The quantum Zeno and anti-Zeno effects with strong system-environment coupling

To date, studies of the quantum Zeno and anti-Zeno effects focus on quantum systems that are weakly interacting with their environment. In this paper, we investigate what happens to a quantum system under the action of repeated measurements if the quantum system is strongly interacting with its environment. We consider as the quantum system a single two-level system coupled strongly to a collection of harmonic oscillators. A so-called polaron transformation is then used to make the problem in the strong system-environment coupling regime tractable. We find that the strong coupling case exhibits quantitative and qualitative differences as compared with the weak coupling case. In particular, the effective decay rate does not depend linearly on the spectral density of the environment. This then means that, in the strong coupling regime that we investigate, increasing the system-environment coupling strength can actually decrease the effective decay rate. We also consider a collection of two-level atoms coupled strongly with a common environment. In this case, we find that there are further differences between the weak and strong coupling cases since the two-level atoms can now indirectly interact with one another due to the common environment.


Results
Spin-boson model with strong system-environment coupling. We start with the paradigmatic spin-boson model Hamiltonian 37,47,48 which we write as (we set ħ = 1 throughout) where the system Hamiltonian is 2 2 , the environment Hamiltonian is k k k k , and the system-environment coupling is ε is the energy level difference of the two-level system, Δ is the tunneling amplitude, ω k are the frequencies of the harmonic oscillators, b k and † b k are the annihilation and creation operators for the harmonic oscillators, and σ x and σ z are the standard Pauli operators. The 'L' denotes the 'lab' frame. If the system-environment coupling is strong, we cannot treat the system-environment coupling perturbatively. Furthermore, the system-environment correlation effects are significant as well in general. To motivate our basic approach in this strong coupling regime, we note that if the system tunneling amplitude is negligible and the initial system state is an eigenstate of σ z , then, even though the system and the environment are strongly interacting, the evolution of the system state is negligible. This then means that we should look to unitarily transform H L such that the effective system-environment coupling contains the tunneling amplitude Δ. This unitary transformation is provided by the 'polaron' transformation, whereby the system-environment Hamiltonian in this new 'polaron' frame becomes , with = χ X e (see Methods for details). Now, if the tunneling amplitude is small, we can use time-dependent perturbation theory, treating V as the perturbation. This is the key idea to deal with the strong system-environment coupling regime. Although the system and the environment are strongly interacting, in the polaron frame, they are effectively interacting weakly. Let us now use this fact in order to calculate the survival probability, and thereby the effective decay rate. For concreteness, we assume that the initial state prepared is |↑〉, where σ z |↑〉 = |↑〉. In other words, we consider the same initial state as that considered in the analysis of the usual population decay model 23,24 . At time t = 0, we prepare the system state |↑〉, and we subsequently perform measurements with time interval τ to check if the system state is still |↑〉 or not. The survival probability after time interval τ is then s(τ) = Tr s,B [|↑〉〈↑|ρ L (τ)], where ρ τ ( ) L is the combined density matrix of the system and the environment at time τ just before the projective measurement. Then, It is important to note that the initial state that we have prepared cannot simply be taken as the usual product H B since the system and the environment are strongly interacting and consquently there will be significant initial system-environment correlations 49,50 . Rather, the initial state that we should consider is ρ = . This is a similar approximation as the usual assumption that the initial system-environment state is ρ ρ ⊗ (0) S B since, in the polaron frame, the system and the environment are weakly interacting. We thus get Our objective then is to find ρ τ ( ) S , given the initial system-environment state . We find that (see the Methods section) where Tr B denotes taking trace over the environment, the environment correlation functions are defined as = , and h.c. denotes the hermitian conjugate. Now, since the system-environment coupling in the polaron frame is weak, we can neglect the build up of correlations between the system and the environment. Thus, we can write the survival probability after time τ = t N , where N is the number of measurements performed after time t = 0, as ( ) , thereby defining the effective decay rate τ Γ( ). It then follows that τ τ Γ = − τ s ( ) ln ( ) 1 . Since we have the system density matrix in the polaron frame, we can work out the survival probability τ s( ) and hence the effective decay rate τ Γ( ). The result is that (see the Methods section for details) and the spectral density of the environment has been introduced as . At this point, it is useful to compare this expression for the effective decay rate for the case of strong system-environment coupling with the case of the usual population decay model where the effective decay rate is 23,24 . It should be clear that for the strong system-environment coupling case, the effective decay rate given by Eq. (6) has a