Using the Quantum Zeno Effect for Suppression of Decoherence

Projective measurements are an essential element of quantum mechanics. In most cases, they cause an irreversible change of the quantum system on which they act. However, measurements can also be used to stabilize quantum states from decay processes, which is known as the quantum Zeno effect (QZE). Here, we demonstrate this effect for the case of a superposition state of a nuclear spin qubit, using an ancilla to perform the measurement. As a result, the quantum state of the qubit is protected against dephasing without relying on an ensemble nature of NMR experiments. We also propose a scheme to protect an arbitrary state by using QZE.

In the simplest case, the evolution that is suppressed by the QZE is driven by a single interaction to a static external degree of freedom. Typical examples include a transition between two states of a single trapped ion [8], nuclear magnetic resonance (NMR) [9], and atomic systems [10,11]. In a more sophisticated case, QZE restrains the decoherence dynamics of a quantum open system [12], instead of suppressing a unitary evolution driven by external fields. The state that was preserved by these QZE experiments was an eigenstate of the static system Hamiltonian. In this paper, we demonstrate that it is equally possible to protect superposition states between eigenstates of the Hamiltonian and that suppression of dephasing can also be achieved by measurements. To show such effect, we use an ensemble of nuclear spins. One nuclear spin is used as the system qubit to be protected, while a second one acts as the measurement device. A similar scheme was proposed by Matsuzaki et al. [13].
Theory: For a description of the system, it is sufficient to consider a qubit whose coherent superposition state is being dephased by a randomly fluctuating classical field [1,2] with the concept of "mixing process" [14]. The relevant Hamiltonian is thus where λ denotes the strength of the fluctuating field, f (t) denotes a normalized time-dependent classical random variable, and σ i is a Pauli matrix. We assume an unbiased noise, i.e., f (t) = 0, where * denotes an ensemble average. The Hamiltonian contains only σ z because only pure dephasing is considered. We prepare a superposition state ρ(0) = |+ +|, where |+ = 1 √ 2 (|0 + |1 ). Formally integrating the von Neu- ]. The natural unit system is used with = 1. By iterating ρ(t 1 ) on the right hand side, taking the ensemble average, and replacing ρ(t 2 ) by ρ(0), we obtain the second order expression If the correlation time of the noise is much shorter than the relevant time scale of the system [15], we can approximate it by a δ-function f (t 1 )f (t 2 ) = 2τ c δ(t 1 − t 2 ), where τ c is the correlation time. Substituting into Eq.
The state fidelity F (t) = +|ρ(t)|+ is then F (t) ≃ 1−2λ 2 τ c t, which corresponds to a linear decay. It is known that we cannot observe the QZE in this linear-decay regime [3].
In the opposite limit of a slowly fluctuating environment, where |t 1 − t 2 | ≪ τ c , the correlation function is constant, f (t 1 )f (t 2 ) = 1, and Eq. (2) becomes for λ 2 τ c t ≪ 1, i.e., the decoherence starts quadratically: F (t) ≃ 1−λ 2 t 2 . This is the regime where the QZE can be observed: if N sequential projective measurements of |+ +| are carried out on the system and the delay between the measurements is short compared to the correlation time, t/N ≪ τ c , the probability that the system remains in the initial state Since the exponent can become arbitrarily small for large N , the decoherence vanishes asymptotically.
We show that the decoherence of one of two nuclear spins in a molecule can be suppressed by QZE in our NMR experiments. Since projective measurements cannot be implemented, we employ non-selective measurements where postselection is not performed after the measurement as suggested in [13]. Unlike other QZE experiments with NMR [9], pulsed gradient fields have not been used to simulate a measurement.
Sample: The experiments are carried out using a 13 Clabeled chloroform (Cambridge Isotopes) diluted in d 6acetone with a 303 mM concentration at room temperature. The spectrometer is a JEOL ECA-500. The phase decoherence of the 13 C nuclear spin (hereafter, the system) could be controlled by flip-flopping randomly the proton nuclear spin (hereafter, the device) [16,17], as shown in [14].
The same idea is employed here by adding a magnetic impurity (47.7 mM of Fe(III)acac) to the solution. The paramagnetic salt mainly introduces a random flip-flop motion of the device since the system is surrounded and shielded by three Cl atoms and the device, as illustrated in the inset of Fig. 1. This is confirmed by measuring an effective T 2 of the system through the sequential application of π-pulses [18,19] to the device. When the interval between the pulses on the device is below 4 ms, the system T 2 becomes longer due to the decoupling of the nuclear spins. Such behavior indicates a finite correlation time.
T 1 and T 2 of the device are both about 7 ms. It is reasonable because the magnetic impurity flips spins and destroys their phase coherence simultaneously. On the other hand, T 1 of the system is 300 ms, while its T 2 is 17 ms. We measured the T 1 values using the standard inversion recovery method, while we obtained the T 2 values applying a Hahn-echo sequence. Note that the T 2 values are also obtained from the FID signal because of a good field homogeneity. This shorter T 2 , compared with T 1 , can be understood using the scalar coupling between heteronuclear spins H = J σz ⊗σz 4 , where J/2π ≈ 215 Hz for our sample. The random flip-flop motions of the device cause the transversal relaxation on the system through the scalar coupling without any influence on its longitudinal relaxation [14]. Because the finite strength interaction dominate the dynamics of the system, τ c must be finite and the QZE should be observable.
