Ramsey fringe measurement of decoherence in a novel superconducting quantum bit based on the Cooper pair box

Ramsey fringe measurement of decoherence in a novel superconducting quantum bit based on the Cooper pair box D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve and M.H. Devoret Quantronics Group Service de Physique de l Etat Condensé, CEA-Saclay F-91191 Gif-sur-Yvette cedex, France April 12, 2002 Abstract We have designed and operated a novel superconducting tunnel junction circuit which behaves as a controllable atom, with a ground and first excited state forming an effective spin 1/2. These two states are the symmetric and antisymmetric combinations of two charge configurations of a Cooper pair box. Any spin orientation corresponding to an arbritary superposition of these two states can be prepared by applying NMR-like microwave pulses to the gate of the box. The spin state is readout by measuring the voltage response to a probe current pulse applied to the bridge formed by the two junctions of the box and an additional large, shunting junction. In a sample whose transition period is 60 ps, coherent superpositions decay with a 0.5 µs time constant, as measured by a Ramsey fringes experiment. This result is a step towards the realization of a solid-state quantum information processor. Among all the practical implementations of quantum bits for quantum information processing [1], those involving solid state electrical circuits are particularly attractive since they could benefit from the parallel fabrication techniques of microelectronics. However, unlike the electric dipoles of isolated atoms or ions, the state variables of a circuit like voltages and currents usually undergo rapid quantum decoherence [2] because, in general, they are coupled to an environment with a large number of uncontrolled degrees of freedom [3]. Tunnel junctioncircuits[4,5,6,7,8]havedisplayedsofarcoherencequalityfactorsq ϕ of several hundreds [9]. Here, the coherence quality factor for a pair of quantum levels ai and bi is defined by Q ϕ = πν ab T ϕ where ν ab is the transition frequency and T ϕ is the inverse of decoherence rate of quantum superpositions of these two states. A particularly advanced circuit is the so-called single Cooper pair box [10, 11] in which coherent oscillations on the nanosecond time scale have 1

been recently observed [5]. In this article, we report an experiment showing that coherence quality factors of more than 10 4 canexperimentallybeachievedina new circuit built around the Cooper pair box, for which two orthogonal access ports are used for preparing and measuring the quantum state [12, 13]. The basic Cooper pair box consists of a low capacitance superconducting electrode, the island, connected to a superconducting reservoir by a Josephson tunnel junction with capacitance C j and Josephson energy E J. The junction is biased by a voltage source U in series with a gate capacitance C g. In addition to E J, the box has a second energy scale which is the Cooper pair Coulomb energy E CP =(2e) 2 /2(C g + C j ). The temperature T and the superconducting gap satisfy k B T / ln N and E CP, wheren is the total number of paired electrons in the island. The number of excess electrons is then even [14, 15] and the system has discrete quantum states which are in general quantum superpositions of several charge states with different number ˆN of excess Cooper pairs in the island. The Hamiltonian of the box is Ĥ = E CP ³ ˆN N g 2 EJ cos ˆθ (1) where N g = C g U/2e is the dimensionless gate charge and ˆθ the phase operator conjugate to the Cooper pair number ˆN [11]. In our experiment E J ' E CP and neither ˆN or ˆθ are good quantum numbers. We can restrict ourselves to the Hilbert space spanned by the ground 0i and first excited energy eigenstate 1i since the system is sufficiently nonharmonic. This Hilbert space corresponds to an effective spin one-half s whose Zeeman energy hν 01 goes to the minimal value E J when N g =1/2. The spin up (s z =1/2, state 0i) and down (s z = 1/2, state 1i) states are there approximately D ( N =0i± N =1i)/2. Both states have the same average charge ˆNE =1/2, and consequently the system is immune to first order fluctuations of the gate charge. With appropriate NMR-like microwave pulses on the gate u(t) =U µw (t)sin2πνt, whereν ' ν 01, any superposition Ψi = α 0i + β 1i can be prepared [16]. D For readout, instead of measuring the charge ˆNE which requires moving away from N g =1/2 [5, 17], we entangle s with a new degree of freedom more advantageous to measure. For this purpose, the single junction of the basic Cooper pair box has been split into two nominally identical junctions inserted into a superconducting loop (see Fig. 1). The new degree of freedom is the phase difference ˆδ across the two junctions, and the Josephson energy E J in Eq. (1) becomes E J cos(ˆδ/2) [18]. The information about s is transferred onto ˆδ through the supercurrent Din the loop E Î =(2e/ h) Ĥ/ ˆδ D E ' i 0 s z sin(ˆδ/2) where i 0 = ee J / h. The currents 0 Î 0 and 1 Î 1 in the two states grow with E opposite signs when δ = Dˆδ moves away from zero. Our readout strategy is to exploit the two corresponding evolutions of δ generated by its entanglement with s. It has been implemented by inserting in 2