very different qualitative behavior. In particular, the effective decay rate can no longer be regarded as simply an overlap integral of the spectral density of the environment with a sinc-squared function. Rather, the effective decay rate now has a very prominent non-linear dependence on the spectral density, leading to very different behavior as compared with the population decay case. For example, as the system-environment coupling strength increases, Φ t ( ) R increases, and thus we expect Γ(τ) to decrease. To make this claim concrete, let us model the spectral density as ω ωω = where G is a dimensionless parameter characterizing the system-environment coupling strength, ω c is the cutoff frequency, and s is the Ohmicity parameter 48 . For concreteness, we look at the Ohmic case (s = 1). In this case, (1 ) The double integral can be worked out numerically. Results are shown in Fig. 1(a) for different system-environment coupling strengths G. For the strong system-environment regime that we are dealing with, it is clear that increasing the system-environment coupling strength G actually decreases the effective decay rate. This is in contrast with what happens in the weak system-environment regime for the paradigmatic population decay model [see Fig. 1 Here it is clear that increasing the system-environment coupling strength increases the effective decay rate as expected. It should also be noted that the behaviour of Γ(τ) as a function of τ allows us to identify the Zeno and anti-Zeno regimes. One approach is to simply say that if Γ(τ) decreases when τ decreases, we are in the Zeno regime, while if Γ(τ) increases if τ decreases, then we are in the anti-Zeno regime 23,30,33,35 . From Fig. 1(b), it should also be noted that increasing the coupling strength does not change the qualitative behavior of the Zeno to anti-Zeno transition, but for the strong coupling regime [see Fig. 1(a)], while we only observe the Zeno effect for G = 1, both the Zeno and anti-Zeno effects are observed for G = 2.5. Similarly, as shown in Fig. 2(a), increasing the cutoff frequency for the strong coupling case decreases the effective decay rate, but the opposite behaviour is observed for the weak coupling case [see Fig. 2 In our treatment until now, we have considered the change in the system state due to the tunneling term. This tunneling term, due to its presence in , leads to the system state changing even if the system and the environment are not coupled to each other. Thus, an alternative way to quantify the effective decay rate would be to remove the evolution due to the system Hamiltonian (in the 'lab' frame) H S L , before performing each measurement since what we are really interested in is the change in the system state due to the system-environment interaction. A similar approach has been followed in refs 35, 36 and 51 Therefore, we now derive an expression for the effective decay rate of the system state when, just before each measurement, we remove the system evolution due to H S L , . The survival probability, after one measurement, is now (starting from the state |↑〉) Notice now the presence of τ e iH S L , and τ − e iH S L , which remove the evolution of the system due to the system Hamiltonian before performing the measurement. Once again transforming to the polaron frame, we obtain Since we are assuming that the tunneling amplitude is small, the unitary operator τ − e iH S P , can be expanded as a perturbation series. At the same time, τ − e iH can also expanded as a perturbation series. Keeping terms to second order in the tunneling amplitude (see the Methods section), we find that now the modified decay rate Γ n (τ) is n mod where the modification to the previous decay rate is Using these expressions, we have plotted the behavior of Γ n (τ) for the strong system-environment coupling regime in Fig. 3(a). It should be clear that once again increasing the system-environment coupling strength generally decreases the effective decay rate Γ n (τ). This is in sharp contrast with what happens in the weak coupling regime. For the weak coupling case, it is known that 36 n 0 where the filter function ω τ 2 . Using these expressions, we can investigate how the decay rate varies as the measurement interval changes for different system-environment coupling strengths in the weak coupling regime. Typical results are illustrated in Fig. 3(b) from which it should be clear that increasing the coupling strength in the weak coupling regime increases the effective decay rate. Furthermore, changing the coupling strength has no effect on the measurement time interval at which the Zeno to anti-Zeno transition takes place for the weak coupling regime as the three curves in Fig. 3(b) achieve their maximum value for the same value of τ. This is not the case for the strong coupling regime [see Fig. 3(a)]. At this point, it is worth pausing to consider where the qualitative difference in the behavior of the effective decay rate in the weak and the strong coupling regime comes from. The effective decay rate is derived from the survival probability after one measurement τ s( ). For both the weak and the strong coupling regimes, the survival . This is not the case for the strong coupling due to the significant system-environment correlations. Thus, we can say that the qualitative difference in the behavior of the effective decay rate is because of the presence of the system-environment correlations. It seems that these correlations can protect the quantum state of the system -as the coupling strength increases, these correlations become more and more significant, and at the same time, the effective decay rate goes down.
