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
共Received 27 October 2004; revised manuscript received 14 February 2005; published 13 May 2005兲
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 共dc-SQUID兲. 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 number共s兲: 03.67.Lx, 03.65.Yz, 85.25.Cp, 85.25.Dq
I. INTRODUCTION
An information processor based on a quantum mechanical system can be used to solve certain problems significantly faster than a classical computer.1 This idea has motivated intense research in recent years on the control and measure-ment of quantum mechanical systems. The basic units in a quantum computer are two level systems, also called quan-tum bits or qubits. Many types of qubits based on various physical systems have been proposed and implemented ex-perimentally.
Qubits based on solid state systems have the advantage of flexibility in design parameters and scalability. An important class of solid state qubits are the superconducting qubits. They are mesoscopic systems formed of superconductor structures containing Josephson junctions. The energy level structure in these systems is the result of the interplay be-tween the charging energy, associated with the electrostatic energy due to distribution of the charge of a single Cooper pair, and of the Josephson energy, associated with the tunnel-ling probability for Cooper pairs across the Josephson junc-tions. Quantum coherent oscillations have been observed for a few versions of qubits with Josephson junctions2–7 and coupling of two qubits was demonstrated.8
A suitable qubit state detection apparatus for individual qubits is an essential ingredient for the implementation of algorithms for a quantum computer. Efficient measurement is necessary to extract all the relevant information on single qubit states within a restricted time. Moreover, for correla-tion type measurements in a multiple qubit system, the un-wanted back-action of the first measurement should not dis-turb the system so strongly, that subsequent measurements will be meaningless. In this paper we discuss a measurement method for superconducting flux qubits. Flux qubits are a qubit variety formed of a superconducting loop interrupted by Josephson junctions. The basis states have oppositely cir-culating persistent currents in the loop. The control param-eter is an external magnetic flux in the qubit loop. The qubit state can be determined by measuring the magnetic flux gen-erated by its persistent current.
A natural candidate for the measurement of the state of a flux qubit is a dc superconducting quantum interference de-vice共dc-SQUID兲. A dc-SQUID is a loop containing two Jo-sephson junctions. Its critical current, which is the maximum supercurrent that it can sustain, depends on the magnetic flux enclosed in the loop.9The state of a flux qubit10,11was mea-sured using an underdamped dc-SQUID.7,12The critical cur-rent of the SQUID, and thus the state of the flux qubit, is determined as the maximum value of the current, where the SQUID switches to a finite voltage state. Due to thermal and quantum fluctuations, switching of the SQUID is a stochastic process.13 The qubit states are distinguishable if the differ-ence between the two average values of the switching cur-rent, corresponding to the qubit flux states, is larger than the statistical spread of the measured values of the switching current. The measurement of a flux qubit using a switching dc-SQUID was characterized by an efficiency as large as 60%.7Further improvement of the measurement efficiency is possible. Also, the back-action on the qubit coupled to the measurement apparatus, in the situation where no measure-ment is performed, can be reduced to acceptable levels.14 Nevertheless, switching to the dissipative state has a few drawbacks. The finite voltage across the dc-SQUID deter-mines the generation of quasiparticles which causes decoher-ence of the qubit.15 The long quasiparticle recombination time is a severe limit to the reset times for the qubits. In the finite voltage state the SQUID generates ac signals with fre-quencies in the microwaves range and broad spectral content, that can induce transitions in a multiple qubit system, con-strained to have energy level spacings in the same region. The mentioned types of back-action will not have an effect on the statistics of the measurements on a single qubit, as long as the repetition rate of the measurements is small. However, in a complex multiple qubit system switching of a dc-SQUID to the finite voltage state is a strong disturbance of the state of the total system which introduces errors in subsequent computations and/or measurements.
A dc-SQUID can be used as a flux detector in an alterna-tive mode of operation, in which switching to the dissipaalterna-tive
state is avoided. This is based on the property of a SQUID to behave as an inductor, with a Josephson inductance that de-pends on the magnetic flux enclosed in the loop.9The 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 super-conducting loop with a single Josephson junction; the imped-ance of a high quality tank circuit inductively coupled to the loop is measured near resonance, where it is very sensitive to the value of the magnetic flux in the loop. For charge mea-surement, a similar device is the RF single electron tunnel-ling transistor共RF-SET兲, with the difference that a dissipa-tive property of a SET transistor is measured directly. The RF-SET was used as a detector for charge qubits by Duty et al.6Motivated by research on superconducting qubits a few flux or charge detectors based on the measurement of a re-active circuit element have been recently implemented. A flux qubit was studied by Grajcar et al.17 by measuring the susceptibility of the qubit loop using a coupled high quality tank circuit. A detector for charge qubits based on the mea-surement of the inductance of a superconducting SET was proposed by Zorin18and implemented by Sillanpää et al.19A sensitive measurement of the critical current of a Josephson junction which exploits the nonlinearity of the current phase relation was demonstrated by Siddiqi et al.20 The state of a charge qubit was read out by Wallraff et al.21by measuring the transmission through a coupled transmission line resona-tor.
