Quantum Zeno effect with a superconducting qubit
Yuichiro Matsuzaki, Shiro Saito, Kosuke Kakuyanagi, Kouichi Semba
QQuantum Zeno effect with a superconducting qubit
Y. Matsuzaki, ∗ S. Saito, K. Kakuyanagi, and K. Semba NTT Basic Research Laboratories, NTT Corporation, Kanagawa, 243-0198, Japan
Detailed schemes are investigated for experimental verification of Quantum Zeno effect with a superconduct-ing qubit. A superconducting qubit is affected by a dephasing noise whose spectrum is /f , and so the decayprocess of a superconducting qubit shows a naturally non-exponential behavior due to an infinite correlationtime of /f noise. Since projective measurements can easily influence the decay dynamics having such non-exponential feature, a superconducting qubit is a promising system to observe Quantum Zeno effect. We havestudied how a sequence of projective measurements can change the dephasing process and also we have sug-gested experimental ways to observe Quantum Zeno effect with a superconducting qubit. It would be possibleto demonstrate our prediction in the current technology. Quantum Zeno effect(QZE) is one of fascinating phenom-ena which quantum mechanics predicts. A sequence of pro-jective measurements to an unstable system can suppress thedecay process of the state [1–3]. This phenomena will be ob-served if the time interval of projective measurements is suf-ficiently small and the decay behavior in the time interval isquadratic. Although it was proved that an unstable systemshows a quadratic behavior in the initial stage of the decay [4], it is difficult to observe such quadratic decay behavior ex-perimentally, because the time region to show such quadraticbehavior is usually much shorter than typical time resolutionof a measurement apparatus in the current technology. Aftershowing the quadratic decay, unstable system shows an expo-nential decay [4] and QZE doesn’t occur through projectivemeasurements to a system which decays exponentially. Dueto such difficulty, in spite of the many effort to observe theQZE, there was only one experimental demonstration to sup-press the decay process of an unstable state [5]. Note that,except this experiment, all previous demonstration of QZEdidn’t focus on a decoherence process caused by a couplingwith environment but focused on a suppression of a unitaryevolution having a finite Poincare time such as Rabi oscil-lation [6–10]. Such approach to change the behavior of theunitary evolution by measurements are experimentally easy tobe demonstrated, but is different from the original suggestionof QZE for the decay process of unstable systems [1–3] witha decoherence process. Throughout this paper, we consideronly such QZE to change decoherence behavior.In this paper, we suggest a way to demonstrate QZE for thedecay process of unstable system experimentally with a su-perconducting qubit. A superconducting qubit is one of can-didates to realize quantum information processing and, for asuperconducting qubit, the quadratic decay has been observedin an experiment [11, 12] , which is necessary condition toobserve QZE experimentally. Moreover, a high fidelity singlequbit measurement has already been constructed in the cur-rent technology [13]. A superconducting flux qubit has beentraditionally measured by superconducting quantum interfer-ence device(SQUID) [14]. The state of a SQUID is switchedfrom zero-voltage state to a finite voltage state for a particu-lar quantum state of the qubit, while no switching occurs forthe other state. Such switching transition produces a macro- scopic signal to construct a measurement for a superconduct-ing flux qubit. Also, entirely-new qubit readout method suchas JBA(:Josephson Bifurcation Amplifier) has been demon-strated [15, 16]. The JBA has advantages in its readout speed,high sensitivity, low backaction [16] and absence of on-chipdissipative process. It is also studied JBA readout mechanism[17] and the projection conditions [18] of the superpositionstate of a qubit. All these properties are prerequisite in ob-serving the QZE. So a superconducting qubit is a promisingsystem to verify QZE for an unstable state.We study a general decay process of unstable system. Al-though a decay behavior of unstable system has been studiedand