the loop a large Josephson junction biased with a current I b [19]. This junction has a Josephson energy E J0 which is about 20 times larger than E J. Its phase difference is denoted by ˆγ. We have designed the loop dimensions to be as small as possible, thus making the loop inductance much smaller than the effective inductance of all junctions. This condition imposes the constraint ˆδ = φ+ˆγ,where φ =2eΦ/ h, Φ being the externally imposed flux through the loop. By placing a large capacitance C in parallel with the large junction so that (2e) 2 /2C E J0, the phase ˆγ, and consequently ˆδ, can be treated in first approximation as classical variables. Under these conditions, δ ' φ +arcsin(i b /I 0 ), neglecting terms of order i 0 /2I 0 ' 0.01 where I 0 =2eE J0 / h. Just as the system is immune to charge noise at N g =1/2, it is immune to flux and current noise at φ =0and I b =0, where Î =0. The preparation of the quantum state and its manipulation are therefore performed at this working point in charge-flux-current space. Readout is then achieved by displacing the system adiabatically along the I b axis to make the loop supercurrent non-zero. In practice, a trapezoidal pulse of current with amplitude slightly below I 0 is applied to the large junction (see Fig. 2). Depending on whether the effective spin s is up or down, a value of order i 0 /2 will be subtracted or added to I b in the large junction. The junction is extremely sensitive to the total current running through it when a value close to γ = π/2 is reached at the top of the pulse. A small excess current induces the switching of the junction from the zero-voltage state to the finite voltage state. With a precise adjustment of the amplitude and duration of the I b (t) pulse, the large junction switches to the voltage state with a large probability if s z = 1/2 and with a small probability if s z =+1/2 [12]. For the parameters given above, the efficiency of this projective measurement should be η = p 1 p 0 =0.95, wherep 1 and p 0 are the switching probabilities in the excited and ground states, respectively, for optimum readout conditions. The readout is also designed so as to minimize the relaxation of s using a Wheatstone-bridge-like symmetry. The large ratios E J 0 /E J and C/C j provide further protection from the environment. It is worthwhile to develop analogies between our experiment on an electrical circuit and experiments on atoms. Our circuit can be viewed as an artificial two-level atom, which we have nicknamed quantronium. The gate voltage and the magnetic flux through the loop, the two bias parameters used to tune the transition frequency, play a role similar to that of the static electric and magnetic fields for atoms. Like in many atomic physics experiments, transitions are induced by microwave pulses. Finally, the readout scheme can be compared with a Stern and Gerlach experiment: s is the analog of the spin of the Ag atom while the applied bias current is the analog of the magnetic field gradient. The loop current response is to be compared with the transverse acceleration of the Ag atom, the phase γ being the analog of the transverse position of the atom. The presence or absence of the voltage pulse corresponds to the impact of the Ag atom in the upper or lower spot of the screen. The actual sample on which measurements have been performed is shown in Fig. 1b. Tunnel junctions have been fabricated using the standard technique 3