The paper is organized as follows. In Sec. II we discuss a few general constraints on the parameter range where the dc-SQUID in the inductive mode can operate. We continue in Sec. III with a general analysis of the response function of this device as a flux detector. The response function is de-rived for a general type of circuit embedding the dc-SQUID. In Sec. IV we discuss the qubit-SQUID interaction and we identify the relevant aspects of the measurement back-action. The energy relaxation rate and the dephasing rate of the qubit during the measurement are derived in Sec. V. Because of ac driving of the SQUID and quadratic coupling of the qubit to the SQUID, the qubit relaxation rate is proportional to the spectral density of circuit noise at frequencies which are the sum and the difference of SQUID ac driving frequency and qubit Larmor frequency. Similarly, the dephasing rate is pro-portional to the spectral density of circuit noise at the fre-quency of the SQUID ac driving. In Sec. VI we discuss the results of the calculations on the measurement back-action. We analyze the measurement efficiency for a specific readout circuit and we find that single shot readout of a flux qubit is possible.
II. GENERAL CONSIDERATIONS
In this paper we focus on the readout of a persistent cur-rent qubit 共PCQ兲,10,11 though the analysis of the
measure-ment is applicable to flux qubits in general. The PCQ is formed of a superconducting loop with three Josephson junc-tions. Two of the three junctions are of equal size, with Jo-sephson energy EJand charging energy Ec, while the third junction is smaller by a factor␣q. Figure 1共a兲 shows a rep-resentation of the energy levels vs the value of the external magnetic flux in the loop, ⌽qb, for a set of typical param-eters. The qubit quantum state can be represented as a super-position of two basis states that are persistent current states in the loop, with values of the current equal to +Ip and −Ip, respectively. Away from the symmetry point⌽qb=⌽0/ 2 the energy eigenstates are almost equal to the persistent current states. When ⌽qb approaches ⌽0/ 2, the energy eigenstates are superpositions of the basis states. For⌽qb=⌽0/ 2 the en-ergy eigenstates are the symmetric and antisymmetric com-binations of the basis persistent current states and are sepa-rated by an energy gap denoted by⌬. A representation of the expectation value of the current for each energy eigenstate is given in Fig. 1共b兲.
The dc-SQUID is characterized by the gauge-invariant phase variables across the two Josephson junctions, denoted by␥1 and␥2. The two variables are connected through the fluxoid quantization condition, ␥1−␥2= −2⌽sq/⌽0, where ⌽0is the flux quantum and⌽sqis the total flux in the SQUID loop. The flux in the SQUID loop contains an external com-ponent ⌽x and a self-generated component, which can be neglected for the typical parameters we will discuss. With this assumption ␥1−␥2= −2fx, where fx=⌽x/⌽0, and the SQUID can be described as a single Josephson junction with a critical current given by Ic共fx兲=2Ic0兩cos共fx兲兩 for a
sym-FIG. 1.共a兲 Ground 共E0兲 and excited 共E1兲 state energy levels for a PCQ with two junctions equal, with EJ= 258 GHz and EC = 6.9 GHz, and the third junction smaller by a factor␣q= 0.75, re-sulting in Ip= 300 nA and⌬=5.5 GHz. 共b兲 Expectation value of the loop current for the ground 共具Iqb典0兲 and excited 共具Iqb典1兲 energy states with the parameters mentioned in共a兲. 共c兲 Schematic represen-tation of the PCQ and of the measuring dc-SQUID, with crosses indicating Josephson junctions. The SQUID junctions have critical currents Ic1and Ic2. The SQUID acts like a variable inductor, with an inductance LJdependent on the state of the coupled flux qubit. The impedance Z of a resonant circuit formed of the dc-SQUID and a shunt capacitor C is plotted vs the frequency 0 for the cases corresponding to the qubit in the ground state共continuous line兲 or in the excited state共dashed line兲. The parameters of the circuit have typical values, as discussed in Sec. VI共R is an equivalent resistance representing the energy loss of the driven circuit兲.
metric SQUID 共Ic1= Ic2= Ic0兲. For symmetric qubit-SQUID coupling共when the two SQUID branches have mutual induc-tances to the qubit loop with opposite value兲 the flux gener-ated by the SQUID in the qubit loop is given by MsIcirc, where Ms is the inductance between the qubit and the SQUID loops and Icirc=共Isq,1− Isq,2兲/2 共with Isq,1 and Isq,1 being the currents in the SQUID junctions兲 is the circulating current in the SQUID loop. The circulating current of the SQUID is given by
Icirc= Ic0sin共fx兲cos␥e, 共1兲 where ␥e=共␥1+␥2兲/2. The current and the voltage of the SQUID are related to the variable␥ethrough the two Joseph-son relations: I = Ic共fx兲sin␥e, 共2兲 and V =⌽0 2 d␥e dt . 共3兲
From Eqs. 共2兲 and 共3兲 it follows that in the linear approxi-mation (sin共␥e兲⯝␥e) the SQUID behaves as a linear inductor with the Josephson inductance,
LJ= ⌽0
2Ic. 共4兲
If an ac current is injected in the SQUID at frequency0 with a small amplitude Iac, the voltage across the SQUID has the amplitude V = 20LJIac=⌽00Iac/ Ic. The maximum voltage across the SQUID is very small 共⬃2V for 0 = 1 GHz兲; very low noise amplification is necessary to detect such a voltage in a short time. Increasing the value of0will result in a proportional increase in the value of the maximum ac voltage. However, from Eqs. 共1兲 and 共2兲 it follows that when the SQUID current varies at frequency0, the circulat-ing current contains a significant frequency component at 20and additional higher harmonics for strong driving in the nonlinear regime. The flux generated by this circulating cur-rent in the qubit loop can cause transitions between the qubit energy levels if the harmonics of the driving current are close to the qubit energy levels splitting. With typical level split-ting of 1–20 GHz, the value of0 is limited toⱗ1 GHz.
