortion of the original resonant lineshape introduced by the strongly driven nonline~ity associated with the SQUID ring. Such cuts indicate bifurcation points, and a cyclic fold [ IO] is clearly revealed by
the bi-directional sweep. Experimentally, the degree of disto~ion in the frequency response varies continuously with the drive current amplitude (and consequently the rf flux ,Gt experienced by the ring). For small drive signal amplitudes, such as those associated with Figs. 3 and 5, we observe no obvious deviation from the Lorentzian lineshape. However, as the signal amplitude is increased to a significant fraction of a flux quantum the frequency response becomes progressively more skewed until it eventually “folds” on itself and the above bifurcations are seen. The voltage output against frequency responses presented in Figs. 5 and 6, for weak (@jr N Qi,/60 rms) and strong (@r =2@,/3 rms) driving levels, respectively, can be computationally modelled by mimick-
R. Whitman et al./Physics Letters A 226 (1997) 275-279
@xstat
=
219
(n+lQ@o
-11 20.7
Driving Frequency (MHz)
21.3
20.6
Driving Frequency (MHz)
21.2
Fig. 7. Computed theoretical SQUID ring-tank circuit driving frequency responses, at integer and half integer static flux bias, for the strongly driven case where the drive current is constant and we find the rms value of @, N 2&/3 , using the SQUID ring-tank circuit parameters of Figs. 3 and 5.
ing the function of the spectrum analyser. Solving the equation of motion (3) numerically, for a given set of parameters (and a particular drive current amplitude), we calculate the power per unit bandwidth by single point Fourier transform at a discrete set of frequencies over the spectral window of interest. The response is then built up by allowing the system to evolve continuously from one side of the spectral window to the other, with the drive frequency being incremented at the end of each transform. In Fig. 7 we show the theoretical, strongly driven frequency response curves at the integer and half integer bias points for the measured circuit parameters calculated from a single point Fourier transform over 15000 cycles of the drive frequency. The solid line here corresponds to sweeping from low to high frequency, whereas the dashed line corresponds to sweeping from high to low frequency. In this figure we see the discontinuous bifurcation points as calculated from the equation of motion, and the cyclic folds are again apparent. As can be seen, the correspondence between the computed frequency responses of Fig. 7, and the ex~rimental data of Fig. 6 is quite clear. The cyclic bifurcations are revealed, and the agreement between these figures supports the inherent physical accuracy of our model. Although such bifurcations are common to many strongly driven nonlinear oscillators [ lo,1 t 1, to our knowledge, this is the first observation of the phenomenon in SQUID systems.
We would like to thank the Engineering and Physical Sciences Research Council and the National Physical Laboratory for their generous support of this work. References [ 11J. Diggins, J.F. Ralph, T.P. Spiller, T.D. Clark, H. Prance and R.J. Prance, in: 6th Quantum l/f noise and other low frequency fluctuations in electronic devices Symposium, St. Louis, MO, eds. PH. Handel and A.L. Chung, AIP Conf. Proc. 371 (AIP Publishing, Woodbury, NY, 1994). [Z] J. Diggins, J.F. Ralph, T.P Spiller, T.D. Clark, H. Prance and R.J. Prance, Phys. Rev E 49 ( 1994) 1854. 131 A. Barone and G. Patemo, Physics and applications of the Josephson effect (Wiley, New York, 1987 ) [ 41 H. Prance, T.D. Clark, R.J. Prance, TP Spiller, J. Diggins and J.F. Ralph, Nucl. Phys. B Proc. Suppl. 33C (1993) 35. [51 J.E. Mutton, R.J. Prance and T.D. Clark, Phys. Len. A 104 ( 1984) 375. j6j R.J. Prance, T.D. Clark, R. Whiter, 1. Diggins, J.F. Ralph, H. Prance, T.P Spiller, Y. Srivastava and A. Widom, in: Mesoscopic superconductivity - Proc. NATO Advanced workshop (North-Holland, Amsterdam, 1994). [ 7) J. Diggins, D. Phil. Thesis, University of Sussex ( 1994). 181 J. Diggins, R. Whiteman, T.D. Clark, R.J. Prance, H. Prance, J.F. Ralph. A. Widom and Y.N. Srivastava, Solutions of the time de~ndent Schrlidinger equation for a SQUID ring, Physica B, accepted for publication. (91 J.E. Zimmerman, P Thieve and J.T. Harding, J. Appi. Phys. 41 (1970) 1572. [ IO1 J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos (Wiley, New York, 1986) p. 125. 1i 11 D.W. Jordan and P Smith, Oxford applied mathematics and computing science series. Nonlinear ordinary di~e~ntial equations, eds. J. Crank, H.G. Martin and D.M. Melluish (Clarendon, Oxford, 1977) p. 188.