Large spin-boson model with strong system-environment coupling. Let us now generalize the usual spin-boson model to deal with N S two-level systems interacting with a common environment. In this case, the system-environment Hamiltonian (in the 'lab' frame) is given by 36,40,50 where J x y z , , are the usual angular momentum operators obeying the commutation relations k l klm m . We now start from the spin coherent state |j〉 such that J z |j〉 = j|j〉 with = j N /2 S . Other eigenstates of J z can be considered as the initial state in a similar manner. Our objective is to again perform repeated projective measurements, described by the projector |j〉〈 j|, with time interval τ and thereby investigate what happens to the effective decay rate. As before, the survival probability after one measurement is τ ρ = , and = ± ± J J iJ x y are the standard raising and lowering operators. Interestingly, the transformed Hamiltonian now contains a term proportional to J z 2 . This term arises because the collection of two-level systems interacting with the collective environment are indirectly interacting with each other. This term is obviously proportional to the identity operator for a single two-level system, and thus has no influence for a single two-level system. If the tunneling amplitude is small, then we again use perturbation theory and assume that, in the polaron frame, the system-environment correlations can be neglected. We find that now the effective decay rate is (see the Methods section) are the same as defined before. This result obviously agrees with the result that we obtained for a single two-level system. Moreover, it is clear from Eq. (20) that increasing the system-environment coupling strength G should reduce the effective decay rate due to the −Φ ′ e t ( ) R factor in the integrand. This is precisely what we observein Fig. 4(a). Furthermore, it may be thought that increasing j (or, equivalently, N S ) increases the effective decay rate. On the other hand, the dependence on j is not so clear because of the presence of the indirect interaction. Namely, increasing j increases the oscillatory behavior of the integrand due to the dependence of the integrand on . Thus, once the integral over this rapidly oscillating integrand is taken, we can again get a small number. Such a prediction is borne out by Fig. 4(b) where the effective decay rate has been plotted for different values of j. It is obvious that there is a big difference between the single two-level j 0 5 (red, dashed curve), j = 1 (dot-dashed, magenta curve), and j = 2 (solid, blue curve). We have s = 1, ε = 1, ω = 10 c , and Δ = 0.05. The initial state is |j〉. system case and the more than one two-level system case. Furthermore, it seems that increasing j can largely reduce the value of the effective decay rate, meaning that in the strong coupling regime, the indirect interaction helps in keeping the quantum state alive.
Let us now consider the situation where the evolution to the system Hamiltonian is removed before each measurement. In the polaron frame H S L , becomes The major difference now compared to the previous single two-level system case is that the total system-environment Hamiltonian in the polaron frame 2 contains a term (namely, κ − J z 2 ) that is not part of the system Hamiltonian in the polaron frame. As a result, when the system evolution is removed just before performing each measurement, the evolution induced by this extra term survives. Keeping this fact in mind, the effective decay rate τ Γ ( ) n is now n mod where τ Γ( ) is given by Eq. (20) and . It should be obvious that we observe multiple Zeno-anti Zeno regimes. Also, increasing the coupling strength does not generally increase the effective decay rate τ Γ ( ) n . This behavior should be contrasted with the weak coupling scenario. For weak coupling, it has been found that the effective decay rate is still given by Eq. (14), but now the filter function is N S times the filter function given by Eq. (15) 36 . Thus, increasing the coupling strength should now increase the effective decay rate. This is precisely what is observed in Fig. 5(b). Consequently, the weak coupling and the strong coupling regimes are very different for the strong and the weak coupling regimes. The difference is again due to the system-environment correlations.

