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Abstract
We discuss a model of two nonlinear quantum oscillators mutually coupled by linear interaction and continuously driven by external coherent excitation. For such a system, we analyze temporal correlations. We examine the violation of the Leggett-Garg inequality analysing various scenarios of measurements. These scenarios are based on the projection onto different Bell states. We show that the possibility of violation of the Leggett-Garg inequalities is associated with the use of different projectors.
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1. Introduction
One of the well-known differences between the classical and quantum worlds is that in quantum systems we can observe such kinds of correlations that cannot be observed in any classical system. Quantum correlations in the space domain, which include quantum entanglement [1–9], Einstein-Podolsky-Rosen (EPR) steering [10–18], and Bell’s nonlocality [19–27], have attracted the vast majority of interest. These types of nonclassical correlations cannot be explained by locality and reality assumptions. It means that the results of measurements of physical quantities are not determined by some local hidden variables. Quantum entanglement, EPR steering, and Bell’s nonlocality have found various implementations in the quantum information theory, especially in quantum communication [28–32], quantum cryptography [33–36], and quantum computations [37–40].
In recent years, the research in the area the quantum correlations has been extended to the time domain. Here, the presence of quantum correlations is related to the assumptions of macro-realism and non-invasive measurements. According to both assumptions, the physical systems can be in one of two or more different states at any moment in time, and the state of the system can be measured by a non-invasive measurement. Such a measurement does not change the current state of the system and does not affect the system’s state in the future. Based on these assumptions, A. J. Leggett and A. Garg derived in 1985 an inequality [41], currently known as the Leggett-Garg inequality (LGI). It is an equivalent of the Bell inequality (BI) for the temporal correlations. Whereas the violation of BI indicates the quantum correlations of two spatially separated systems, the violation of LGI reveals nonclassical temporal correlations, i.e. temporal quantum entanglement [42,43]. LGI has been found the most useful in the analysis of many systems as it allows to distinguish between the quantum and classical dynamics [44,45].
In the space domain, we can distinguish different types of correlations, such as entanglement, EPR steering, and the Bell nonlocality. In 2007, Wiseman et al. [46,47] showed that there is a certain hierarchy among these types of spatial correlations: Every nonlocal state in the Bell sense is steerable, and every steerable state is nonseparable. In other words, entanglement as a measure of quantumness is weaker than steering, and steering is weaker than the Bell nonlocality. Similarly as the quantum spatial correlations, also the temporal correlations form a hierarchy. The relation among temporal inseparability, temporal steering, and macrorealism is demonstrated in the papers [48,49]. The authors have proven that the hierarchy of temporal correlations exists and showed that the unsteerable temporal correlations are the subset of non-macrorealistic correlations.
Using the LGIs, we can analyze correlations between the measurement results performed at different times. If a non-invasive measurement is performed, the violation of LGI is associated with the existence of quantum temporal correlation that do not allow for the classical description. Therefore, for testing these correlations in physical systems, the preparation of ideal negative measurements giving the information about the system state without any direct interaction with this system plays a crucial role [41,50,51].
The violation of the LGI in quantum-mechanical systems was first experimentally observed in 2010 [52]. In the next years, other experimental implementations were also proposed [53–59]. The violation of the LGI has been recently investigated in various systems including nuclear magnetic resonance [60,61], photonics systems [62–65], a driven two-level system [66], spin systems [67], multi-level systems [68], quantum-electrodynamical systems in cavities (QED) [69], and many others. Additionally, the relations among the presence of nonclassical time correlations and existence of various phenomena have been analyzed, involving squeezing [70], coherence [71], or chaos [72]. In general, the temporal correlations have been a major subject of study in recent years, and these studies have developed into a rapidly growing branch of quantum mechanics [73–76].
In this paper, we study temporal correlations in the system of two coupled nonlinear oscillators driven by a continuous coherent excitation. Such a system is a very versatile source of entangled states. Under suitable conditions, it even behaves as the quantum scissors [77–79]. It means that, for the analyzed mutually interacting nonlinear oscillators, the evolution is limited to four two-mode states, and we have a source of maximally or almost maximally entangled states of two qubits.