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 226 ( 1997) 275-279
Bistability in the frequency response of a driven SQUID ring-radio frequency resonator system R. Whiteman I, J. Diggins, V. SchZjllmann, G. Buckling, T.D. Clark*+2, R.J. Prance, H, Prance, J.F. Ralph, A. Widom3 Physical electronics Group, School of Engineering. Universi~
of Sussex,Brighton, Sussex BNI 9QT, UK
Received 28 August 1996; revised manuscript received 3 December 1996; accepted for publication 12 December 1996 Communicated by A.P. Fordy
Abstract
We discuss the experimental observation of the frequency response of a parallel LC tank circuit inductively coupled to a SQUID ring. We show that when the rms rf flux that couples to the ring is a significant fraction of the superconducting flux quantum, (p, (= h/Ze), the resonance exhibits a cyclic fold bis~bility. PACS: 85.25.Dq; 47.2O.K~ Keywords: SQUID, Nonlinear dynamics; Bifurcations
In a series of theoretically based articles (for example Refs. [I] and [2]) we have reported on the dynamical behaviour of a radio frequency (rf) resonant “tank” circuit, inductively coupled to a single weak link SQUID ring, i.e. one Josephson weak link enclosed by a thick superconducting ring (for a comprehensive review of SQUID behaviour see Ref. [ 31). The dynamical behaviour discussed in these references is calculated from the nonlinear equation of motion of the coupled system, where the screening current response has the form of an almost discontinuous sawtooth with respect to an applied external flux,
* Corresponding author. ’ E-mail: r.r.whitem~~sussex.ac.uk. a E-mail: [email protected]. s Permanent address: Department of Physics, Northeastern University, Boston, MA 02115, USA.
cent improvements in electronic technique the situation has changed and we are now in a position to explore some of the experimental consequences of such nonlinear responses. It is convenient to describe the screening current in terms of its quantum mechanical expectation value, which is given by the derivative [4] (I,( Qi,)), = -dE,( @,)/&I&, where E,(@,) is the @,-dependent (and @,-periodic) energy of the SQUID ring (geometric inductance A and weak link capacitance C) in the eigenstate K (with K = 0, 1,2, . . .) , found by solving the time independent Schriidinger equation [ 51. The SQUID ring behaviour here is parameterized in terms of the matrix element, kv/2 (weak link critical current 2ev), for coherent Josephson pair tunnelling through the weak link and the characteristic ring oscillator frequency w, = l/&E. The ring magnetic susceptibility in state K is [ 4,151 xx(@x)
=
0375-9601/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PIf SO375-9601(96)00945-O
-~~2&(@x)/@:.
(1)
276
R. Whitenran et al. /Phgics
IinCt)
Fig. 1. Schematic of the coupled SQUID ring-tank circuit system including a separate static magnetic flux (c&r) bias coil, where A is the ring inductance. L, and Cr are, respectively, the tank circuit inductance and capacitance, R, is the tank circuit resistance on resonance, @, is the rf flux in the tank circuit coil, M is the mutual inductance between this coil and the SQUID ring, lin( t) is the rf drive signal. V,,, is the rf voltage response of the coupled system to Iin ( 1).