conditions for quadratic decay have been shown by sev-eral authors[4, 19–21] , we introduce a simpler solvable modeland we confirm the conditions for the exponential decay andthe quadratic decay, respectively. Also, from the analytical so-lution of the model, we derive a master equation for /f noise.We consider an interaction Hamiltonian to denote a couplingwith an environment such as H I = λf ( t ) ˆ A where f ( t ) is aclassical normalized Gaussian noise, ˆ A is an operator of thesystem, and λ denote a coupling constant. Also, we assumenon-biased noise and therefore f ( t ) = 0 is satisfied where thisover-line denotes the average over the ensemble of the noises.In an interaction picture, by solving the Schrodinger equationand taking the average over the ensembles, we obtain ρ I ( t ) − ρ = ∞ (cid:88) n =1 ( − iλ ) n (cid:90) t dt (cid:90) t dt · · · (cid:90) t n − dt n f ( t ) f ( t ) · · · f ( t n )[ ˆ A, [ ˆ A, · · · , [ ˆ A, ρ ] , · · · ]] (1)where ρ = | ψ (cid:105)(cid:104) ψ | is an initial state and ρ I ( t ) is a state in theinteraction picture. Throughout this paper, we restrict ourselfto a case that the system Hamiltonian commutes with the op-erator of /f noise as [ H s , ˆ A ] = 0 . Firstly, we consider a casethat the correlation time of the noise τ c ≡ (cid:82) ∞ f ( t ) f (0) dt ismuch shorter than the time resolution of experimental appara-tus, which is valid condition for the most of unstable systems.Since the correlation time of the noise is short, we obtain (cid:82) t (cid:82) t (cid:48) f ( t (cid:48) ) f ( t (cid:48)(cid:48) ) dt (cid:48) dt (cid:48)(cid:48) = (cid:82) t dτ ( t − τ ) f ( τ ) f (0) (cid:39) tτ c . Also,since the noise f ( t ) is Gaussian, f ( t ) f ( t ) · · · f ( t n ) can bedecomposed of a product of two-point correlation f ( t i ) f ( t j ) , a r X i v : . [ qu a n t - ph ] O c t and so we obtain ρ I ( t ) (cid:39) (cid:88) A,A (cid:48) ,ν,ν (cid:48) | Aν (cid:105)(cid:104) Aν | ρ | A (cid:48) ν (cid:48) (cid:105)(cid:104) A (cid:48) ν (cid:48) | e − λ τ c | A − A (cid:48) | t (2)where | Aν (cid:105) is an eigenstate of the operator ˆ A and ν denote a degeneracy. So a dynamical fidelity F ≡(cid:104) ψ | e iH s t ρ ( t ) e − iH s t | ψ (cid:105) , a distance between the state ρ ( t ) anda state e − iH s t ρ e iH s t , becomes a sum of exponential decays. F (cid:39) (cid:88) A,A (cid:48) ,ν,ν (cid:48) |(cid:104) Aν | ψ (cid:105)| |(cid:104) A (cid:48) ν (cid:48) | ψ (cid:105)| e − λ τ c | A − A (cid:48) | t (3)Secondly, when the correlation time of the noise is muchlonger than the time resolution of the apparatus such as /f noise having an infite correlation time, we obtain (cid:82) t (cid:82) t (cid:48) f ( t (cid:48) ) f ( t (cid:48)(cid:48) ) dt (cid:48) dt (cid:48)(cid:48) (cid:39) t . So, by taking average overthe ensemble of noise in (1), we obtain ρ I ( t ) (cid:39) (cid:88) A,A (cid:48) ,ν,ν (cid:48) | Aν (cid:105)(cid:104) Aν | ρ | A (cid:48) ν (cid:48) (cid:105)(cid:104) A (cid:48) ν (cid:48) | e − λ | A − A (cid:48) | t (4)So we obtain a master equation for /f noise as dρ I ( t ) dt = − λ t [ ˆ A, [ ˆ A, ρ I ( t )]] . The behavior of the dynamical fidelitybecomes quadratic in the early stage of the decay ( t (cid:28) λ ) as F (cid:39) (cid:88) A,A (cid:48) ,ν,ν (cid:48) |(cid:104) Aν | ψ (cid:105)| |(cid:104) A (cid:48) ν (cid:48) | ψ (cid:105)| e − λ | A − A (cid:48) | t (cid:39) − λ t (cid:88) A,A (cid:48) ,ν,ν (cid:48) | A − A (cid:48) | |(cid:104) Aν | ψ (cid:105)| |(cid:104) A (cid:48) ν (cid:48) | ψ (cid:105)| (5)These results show that an unstable system has an exponentialdecay for t (cid:29) τ c , while a quadratic decay occurs for t (cid:28) τ c .Let us summarize the QZE. Usually, to observe QZE, sur-vival probability is chosen as a measure for the decay. How-ever, we use a dynamical fidelity to observe the QZE ratherthan a survival probability to take into account of the effect ofa system Hamiltonian. We consider a sequence of projectivemeasurements ˆ P ( k ) = e − iH s kτ | ψ (cid:105)(cid:104) ψ | e iH s kτ with τ = tN and k = 1 , , . . . , N to an unstable state where N denotesthe number of the measurements performed during the time t . For noises whose correlation time is short, a dynamical fi-delity without measurements becomes a