The paper is organized as follows. In Sec. 2, we describe the system composed of two nonlinear mutually coupled quantum oscillators continuously driven by an external field. In Sec. 3, we introduce the LGIs and describe various types of measurements based on the projections onto the Bell states. In Sec. 4, we reveal the relationship among values of correlators, time intervals between the measurements, and system parameters from the point of view of the violation of LGIs. Finally, we bring our conclusions in Sec. 5.
2. Model
The two-mode bosonic system is composed of two nonlinear oscillators (characterized by the third-order susceptibility) mutually coupled by the linear interaction and externally driven in both modes by the continuous coherent excitation. In the interaction picture, the system Hamiltonian is given by the following formulas:
Temporal quantum correlations were discussed in a nonlinear coupler externally driven by ultra-short pulses in [72].
3. Leggett-Garg Inequality
To analyze temporal correlations, we will use the LGIs. Assuming an experiment in which $n$ measurements $(n> 2)$ are made at different times, and the measured observable has two eigenvalues $Q =\pm 1$, the LGI can be written as [41,83,84]
In Eq. (4), $K_n$ is the $n$-th order correlator
where $C_{ij}$ are the correlation functions defined with the use of the joint probability [68,83]:The quantity $p\left (q_i,t_i;q_j,t_j \right )$ is the probability of obtaining the results $q_i$ from the measurements performed at the time instant $t_i$, and the results $q_j$ at the time instant $t_j$. Such a probability is defined by the following equation [68,83]:
In what follows, we assume that the system wave function is projected onto one of the Bell states:
The two-mode coupled oscillatory system for weak excitations can be treated as a two-qubit system. For such case, the system dynamics is closed within the following four two-mode states $\vert n_a\rangle \vert n_b\rangle$ with $n_a(n_b)=\lbrace 0,1\rbrace$, that are used in Eq. (10).
For two-qubit systems, one can perform several types of measurements that differ in their sensitivity and the information provided by analyzing their outcomes. For our system we can define two types of measurements in which we differentiate between the specific single Bell state (A-type measurement) and a superposition of the Bell states (B-type measurement). We have to point out that, in the analyzed system of coupled oscillators, there occur several populated two-mode states. It is not a simple two-qubit system, and therefore, for the measurements performed, projectors defined in $\Pi _{-}$ in Eq. (9) have to include all the states considered in the system. Thus:
- (A) An observable Q in Eq. (8) is constructed through the projections (9) onto only one of the Bell states $\{|B_1\rangle,|B_2\rangle,|B_3\rangle,|B_4\rangle \}$ written in Eq. (10). Such measurement, that distinguishes only one of the Bell states, gives us information whether the system is in that specified Bell state or not;
- (B) An observable in Eq. (8) is constructed through the projection (9) onto the sum of two Bell states ($\vert B_1\rangle$ and $\vert B_3\rangle$) described by the projection:
This measurement distinguishes only between the appropriate superposition of the Bell states and, therefore, it is less sensitive than the measurement (A), which gives us the information about the specific Bell state found in the system.
The LGIs are related to the quantum correlations and they were introduced to verify whether the concept of macrorealism or the possibility of performing noninvasive measurements are plausible.
Any violation of these inequalities is a signature of the nonclassicality of the analyzed system state as well as inconsistency in the assumptions made when formulating the inequalities. In particular, the violation of LGIs indicates that either the macrorealism assumption is not correct, or the measurements made change the system state.
4. Results
In the present paper, we address the problem whether the continuously excited coupled system, that generates the entanglement in the two-qubit subsystem, violates the LGIs. We analyze the violation by inspecting the correlators of different orders for various values of the external excitation.
The problem of LGI violation can be analyzed by checking the inequalities with various types of measurements performed on the coupled system. For all types of measurements considered, we assume the system in its vacuum state at the beginning of the interaction.
In Figs. 1 and 2, we present the dependence of correlators $K_3$ — $K_6$ on the time delay $\tau$ for A-type and B-type measurements introduced in the previous section. The measurement (A) is performed with two different projection operators: in Fig. 1(a) we project onto the $\vert B_3\rangle$ state and in Fig. 1(b) onto the $\vert B_1\rangle$ state. In this analysis, we have chosen a small excitation strength $\beta /\chi =0.03$ to guarantee that, for the whole evolution period, we deal with the system states without additional two-mode states extending the original Bell states.
