Quantum State Detection of a Superconducting Flux Qubit Using A

PHYSICAL REVIEW B 71, 184506 ͑2005͒

Quantum state detection of a superconducting ﬂux qubit using a dc-SQUID in the inductive mode

A. Lupașcu, C. J. P. M. Harmans, and J. E. Mooij Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands ͑Received 27 October 2004; revised manuscript received 14 February 2005; published 13 May 2005͒

We present a readout method for superconducting flux qubits. The qubit quantum flux state can be measured by determining the Josephson inductance of an inductively coupled dc superconducting quantum interference device ͑dc-SQUID͒. We determine the response function of the dc-SQUID and its back-action on the qubit during measurement. Due to driving, the qubit energy relaxation rate depends on the spectral density of the measurement circuit noise at sum and difference frequencies of the qubit Larmor frequency and SQUID driving frequency. The qubit dephasing rate is proportional to the spectral density of circuit noise at the SQUID driving frequency. These features of the back-action are qualitatively different from the case when the SQUID is used in the usual switching mode. For a particular type of readout circuit with feasible parameters we find that single shot readout of a superconducting flux qubit is possible.

DOI: 10.1103/PhysRevB.71.184506 PACS number͑s͒: 03.67.Lx, 03.65.Yz, 85.25.Cp, 85.25.Dq

I. INTRODUCTION A natural candidate for the measurement of the state of a flux qubit is a dc superconducting quantum interference de- An information processor based on a quantum mechanical ͑ ͒ system can be used to solve certain problems significantly vice dc-SQUID . A dc-SQUID is a loop containing two Jo- faster than a classical computer.1 This idea has motivated sephson junctions. Its critical current, which is the maximum intense research in recent years on the control and measure- supercurrent that it can sustain, depends on the magnetic flux 9 10,11 ment of quantum mechanical systems. The basic units in a enclosed in the loop. The state of a flux qubit was mea- 7,12 quantum computer are two level systems, also called quan- sured using an underdamped dc-SQUID. The critical cur- tum bits or qubits. Many types of qubits based on various rent of the SQUID, and thus the state of the flux qubit, is physical systems have been proposed and implemented ex- determined as the maximum value of the current, where the perimentally. SQUID switches to a finite voltage state. Due to thermal and Qubits based on solid state systems have the advantage of quantum fluctuations, switching of the SQUID is a stochastic flexibility in design parameters and scalability. An important process.13 The qubit states are distinguishable if the differ- class of solid state qubits are the superconducting qubits. ence between the two average values of the switching cur- They are mesoscopic systems formed of superconductor rent, corresponding to the qubit flux states, is larger than the structures containing Josephson junctions. The energy level statistical spread of the measured values of the switching structure in these systems is the result of the interplay be- current. The measurement of a flux qubit using a switching tween the charging energy, associated with the electrostatic dc-SQUID was characterized by an efficiency as large as energy due to distribution of the charge of a single Cooper 60%.7 Further improvement of the measurement efficiency is pair, and of the Josephson energy, associated with the tunnel- possible. Also, the back-action on the qubit coupled to the ling probability for Cooper pairs across the Josephson junc- measurement apparatus, in the situation where no measure- tions. Quantum coherent oscillations have been observed for ment is performed, can be reduced to acceptable levels.14 a few versions of qubits with Josephson junctions2–7 and Nevertheless, switching to the dissipative state has a few coupling of two qubits was demonstrated.8 drawbacks. The finite voltage across the dc-SQUID deter- A suitable qubit state detection apparatus for individual mines the generation of quasiparticles which causes decoher- qubits is an essential ingredient for the implementation of ence of the qubit.15 The long quasiparticle recombination algorithms for a quantum computer. Efficient measurement is time is a severe limit to the reset times for the qubits. In the necessary to extract all the relevant information on single finite voltage state the SQUID generates ac signals with fre- qubit states within a restricted time. Moreover, for correla- quencies in the microwaves range and broad spectral content, tion type measurements in a multiple qubit system, the un- that can induce transitions in a multiple qubit system, con- wanted back-action of the first measurement should not dis- strained to have energy level spacings in the same region. turb the system so strongly, that subsequent measurements The mentioned types of back-action will not have an effect will be meaningless. In this paper we discuss a measurement on the statistics of the measurements on a single qubit, as method for superconducting flux qubits. Flux qubits are a long as the repetition rate of the measurements is small. qubit variety formed of a superconducting loop interrupted However, in a complex multiple qubit system switching of a by Josephson junctions. The basis states have oppositely cir- dc-SQUID to the finite voltage state is a strong disturbance culating persistent currents in the loop. The control param- of the state of the total system which introduces errors in eter is an external magnetic flux in the qubit loop. The qubit subsequent computations and/or measurements. state can be determined by measuring the magnetic flux gen- A dc-SQUID can be used as a flux detector in an alterna- erated by its persistent current. tive mode of operation, in which switching to the dissipative