sum of exponentialdecay terms such as F ( t ) = (cid:80) mj =1 c j e − Γ j t . The successprobability to project the unstable state into the target statesbecomes P ( N ) = (cid:0) (cid:80) mj =1 c j e − Γ j τ (cid:1) N (cid:39) − t (cid:80) mj =1 c j Γ j ,and so the success probability decreases linearly as the timeincreases. On the other hand, if the dynamical fidelity hasa quadratic decay without projective measurements such as F = e − Γ t , we obtain the success probability to project theunstable state into the state e − iH s t | ψ (cid:105) becomes as following. P ( N ) = (cid:16) − Γ τ + O ( τ ) (cid:17) N (cid:39) − Γ t N . So, by increasingthe number of the measurements, the success probability goesto unity, and this means that the time evolution of this stateis confined into e − iH s t | ψ (cid:105) which is a purely unitary evolutionwithout noises, and so one can observe the QZE. It is known that a superconducting qubit is mainly affectedby two decoherence sources, a dephasing whose spectrum is /f and a relaxation whose spectrum is white. The /f noisecauses a quadratic decay to the quantum states as we haveshown. Moreover, such quadratic decay has already been ob-served experimentally [11, 12] . On the other hand, since therelaxation process from an excited state | (cid:105) to a ground state | (cid:105) where a high frequency is cut off, the correlation time ofthe environment is extremely small and so only an exponentialdecay can be observed for a relaxation process in the currenttechnology. Therefore, when the dephasing is relevant and therelaxation is negligible, it should be possible to observe QZEwith a superconducting qubit as following. Firstly, one pre- FIG. 1: A schematic of quantum states in a Bloch sphere to show howQZE is observed with a superconducting qubit. An initial state is pre-pared in | + (cid:105) , and the state has an unknown rotation around z axis dueto a dephasing. To construct a measurement | + (cid:105)(cid:104) + | , one performsa π rotation U y around y axis, performs a measurement | (cid:105)(cid:104) | , andperforms a π rotation U † y . If a measurement interval is much smallerthan a dephasing time, this measurement | + (cid:105)(cid:104) + | recovers a state intothe initial state with almost unity success probability. pares an initial state | + (cid:105) = √ | (cid:105) + √ | (cid:105) which is an eigen-state of ˆ σ x . Secondly, in a time interval τ = tN , one continuesto perform projective measurements | + (cid:105)(cid:104) + | = (ˆ + ˆ σ x ) to the superconducting qubit where N is the number of themeasurement performed. For simplisity, let us make an as-sumption that an effect of a system Hamiltonian is negligiblecompared with the dephasing effect. (Since this assumptioncould be unrealistic for a superconducting qubit, we will relaxthis condtion and discuss more rigorous case later.) Note thatwe perform a selective measurement here to consider only acase to project the state into | + (cid:105)(cid:104) + | and, if the state is pro-jected into the other state, we discard the state as a failurecase. As a result, due to the quadratic decay behavior causedby /f noise, the success probability to project the state intoa target state N times goes to a unity as the number of themeasurements becomes larger, and therefore one can observeQZE(Fig.1). Note that a direct measurement of ˆ σ x with a su-perconducting qubit has not been constructed yet experimen-tally. So, in order to know a measurement result of ˆ σ x in thecurrent technology, one has to perform a π rotation around y axis before and after performing a projective measurementabout ˆ σ z . However, recently, a coupling about ˆ σ x betweena superconducting qubit and a flux bias control line has beendemonstrated [22, 23], which shows a possibility to realize adirect measurement of ˆ σ x in the near future. Since it is notnecessary to perform preliminary rotations around y axis, thisdirect measurement of ˆ σ x has advantage in its readout speed.In the above discussion, the effect of the relaxation and sys-tem Hamiltonian is not taken into account. Since they are notalways negligible in a superconducting qubit, it is necessaryto investigate whether one can observe QZE or not under theinfluence of them. When considering the effect of dephas-ing and relaxation whose spectrum are /f and white respec-tively, we use a master equation as following dρ I ( t ) dt = −