Non-selective measurements: First of all, the thermal state density matrix of the two qubit system is well approximated as ρ = σ0 where ǫ i = ω i /2k B T , ω i is the Larmor frequency of the i-th spin, and σ 0 is the identity matrix of dimension 2. The suffixes s and d denote system and device, respectively. Since the NMR observable has trace zero, the identity term cannot be detected. If only the system is observed, then the above density matrix can be regarded as ǫρ 0 , where ρ 0 = |0 0| ⊗ σ0 2 . This state can be normalized and a pseudopure state |0 0| is obtained for the system without any initialization operations. We perform non-selective measurements on the system via the scalar coupling and the bit-phase flip noise on the device. The performance of this non-selective measurement does not depend on the initial state of the device, and so we do not need any state preparation for the device. Thus, it is sufficient to consider the state ρ 0 as the initial density matrix of the system-device in our experiments.
The pulse sequence that implements a non-selective measurement is where (θ) s φ is a θ rotation of the system along the axis (cos φ, sin φ, 0). τ z is a waiting time. During this waiting time the system and the device are coupled via scalar coupling and the device looses its coherence. To reduce possible complications caused by imperfect pulses, composite pulses (SCROFULOUS [20]) are employed for the (∓π/2) s π 2 rotations. Note that M ± = e ±(π/2)(σy/2)σ0 e −Jτz(σz ⊗σz)/4 e ∓(π/2)(σy/2)σ0 = e ∓Jτz(σx⊗σz )/4 , if we ignore the decoherence of the device. The pulse sequence that implements QZE experiment is where τ xy is a time delay to let the system dephase.
Simulation with operator sum formalism: In order to evaluate an implemented non-selective measurement, the dynamics of both the system and device has been simulated, including the effects of longitudinal relaxations, using the operator sum formalism [21].
First, we consider the device. It is assumed that the device is subjected to a bit-phase flip channel [22], whose time constant is T d , because of the paramagnetic salt in the sample. This parameter is determined by the densities of chloroform and magnetic impurity in the sample and, thus, it is controllable. If the evolution time t is divided into N t equal steps, then the device's state after the i-th step is obtained iteratively according to the equation where p d = t/(2N t T d ) and Ad( * , ρ) = * ρ * † . Then N t → ∞ gives the time development under this assumption. Second, we consider the system that is subjected to a longitudinal relaxation with a characteristic time T 1s . This process is represented similarly as Eq. (4), with p s = t/(N t T 1s ). Note that the longitudinal relaxation is a process toward the thermal state, well approximated as |0 0| in our case. If the dynamics of the whole system-device is considered under a Hamiltonian H with a general initial state ρ(0), the state after the i-th iteration is where U (H, t) = T exp −i t 0 dt ′ H(t ′ ) and T stands for the time-ordering operator. In short, we write this time development as follows, ρ(t) = lim Nt→∞ ρ Nt = V(H, T d , T 1s , t, ρ(0)). Let us consider the Hamiltonian for the scalar coupling between the system and device: H = Jσ z ⊗ σ z /4, where J is the coupling strength. This Hamiltonian becomes Eq. (1) when tracing out the device, with λ corresponding to J 2 . The system state is determined by the equation The signal from the system is obtained as T r((σ x ⊗ σ 0 )ρ). Fig. 2 shows the measured and simulated FID signals. Taking a proper parameter set of (T d , T 1s ) = (6.5, 300) ms reproduces the measured FID well. T d and T 1s are equal to T 1 = T 2 of the device and T 1 of the system within experimental errors, respectively. Fig. 2 implies the validity of our simulation with operator sum formalism.
To simulate QZE experiments described by the pulse sequence given in Eq. both τ xy and τ z are computed using Eq. (7). Additionally, the time development during each pulse (±π/2) s π 2 with duration τ and strength ω 1 = π/2τ is described by Using both Eqs. (7) and (8) simulates the QZE experiments. Fig. 3 illustrates this numerical simulation, and a good agreement with the experimental results for both M ± can be seen. T d and T 1s are the same for Fig. 2. Note that the simulations with M + are very similar to those with T 1s = ∞ (without longitudinal relaxation). This is quite reasonable since the state during τ z in the case of M + is very close to ρ 0 . Therefore, we conclude that M + can be employed as a non-selective measurement. QZE experiments: Experiments with various τ z = 0.8, 1.0, 1.2, 1.5, 2.0, and 2.5 ms and τ xy = 0.3 ms had been performed. Note that we employ M + as a non-selective measurement. We observe that the signals decay slower than the FID signal for all τ z 's, which is QZE. We only show the two cases for τ z = 0.8 and 2.5 ms in Fig. 4.
Conclusion: We successfully suppressed the phase decoherence of an ensemble of nuclear spins through the application of sequential non-selective measurements. It is a proof-of-principle demonstration of the QZE suppressing non-unitary evolutions. One interesting extension of this work would be the implementation of protection an arbitrary unknown state of a quantum system, using QZE, as discussed in the very early stage of the development of quantum information processing [23].