state is avoided. This is based on the property of a SQUID to behave as an inductor, with a Josephson inductance that depends on the magnetic flux enclosed in the loop.9 The value of the Josephson inductance can be determined by measuring the impedance of the SQUID. The flux sensitivity in this operation mode is increased if the junction is shunted by a capacitor and the circuit is excited with an ac signal at a frequency close to the resonance frequency. A SQUID in the inductive mode integrated in a resonant circuit was used for the measurement of spectroscopy of a flux qubit.16 The inductive operation mode resembles the RF-SQUID in the dispersive mode.9 The RF-SQUID contains a superconducting loop with a single Josephson junction; the impedance of a high quality tank circuit inductively coupled to the ͑ ͒ ͑ ͒ ͑ ͒ FIG. 1. a Ground E0 and excited E1 state energy levels for loop is measured near resonance, where it is very sensitive to a PCQ with two junctions equal, with EJ =258 GHz and EC ␣ the value of the magnetic flux in the loop. For charge mea- =6.9 GHz, and the third junction smaller by a factor q =0.75, re- ⌬ ͑ ͒ surement, a similar device is the RF single electron tunnel- sulting in Ip =300 nA and =5.5 GHz. b Expectation value of the ͑ ͒ ͑͗ ͘ ͒ ͑͗ ͘ ͒ ling transistor RF-SET , with the difference that a dissipa- loop current for the ground Iqb 0 and excited Iqb 1 energy tive property of a SET transistor is measured directly. The states with the parameters mentioned in ͑a͒. ͑c͒ Schematic represen- RF-SET was used as a detector for charge qubits by Duty et tation of the PCQ and of the measuring dc-SQUID, with crosses al.6 Motivated by research on superconducting qubits a few indicating Josephson junctions. The SQUID junctions have critical flux or charge detectors based on the measurement of a re- currents Ic1 and Ic2. The SQUID acts like a variable inductor, with active circuit element have been recently implemented. A an inductance LJ dependent on the state of the coupled flux qubit. flux qubit was studied by Grajcar et al.17 by measuring the The impedance Z of a resonant circuit formed of the dc-SQUID and ␯ susceptibility of the qubit loop using a coupled high quality a shunt capacitor C is plotted vs the frequency 0 for the cases ͑ ͒ tank circuit. A detector for charge qubits based on the mea- corresponding to the qubit in the ground state continuous line or ͑ ͒ surement of the inductance of a superconducting SET was in the excited state dashed line . The parameters of the circuit have 18 19 typical values, as discussed in Sec. VI ͑R is an equivalent resistance proposed by Zorin and implemented by Sillanpää et al. A ͒ sensitive measurement of the critical current of a Josephson representing the energy loss of the driven circuit . junction which exploits the nonlinearity of the current phase ment is applicable to flux qubits in general. The PCQ is 20 relation was demonstrated by Siddiqi et al. The state of a formed of a superconducting loop with three Josephson junc- 21 charge qubit was read out by Wallraff et al. by measuring tions. Two of the three junctions are of equal size, with Jo- the transmission through a coupled transmission line resona- sephson energy EJ and charging energy Ec, while the third tor. ␣ ͑ ͒ junction is smaller by a factor q. Figure 1 a shows a rep- The paper is organized as follows. In Sec. II we discuss a resentation of the energy levels vs the value of the external few general constraints on the parameter range where the ⌽ magnetic flux in the loop, qb, for a set of typical param- dc-SQUID in the inductive mode can operate. We continue eters. The qubit quantum state can be represented as a super- in Sec. III with a general analysis of the response function of position of two basis states that are persistent current states this device as a flux detector. The response function is de- in the loop, with values of the current equal to +Ip and −Ip, rived for a general type of circuit embedding the dc-SQUID. ⌽ ⌽ respectively. Away from the symmetry point qb= 0 /2 the In Sec. IV we discuss the qubit-SQUID interaction and we energy eigenstates are almost equal to the persistent current identify the relevant aspects of the measurement back-action. ⌽ ⌽ states. When qb approaches 0 /2, the energy eigenstates The energy relaxation rate and the dephasing rate of the qubit ⌽ ⌽ are superpositions of the basis states. For qb= 0 /2 the en- during the measurement are derived in Sec. V. Because of ac ergy eigenstates are the symmetric and antisymmetric com- driving of the SQUID and quadratic coupling of the qubit to binations of the basis persistent current states and are sepa- the SQUID, the qubit relaxation rate is proportional to the rated by an energy gap denoted by ⌬. A representation of the spectral density of circuit noise at frequencies which are the expectation value of the current for each energy eigenstate is sum and the difference of SQUID ac driving frequency and given in Fig. 1͑b͒. qubit Larmor frequency. Similarly, the dephasing rate is pro- The dc-SQUID is characterized by the gauge-invariant portional to the spectral density of circuit noise at the fre- phase variables across the two Josephson junctions, denoted quency of the SQUID ac driving. In Sec. VI we discuss the ␥ ␥ by 1 and 2. The two variables are connected through the results of the calculations on the measurement back-action. ␥ ␥ ␲⌽ ⌽ fluxoid quantization condition, 1 − 2 =−2 sq/ 0, where We analyze the measurement efficiency for a specific readout ⌽ ⌽ 0 is the flux quantum and sq is the total flux in the SQUID circuit and we find that single shot readout of a flux qubit is loop. The flux in the SQUID loop contains an external com- possible. ⌽ ponent x and a self-generated component, which can be neglected for the typical parameters we will discuss. With II. GENERAL CONSIDERATIONS ␥ ␥ ␲ ⌽ ⌽ this assumption 1 − 2 =−2 fx, where fx = x / 0, and the In this paper we focus on the readout of a persistent cur- SQUID can be described as a single Josephson junction with ͑ ͒ 10,11 ͑ ͒ ͉ ͑␲ ͉͒ rent qubit PCQ , though the analysis of the measure- a critical current given by Ic fx =2Ic0 cos fx for a sym-

184506-2 QUANTUM STATE DETECTION OF A… PHYSICAL REVIEW B 71, 184506 ͑2005͒

͑ ͒ metric SQUID Ic1 =Ic2 =Ic0 . For symmetric qubit-SQUID limit on the quality factor Q will be set by the fact that the ␻ coupling ͑when the two SQUID branches have mutual induc- response time of the resonator, Q/ 0, has to be smaller than tances to the qubit loop with opposite value͒ the flux gener- the intrinsic qubit relaxation time, which is in the microsec- 7,16 Ͼ ␦ ated by the SQUID in the qubit loop is given by MsIcirc, onds range. When Q 2LJ / LJ the two circuit resonance where Ms is the inductance between the qubit and the peaks, corresponding to the different qubit states, are sepa- ͑ ͒ ͑ SQUID loops and Icirc= Isq,1−Isq,2 /2 with Isq,1 and Isq,1 rated and a further increase of Q will not contribute to an being the currents in the SQUID junctions͒ is the circulating increase in the ac voltage difference. current in the SQUID loop. The circulating current of the The above considerations show that, given the typical qu- SQUID is given by bit energy level splitting and relaxation time, the constraints ␯ Շ Ͻ ͑␲ ͒ ␥ ͑ ͒ on the circuit parameters are 0 1 GHz and Q 100. Icirc = Ic0 sin fx cos e, 1 ␥ ͑␥ ␥ ͒ where e = 1 + 2 /2. The current and the voltage of the ␥ III. DETECTOR RESPONSE FUNCTION SQUID are related to the variable e through the two Joseph- son relations: In this section we analyze the dc-SQUID in the inductive ͑ ͒ ␥ ͑ ͒ I = Ic fx sin e, 2 mode as a flux detector. We consider the case of moderate ac driving, when the SQUID behaves as a linear inductor. The and function describing the conversion of flux in the SQUID loop ⌽ d␥ to ac voltage is determined for a general type of circuit in V = 0 e . ͑3͒ 2␲ dt which the SQUID is embedded. If the magnetic flux in the SQUID loop varies in time, the From Eqs. ͑2͒ and ͑3͒ it follows that in the linear approxi- relation between the transport current and the voltage across ͑␥ ͒Ӎ␥ mation (sin e e) the SQUID behaves as a linear inductor the terminals of the SQUID is given by with the Josephson inductance, 1 t ⌽ I͑t͒ = ͵ V͑tЈ͒dtЈ, ͑6͒ 0 ͑ ͒ L = . ͑4͒ LJ t J ␲ 2 Ic where L ͑t͒ is the time-dependent Josephson inductance. Let ␯ J If an ac current is injected in the SQUID at frequency 0 us consider with a small amplitude Iac, the voltage across the SQUID has the amplitude V=2␲␯ L I =⌽ ␯ I /I . The maximum 1 1 0 J ac 0 0 ac c = 1+␣͑t͒ , ͑7͒ ͑ϳ ␮ ␯ ͑ ͒ „ … voltage across the SQUID is very small 2 V for 0 LJ t LJ0 =1 GHz͒; very low noise amplification is necessary to detect ␯ where ␣͑t͒ parametrizes the variations of the magnetic flux such a voltage in a short time. Increasing the value of 0 will result in a proportional increase in the value of the maximum in the SQUID. The time-dependent Josephson inductance ͑ ͒ ac voltage. However, from Eqs. ͑1͒ and ͑2͒ it follows that LJ t can be represented as the parallel combination of the ␯ inductances L and L /␣͑t͒͑see Fig. 2͑a͒͒. In the case of when the SQUID current varies at frequency 0, the circulat- J0 J0 ing current contains a significant frequency component at the qubit measurement, ␣͑t͒ describes the dynamics of the ␯ ␣͑ ͒ 2 0 and additional higher harmonics for strong driving in the qubit generated flux. The extreme values of t in this case nonlinear regime. The flux generated by this circulating cur- correspond to the qubit in a clockwise or anticlockwise per- rent in the qubit loop can cause transitions between the qubit sistent current state and are given approximately by ␲ ͑␲ ͒ ⌽ ͑ ͑ ͒͒ ͉␣͑ ͉͒Ӷ energy levels if the harmonics of the driving current are close ± tan fx MIp / 0 see Eq. 5 . We assume t 1, con- to the qubit energy levels splitting. With typical level split- sistent with the usual value of the flux generated by the ␯ Շ ting of 1–20 GHz, the value of 0 is limited to 1 GHz. coupled qubit in the SQUID loop which is of the order of 1% ⌽ 7,12,16 ͑ ͒ ͑ ͒ The relative change in Josephson inductance when the of 0. From Eqs. 6 and 7 , it follows that the current qubit evolves from the ground to the excited state is given by in the SQUID can be written as ␦ ␦ LJ Ic MsIp I = I + I , ͑8͒ ͯ ͯ Ӎ ͯ ͯ Ӎ 2␲͉tan͑␲f ͉͒ , ͑5͒ 0 1 L I x ⌽ J c 0 with