The relative change in Josephson inductance when the qubit evolves from the ground to the excited state is given by
冏
␦LJ LJ冏
⯝冏
␦Ic Ic冏
⯝ 2兩tan共fx兲兩MsIp ⌽0 , 共5兲where it was assumed that the measurement is performed at a bias flux in the qubit away from⌽0/ 2, so that the expecta-tion value of the qubit current in each energy eigenstate ap-proaches in absolute value Ip 共see Fig. 1共b兲兲. The typical 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, the maximum difference in ac voltage corresponding to a qubit state change, from the ground to the excited state, is ⬃⌽00␦LJ/ LJ. This can be increased if the SQUID is placed in a resonant circuit and the driving frequency 0 is taken close to the circuit resonance frequency 共see Fig. 1共c兲兲. A
limit on the quality factor Q will be set by the fact that the response time of the resonator, Q /0, has to be smaller than the intrinsic qubit relaxation time, which is in the microsec-onds range.7,16When Q⬎2LJ/␦L
Jthe two circuit resonance peaks, corresponding to the different qubit states, are sepa-rated and a further increase of Q will not contribute to an increase in the ac voltage difference.
The above considerations show that, given the typical qu-bit energy level splitting and relaxation time, the constraints on the circuit parameters are0ⱗ1 GHz and Q⬍100.
III. DETECTOR RESPONSE FUNCTION
In this section we analyze the dc-SQUID in the inductive mode as a flux detector. We consider the case of moderate ac driving, when the SQUID behaves as a linear inductor. The function describing the conversion of flux in the SQUID loop to ac voltage is determined for a general type of circuit in which the SQUID is embedded.
If the magnetic flux in the SQUID loop varies in time, the relation between the transport current and the voltage across the terminals of the SQUID is given by
I共t兲 = 1 LJ共t兲
冕
t
V共t
⬘
兲dt⬘
, 共6兲 where LJ共t兲 is the time-dependent Josephson inductance. Let us consider 1 LJ共t兲= 1 LJ0„1 + ␣共t兲…, 共7兲 where␣共t兲 parametrizes the variations of the magnetic flux in the SQUID. The time-dependent Josephson inductance LJ共t兲 can be represented as the parallel combination of the inductances LJ0 and LJ0/␣共t兲 共see Fig. 2共a兲兲. In the case of the qubit measurement, ␣共t兲 describes the dynamics of the qubit generated flux. The extreme values of␣共t兲 in this case correspond to the qubit in a clockwise or anticlockwise per-sistent current state and are given approximately by ±tan共fx兲MIp/⌽0共see Eq. 共5兲兲. We assume 兩␣共t兲兩Ⰶ1, con-sistent with the usual value of the flux generated by the coupled qubit in the SQUID loop which is of the order of 1% of⌽0.7,12,16From Eqs.共6兲 and 共7兲, it follows that the current in the SQUID can be written asI = I0+ I1, 共8兲 with I0共t兲 = 1 LJ0
冕
t V共t⬘
兲dt⬘
共9兲 and I1共t兲 =␣共t兲 LJ0冕
t V共t⬘
兲dt⬘
. 共10兲 As discussed in Sec. II, the measurement of the Josephson inductance is more efficient if the dc-SQUID is integrated in a resonant circuit. In this case the circuit is driven with an accurrent source at a frequency close to the resonance fre-quency and the ac voltage is suitably amplified. The output ac voltage depends on␣共t兲. Since there is a certain freedom in the design of the resonant circuit, we calculate here the dependence of the output voltage on␣共t兲 for a general type of circuit in which the dc-SQUID is embedded 共see Fig. 2共b兲兲. We consider a linear network with three ports that contains LJ0 共the constant component of the SQUID induc-tance兲, the impedance of the driving current source, the am-plifier input impedance, complemented by other linear circuit elements. The port Pi has its terminals across the current source. The port Pmis connected across the Josephson induc-tance LJ0. Finally, the port Pohas its connections at the input of the amplifier. Three elements are connected to the ports Pi, Po, and Pm, respectively: the ideal current source Ii共t兲, an ideal voltage amplifier, and the inductance LJ0/␣共t兲. A spe-cific electrical circuit described in the way indicated here is shown in Fig. 3共a兲.