of Al evaporation through a shadow-mask obtained by e-beam lithography [20]. With the external microwave circuit capacitor C =1pF, the plasma frequency of the large junction with I 0 =0.77 µa is ω p /2π ' 8GHz. The sample and last filtering stage were anchored to the mixing chamber of a dilution refrigerator with 15 mk base temperature. The switching of the large junction to the voltage state is detected by measuring the voltage across it with a room temperature preamplifier followed by a discriminator with a threshold voltage V th well above the noise level (Fig. 2). By repeating the experiment, we can determine the switching probability, and hence, the occupation probabilities α 2 and β 2. We have first tested the readout part of the circuit by measuring at thermal equilibrium, for a current pulse duration of τ r = 100 ns, the switching probability p as a function of the pulse height I p. Wehavefoundthatthediscrimination between the currents corresponding to the 0i and 1i stateshadanefficiency of η =0.6, lower than the expected η =0.95. Measurements of the switching probability as a function of temperature and repetition rate indicate that the discrepancy between the theoretical and experimental readout efficiency could be due to an incomplete thermalization of our last filtering stage in the bias current line. We have then performed spectroscopic measurements of ν 01 by applying to the gate a weak continuous microwave irradiation suppressed just before the readout current pulse. The variations of the switching probability as a function of the irradiation frequency display a resonance whose center frequency evolves as a function of the dc gate voltage and flux as the Hamiltonian (1) predicts, reaching ν 01 ' 16.5 GHzat the optimal working point (see Fig. 3). A small discrepancy between theory and experiment can be observed in the variations of the center frequency with flux, but we attribute it to a residual flux penetration in the small junctions not taken into account in the model. We have used these spectroscopic data to determine precisely the relevant circuit parameters and found i 0 =18.1 naand E J /E CP =1.27. At the optimal working point, the linewidth was found to be minimal with a 0.8 MHz full width at half-maximum. When varying the delay between the end of the irradiation and the measurement pulse, the peak height decays with a time constant T 1 =1.8 µs (see Fig. 4). Supposing that the energy relaxation of the system is only due to the bias circuitry, a calculation along the lines of Ref. [21] predicts that T 1 10 µs for a crude discrete element model. This result shows that no detrimental sources of dissipation have been seriously overlooked in our circuit design. We have then addressed the fidelity of controlled rotations of s around an axis x perpendicular to the quantization axis z. Prior to readout, a single pulse at the transition frequency with variable amplitude U µw and duration τ was applied. The resulting change in switching probability is an oscillatory function of the product U µw τ (see Fig. 5), in agreement with the theory of Rabi oscillations [22, 16]. It provides direct evidence that the resonance indeed corresponds to an effective spin rather than to a spurious harmonic oscillator resonance in the circuit. The proportionality ratio between the Rabi period and U µw τ was used to calibrate microwave pulses for the application of controlled rotations of s. 4

The main result of this paper, the coherence time of s during free evolution, was obtained by performing the classic Ramsey fringes experiment [23] on which atomic clocks are based. One applies on the gate two phase coherent microwave pulses corresponding each to a π/2 rotation around x [24] and separated by a delay t during which the spin precesses freely around z. For a given detuning of the pulse center frequency, we have observed decaying oscillations of the switching probability as a function of t (see Fig. 6), which correspond to the beating of the spin precession with the external microwave field. The oscillation period agrees exactly with the inverse of the detuning, allowing a measurement of the transition frequency with an accuracy of 6 10 6 [25]. The envelope of the oscillations yields the decoherence time T ϕ ' 0.5 µs. Giventhe transition period 1/ν 01 ' 60 ps, thismeansthat s can perform on average 8000 coherent free precession turns. In all our time domain experiments, the oscillation period of the switching probability closely agrees with theory, meaning a precise control of the preparation of s and of its evolution. However, the amplitude of the oscillations is smaller than expected by a factor of three to four. This loss of contrast is likely to be due to a relaxation of the level population during the measurement itself. In principle the current pulse, whose rise time is 50 ns, is sufficiently adiabatic not to induce transitions directly between the two levels. Nevertheless, it is possible that the readout or even the preparation pulses excite resonances in the bias circuitry which in turn could induce transitions in our two-level manifold. Experiments using better shaped readout pulses and a bias circuitry with better controlled high-frequency impedance are needed to clarify this point. In order to understand what limits the coherence time of the circuit, we have performed measurements of the linewidth ν 01 of the resonant peak as a function of U and Φ. The linewidth varies linearly when departing from the optimal point (N g =1/2, φ =0,I b =0), the proportionality coefficients being ν 01 / N g ' 250 MHz and ν 01 / (φ/2π) ' 430 MHz (see Fig. 7). These values can be translated into RMS deviations N g =0.004 and (φ/2π) = 0.002 of the transition frequency during the time needed to record the resonance. The residual linewidth at the optimal working point is well-explained by the second order contribution of these noises and is not therefore limited by any fundamental factor or unknown noise sources. The amplitude of the charge noise is in agreement with measurements of 1/f charge noise [26], and its effect could be minimized by increasing the E J /E C ratio. By contrast, the amplitude of the flux noise is unusually large [27], and we think that implementation of magnetic shielding and better Al layer configuration will significantly reduce it. An improvement of Q ϕ by an order of magnitude seems thus possible [28]. In conclusion, we have designed and operated a superconducting tunnel junction circuit which behaves as a tunable two-level atom that can be decoupled from its environment. When the readout is off, the coherence of this quantronium atom is of sufficient quality (Q ϕ =2.5 10 4 ) that an arbitrary quantum evolution can be programmed with a series of NMR-like microwaves pulses. Coupling several of these circuits can be achieved using on-chip capacitors. Since we could tune and address them individually, we could produce entangled states 5