where it was assumed that the measurement is performed at a t ⌽ 1 bias ﬂux in the qubit away from 0 /2, so that the expecta- ͑ ͒ ͵ ͑ Ј͒ Ј ͑ ͒ I0 t = V t dt 9 tion value of the qubit current in each energy eigenstate ap- LJ0 ͑ ͑ ͒͒ proaches in absolute value Ip see Fig. 1 b . The typical ␦ and values for Ms and Ip limit the value of LJ /LJ to a few percent. If the SQUID is driven with a constant ac current, ␣͑t͒ t I ͑t͒ = ͵ V͑tЈ͒dtЈ. ͑10͒ the maximum difference in ac voltage corresponding to a 1 L qubit state change, from the ground to the excited state, is J0 ϳ⌽ ␯ ␦ 0 0 LJ /LJ. This can be increased if the SQUID is placed As discussed in Sec. II, the measurement of the Josephson ␯ in a resonant circuit and the driving frequency 0 is taken inductance is more efﬁcient if the dc-SQUID is integrated in close to the circuit resonance frequency ͑see Fig. 1͑c͒͒.A a resonant circuit. In this case the circuit is driven with an ac

184506-3 A. LUPAȘCU, C. J. P. M. HARMANS, AND J. E. MOOIJ PHYSICAL REVIEW B 71, 184506 ͑2005͒

͑ ͒ ␣͑ ͒ ͑ ͒ FIG. 2. a The time-dependent Josephson inductance can be represented as a parallel combination of the inductors LJ0 and LJ0 / t . b A schematic representation of the circuit in which the dc-SQUID is inserted ͑see the text for explanations͒.

current source at a frequency close to the resonance fre- * Vm,n ͑␻ ␻͒ Vm,n ͑␻ ␻͒ ͑ ͒ ͩ −i 0+n t i 0+n tͪ ͑ ͒ quency and the ac voltage is suitably amplified. The output Vm t = ͚ e + e . 13 2 2 ac voltage depends on ␣͑t͒. Since there is a certain freedom n in the design of the resonant circuit, we calculate here the ␣͑ ͒ A similar expression for Im can be written if V is replaced by dependence of the output voltage on t for a general type ͑ ͒ ͑ I in expression 13 . of circuit in which the dc-SQUID is embedded see Fig. ͑ ͒ ␣ ͑ ͒͒ From Eq. 12 written for =m, one obtains the Fourier 2 b . We consider a linear network with three ports that ͑ ͒ ͑ components of Vm t as a function of the Fourier coefficients contains LJ0 the constant component of the SQUID induc- ͑ ͒͑ ͑ ͒ ͒ of Im t note that Ii t is imposed and has frequency compo- tance , the impedance of the driving current source, the am- ␻ ͒ ͑ ͒ nents at ± 0 . These values can be replaced in 11 , and the plifier input impedance, complemented by other linear circuit ␻ ␻ elements. The port P has its terminals across the current terms corresponding to the frequencies 0 +n are separated. i Using the equations corresponding to n=0,1,Ϫ1 in expan- source. The port Pm is connected across the Josephson induc- ͑ ͒ sions of the form 13 and neglecting the terms Im,2 and Im,−2, tance LJ0. Finally, the port Po has its connections at the input the values for Im,0, Im,1, and Im,−1 can be obtained in lowest of the amplifier. Three elements are connected to the ports ͉␣ ͉ ͑ ͒ ␤ ͑ ͒ order in 0 . Using these values in Eq. 12 for =o leads to Pi, Po, and Pm, respectively: the ideal current source Ii t ,an ideal voltage amplifier, and the inductance L /␣͑t͒. A spe- the following expression for the components of the output J0 voltage at frequencies ␻ +␻ and ␻ −␻: cific electrical circuit described in the way indicated here is 0 0 shown in Fig. 3͑a͒. ␣ I The relation between the current and the voltage at port V = 0 Z ͑␻ ͒ e Z ͑␻ + ␻͒͑14͒ ␣͑ ͒ o,1 ␻ mi 0 om 0 Pm is determined by the inductance LJ0 / t : i 0LJ0 2 and ␣͑t͒ t − I ͑t͒ = ͵ V ͑tЈ͒dtЈ, ͑11͒ m m * LJ0 ␣ I V = 0 Z ͑␻ ͒ e Z ͑␻ − ␻͒. ͑15͒ o,−1 ␻ mi 0 om 0 i 0LJ0 2 which is Eq. ͑10͒ with changed sign in order to preserve the The expressions ͑14͒ and ͑15͒ for the up- and down- sign convention for the three-ports network.22 The voltage at converted voltage at the output of the circuit are proportional the port P␣, ␣=o,m can be written as to the amplitude of the driving current Ie and to the ampli- ͉␣ ͉ tude of the flux modulation 0 . These expressions are us- ϱ ϱ able only when the driving conditions ͑I and ␻ ͒ are such ͑ ͒ ͵ ͑ Ј͒ ͑ Ј͒ Ј ͵ ͑ Ј͒ ͑ Ј͒ Ј e 0 V␣ t = Z␣i t Ii t − t dt + Z␣m t Im t − t dt . that the maximum amplitude of the current in the SQUID is 0 0 not close to the SQUID critical current. Besides the up- and ͑ ͒ 12 down-converted components Vo,1 and Vo,−1, the output volt- ␻ age contains a strong component Vo,0 at frequency o. Vo ͉␣ ͉ Here Z␣␤͑t͒ is the impedance matrix for the three-port net- depends only quadratically on 0 and thus it cannot be used work, with ␣,␤=i,m,o. We assume that the ac driving current for an efficient detection of the flux. ͑ ͒ ͑␻ ͒ is Ii t =Ie cos 0t and the flux variations in the dc-SQUID ͑ ͑ ͒͒ ␣͑ ͒ ␣ ͑ ␻ ͒ loop see Eq. 7 are described by t =Re( 0 exp −i t ). IV. QUBIT-SQUID INTERACTION ͑ ͒ Equation 11 implies that Vm and Im have components at ␻ ␻ frequencies 0 +n , with n being an integer. The voltage Vm In a basis formed of two persistent current states, the can be written as Hamiltonian of the flux qubit can be written as11