The relation between the current and the voltage at port Pmis determined by the inductance LJ0/␣共t兲:
− Im共t兲 =␣共t兲 LJ0
冕
t
Vm共t
⬘
兲dt⬘
, 共11兲 which is Eq.共10兲 with changed sign in order to preserve the sign convention for the three-ports network.22The voltage at the port P␣,␣= o,m can be written asV␣共t兲 =
冕
0 ⬁ Z␣i共t⬘
兲Ii共t − t⬘
兲dt⬘
+冕
0 ⬁ Z␣m共t⬘
兲Im共t − t⬘
兲dt⬘
. 共12兲 Here Z␣共t兲 is the impedance matrix for the three-port net-work, with␣,= i,m,o. We assume that the ac driving current is Ii共t兲=Iecos共0t兲 and the flux variations in the dc-SQUID loop共see Eq. 共7兲兲 are described by ␣共t兲=Re(␣0exp共−it兲). Equation 共11兲 implies that Vm and Im have components at frequencies0+ n, with n being an integer. The voltage Vm can be written as Vm共t兲 =兺
n冉
Vm,n 2 e −i共0+n兲t+ Vm,n * 2 e i共0+n兲t冊
. 共13兲 A similar expression for Imcan be written if V is replaced by I in expression共13兲.From Eq. 共12兲 written for␣= m, one obtains the Fourier components of Vm共t兲 as a function of the Fourier coefficients of Im共t兲 共note that Ii共t兲 is imposed and has frequency compo-nents at ±0兲. These values can be replaced in 共11兲, and the terms corresponding to the frequencies0+ nare separated. Using the equations corresponding to n = 0,1,⫺1 in expan-sions of the form共13兲 and neglecting the terms Im,2and Im,−2, the values for Im,0, Im,1, and Im,−1 can be obtained in lowest order in兩␣0兩. Using these values in Eq. 共12兲 for= o leads to the following expression for the components of the output voltage at frequencies0+and0−:
Vo,1= ␣0 i0LJ0 Zmi共0兲 Ie 2Zom共0+兲 共14兲 and Vo,−1= ␣0 * i0LJ0 Zmi共0兲 Ie 2Zom共0−兲. 共15兲 The expressions 共14兲 and 共15兲 for the up- and down-converted voltage at the output of the circuit are proportional to the amplitude of the driving current Ie and to the ampli-tude of the flux modulation 兩␣0兩. These expressions are us-able only when the driving conditions 共Ie and0兲 are such that the maximum amplitude of the current in the SQUID is not close to the SQUID critical current. Besides the up- and down-converted components Vo,1and Vo,−1, the output volt-age contains a strong component Vo,0 at frequency o. Vo depends only quadratically on兩␣0兩 and thus it cannot be used for an efficient detection of the flux.
IV. QUBIT-SQUID INTERACTION
In a basis formed of two persistent current states, the Hamiltonian of the flux qubit can be written as11
FIG. 2.共a兲 The time-dependent Josephson inductance can be represented as a parallel combination of the inductors LJ0and LJ0/␣共t兲. 共b兲 A schematic representation of the circuit in which the dc-SQUID is inserted共see the text for explanations兲.
Hˆqb=⑀ 2ˆz+
⌬
2ˆx, 共16兲
where ˆi, i = x , y , z, have the Pauli matrices representation. The coefficient of the first term in 共16兲 is ⑀= 2Ip共⌽qb −⌽0/ 2兲, where ⌽qbis the flux in the qubit loop. The maxi-mum persistent current Ip and the minimum energy level splitting⌬ 共see Figs. 1共a兲 and 1共b兲兲 are parameters fixed by the qubit junctions design. The average flux induced in the SQUID loop by the qubit is MIp具ˆz典. The flux-dependent term in the energy of a SQUID is the Josephson energy given by −2⌽0/共2兲Ic0cos共␥e兲cos共⌽sq/⌽0兲. The total flux ⌽sqin the SQUID loop contains the external flux ⌽x and qubit-induced flux MIp具ˆz典. It follows that the interaction Hamil-tonian can be written as
Hˆc= MIpIc0cos共␥ˆe兲sin共fx兲ˆz, 共17兲 where we assumed that the flux generated by the qubit is small and thus a linear approximation could be used. A rig-orous derivation of the interaction term in the Hamiltonian for a coupled dc-SQUID and a three Josephson junctions qubit, assuming a SQUID with a small self inductance and using the two level approximation for the three Josephson junctions loop leads to the same result as共17兲.
If 共17兲 is compared to 共16兲 with ⑀/ 2 = Ip共⌽qb−⌽0/ 2兲, it becomes clear that the back-action due to the measurement is described by an equivalent flux operator, expressed as ⌽ˆb = MIˆcirc. Iˆcircis the operator corresponding to the circulating current in the dc-SQUID and is given by Iˆcirc= Ifcos共␥ˆ 兲,e where If= Ic0sin共fx兲 共see Eq. 共1兲兲. Classically, qubit deco-herence can be understood as a result of the fluctuations in the flux bias, due to the SQUID. If␥eis treated as a classical variable, its time evolution is given by
␥e共t兲 =␥e,coh共t兲 +␥e,n共t兲, 共18兲 where ␥e,coh共t兲=Re(␥e0exp共−i0t兲) with ␥e0 = 2IeZim共0兲/共i⌽00兲 the response to circuit driving and ␥e,n共t兲 is a random term, corresponding to, e.g., thermal fluc-tuations. The “classical” flux is given by ⌽b共t兲 = MIfcos(␥e共t兲), which can be approximated by
⌽b共t兲 = MIf
冉
1 −␥e 2共t兲2
冊
. 共19兲The statistical properties of ⌽b共t兲 are thus determined by
␥e
2共t兲. From 共18兲 it follows that for the case when the phase oscillations amplitude is large compared to the typical values of␥en共t兲, the most important contribution will be the mixing term ␥e,coh共t兲␥e,n共t兲. This results in frequency conversion of the circuit noise.
The analog of Eq.共18兲 for the quantized system is
␥ˆeI共t兲 =␥e,coh共t兲 +␥ˆen共t兲, 共20兲 in which␥ˆeI共t兲 is the phase operator in the interaction repre-sentation with respect to the qubit-SQUID interaction, which is thus equivalent to the Heisenberg representation for the SQUID system.␥ˆe,n共t兲 is the phase operator representing the
intrinsic evolution, in the absence of circuit driving. The first term on the right-hand side of Eq.共20兲 corresponds to the evolution due to driving, and it is the same as the first term on the right-hand side of Eq.共18兲.