Discussion
We have investigated the quantum Zeno and anti-Zeno effects for a single two-level system interacting strongly with an environment of harmonic oscillators. Although it seems that perturbation theory cannot be applied, we have applied a polaron transformation that can make the coupling strength effectively small in the transformed frame and thereby validate the use of perturbation theory. We have obtained general expressions for the effective decay rate, independent of any particular form of the spectral density of the environment. Thereafter, we have shown that the strong coupling regime shows both qualitative and quantitative differences in the behavior of the effective decay rate as a function of the measurement interval and the QZE to QAZE transitions as compared with the weak system-environment coupling scenario. The effective decay rate is no longer an overlap integral of the spectral density of the environment and some other function. Rather, there is a very pronounced non-linear dependence on the spectral density of the environment. Most importantly, increasing the coupling strength in the strong coupling regime can actually reduce the effective decay rate. These differences can be understood in terms of the significant role played by the system-environment correlations. Moreover, we have extended our results to many two-level systems interacting with a common environment. Once again, we obtained expressions for the effective decay rate that are independent of the spectral density of the environment. We illustrated that in this case as well the behavior of the effective decay rate is very different from the commonly considered weak coupling regime. Our results should be important for understanding the role of repeated measurements in quantum systems that are interacting strongly with their environment.

The polaron transformation. For completeness, let us sketch how to transform the spin-boson
Hamiltonian to the polaron frame 39,40,[42][43][44]46 . We need to find = χσ χσ − H e H e L /2 /2 z z . We use the identity This is simply a c-number, so the higher-order commutators are zero. Furthermore, this c-number leads to a constant shift in the transformed Hamiltonian, and can thus be dropped. Putting all the commutators together, we find that . Thus, we finally have the required Hamiltonian in the polaron frame. For the large spin case, the calculation is very similar 40 . The major difference is that now the c-number term that we dropped before cannot be dropped any longer since this term is proportional to J z 2 (for the spin half case, this is proportional to the identity operator, so this is just a constant shift for the spin half case). Namely, we now find that , where τ U ( ) 0 is the unitary time-evolution operator corresponding to H S and H B , while τ U ( ) I is the 'left over' part that we can find using time-dependent perturbation theory. Writing the system-environment coupling in the polaron frame as ∑ ⊗ Correct to second order in the tunneling amplitude Δ (in particular, we assume that Δτ is small enough such that higher order terms can be ignored), we can then write . Eq. (27) can now be simplified term by term. First, we find that is the system density matrix if the tunneling amplitude is zero. Next, we find that . S i m i l a r l y, . Carrying on,    Finding the effective decay rate. We now explain how to find the effective decay rate given by Eq. (6).
With the system density matrix at time τ available, we first calculate the survival probability s(τ). This can be done via τ = − s( ) 1 〈↓|ρ s (τ)|↓〉. Since the state |↓〉 is an eigenstate of H S , and ρ S (0) = |↑〉〈↑|, it is straightforward to see that . Therefore, What remains to be worked out is the environment correlation function . Substituting these expressions in the expression for the survival probability as well as the perturbation expansions for τ e iH and τ − e iH , and keeping terms up to second order in Δ, we find that the new survival probability consists of the previous survival probability plus some additional terms. It can be be easily seen that most of these additional terms, once the trace with the projector |↓〉〈↓| is taken, give zero. The additional terms that need to be worked out are to simply the inner products. Putting all the pieces together, we arrive at Eq. (12). The calculation of τ Γ ( ) n for the large spin case is quite similar. One only needs to be careful about the fact that the system-environment Hamiltonian, in the polaron frame, contains a term, namely κ − J z 2 , that is not a part of the transformed system Hamiltonian H S P , .