184506-4 QUANTUM STATE DETECTION OF A… PHYSICAL REVIEW B 71, 184506 ͑2005͒

⑀ ⌬ intrinsic evolution, in the absence of circuit driving. The first Hˆ = ␴ˆ + ␴ˆ , ͑16͒ ͑ ͒ qb 2 z 2 x term on the right-hand side of Eq. 20 corresponds to the evolution due to driving, and it is the same as the first term ␴ where ˆ i, i=x,y,z, have the Pauli matrices representation. on the right-hand side of Eq. ͑18͒. ͑ ͒ ⑀ ͑⌽ The coefficient of the first term in 16 is =2Ip qb The essential feature of the interaction Hamiltonian given ⌽ ͒ ⌽ ͑ ͒ − 0 /2 , where qb is the flux in the qubit loop. The maxi- by 17 is that the coupling to the external phase operator mum persistent current Ip and the minimum energy level does not have a linear part. Recent work on the influence of splitting ⌬ ͑see Figs. 1͑a͒ and 1͑b͒͒ are parameters fixed by nonlinear coupling of the noise on the evolution of a two the qubit junctions design. The average flux induced in the level system has been done,23 motivated by results reported ͗␴ ͘ 3 SQUID loop by the qubit is MIp ˆ z . The flux-dependent by Vion et al., where long coherence times were obtained term in the energy of a SQUID is the Josephson energy given for the operation of a qubit at settings where the energy level ⌽ ͑ ␲͒ ͑␥ ͒ ͑␲⌽ ⌽ ͒ ⌽ by −2 0 / 2 Ic0 cos e cos sq/ 0 . The total flux sq in separation was insensitive in the first order to external noise. ⌽ In the second order approximation, the back-action flux noise the SQUID loop contains the external flux x and qubit- ͗␴ ͘ is described by induced flux MIp ˆ z . It follows that the interaction Hamil- tonian can be written as ␥ˆ 2 ⌽ˆ ͩ e ͪ ͑ ͒ ˆ ͑␥ ͒ ͑␲ ͒␴ ͑ ͒ b = MIf 1− , 21 Hc = MIpIc0 cos ˆ e sin fx ˆ z, 17 2 where we assumed that the flux generated by the qubit is which can be separated in three parts as small and thus a linear approximation could be used. A rig- orous derivation of the interaction term in the Hamiltonian ␥2 ͑t͒ ␥ˆ 2 ͑t͒ for a coupled dc-SQUID and a three Josephson junctions ⌽ˆ ͫͩ e,coh ͪ en ␥ ͑ ͒␥ ͑ ͒ͬ b1 = MIf 1− − − e,coh t ˆ en t . qubit, assuming a SQUID with a small self inductance and 2 2 using the two level approximation for the three Josephson ͑22͒ junctions loop leads to the same result as ͑17͒. ͑ ͒ ͑ ͒ ⑀ ͑⌽ ⌽ ͒ If 17 is compared to 16 with /2=Ip qb− 0 /2 ,it The first term on the right-hand side of ͑22͒ can cause tran- becomes clear that the back-action due to the measurement is sitions between the qubit energy levels. As we discussed in ⌽ˆ described by an equivalent flux operator, expressed as b Sec. II, resonant transitions occur when the qubit energy lev- ˆ ˆ els splitting is close to the harmonics of the driving ac fre- =MIcirc. Icirc is the operator corresponding to the circulating ˆ ͑␥ˆ ͒ quency, and in particular to the second harmonic. The time current in the dc-SQUID and is given by Icirc=If cos e , ͑ ͒ ͑␲ ͒͑ ͑ ͒͒ average of the first term in 22 , dependent on the amplitude where If =Ic0 sin fx see Eq. 1 . Classically, qubit deco- of the ac driving current, will be considered a part of the herence can be understood as a result of the fluctuations in qubit flux bias ⌽ . The effects of the second term in ͑22͒ ␥ qb the flux bias, due to the SQUID. If e is treated as a classical were analyzed by Makhlin et al.23 for Ohmic and 1/␻ type variable, its time evolution is given by spectral densities. In this paper we focus on the calculation ␥ ͑ ͒ ␥ ͑ ͒ ␥ ͑ ͒ ͑ ͒ of decoherence determined by the third term in ͑22͒.Inthe e t = e,coh t + e,n t , 18 second order perturbation theory, used for the calculation of ␥ ͑ ͒ ␥ ͑ ␻ ͒ ␥ where e,coh t =Re( e0 exp −i 0t ) with e0 the decoherence rates in the next section, the contributions ␲ ͑␻ ͒ ͑ ⌽ ␻ ͒ =2 IeZim 0 / i 0 0 the response to circuit driving and from the different terms in ͑22͒ can be treated independently. ␥ ͑ ͒ e,n t is a random term, corresponding to, e.g., thermal fluc- ⌽ ͑ ͒ tuations. The “classical” flux is given by b t ␥ ͑ ͒ V. CALCULATION OF THE DECOHERENCE RATES =MIf cos( e t ), which can be approximated by ␥2͑t͒ In this section we calculate the relaxation and dephasing ⌽ ͑ ͒ ͩ e ͪ ͑ ͒ b t = MIf 1− . 19 rates of a flux qubit during the measurement by a dc-SQUID 2 in the inductive mode. It is assumed that the external qubit ⌽ ͑ ͒ flux ⌽ is fixed. However, the calculations can be extended The statistical properties of b t are thus determined by qb ␥2͑t͒. From ͑18͒ it follows that for the case when the phase to include the case of the measurement performed during e 24,25 oscillations amplitude is large compared to the typical values induced Rabi oscillations or other control sequences. The ␥ ͑ ͒ model Hamiltonian used for the combined system qubit- of en t , the most important contribution will be the mixing ␥ ͑ ͒␥ ͑ ͒ SQUID is term e,coh t e,n t . This results in frequency conversion of the circuit noise. ˆ ͑ ͒ ˆ ˆ ˆ ͑ ͒ ͑ ͒ The analog of Eq. ͑18͒ for the quantized system is H t = Hqb + Hc + HSQUID t , 23 ␥I ͑ ͒ ␥ ͑ ͒ ␥ ͑ ͒ ͑ ͒ ˆ e t = e,coh t + ˆ en t , 20 ˆ ͑ ͒ ˆ where Hqb is the qubit Hamiltonian given by 16 , Hc is the ␥I ͑ ͒ ͑ ͒ ˆ ͑ ͒ in which ˆ e t is the phase operator in the interaction repre- interaction term given by 17 and HSQUID t is the SQUID sentation with respect to the qubit-SQUID interaction, which Hamiltonian, which is time-dependent due to driving. If a is thus equivalent to the Heisenberg representation for the transformation is made to the qubit energy eigenstates, the ␥ ͑ ͒ ͑ ͒ SQUID system. ˆ e,n t is the phase operator representing the first two terms in 23 become