The essential feature of the interaction Hamiltonian given by 共17兲 is that the coupling to the external phase operator does not have a linear part. Recent work on the influence of nonlinear coupling of the noise on the evolution of a two level system has been done,23motivated by results reported by Vion et al.,3 where long coherence times were obtained for the operation of a qubit at settings where the energy level separation was insensitive in the first order to external noise. In the second order approximation, the back-action flux noise is described by
⌽ˆb= MIf
冉
1 −␥ˆe2
2
冊
, 共21兲which can be separated in three parts as
⌽ˆb1= MIf
冋
冉
1 − ␥e,coh 2 共t兲 2冊
− ␥ˆen2 共t兲 2 −␥e,coh共t兲␥ˆen共t兲册
. 共22兲 The first term on the right-hand side of共22兲 can cause tran-sitions between the qubit energy levels. As we discussed in Sec. II, resonant transitions occur when the qubit energy lev-els splitting is close to the harmonics of the driving ac fre-quency, and in particular to the second harmonic. The time average of the first term in共22兲, dependent on the amplitude of the ac driving current, will be considered a part of the qubit flux bias⌽qb. The effects of the second term in 共22兲 were analyzed by Makhlin et al.23 for Ohmic and 1 /type spectral densities. In this paper we focus on the calculation of decoherence determined by the third term in共22兲. In the second order perturbation theory, used for the calculation of the decoherence rates in the next section, the contributions from the different terms in共22兲 can be treated independently.V. CALCULATION OF THE DECOHERENCE RATES In this section we calculate the relaxation and dephasing rates of a flux qubit during the measurement by a dc-SQUID in the inductive mode. It is assumed that the external qubit flux⌽qbis fixed. However, the calculations can be extended to include the case of the measurement performed during induced Rabi oscillations or other control sequences.24,25The model Hamiltonian used for the combined system qubit-SQUID is
Hˆ 共t兲 = Hˆqb+ Hˆc+ HˆSQUID共t兲, 共23兲 where Hˆqbis the qubit Hamiltonian given by共16兲, Hˆcis the interaction term given by 共17兲 and HˆSQUID共t兲 is the SQUID Hamiltonian, which is time-dependent due to driving. If a transformation is made to the qubit energy eigenstates, the first two terms in共23兲 become
Hˆqb=
冑
⑀ 2+⌬22 ˆz 共24兲
and
Hˆc= MIpIfcos共␥ˆe兲„cos共兲ˆz− sin共兲ˆx…, 共25兲 where theˆi, i = x , y , z are Pauli matrices in the energy eigen-state basis and tan共兲=⌬/⑀. In the interaction representation with respect to the qubit-SQUID interaction, the operatorsiI evolve in time according to
ˆiI共t兲 = Aij共t兲ˆj 共26兲 with the matrix A given by
A共t兲 =
冢
cos共01t兲 − sin共01t兲 0 sin共01t兲 cos共01t兲 0
0 0 1
冣
, 共27兲
in which01=
冑
⑀2+⌬2/ប is the frequency corresponding to the qubit energy level separation冑
⑀2+⌬2. The evolution of the operatorsˆiHin the Heisenberg picture is obtained using time-dependent second-order perturbation theory. As ex-plained in Sec. IV, the relevant part of the interaction Hamil-tonian共Eq. 共25兲兲 for the calculation of decoherence is given in the interaction picture byHˆcI共t兲 = − MIpIf␥e,coh共t兲␥ˆe,n共t兲„cos共兲ˆz I
− sin共兲ˆxI…. 共28兲 The evolution of the operators ˆiH, which allows us to de-scribe the qubit operators expectation values if the initial state is known, is given by
ˆiH共t兲 =ˆiI共t兲 − 1 ប2
冕冕
t⬎t1⬎t2⬎0 dt1dt2†关ˆi I共t兲,HˆcI共t 1兲兴,Hˆc I共t 2兲‡. 共29兲 In the following, the second term on the right-hand side of Eq.共29兲 is calculated. For the initial calculation we assume the most general interaction Hamiltonian with linear cou-pling to the bath, which can be written asHˆc I共t兲 =
兺
i=x,y,z fˆi I共t兲ˆ i I共t兲, 共30兲where fˆi are bath operators 共note that the interaction repre-sentation is used in共30兲兲. In the end the form of the operators fˆiI共t兲 corresponding to our case, as given by Eq. 共28兲, will be considered:
fˆxI共t兲 = MIpIfsin共兲␥e,coh共t兲␥ˆen共t兲,
fˆyI共t兲 = 0,
fˆzI共t兲 = − MIpIfcos共兲␥e,coh共t兲␥ˆen共t兲. 共31兲 If the commutation and anticommutation relations for theˆi operators are used, Eq.共29兲 results in
ˆiH共t兲 =
兺
j,l,m=x,y,z Aij共t兲冉
ˆj+ 2 ប2冕冕
t⬎t1,t2⬎0 dt1dt2„Oˆlj+共t1,t2兲ˆl − Oˆll+共t1,t2兲ˆj−⑀jlmOˆlm −共t 1,t2兲…冊
. 共32兲 In the last expression,Oˆij±共t1,t2兲 =
兺
k,l=x,y,z Aik共− t1兲Cˆkl ±共t 1,t2兲Alj共t2兲, 共33兲 with Cˆkl+共t1,t2兲 = 1 2„fˆk I共t 1兲fˆl I共t 2兲 + fˆl I共t 2兲fˆk I共t 1兲… 共34兲 and Cˆkl−共t1,t2兲 = i 2„fˆk I共t 1兲fˆl I共t 2兲 − fˆl I共t 2兲fˆk I共t 1兲…. 共35兲 The last two expressions are symmetrized and antisymme-trized products of operators at different times. Their expec-tation values calculated for a thermal equilibrium state are connected with the linear response functions by the fluctuation-dissipation theorem.26 Note that, to obtain 共32兲, the integral in共29兲 was extended to the region t1⬍t2because the integrand is symmetric under the interchange of t1and t2. We assume that initially the qubit and SQUID states were separable and the SQUID is described by the thermal equi-librium density matrix. For the case of the coupling Hamil-tonian given in Eq. 共28兲, the relevant correlation functions are C␥ e +共t 1,t2兲 =冓
1 2兵␥ˆe I共t 1兲,␥ˆe I共t 2兲其+冔
0 共36兲 and C␥ e −共t 1,t2兲 =冓
i 2兵␥ˆe I共t 1兲,␥ˆe I共t 2兲其−冔
0 , 共37兲where⫹/⫺ denote the anticommutator/commutator, and the expectation value is taken for the SQUID thermal equilib-rium density matrix. From 共32兲–共37兲 we see that the time evolution of the operatorsˆi, i = x , y , z, depends on their ex-pectation values for the initial qubit state and on a two-dimensional integral involving the expectation values of op-erators of type共36兲 and 共37兲.