and probe their quantum correlations. These fundamental physics experiments would then lead to the realization of solid state quantum logic gates, an important step towards the practical implementation of quantum processors. Acknowledgements: The indispensable technical work of Pief Orfila is gratefully acknowledged. This work has greatly benefited from direct inputs from J. M. Martinis and Y. Nakamura. The authors acknowledge discussions with P. Delsing, G. Falci, D. Haviland, H. Mooij, R. Schoelkopf, G. Schön and G. Wendin. This work is partly supported by the European Union through contract IST-10673 SQUBIT. References [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). [2] W.H.Zurek,PhysicsToday44, 36(1991);W.H.ZurekandJ.P.Paz,in Coherent atomic matter waves, edited by R. Kaiser, C. Westbrook and F. David, (Springer-Verlag Heidelberg 2000) [quant-ph/0010011]. [3] Y. Makhlin, G. Schön, A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). [4]J.M.Martinis,M.H.Devoret,andJ.Clarke,Phys.Rev.B35, 4682-4698 (1987); M. H. Devoret, D. Esteve, C. Urbina, J. M. Martinis, A. N. Cleland, and J. Clarke, in Quantum Tunneling in Condensed Media edited by Y. Kagan and A.J. Leggett (Elsevier Science Publishers, 1992). [5] Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature 398, 786-788, (1999); Phys. Rev. Lett. 87, 246601 (2001); ibid. to be published [condmat/0111402]. [6] CasparH.vanderWal,A.C.J.terHaar,F.K.Wilhelm, R.N.Schouten, C. J. P. M. Harmans, T. P. Orlando, Seth Lloyd and E. Mooij, Science 290, 773 (2000). [7] Siyuan-Han, R. Rouse, J. E. Lukens, Phys. Rev. Lett. 84, 1300 (2000). [8] Siyuan-Han, Yang-Yu, Xi-Chu, Shih-I-Chu, Zhen-Wang, Science. 293, 1457 (2001). [9] J. M. Martinis, Sae woo Nan, J. Aumentado, and C. Urbina (unpublished) have recently obtained Q ϕ s of the order of 500 for a current biased Josephson junction. [10] M. Büttiker, Phys. Rev. B 36, 3548 (1987). [11] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M. H. Devoret, Phys. Scr. T76, 165-170 (1998). 6

[12] A.Cottet,D.Vion,P.Joyez,A.Aassime,D.Esteve,andM.H.Devoret, to be published in Physica C. [13] Another two-port design has been proposed by A. B. Zorin (condmat/0112351; to be published in Physica C) [14] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and M. Tinkham, Phys. Rev. Lett. 69, 1997 (1992). [15] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M.H. Devoret, Nature 365, 422 (1993). [16] A. Abragam, The principles of nuclear magnetism (Oxford University Press, 1961). [17] A. Aassime, G. Johansson, G. Wendin, R. J. Schoelkopf, and P. Delsing, Phys. Rev. Lett. 86, 3376 (2001). [18] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1991). [19] A different Cooper pair box readout scheme using a large Josephson junctionisdiscussedbyf.w.j.hekking,o.buisson,f.balestro,andm.g. Vergniory, in Electronic Correlations: from Meso- to Nanophysics, T. Martin, G. Montambaux and J. Trân Thanh Vân, eds. (EDPSciences, 2001), p. 515. [20] G. J. Dolan and J. H. Dunsmuir, Physica (Amsterdam) 152B, 7 (1988). [21] A.Cottet,A.H.Steinbach,P.Joyez,D.Vion,H.Pothier,D.Esteve,and M. E. Huber, in Macroscopic quantum coherence and quantum computing, edited by D. V. Averin, B. Ruggiero, and P. Silvestrini, Kluwer Academic, Plenum publishers, New-York, 2001, p. 111-125. [22] I. I. Rabi, Phys. Rev. 51, 652 (1937). [23] N. F. Ramsey, Phys. Rev. 78, 695 (1950). [24] In practice, the rotation axis does not need to be x, but the rotation angle of the two pulses is always adjusted so as to bring a spin initially along z into a plane perpendicular to z. [25] At fixed t, the switching probability displays a decaying oscillation as a function of detuning, the maximum corresponding to zero detuning. [26] H. Wolf, F.-J. Ahlers, J. Niemeyer, H. Scherer, Th. Weimann, A. B. Zorin, V.A.Krupenin,S.V.Lotkhov,D.E.Presnov,IEEETrans.onInstrum. and Measurement 46, 303 (1997). 7