184506-5 A. LUPAȘCU, C. J. P. M. HARMANS, AND J. E. MOOIJ PHYSICAL REVIEW B 71, 184506 ͑2005͒

ͱ⑀2 + ⌬2 2 ˆ ␶ ͑ ͒ ␶ˆH͑t͒ = ͚ A ͑t͒ͩ␶ˆ + ͵͵ dt dt Oˆ +͑t ,t ͒␶ˆ Hqb = ˆz 24 i ij j 2 1 2 lj 1 2 l ប Ͼ Ͼ „ 2 j,l,m=x,y,z t t1,t2 0 and − Oˆ +͑t ,t ͒␶ˆ − ⑀ Oˆ − ͑t ,t ͒ ͪ. ͑32͒ ll 1 2 j jlm lm 1 2 … ˆ ͑␥ˆ ͒ ͑␪͒␶ˆ ͑␪͒␶ˆ ͑ ͒ Hc = MIpIf cos e „cos z − sin x…, 25 ␶ In the last expression, where the î, i=x,y,z are Pauli matrices in the energy eigen- ͑␪͒ ⌬ ⑀ state basis and tan = / . In the interaction representation ˆ ±͑ ͒ ͑ ͒ ˆ ± ͑ ͒ ͑ ͒ ͑ ͒ ␶I Oij t1,t2 = ͚ Aik − t1 Ckl t1,t2 Alj t2 , 33 with respect to the qubit-SQUID interaction, the operators i k,l=x,y,z evolve in time according to with ␶I͑ ͒ ͑ ͒␶ ͑ ͒ î t = Aij t ˆ j 26 1 with the matrix A given by Cˆ + ͑t ,t ͒ = ˆfI ͑t ͒ˆfI͑t ͒ + ˆfI͑t ͒ˆfI ͑t ͒ ͑34͒ kl 1 2 2„ k 1 l 2 l 2 k 1 … ͑␻ ͒ ͑␻ ͒ 0 cos 01t − sin 01t and ͑␻ ͒ ͑␻ ͒ A͑t͒ = sin 01t cos 01t 0 , ͑27͒ ΂ ΃ − i I I I I 001 Cˆ ͑t ,t ͒ = ˆf ͑t ͒ˆf ͑t ͒ − ˆf ͑t ͒ˆf ͑t ͒ . ͑35͒ kl 1 2 2„ k 1 l 2 l 2 k 1 … ␻ ͱ⑀2 ⌬2 ប in which 01= + / is the frequency corresponding to The last two expressions are symmetrized and antisymme- the qubit energy level separation ͱ⑀2 +⌬2. The evolution of trized products of operators at different times. Their expec- ␶H the operators î in the Heisenberg picture is obtained using tation values calculated for a thermal equilibrium state are time-dependent second-order perturbation theory. As ex- connected with the linear response functions by the plained in Sec. IV, the relevant part of the interaction Hamil- fluctuation-dissipation theorem.26 Note that, to obtain ͑32͒, ͑ ͒ Ͻ tonian ͑Eq. ͑25͒͒ for the calculation of decoherence is given the integral in 29 was extended to the region t1 t2 because in the interaction picture by the integrand is symmetric under the interchange of t1 and t2. We assume that initially the qubit and SQUID states were ˆ I ͑ ͒ ␥ ͑ ͒␥ˆ ͑ ͒ ͑␪͒␶Î ͑␪͒␶Î Hc t =−MIpIf e,coh t e,n t „cos z − sin x…. separable and the SQUID is described by the thermal equi- ͑28͒ librium density matrix. For the case of the coupling Hamil- tonian given in Eq. ͑28͒, the relevant correlation functions ␶H The evolution of the operators î , which allows us to de- are scribe the qubit operators expectation values if the initial 1 state is known, is given by + ͑ ͒ ͳ ͕␥I ͑ ͒ ␥I ͑ ͖͒ ʹ ͑ ͒ C␥ t1,t2 = ˆ t1 , ˆ t2 + 36 e 2 e e 1 0 ␶H͑ ͒ ␶I͑ ͒ ͵͵ ͓␶I͑ ͒ ˆ I ͑ ͔͒ ˆ I ͑ ͒ î t = î t − 2 dt1 dt2 î t ,Hc t1 ,Hc t2 . and ប Ͼ Ͼ Ͼ † ‡ t t1 t2 0 i ͑29͒ − ͑ ͒ ͳ ͕␥I ͑ ͒ ␥I ͑ ͖͒ ʹ ͑ ͒ C␥ t1,t2 = ˆ t1 , ˆ t2 − , 37 e 2 e e In the following, the second term on the right-hand side of 0 Eq. ͑29͒ is calculated. For the initial calculation we assume where ϩ/Ϫ denote the anticommutator/commutator, and the the most general interaction Hamiltonian with linear cou- expectation value is taken for the SQUID thermal equilib- pling to the bath, which can be written as rium density matrix. From ͑32͒–͑37͒ we see that the time ␶ evolution of the operators î, i=x,y,z, depends on their ex- ˆ I ͑ ͒ Î͑ ͒␶I͑ ͒ ͑ ͒ Hc t = ͚ fi t î t , 30 pectation values for the initial qubit state and on a two- i=x,y,z dimensional integral involving the expectation values of operators of type ͑36͒ and ͑37͒. where ˆf are bath operators ͑note that the interaction repre- i The interaction between the qubit and the measurement sentation is used in ͑30͒͒. In the end the form of the operators dc-SQUID has the consequence that the qubit quantum state Î͑ ͒ ͑ ͒ fi t corresponding to our case, as given by Eq. 28 , will be becomes a mixed state. In general one distinguishes between considered: energy relaxation, corresponding to a change in the qubit energy expectation value, and dephasing, corresponding to Î ͑ ͒ ͑␪͒␥ ͑ ͒␥ˆ ͑ ͒ fx t = MIpIf sin e,coh t en t , randomization of the phase of a coherent superposition of energy eigenstates.24 To calculate the energy relaxation, we Î ͑ ͒ fy t =0, determine the transition rates between energy eigenstates by ͗␶H͑ ͒͘ determining the evolution of ˆz t . If in this calculation the ˆfI͑t͒ =−MI I cos͑␪͒␥ ͑t͒␥ˆ ͑t͒. ͑31͒ initial qubit state is chosen to be the ground or the excited z p f e,coh en state, these rates will represent the absorption and emission ␶ If the commutation and anticommutation relations for the î rates, respectively. To calculate the dephasing rate, we operators are used, Eq. ͑29͒ results in determine the decay of the expectation values