The interaction between the qubit and the measurement dc-SQUID has the consequence that the qubit quantum state becomes a mixed state. In general one distinguishes between energy relaxation, corresponding to a change in the qubit energy expectation value, and dephasing, corresponding to randomization of the phase of a coherent superposition of energy eigenstates.24 To calculate the energy relaxation, we determine the transition rates between energy eigenstates by determining the evolution of具ˆzH共t兲典. If in this calculation the initial qubit state is chosen to be the ground or the excited state, these rates will represent the absorption and emission rates, respectively. To calculate the dephasing rate, we determine the decay of the expectation values
具1/2(ˆxH共t兲±iˆHy共t兲)典, with the qubit initial state being a ˆx eigenstate.
The calculation of the integral on the right-hand side of 共32兲 involves a product of the functions cos共01t1,2兲 and sin共01t1,2兲, resulting from the expression for the free evolu-tion matrix A共see Eq. 共27兲兲 and of the functions cos共0t1,2兲 and sin共0t1,2兲, resulting from the time dependence of the coupling operators fˆiI共t兲 共see 共31兲兲. The correlation functions appearing in共36兲 and 共37兲 only depend on the time differ-ence t1− t2. For times that are long compared to 2/0and 2/01 the relaxation 共decay of 具ˆz
H共t兲典兲 and the dephasing 共decay of 具1/2(ˆxH共t兲±iˆ
y
H共t兲)典兲 can be described as an inte-gral of the product of the Fourier transform of one of the spectral functions C␥
e
±共t
1, t2兲 and a weight function that has a width depending on the integration time t. This weight func-tion is given by
W共,t兲 = 1
2t
冕冕
0⬍t1,t2⬍t
dt1dt2e−i共t1−t2兲, 共38兲
and has the property limt→⬁W共, t兲=␦共兲. The expressions below for the relaxation and dephasing rates are given as-suming that the spectral density of the circuit noise, given by the Fourier transform of the correlation functions,共36兲 and 共37兲, does not have significant variations over the frequency range, where the weight function has substantial values. In this case, the relaxation and the dephasing of the qubit state are proportional to the time from the beginning of the mea-surement.
The transitions rates between the two energy eigenstates depend on the initial state. The transition rate from the ex-cited state to the ground state⌫↓共emission兲 and the transition rate form the ground state to the excited state⌫↑共absorption兲 are given by ⌫↓= 1 2ប2sin 2共兲k2共␥e0兲关S ␥e共01+0兲 + S␥e共01−0兲兴 共39兲 and ⌫↑= 1 2ប2sin 2共兲k2共␥e0兲关S ␥e共−01+0兲 + S␥e共−01−0兲兴 共40兲 in which relations
k共␥e0兲 = MIpIf兩␥e0兩 共41兲 are a measurement coupling factor and
S␥ e共兲 = S␥e +共 兲 − iS␥ e −共 兲. 共42兲 S␥ e ±
are the Fourier transforms of the correlation functions C␥
e
±共t,0兲 given by 共36兲 and 共37兲. The corresponding expres-sion for the dephasing rate⌫is
⌫= 1 4ប2sin 2共兲k2共␥e0兲关S ␥e +共 01+0兲 + S␥e −共 01−0兲兴 + 1 ប2cos 2共兲k2共␥e0兲S ␥e +共 0兲. 共43兲 VI. DISCUSSION
In this section we discuss the results of the calculations of the parameters characterizing qubit decoherence and we ana-lyze a practical circuit which can be used for single-shot readout of a flux qubit.