[27] F. C. Wellstood, C. Urbina, and J. Clarke, Appl. Phys. Lett. 50, 772-774 (1987). [28] Criticalcurrentnoise[B.Savo,F.C.Wellstood,andJ.Clarke,Appl.Phys. Lett. 50, 1758-1760 (1987)] seems to be of a lesser concern since none of our results forces us to invoke it. 8

Figure 1: Top: schematic diagram of our quantum coherent circuit with its tuning, preparation and readout blocks. The coherent circuit, nicknamed quantronium, consists of a Cooper pair box island (black node) delimited by two small Josephson junctions (cross symbols) in a superconducting loop. The loop also includes a third, much larger Josephson junction shunted by a capacitance C. The Josephson energy of the box and the large junction are E J and E J0.The Cooper pair number N and the phases δ and γ are the degrees of freedom of the circuit. A dc voltage U applied to the gate capacitance C g and a dc current I φ applied to a coil producing a flux Φ in the circuit loop tune the quantum energy levels. Microwave pulses u(t) applied to the gate prepare arbitrary quantum states of the circuit. The states are readout by applying a current pulse I b (t) to the large junction and by monitoring the voltage V (t). Filters in the microwave and dc lines have not been represented for simplicity. Bottom: scanning electron micrograph of a quantronium sample. 9

Figure 2: Signals involved in quantum state manipulations and measurement of the quantronium. Top: microwave voltage pulses are applied to the gate for state manipulation. Middle: a readout current pulse I b (t) with amplitude I p is applied to the large junction t d after the last microwave pulse. Bottom: voltage V (t) across the junction. The occurence of a pulse depends on the occupation probabilities of the energy eigenstates. A discriminator with threshold V th converts V (t) into a boolean 0/1 output for statistical analysis. 10

Figure 3: Measured center frequency (symbols) of the resonance as a function of reduced gate charge N g for reduced flux φ =0(right panel) and as a function of φ for N g =0.5 (left panel), at 15 mk. Spectroscopy is performed by measuring the switching probability p (10 5 events) when a continuous microwave irradiation of variable frequency is applied to the gate before readout (t d < 100 ns). Continuous line: theoretical best fit (see text). Inset: minimum width lineshape measured at the optimal working point φ =0and N g =0.5 (dots). Lorentzian fit withafwhm ν 01 =0.8 MHzand a center frequency ν 01 = 16463.5 MHz (curve). 11

s w it c h in g p r o b a b ilit y 0.32 0.28 0.24 WON 0.20 0.16 W O F F 0 1 2 3 4 5 6 7 timedelay( s) Figure 4: Decay of the switching probability as a function of the delay time after a continuous excitation at the center frequency of the line shown in the inset of Fig. 3 (data points labeled µw ON ). A control experiment has been performed by doing the same measurement as a function of the delay time, but without excitation (data points labeled µw OFF ). 12

Figure 5: Top: Rabi oscillations of the switching probability p (5 10 4 events) measured just after a resonant microwave pulse of duration τ. Data taken at 15mK for a nominal pulse amplitude U µw =10µV (dots). The Rabi frequency is extracted from an exponentially damped sinusoidal fit (continuous line). Bottom: the measured Rabi frequency (dots) varies linearly with U µw,asexpected. 13

Figure 6: Ramsey fringes of the switching probability p (5 10 4 events) after two phase coherent microwave pulses separated by t. Dots: data at 15mK; The total acquisition time was 5 mn. Continuous line: fit by exponentially damped sinusoid with time constant T ϕ = 500 ± 50 ns. The oscillation corresponds to the beating of the free evolution of the spin with the external microwave field. Its period indeed coincides with the inverse of the detuning frequency (here ν ν 01 =20.6 MHz). 14

r e s o n a n c e p e a k F W H M ( M H z ) 60 60 40 40 20 20 0 0.2 0.1 /2 0 0.5 0.6 N g -1 /2 Figure 7: Variations of the resonance width with deviations of bias point from the optimal bias point. The dashed lines indicate fits of the data assuming gaussian charge and flux fluctuations in bias with RMS deviations 4 10 3 Cooper pair and 2 10 3 flux quantum. 15