184506-6 QUANTUM STATE DETECTION OF A… PHYSICAL REVIEW B 71, 184506 ͑2005͒

͗1/2(␶ˆH͑t͒±i␶ˆH͑t͒)͘, with the qubit initial state being a ␶ˆ 1 x y x ⌫ 2͑␪͒ 2͑␥ ͓͒ + ͑␻ ␻ ͒ − ͑␻ ␻ ͔͒ ␾ = sin k e0 S␥ 01 + 0 + S␥ 01 − 0 eigenstate. 4ប2 e e The calculation of the integral on the right-hand side of ͑ ͒ ͑␻ ͒ 1 32 involves a product of the functions cos 01t1,2 and 2͑␪͒ 2͑␥ ͒ + ͑␻ ͒ ͑ ͒ + cos k e0 S␥ 0 . 43 ͑␻ ͒ ប2 e sin 01t1,2 , resulting from the expression for the free evolu- ͑ ͑ ͒͒ ͑␻ ͒ tion matrix A see Eq. 27 and of the functions cos 0t1,2 ͑␻ ͒ and sin 0t1,2 , resulting from the time dependence of the Î͑ ͒͑ ͑ ͒͒ coupling operators fi t see 31 . The correlation functions VI. DISCUSSION appearing in ͑36͒ and ͑37͒ only depend on the time differ- In this section we discuss the results of the calculations of ence t −t . For times that are long compared to 2␲/␻ and 1 2 0 the parameters characterizing qubit decoherence and we ana- 2␲/␻ the relaxation ͑decay of ͗␶ˆH͑t͒͒͘ and the dephasing 01 z lyze a practical circuit which can be used for single-shot ͑ ͗ ␶ˆH͑ ͒ ␶ˆH͑ ͒ ͒͘ decay of 1/2( x t ±i y t ) can be described as an inte- readout of a flux qubit. gral of the product of the Fourier transform of one of the ± We start with a discussion on the emission and absorption spectral functions C␥ ͑t ,t ͒ and a weight function that has a 27 e 1 2 rates, along the lines of similar analysis done for charge 24 width depending on the integration time t. This weight func- and charge-phase qubits. Both ⌫↓ and ⌫↑ are proportional 2͑␪͒ ␶ ͑ ͑ ͒͒ tion is given by to sin , due to the fact that the operator ˆx see 25 causes transitions between the energy eigenstates. The differ- 1 ␻͑ ͒ ence between the two rates given by ͑39͒ and ͑40͒ is due to W͑␻,t͒ = ͵͵ dt dt e−i t1−t2 , ͑38͒ ␲ 1 2 ͑ ͒ 2 t 0Ͻt ,t Ͻt the last term in the integrand in expression 32 , connected 1 2 with the fact that a commutator is nonvanishing, so it can be ͑␻ ͒ ␦͑␻͒ attributed to quantum noise. Let p and p be the probabili- and has the property limt→ϱ W ,t = . The expressions g e below for the relaxation and dephasing rates are given as- ties for the qubit to be, respectively, in the ground and in the suming that the spectral density of the circuit noise, given by excited state. The time evolution of these probabilities is the Fourier transform of the correlation functions, ͑36͒ and determined by the rate equations ͑37͒, does not have significant variations over the frequency ⌫ ⌫ dpg ↑ ↓ range, where the weight function has substantial values. In =− pg + pe, this case, the relaxation and the dephasing of the qubit state dt 2 2 are proportional to the time from the beginning of the mea- ͑44͒ dp ⌫↓ ⌫↑ surement. e = p − p , The transitions rates between the two energy eigenstates dt 2 g 2 e depend on the initial state. The transition rate from the ex- and the normalization condition p +p =1. Since ⌫↓ and ⌫↑ cited state to the ground state ⌫↓ ͑emission͒ and the transition g e in ͑39͒ and ͑40͒ describe the decay of ͗␶ˆ ͘, they appear di- rate form the ground state to the excited state ⌫↑ ͑absorption͒ z ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ are given by vided by 2 in 44 . The polarization P t =pg t −pe t tends to the equilibrium value 1 ⌫ 2͑␪͒ 2͑␥ ͓͒ ͑␻ ␻ ͒ ͑␻ ␻ ͔͒ ⌫↓ − ⌫↑ ↓ = sin k e0 S␥ 01 + 0 + S␥ 01 − 0 ͑ ͒ 2ប2 e e Ps = , 45 ⌫↓ + ⌫↑ ͑39͒ with a relaxation rate and ⌫↓ + ⌫↑ ⌫ = . ͑46͒ r 2 1 2 2 ⌫↑ = sin ͑␪͒k ͑␥ ͓͒S␥ ͑− ␻ + ␻ ͒ + S␥ ͑− ␻ − ␻ ͔͒ 2ប2 e0 e 01 0 e 01 0 The spectral densities of the symmetrized and antisymme- ͑ ͒ ͑ ͒ ͑ ͒ trized correlation functions 36 and 37 depend on the im- 40 ͑ ͒ pedance at the port Pm see Sec. III as in which relations 8␲ ប␻ Re Z ͑␻͒ + ͑␻͒ ͩ ͪ „ mm … ͑ ͒ S␥ = coth 47 e ␻ ͑␥ ͒ ͉␥ ͉͑͒ 2kBT RK k e0 = MIpIf e0 41 and are a measurement coupling factor and ␲ ͑␻͒ − 8 Re Zmm − iS␥ ͑␻͒ = „ … , ͑48͒ + − e ␻ S␥ ͑␻͒ = S␥ ͑␻͒ − iS␥ ͑␻͒. ͑42͒ RK e e e where R =h/e2 ͑see Devoret28͒. Given the relations ͑46͒, ± K S␥ are the Fourier transforms of the correlation functions ͑ ͒ ͑ ͒ ͑ ͒ + ͑␻͒ − ͑␻͒ e 39 , 40 , and 42 and the properties of S␥ and S␥ to ± e e C␥ ͑t,0͒ given by ͑36͒ and ͑37͒. The corresponding expres- e be, respectively, even and odd functions, the relaxation rate sion for the dephasing rate ⌫␾ is can be written as