12 Γ (cid:16) ˆ σ + ˆ σ − ρ I ( t ) + ρ I ( t )ˆ σ + ˆ σ − − σ − ρ I ( t )ˆ σ + (cid:17) −
12 (Γ ) t [ˆ σ z , [ˆ σ z , ρ I ( t )]] (6)where Γ and Γ denote a decoherence rate of relaxation anddephasing respectively. In this master equation, the first partis a Lindblad type master equation to denote a relaxation, andthe second part denotes a dephasing whose spectrum is /f coming from the fluctuation of (cid:15) . Also, we assume that a sys-tem Hamiltonian is H s = (cid:15) ˆ σ z + ∆ˆ σ x (cid:39) (cid:15) ˆ σ z for (cid:15) (cid:29) ∆ ,because we have derived a master equation for /f noise onlywhen the system Hamiltonian commutes with the noise oper-ator of /f fluctuation. We find an analytical solution of thisequation, and when the initial state is | ψ (cid:105) = | + (cid:105) , we obtain ρ ( t ) = e − iH s t (cid:16) e − Γ t | (cid:105)(cid:104)| e − Γ t − (Γ ) t | (cid:105)(cid:104) | + 12 e − Γ t − (Γ ) t | (cid:105)(cid:104) | + (1 − e − Γ t ) | (cid:105)(cid:104) | (cid:17) e iH s t (7)Note that, while the /f noise causes a quadratic dephasing,the relaxation causes an exponential decay, which cannot besuppressed by projective measurements. Here, we considerthe effect of system Hamiltonian, and so we perform a pro-jective measurement to the state e − iH s t | + (cid:105) . Since there al-ways exists a time-dependent single qubit rotation U t to sat-isfy U t e − iH s t | + (cid:105) = | (cid:105) , this measurement can be realized byperforming the single qubit rotation before and after a mea-surement of ˆ σ z Note that this single qubit rotation U t can beperformed in a few ns by using a resonant microwave[24]. Inthis paper, we call the entire process including U t as “mea-surement” for simplicity. The success probability P ( N ) toproject the state into the target state is calculated as P ( N ) = ( 12 + 12 · e − τ T − τ T ) N (8)where T = (Γ ) − and T = (Γ ) − denote a relax-ation time and a dephasing time respectively. So we ob-tain P ( N ) = ( e τ T + e − τ T − τ T ) N e − t N ( T (cid:39) (1 − t T ) e − t N ( T for tT , tT (cid:28) . So, as long as the T is much larger than T , one can observe that the success prob-ability increases as one increases the number of the projective N P FIG. 2: A success probability to perform projective measurementsinto a target state under the effect of dephasing and relaxation is plot-ted. The horizontal axis and the vertical axis denote the success prob-ability and the number of measurements respectively The lowest lineis for t = 35 (ns), and the other lines are for t = 30 , , (ns),respectively. As one increases the number of measurements, thesuccess probability increases. Here, we assume a relaxation time T = 1 µ s and a dephasing time T = 20 ns, respectively. measurements(see Fig.2). Note that we assume a Hamiltonianas H s (cid:39) (cid:15) ˆ σ z , far from the optimal point for a supercon-ducting flux qubit, and so a coherence time T of this qubitbecomes as small as tens of ns. In the current technology, ittakes tens of ns to perform JBA[16] and so one has to use aswitching measurement to utilize a SQUID to be performed ina few ns. The state of a SQUID remains a zero-voltage whenthe state of a qubit is | (cid:105) , while a SQUID makes a transitionto a finite voltage state to produce a macroscopic signal for | (cid:105) . One of the problems of the SQUID measurements is thata transition to a finite voltage state destroys quantum states ofthe qubit and following measurements are not possible afterthe transition. However, as long as the state is | (cid:105) , the state ofa SQUID remains a zero-voltage state and so sequential mea-surements are possible. Since one postselects a case that allmeasurement results are | (cid:105) while one discards the other caseas a failure, the SQUID can be utilized to observe QZE withthe selective measurements.Importantly, it is also possible to observe QZE at the op-timal point where T can be as large as µ s. A recent demon-stration of coupling about ˆ σ x between a superconducting qubitand a flux bias control line shows a possibility to have a rele-vant /f noise