We start with a discussion on the emission and absorption rates, along the lines of similar analysis done for charge27 and charge-phase24 qubits. Both⌫
↓ and⌫↑ are proportional
to sin2共兲, due to the fact that the operator ˆ
x 共see 共25兲兲 causes transitions between the energy eigenstates. The differ-ence between the two rates given by共39兲 and 共40兲 is due to the last term in the integrand in expression共32兲, connected with the fact that a commutator is nonvanishing, so it can be attributed to quantum noise. Let pg and pebe the probabili-ties for the qubit to be, respectively, in the ground and in the excited state. The time evolution of these probabilities is determined by the rate equations
dpg dt = − ⌫↑ 2 pg+ ⌫↓ 2 pe, 共44兲 dpe dt = ⌫↓ 2 pg− ⌫↑ 2 pe,
and the normalization condition pg+ pe= 1. Since ⌫↓ and⌫↑ in共39兲 and 共40兲 describe the decay of 具ˆz典, they appear di-vided by 2 in共44兲. The polarization P共t兲=pg共t兲−pe共t兲 tends to the equilibrium value
Ps= ⌫↓−⌫↑
⌫↓+⌫↑
, 共45兲
with a relaxation rate
⌫r=⌫↓+⌫↑
2 . 共46兲
The spectral densities of the symmetrized and antisymme-trized correlation functions共36兲 and 共37兲 depend on the im-pedance at the port Pm共see Sec. III兲 as
S␥ e +共 兲 =8 coth
冉
ប 2kBT冊
Re„Zmm共兲… RK 共47兲 and − iS␥ e −共兲 =8 Re„Zmm共兲… RK , 共48兲where RK= h / e2 共see Devoret28兲. Given the relations 共46兲, 共39兲, 共40兲, and 共42兲 and the properties of S␥
e
+共兲 and S
␥e
−共兲 to be, respectively, even and odd functions, the relaxation rate can be written as
⌫r= 1 2ប2sin 2共兲k2共␥e0兲关S ␥e +共 01+0兲 + S␥e +共 01−0兲兴. 共49兲 For a flux qubit coupled to a dc-SQUID biased with a con-stant current, van der Wal et al.14found that qubit relaxation is proportional to S␥
e
+共
01兲. Our results show that because of driving the SQUID with an ac bias current at frequency0 the qubit relaxation rate is proportional to the sum of S␥
e
+共
01+0兲 and S␥e +共
01−0兲, and multiplied by the cou-pling factor of Eq.共41兲. In practice0Ⰶ01, which implies that for a spectral density of the noise which is reasonably flat at large frequencies we can take S␥
e +共 01+0兲⬃S␥ e +共 01 −0兲⬃S␥ e +共
01兲, and our results are not significantly differ-ent from the case of a dc currdiffer-ent biased SQUID.14
From共42兲, 共47兲, and 共48兲 it follows that S␥
e共−兲
S␥
e共兲
= e−ប, 共50兲
where= 1 /共kBT兲. For a qubit coupled to a dc-SQUID bi-ased with a constant current, which is the case analyzed by van der Wal et al.,14 ⌫↓and⌫↑ are proportional to the spec-tral density of the noise at the frequencies 01 and −01, respectively. For that case Eq. 共50兲 implies that ⌫↑/⌫↓ = e−ប01. This is the detailed balance condition and implies
that in a stationary situation the qubit is in thermal equilib-rium with the environment at temperature T 共see also the results of Schoelkopf et al.27兲. In contrast, the relations 共39兲 and共40兲 show that for the case analyzed here the detailed balance condition is in general not satisfied. This is a non-equilibrium situation generated by the presence of the ac driving of the SQUID.
The dephasing rate in共43兲 can be written, if 共46兲 is used, as
⌫=⌫↓
+⌫↑ 4 +⌫
*, 共51兲
where the pure dephasing rate⌫* is given by
⌫* = 1 ប2cos 2共兲k2共␥e0兲S ␥e +共 0兲. 共52兲 The factor cos2共兲 is due to the coefficient of the operatorˆ
z in共25兲. Dephasing is a result of the random modulation of energy level separation due to noise in the SQUID circulat-ing current. The fact that the SQUID is driven with an ac current has the consequence that the pure dephasing rate de-pends on noise at0, which is qualitatively different of the result obtained by van der Wal et al.14 For the radio-frequency Bloch-transistor electrometer18a similar contribu-tion of the converted noise to back-accontribu-tion was found.18,29
We compare the pure dephasing rate ⌫* given by 共52兲 with a similar contribution due to the second term in Eq.共21兲 as calculated by Shnirman et al.,30that we denote by⌫˜
* . We consider the simple case where the dc-SQUID is shunted by
a resistor Rsh, corresponding to Ohmic dissipation, when the result of Shnirman et al. can be used. The following relation is valid for0LJ0ⰆRsh: ⌫* ⌫˜* =
冉
ប0 kBT冊
3 coth冉
ប0 kBT冊
␥e0 2 2 RshRK 共0LJ0兲2 . 共53兲For the case 0LJ0ⰆRK and ប0⬃kBT, the dephasing rate ⌫˜*
is dominant even at small SQUID driving amplitudes. The reliable measurement of the qubit state requires that the ac voltage at the output of the circuit is averaged for a long enough time, such that the noise due to the amplifier is less that the difference between the voltage values corre-sponding to the two qubit flux states. We define the discrimi-nation time as the time necessary to have a measurement signal to noise ratio equal to 1. It is thus given by
Tdiscr= SV共0兲
共⌬Vqb兲2, 共54兲
where SV共0兲 is the spectral density of the voltage noise and ⌬Vqb is the difference in the output voltage values corre-sponding to the two qubit states. The value of ⌬Vqb is pro-portional to␥e0. The discrimination time Tdiscr, the relaxation time Tr= 1 /⌫r, and the dephasing time T= 1 /⌫ are in-versely proportional to ␥e02 共see 共46兲 with 共39兲, 共40兲, and 共43兲兲. Increasing the amplitude of the ac driving leads to a decrease in the discrimination time. However, this is accom-panied by a proportional decrease of the qubit decoherence times Trand T. This illustrates the tradeoff between obtain-ing information about a quantum system and the state distur-bance. The measurement is efficient if the ratio Tr/ Tdiscr is large. The ratio Tr/ Tdiscrdoes not depend on the amplitude of the ac driving. However, a fast measurement is necessary if we take into account the fact that, besides the measurement back-action, there are also other sources of decoherence that will increase the total relaxation rate.