184506-7 A. LUPAȘCU, C. J. P. M. HARMANS, AND J. E. MOOIJ PHYSICAL REVIEW B 71, 184506 ͑2005͒

1 a resistor R , corresponding to Ohmic dissipation, when the ⌫ 2͑␪͒ 2͑␥ ͓͒ + ͑␻ ␻ ͒ + ͑␻ ␻ ͔͒ sh r = sin k e0 S␥ 01 + 0 + S␥ 01 − 0 . ប2 e e result of Shnirman et al. can be used. The following relation 2 ␻ Ӷ is valid for 0LJ0 Rsh: ͑49͒ * 3 2 ⌫␾ ប␻ ប␻ ␥ R R = ͩ 0 ͪ cothͩ 0 ͪ e0 sh K . ͑53͒ For a flux qubit coupled to a dc-SQUID biased with a con- ͑␻ ͒2 ˜⌫* kBT kBT 2 0LJ0 stant current, van der Wal et al.14 found that qubit relaxation ␾ + is proportional to S␥ ͑␻ ͒. Our results show that because of For the case ␻ L ӶR and ប␻ ϳk T, the dephasing rate e 01 0 J0 K 0 B ␻ ˜ * driving the SQUID with an ac bias current at frequency 0 ⌫␾ is dominant even at small SQUID driving amplitudes. the qubit relaxation rate is proportional to the sum of The reliable measurement of the qubit state requires that + ͑␻ ␻ ͒ + ͑␻ ␻ ͒ S␥ 01+ 0 and S␥ 01− 0 , and multiplied by the cou- the ac voltage at the output of the circuit is averaged for a e ͑ e ͒ ␻ Ӷ␻ pling factor of Eq. 41 . In practice 0 01, which implies long enough time, such that the noise due to the amplifier is that for a spectral density of the noise which is reasonably less that the difference between the voltage values corre- + + flat at large frequencies we can take S␥ ͑␻ +␻ ͒ϳS␥ ͑␻ sponding to the two qubit flux states. We define the discrimi- e 01 0 e 01 + −␻ ͒ϳS␥ ͑␻ ͒, and our results are not significantly differ- nation time as the time necessary to have a measurement 0 e 01 ent from the case of a dc current biased SQUID.14 signal to noise ratio equal to 1. It is thus given by From ͑42͒, ͑47͒, and ͑48͒ it follows that S ͑␻ ͒ T = V 0 , ͑54͒ discr ͑⌬ ͒2 Vqb S␥ ͑− ␻͒ e −␤ប␻ ͑ ͒ ͑␻ ͒ = e , 50 where SV 0 is the spectral density of the voltage noise and S␥ ͑␻͒ e ⌬ Vqb is the difference in the output voltage values corre- ⌬ ␤ ͑ ͒ sponding to the two qubit states. The value of Vqb is pro- where =1/ kBT . For a qubit coupled to a dc-SQUID bi- ␥ portional to e0. The discrimination time Tdiscr, the relaxation ased with a constant current, which is the case analyzed by ⌫ ⌫ 14 time Tr =1/ r, and the dephasing time T␾ =1/ ␾ are in- van der Wal et al., ⌫↓ and ⌫↑ are proportional to the spec- ␥2 ͑ ͑ ͒ ͑ ͒ ͑ ͒ ␻ ␻ versely proportional to e0 see 46 with 39 , 40 , and tral density of the noise at the frequencies 01 and − 01, ͑ ͒͒ ͑ ͒ ⌫ ⌫↓ 43 . Increasing the amplitude of the ac driving leads to a respectively. For that case Eq. 50 implies that ↑ / decrease in the discrimination time. However, this is accom- −␤ប␻01 =e . This is the detailed balance condition and implies panied by a proportional decrease of the qubit decoherence that in a stationary situation the qubit is in thermal equilib- ͑ times Tr and T␾. This illustrates the tradeoff between obtain- rium with the environment at temperature T see also the ing information about a quantum system and the state distur- 27͒ ͑ ͒ results of Schoelkopf et al. . In contrast, the relations 39 bance. The measurement is efficient if the ratio T /T is and ͑40͒ show that for the case analyzed here the detailed r discr large. The ratio Tr /Tdiscr does not depend on the amplitude of balance condition is in general not satisfied. This is a non- the ac driving. However, a fast measurement is necessary if equilibrium situation generated by the presence of the ac we take into account the fact that, besides the measurement driving of the SQUID. ͑ ͒ ͑ ͒ back-action, there are also other sources of decoherence that The dephasing rate in 43 can be written, if 46 is used, will increase the total relaxation rate. as We now analyze the measurement of a flux qubit using ⌫ ⌫ our particular SQUID embedding network presented in Fig. ↓ + ↑ * ͑ ͒ ⌫␾ = + ⌫␾, ͑51͒ 3 a . The driving source is represented as an ideal current 4 source with impedance Zs. The amplifier is described as the * combination of the input impedance Za and ideal voltage where the pure dephasing rate ⌫␾ is given by amplifier with gain G. The bias resistor Rb has the purpose of increasing the impedance of the current source. The inductor 1 ⌫* 2͑␪͒ 2͑␥ ͒ + ͑␻ ͒ ͑ ͒ ␾ = cos k e0 S␥ 0 . 52 Ls is a small stray contribution, unavoidable in the design of ប2 e the circuit. The combination of the capacitors C1 and C2 is an impedance transformer that will increase the effective im- 2͑␪͒ ␶ The factor cos is due to the coefficient of the operator ˆz pedance of the amplifier input, at the cost of a division of the ͑ ͒ in 25 . Dephasing is a result of the random modulation of total voltage across the inductors; they also provide the ca- energy level separation due to noise in the SQUID circulat- pacitive part necessary to create a resonant circuit. The dc- ing current. The fact that the SQUID is driven with an ac SQUID has Josephson junctions with a critical current Ic0 current has the consequence that the pure dephasing rate de- =200 nA. The external magnetic flux in the SQUID loop ␻ pends on noise at 0, which is qualitatively different of the corresponds to f =3.35, resulting in a critical current I 14 x c result obtained by van der Wal et al. For the radio- =187 nA. The measured persistent current qubit has I 18 p frequency Bloch-transistor electrometer a similar contribu- =300 nA and ⌬=5.5 GHz. Figures 1͑a͒ and 1͑b͒ and show 18,29 tion of the converted noise to back-action was found. plots of the energy eigenvalues and persistent current expec- ⌫* ͑ ͒ We compare the pure dephasing rate ␾ given by 52 tation value vs bias flux for these qubit parameters. If the ͑ ͒ with a similar contribution due to the second term in Eq. 21 mutual inductance between the qubit and the dc-SQUID is 30 ˜ * as calculated by Shnirman et al., that we denote by ⌫␾.We M =40 pH, the relative change in Josephson inductance is consider the simple case where the dc-SQUID is shunted by given by ␣=3.4%. A plot of the expression S␥ ͑␻͒ given by e

184506-8 QUANTUM STATE DETECTION OF A… PHYSICAL REVIEW B 71, 184506 ͑2005͒

␻ ⌽ 0. The relaxation time away from the symmetry point q ⌽ = 0 /2 increases as a result of both the decrease in the trans- verse coupling term sin2͑␪͒ and the decrease in the real part ͑ of the impedance Zmm away from the resonance peak see Eqs. ͑49͒ and ͑47͒͒. Over a wide range of parameters the relaxation time is considerably higher than the discrimination time, which allows for very efficient readout of the qubit state. Using a SQUID amplifier32 with a noise temperature less than 100 mK would allow for reducing the discrimination time by more than one order of magnitude. The measurement of the qubit state can be performed by applying the ac current to the SQUID for a time Tm and by measuring the average ac voltage during this time interval. Note that the readout does not have to be performed at the same qubit bias flux where qubit manipulation prior to measurement is performed. It is possible to perform operations ⌽ ⌽ on the qubit at qb= 0 /2, where the qubit is insensitive to magnetic flux fluctuations. Afterwards the flux in the qubit can be changed adiabatically to a different value, where the two energy eigenstates have sufficiently different values of the persistent and the qubit relaxation time is larger, allowing for efficient measurement. Figure 3 shows that away from the symmetry point the discrimination time is Tdiscr Ϸ 100 ns, which implies that a measurement time Tm =300 ns ensures a measurement fidelity larger than 80%. This measurement time is not only much smaller than the relaxation time due to readout, but also appreciably smaller FIG. 3. ͑a͒ Schematic representation of the measurement circuit, than the presently attained relaxation times of flux qubits with notations according to Sec. III. The values of the circuit ele- with similar design parameters, which ensures that qubit re- ⍀ ⍀ ments are Rb =4.7 k , Zs =Za =50 , LJ0 =1.76 nH, Ls =0.18 nH, laxation during readout is negligible. ͑ ͒ C1 =60.7 pF, and C2 =60.6 pF. b A representation of the Fourier The dephasing time depends on the Fourier transform of transform of the correlation function for the SQUID phase operator the symmetrized correlation function at the frequency of the vs frequency. ͑c͒ A plot of the measurement discrimination time ͑ ͒ + ͑␻ ͒ ac driving. It follows from 47 that S␥ 0 is large, because ͑ ͒ ͑ ͒ e continuous line and qubit relaxation time Tr dashed line vs qubit ␻ bias flux. The measurement is performed with an amplitude of the 0 has to be close to the resonance frequency of the circuit ␥ ␯ for efficient state readout. Even a small amplitude of the ac ac driving such that e0 =0.5 at a frequency 0 =672 MHz. signal can cause significant dephasing of the qubit. During qubit manipulation, when no measurement is performed, the ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ SQUID ac driving current has to be suppressed very strongly. 42 with 47 and 48 is shown in Fig. 3 b , assuming a ␻ ⌬ For operation at a qubit energy level splitting 01=2 the temperature T=30 mK. ␮ ͑ ͒ decoherence time due to the SQUID is 10 s if the ampli- To calculate the discrimination time given by Eq. 54 ,we ͉␥ ͉ tude of the phase oscillations is e0 =0.003. assume that the voltage noise is dominated by the voltage The continuous nature of the flux detection makes this amplifier. We assume that a low noise cryogenic amplifier readout method suitable for fundamental studies of the dy- 31 ͑ ͒ with a noise temperature of4Kisused. Equations 14 and namics of the measurement process. Further analysis will be ͑ ͒ ⌬ ␻ϳ 15 allow the calculation of Vqb. We assume that 0, necessary for understanding the dynamics of the coupled since qubit relaxation is slow compared to the detector band- qubit-SQUID system and for an evaluation of possible direct width ͑which will be confirmed by our calculation of the observation of qubit coherent evolution, similar to the situa- ͒ ␻ ␲ 33 relaxation time and we choose the value 0 /2 tion described by Korotkov and Averin. ⌬ =672 MHz that gives a maximum amplitude Vqb=189 nV for an ac driving current such that the amplitude of the ␥ SQUID phase oscillations is e0 =0.5. The discrimination VII. CONCLUSIONS time is plotted in Fig. 3͑c͒. The discrimination time increases ⌽ ⌽ when the qubit bias flux q approaches 0 /2, because the In this paper we analyzed the dc-SQUID in the inductive difference between the expectation values of the qubit cur- mode as a readout method for superconducting flux qubits. rent for the two energy eigenstates decreases ͑see Fig. 1͑b͒͒. We characterized the response function of the dc-SQUID as The relaxation time is calculated using the expression ͑49͒ a flux detector. We described the back-action of the measure- and plotted in Fig. 3͑b͒ for the chosen operating frequency ment circuit on the qubit. The relaxation and dephasing rates