Experimentally, the ring magnetic susceptibility is the quantity of primary interest since, in the limit of weak coupling between the SQUID ring and the tank circuit (i.e. where K2x(Qx) < 1, with a coupling parameter K as defined below) the resonant frequency, jr, of this coupled system is related (to first order) to the magnetic susceptibility by the expression [ 61
fr(@x) =
fR
&+ K2xx(@x)’
Here, the coupling parameter K = (M*/L,A) ‘/2, fR is the resonant frequency of the bare tank circuit, Lt is the inductance of the tank circuit coil, and M is the mutual inductance between this coil and the SQUID ring. From Eq. (2)) it is clear that if we monitor fr (@,) we can infer xK (@,) and, in turn, from calculation infer (I,(@,)),. Since the dependence of (Zs)K on QX is generally very nonlinear, the precise behaviour of the ring-tank circuit system has to be found by solving the full nonlinear system equation of motion [ 1,2]
Here, C, is the tank circuit capacitance, Rt is the resistance of the tank circuit on resonance, @+is the rf flux
Letters A 226 (I 997) 275-279
in the tank circuit inductor (coil) as shown schematically in Fig. 1. The fraction of the rf flux coupled between the coil and the SQUID ring is p = M/L, and the total flux Cpy applied to the ring is the sum of the static bias flux @,,,,, and the rf flux @rf = &. The forcing term, 1i” (t) , is the time dependent drive exciting the tank circuit which may contain both coherent and noise contributions [ 71. In this paper we restrict ourselves to the ground state (K = 0) only where, typically, the ground state screening current (Is (a,))~ has the form of a slightly clipped sawtooth. It is, of course, highly relevant to discuss the ground state adiabaticity (or loss) of the SQUID ring in large amplitude (N @,) rf fields. Although not directly part of the work described in this paper, we have solved the time dependent Schrodinger equation for the quantum mechanical SQUID ring in a time dependent sinusoidal electromagnetic field [ 81. Given typical minimum ground to first excited state energy differences (in frequency units) of four to five hundred GHz, which can be inferred from the ground state frequency shift data, the effect of rf (20 MHz), at an amplitude of N QO, on transitions between these states is entirely negligible. In order to follow fr( @,) experimentally we needed to observe the frequency response of the ring-tank circuit system as a function of the static (i.e. incremented, point by point) flux exsbt, and plot the resonant frequency, fr, as a function of &rat. In practice, this was achieved by using a spectrum analyser; in our case, given our special requirement, one of our own design. This spectrum analyser, shown in block diagram form in Fig. 2, provided a constant current amplitude drive signal, which could be swept backwards and forwards over the frequency range of interest. This drive signal was used to excite the tank circuit. External program routines, written in National Instruments Labview software, allowed us to set QXstat, acquire data on the frequency response of the system for each value of this flux and find fr. The rf voltage response V,,, across the tank circuit due to the drive current was first boosted by a liquid helium cooled GaAsFET preamplifier which was followed by further amplification using a low noise room temperature receiver. The output from this amplifier chain was then fed into the spectrum analyser system for processing. In our experiments we used Zimmerman-type, niobium point contact SQUID rings [ 93. With in situ me-
R. Whiteman et al./Physics
Letters A 226 (1997) 275-279
277
’ Spectrum Analyser
’
-I
jRoom temperature;
i_...___ receiver
t
i
i
3
Fig. 4. The ground state screening current expectation value for the parameters rio,, = O.O43@i/A and fiv = 0.047@$/.4.
Frequency modulated RFdrive
T=4?Ki Fig. 2. Block diagram of the spectrum SQUID-tank circuit arrangement.
21
analyser
and experimental
o-
;;
$
cc!
20.9.
20.8 a
xstat
Fig. 3. Experimental pattern, fr versus Gut, for a niobium point contact SQUID ring at T = 4.2 K with K2 = 0.0054,Qi = 1200, ,fa = 20.988 MHz, and a drive signal CL@,of && rms.
adjustment of the point contact at liquid helium temperatures, we found that we were routinely able to make point contacts that exhibited a sharp downward shift in fr at half integer values of the static bias flux, i.e. Qxstat = (n + l/2)@,, (where n is integer) . Such frequency shift behaviour is characteristic of SQUID rings with an almost sawtooth screening current response. To illustrate this experimentally, we chanical
show in Fig. 3 a sharp fr( @jXxstat) pattern for a niobium point contact SQUID ring-rf tank circuit system taken at 4.2 K. Our computational model of the system indicates that such frequency responses are consistent with an rms flux ,!A+ at the ring of N &QO [ 71. Here, fa = 20.988 MHz, K* = 0.0054 and the quality factor Qt of the coupled system at axstat = n@, is 1200. We note that with this value of K* ,and with the maximum ~(experimental) 5 10 at &,,,, = (n + l/2)@,, the condition for weak coupling, K*,y(@y) < 1, is satisfied. In Fig. 4 we show a calculated screening current pattern, consistent with the data of Fig. 3, where we have used the physically reasonable SQUID ring parameters fiiv = O.O47@z/A and b, = O.O43@z/A [ 61. Clearly, for such levels of @r compared with Qp, we might expect a Lorentzian frequency response for any value of bias flux n@, < Gxstat < (n + 1/2)Qp,. This is borne out by the integer and half integer bias state frequency response curves corresponding to the data of Fig. 3. These frequency responses are shown in Fig. 5 with the driving signal being swept in the conventional manner from low to high frequency. It is apparent that the half integer response is markedly reduced in peak amplitude compared with that for the integer bias. This amplitude reduction is entirely consistent with numerical simulations, as calculated from the nonlinear equation of motion for the coupled system (3)) and is discussed in Ref. [ 71. The inherent nonlinear nature of the screening current (Fig. 4) suggests that we might also expect decidedly nonlinear features to appear in the frequency response at larger levels of drive current. Indeed, experimentally we find that this is the case and the system does in fact behave like a classic, strongly driven
278
R. Whitman
et al. /Physics
Metiers A 226 (1997) 27.5-279
CD xstat= (n+l@o
2075
20.80
20.85
20.90
20.95
11M
21.05
:
5
Driving Frequency (MHz)
20.80
zJ.85
2o.vo
2095
t1.00
11.05
:
Driving Frequency (MHz)
Fig. 5. Experimental output voltage (amplitude) versus frequency for the coupled SQUID ring-tank circuit system of Fig. 3 at T = 4.2 K for integer and half integer static flux bias, measured at small levels of driving signal (@t = &&, rms).