caused by a fluctuation of ∆ due to a replace-ment of a Josephson junction with a SQUID [22, 23], and thenoise operator from the /f fluctuation becomes ˆ σ x to com-mute a system Hamiltonian at the optimal point as H = ∆ˆ σ x .So, by replacing the notation from ˆ σ z to ˆ σ x and from | + (cid:105) to | (cid:105) , one can apply our analysis in this paper to a case observ-ing QZE at the optimal point. (For example, in this replacednotation, an initial state should be prepared in | (cid:105) and frequentmeasurements in the zy plane will be performed.) Moreover,since T at the optimal point is much longer than a necessarytime to perform JBA, a sequence of measurements is possi-ble for all measurement results. This motivates us to study averification of QZE without postselection as following.Finally, we discuss how to observe QZE without postselec-tion of measurement results, which can be realized by JBA.We perform frequent non-selective measurements in the xy plane to the state which was initially prepared in | + (cid:105) . Suchnon-selective measurements to a single qubit is modeled as ˆ E ( ρ ) = | φ + (cid:105)(cid:104) φ + | ρ | φ + (cid:105)(cid:104) φ + | + | φ − (cid:105)(cid:104) φ − | ρ | φ − (cid:105)(cid:104) φ − | where | φ + (cid:105) = e − iH s t | + (cid:105) and | φ − (cid:105) = e − iH s t |−(cid:105) are orthogonalwith each other. So, when performing this non-selective mea-surement with a time interval τ = tN under the influence ofdephasing and relaxation, we obtain ρ ( N, t ) = e − iH s t (cid:16) | (cid:105)(cid:104) | + 12 e − t T − t N ( T | (cid:105)(cid:104) | + 12 e − t T − t N ( T | (cid:105)(cid:104) | + 12 | (cid:105)(cid:104) | (cid:17) e iH s t (9)where we use a result in (7). Since we consider a state justafter performing a measurement in the xy plane (not along zaxis), the population of a ground state becomes equivalent asthe population of an excited state. Note that a non-diagonalterm is decayed by the white noise and /f noise, and onlythe decay from /f noise is suppressed by the measurements.In Fig. 3, we show this decay behavior of the non-diagonalterm. A possible experimental way to remove out the effect FIG. 3: A behavior of a phase term |(cid:104) | ρ | (cid:105)| under the effect of non-selective measurements realized by the JBA is plotted. This shows asuppression of the decay by measurements. Here t and N denote thetime(ns) and the number of measurements, respectively. We assumea relaxation time T = 1 µ s and a dephasing time T = 400 ns,respectively. These conditions can be realized at the optimal pointwhere the sytem Hamiltonian is H s = ∆ˆ σ x and ∆ has a /f fluctu-ation due to a replacement of a Josephson junction with a SQUID. of the white noise is measuring (cid:104) | ρ ( N, t ) | (cid:105) and (cid:104) | ρ (1 , t ) | (cid:105) separately by performing a tomography, and plotting the valueof (cid:104) | ρ ( N, t ) | (cid:105) / (cid:104) | ρ (1 , t ) | (cid:105) = e − t N ( T for a fixed time t .As a result one can observe the suppression of the dephasingcaused by /f noise through measurements.In conclusion, we have studied detailed schemes for exper-imental verification of QZE to a decay process with a super-conducting qubit. Since a superconducting qubit is affectedby the dephasing with a /f spectrum, the dynamics show aquadratic decay which is suitable for an experimental demon-stration for QZE, while the relaxation process has an expo- nential decay to cause unwanted noise for QZE. We have sug-gested a way to observe QZE even under an influence of re-laxation. Our prediction is feasible in the current technology.Authors thank H. Nakano and S. Pascazio for valuable dis-cussions on QZE. This work was done during Y. Matsuzaki’sshort stay at NTT corporation and was also supported in partby Funding Program for World-Leading Innovative R&D onScience and Technology(FIRST), Scientific Research of Spe-cially Promoted Research 18001002 by MEXT, and Grant-in-Aid for Scientific Research (A) 22241025 by JSPS. ∗ Electronic address: [email protected][1] B. Misra and ECG. Sudarshan, J. Math. Phys. , 3491 (1998).[2] R. J. Cook, Phys. SCR. T21 , 49 (1988).[3] P. Facchi, H. Nakazato, and S. Pascazio, Phys. Rev. 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