184506-9 A. LUPAȘCU, C. J. P. M. HARMANS, AND J. E. MOOIJ PHYSICAL REVIEW B 71, 184506 ͑2005͒ are proportional to circuit noise at frequencies that are ACKNOWLEDGMENTS shifted by the SQUID ac driving frequency, which is a result qualitatively different of the case of a the measurement done This work was supported by the Dutch Organization for with a switching dc-SQUID. For a realistic measurement cir- Fundamental Research on Matter ͑FOM͒, the European cuit, we found that single shot measurement of a ﬂux qubit is Union SQUBIT-2 project, and the U.S. Army Research Of- possible. ﬁce ͑Grant No. DAAD 19-00-1-0548͒.

1 M. A. Nielsen and I. L. Chuang, Quantum Computation and U. Hubner, T. May, I. Zhilyaev, H. E. Hoenig, Ya. S. Greenberg, Quantum Information ͑Cambridge University Press, Cambridge, V. I. Shnyrkov, D. Born, W. Krech, H.-G. Meyer, Alec Maassen 2000͒. van den Brink, and M. H. S. Amin, Phys. Rev. B 69, 060501͑R͒ 2 Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature ͑London͒ 398, ͑2004͒. 786 ͑1999͒. 18 A. B. Zorin, Phys. Rev. Lett. 86, 3388 ͑2001͒. 3 D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, 19 M. A. Sillanpää, L. Roschier, and P. J. Hakonen, Phys. Rev. Lett. D. Esteve, and M. H. Devoret, Science 296, 886 ͑2002͒. 93, 066805 ͑2004͒. 4 Y. Yu, S. Y. Han, X. Chu, S. I. Chu, and Z. Wang, Science 296, 20 I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. 889 ͑2002͒. Rigetti, L. Frunzio, and M. H. Devoret, Phys. Rev. Lett. 93, 5 J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev. 207002 ͑2004͒. Lett. 89, 117901 ͑2002͒. 21 A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. 6 T. Duty, D. Gunnarsson, K. Bladh, and P. Delsing, Phys. Rev. B Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 69, 140503͑R͒͑2004͒. ͑London͒ 431, 162 ͑2004͒. 7 I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, 22 B. Peikari, Fundamentals of Network Analysis and Synthesis Science 299, 1869 ͑2003͒. ͑Prentice Hall, Englewood Cliffs, NJ, 1974͒. 8 Y. A. Pashkin, T. Yamamoto, O. Astaﬁev, Y. Nakamura, D. V. 23 Y. Makhlin and A. Shnirman, Phys. Rev. Lett. 92, 178301 ͑2004͒. Averin, and J. S. Tsai, Nature ͑London͒ 421, 823 ͑2003͒. 24 J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Ur- 9 A. Barone and G. Paterno, Physics and Applications of the Jo- bina, Phys. Rev. B 67, 094510 ͑2003͒. sephson Effect ͑Wiley, New York, 1982͒. 25 A. Y. Smirnov, Phys. Rev. B 68, 134514 ͑2003͒. 10 J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, 26 R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics, Vol. 2 and S. Lloyd, Science 285, 1036 ͑1999͒. of Springer Series in Solid-State Sciences ͑Springer-Verlag, Ber- 11 T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. Levitov, lin, 1991͒, 2nd ed. S. Lloyd, and J. J. Mazo, Phys. Rev. B 60, 15 398 ͑1999͒. 27 R. J. Schoelkopf, A. A. Clerk, S. M. Girvin, K. W. Lehnert, and 12 C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. M. H. Devoret, cond-mat/0210247 ͑unpublished͒. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. 28 M. H. Devoret, in Quantum Fluctuations, edited by S. Reynaud, Mooij, Science 290, 773 ͑2000͒. E. Giacobino, and J. Zinn-Justin ͑Elsevier, Amsterdam, 1995͒,p. 13 M. H. Devoret, J. M. Martinis, and J. Clarke, Phys. Rev. Lett. 55, 351. 1908 ͑1985͒. 29 A. B. Zorin, Physica C 368, 284 ͑2002͒. 14 C. H. van der Wal, F. K. Wilhelm, C. J. P. M. Harmans, and J. E. 30 A. Shnirman, Y. Makhlin, and G. Schön, Phys. Scr. T102, 147 Mooij, Eur. Phys. J. B 31, 111 ͑2003͒. ͑2002͒. 15 K. M. Lang, S. Nam, J. Aumentado, C. Urbina, and J. M. Marti- 31 R. F. Bradley, Nucl. Phys. B 72, 137 ͑1999͒. nis, IEEE Trans. Appl. Supercond. 13, 989 ͑2003͒. 32 M. Mück, J. B. Kycia, and J. Clarke, Appl. Phys. Lett. 78, 967 16 A. Lupașcu, C. J. M. Verwijs, R. N. Schouten, C. J. P. M. Har- ͑2001͒. mans, and J. E. Mooij, Phys. Rev. Lett. 93, 177006 ͑2004͒. 33 A. N. Korotkov and D. V. Averin, Phys. Rev. B 64, 165310 17 M. Grajcar, A. Izmalkov, E. Il’ichev, Th. Wagner, N. Oukhanski, ͑2001͒.

184506-10