20.7
Driving Frequency (MHz)
21.3
20.6
Driving Frequency (MHz)
21.2
Fig. 6. Experimental strongly driven SQUID ring-tank circuit output voltage (amplitude) versus frequency responses, for the system of Figs. 3 and 5, at the integer and half integer static flux bias states, with @, = 2&,/3 rms and T = 4.2 K.
nonlinear oscillator. As an example, again for the experimental run shown in Fig. 3, we present in Fig. 6 the integer (PI@,) and half integer (n + 1/2)Q0bias flux frequency responses, where the amplitude of the drive signal flux @+ is 40 times greater than that of Fig. 5. The solid triangles in Fig. 6 correspond to sweeping from low to high frequency, whilst the dashed triangles correspond to sweeping from high to low frequency. It is apparent that what are essentially discontinuous amplitude jumps (cuts) have appeared in the frequency responses. These cuts reflect a gross distortion of the original resonant lineshape introduced by the strongly driven nonline~ity associated with the SQUID ring. Such cuts indicate bifurcation points, and a cyclic fold [ IO] is clearly revealed by
the bi-directional sweep. Experimentally, the degree of disto~ion in the frequency response varies continuously with the drive current amplitude (and consequently the rf flux ,Gt experienced by the ring). For small drive signal amplitudes, such as those associated with Figs. 3 and 5, we observe no obvious deviation from the Lorentzian lineshape. However, as the signal amplitude is increased to a significant fraction of a flux quantum the frequency response becomes progressively more skewed until it eventually “folds” on itself and the above bifurcations are seen. The voltage output against frequency responses presented in Figs. 5 and 6, for weak (@jr N Qi,/60 rms) and strong (@r =2@,/3 rms) driving levels, respectively, can be computationally modelled by mimick-
R. Whitman et al./Physics Letters A 226 (1997) 275-279
@xstat
=
219
(n+lQ@o
-11 20.7
Driving Frequency (MHz)
21.3
20.6
Driving Frequency (MHz)
21.2
Fig. 7. Computed theoretical SQUID ring-tank circuit driving frequency responses, at integer and half integer static flux bias, for the strongly driven case where the drive current is constant and we find the rms value of @, N 2&/3 , using the SQUID ring-tank circuit parameters of Figs. 3 and 5.
ing the function of the spectrum analyser. Solving the equation of motion (3) numerically, for a given set of parameters (and a particular drive current amplitude), we calculate the power per unit bandwidth by single point Fourier transform at a discrete set of frequencies over the spectral window of interest. The response is then built up by allowing the system to evolve continuously from one side of the spectral window to the other, with the drive frequency being incremented at the end of each transform. In Fig. 7 we show the theoretical, strongly driven frequency response curves at the integer and half integer bias points for the measured circuit parameters calculated from a single point Fourier transform over 15000 cycles of the drive frequency. The solid line here corresponds to sweeping from low to high frequency, whereas the dashed line corresponds to sweeping from high to low frequency. In this figure we see the discontinuous bifurcation points as calculated from the equation of motion, and the cyclic folds are again apparent. As can be seen, the correspondence between the computed frequency responses of Fig. 7, and the ex~rimental data of Fig. 6 is quite clear. The cyclic bifurcations are revealed, and the agreement between these figures supports the inherent physical accuracy of our model. Although such bifurcations are common to many strongly driven nonlinear oscillators [ lo,1 t 1, to our knowledge, this is the first observation of the phenomenon in SQUID systems.
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