We now analyze the measurement of a flux qubit using our particular SQUID embedding network presented in Fig. 3共a兲. The driving source is represented as an ideal current source with impedance Zs. The amplifier is described as the combination of the input impedance Za and ideal voltage amplifier with gain G. The bias resistor Rbhas the purpose of increasing the impedance of the current source. The inductor Lsis a small stray contribution, unavoidable in the design of the circuit. The combination of the capacitors C1and C2is an impedance transformer that will increase the effective im-pedance of the amplifier input, at the cost of a division of the total voltage across the inductors; they also provide the ca-pacitive part necessary to create a resonant circuit. The dc-SQUID has Josephson junctions with a critical current Ic0 = 200 nA. The external magnetic flux in the SQUID loop corresponds to fx= 3.35, resulting in a critical current Ic = 187 nA. The measured persistent current qubit has Ip = 300 nA and⌬=5.5 GHz. Figures 1共a兲 and 1共b兲 and show plots of the energy eigenvalues and persistent current expec-tation value vs bias flux for these qubit parameters. If the mutual inductance between the qubit and the dc-SQUID is M = 40 pH, the relative change in Josephson inductance is given by␣= 3.4%. A plot of the expression S␥
共42兲 with 共47兲 and 共48兲 is shown in Fig. 3共b兲, assuming a temperature T = 30 mK.
To calculate the discrimination time given by Eq.共54兲, we assume that the voltage noise is dominated by the voltage amplifier. We assume that a low noise cryogenic amplifier with a noise temperature of 4 K is used.31Equations共14兲 and 共15兲 allow the calculation of ⌬Vqb. We assume that ⬃0, since qubit relaxation is slow compared to the detector band-width 共which will be confirmed by our calculation of the relaxation time兲 and we choose the value 0/ 2 = 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 time is plotted in Fig. 3共c兲. The discrimination time increases when the qubit bias flux ⌽q approaches ⌽0/ 2, because the difference between the expectation values of the qubit cur-rent for the two energy eigenstates decreases共see Fig. 1共b兲兲. The relaxation time is calculated using the expression共49兲 and plotted in Fig. 3共b兲 for the chosen operating frequency
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 sin2共兲 and the decrease in the real part of the impedance Zmm away from the resonance peak 共see Eqs. 共49兲 and 共47兲兲. 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 discrimina-tion 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 Tmand 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 mea-surement 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 ⬇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 than the presently attained relaxation times of flux qubits with similar design parameters, which ensures that qubit re-laxation during readout is negligible.
The dephasing time depends on the Fourier transform of the symmetrized correlation function at the frequency of the ac driving. It follows from共47兲 that S␥
e
+共
0兲 is large, because 0 has to be close to the resonance frequency of the circuit for efficient state readout. Even a small amplitude of the ac 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. For operation at a qubit energy level splitting 01= 2⌬ the decoherence time due to the SQUID is 10s if the ampli-tude of the phase oscillations is兩␥e0兩=0.003.
The continuous nature of the flux detection makes this readout method suitable for fundamental studies of the dy-namics of the measurement process. Further analysis will be necessary for understanding the dynamics of the coupled qubit-SQUID system and for an evaluation of possible direct observation of qubit coherent evolution, similar to the situa-tion described by Korotkov and Averin.33
VII. CONCLUSIONS
In this paper we analyzed the dc-SQUID in the inductive mode as a readout method for superconducting flux qubits. We characterized the response function of the dc-SQUID as a flux detector. We described the back-action of the measure-ment circuit on the qubit. The relaxation and dephasing rates
FIG. 3.共a兲 Schematic representation of the measurement circuit, with notations according to Sec. III. The values of the circuit ele-ments are Rb= 4.7 k⍀, Zs= Za= 50⍀, LJ0= 1.76 nH, Ls= 0.18 nH,
C1= 60.7 pF, and C2= 60.6 pF.共b兲 A representation of the Fourier transform of the correlation function for the SQUID phase operator vs frequency. 共c兲 A plot of the measurement discrimination time 共continuous line兲 and qubit relaxation time Tr共dashed line兲 vs qubit bias flux. The measurement is performed with an amplitude of the ac driving such that␥e0= 0.5 at a frequency0= 672 MHz.
are proportional to circuit noise at frequencies that are shifted by the SQUID ac driving frequency, which is a result qualitatively different of the case of a the measurement done with a switching dc-SQUID. For a realistic measurement cir-cuit, we found that single shot measurement of a flux qubit is possible.
ACKNOWLEDGMENTS
This work was supported by the Dutch Organization for Fundamental Research on Matter 共FOM兲, the European Union SQUBIT-2 project, and the U.S. Army Research Of-fice共Grant No. DAAD 19-00-1-0548兲.
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