Fig. 1. Measurement (A). $K_3 - K_6$ correlators as a function of times between projective measurements $\tau = t_j - t_i$ for $\epsilon /\chi = 0.04$, $\beta /\chi = 0.03$ and projection operator $\Pi _+$: (a) $\Pi _+ = \vert B_3 \rangle \langle B_3\vert$, (b) $\Pi _+ = \vert B_1 \rangle \langle B_1\vert$
Fig. 2. Measurement (B). $K_3 - K_6$ correlators as a function of time delay $\tau = t_j - t_i$ between the projective measurements for $\epsilon /\chi = 0.04$, $\beta /\chi = 0.03$ and $\Pi _+ =\vert B_1 \rangle \langle B_1\vert +\vert B_3 \rangle \langle B_3\vert$
$k-$th order correlator (5) is associated with $k$ subsequent measurements made on the system at different times. In our case, the measurements are separated in time by $\tau$. Increasing the number of subsequent measurements of the same type we analyse the LGIs of higher order.
In two measurements of A-type, all the correlators indicate the violation of LGIs but with various values of the violation degree. There is a small shift in the times at which different correlators attain their maximal values. But we can see the difference in the correlators degrees for the two projectors. When we project onto the $\vert B_1\rangle$ state the correlators values are smaller than when projecting onto the $\vert B_3\rangle$ state and, for the increasing $\tau$, the higher-order projectors do not show the violation. That demonstrates the importance of the appropriate choice of projection in the measurement of the A-type.
When in the B-type measurement we are able to distinguish between pairs of the Bell states, for the projection operators like those written in Eq. (11), we obtain the violation of the LGIs by correlators up to the sixth order and with much larger values, as shown in Fig. 2. The maximal violation for all of the correlators takes place when the state $\vert B\rangle _3$ is generated in the system, Then the result of the measurement (in fact several measurements were performed on the system at different times) suggests that the system was in the state which belongs to the pair given in $\Pi _+$ in Eq. (11).
Analyzing different types of measurements performed on the same system, we can find the situations that are more likely to show the violation of an appropriate LGI even using high-order correlators. On the other hand, we can also identify the types of measurement with which the LGI violation would be more demanding.
In the situations analyzed above, the projection operators are defined by the Bell states. Therefore, a natural question arises how the violation of the LGI is related to the presense of spatial entanglement. It should be emphasized that, in the analyzed system with $\epsilon /\chi = 0.04$ and $\beta /\chi = 0.03$, strong entanglement in the generated states is present [85,86] and the accompanying negativity reaches the values up to $0.81$.
However, it is difficult to determine specific relation between the presence of spatial entanglement and the LGI violation. From Figs. 1(a) and 1(b), we can see that, using the same type of measurement with the projective operators defined by different Bell states, we obtain the LGI violations for different values of the time delay $\tau$.
For example, for the projective operator $\Pi _{+}= \vert B_3\rangle \langle B_3 \vert$ and time delay $\tau = \pi$ [see Fig. 1(a)], we observe the violation of LGIs. However, at the same time but for the projective operator $\Pi _{+}= \vert B_1\rangle \langle B_1 \vert$ [see Fig. 1(b)], any LGI violation is indicated, even though we are dealing with the same evolution of the system. Thus, the relation between the LGI violation and the presence of entanglement is difficult to formulate. It needs further investigation.
It should be emphasized that the LGIs is the temporal equivalent of the spatial Bell inequalities. The LGIs allow to test temporal correlations even in single systems, contrary to the Bell inequalities, that need quantum correlations between multipartite spatially separated systems. The difference between temporal and spatial entanglement is also manifested in the monogamy relation. Spatial entanglement satisfies the monogamy relation [2,87]. Two maximally spatially entangled qubits cannot be maximally entangled with others. On the other hand, temporal entanglement can be said to be polygamous. So, two maximally entangled events can still be maximally temporary entangled with two other events [88,89].
In Figs. 3–6, the maps presenting the maximal values of correlators for different types of performed measurements are presented depending on the excitation strength $\beta$ and the time delay $\tau$ between subsequent measurements $\tau$. The change of time delay $\tau$ means that the system starts from its initial state (the vacuum state of both modes) and evolves according to the Hamiltonian written in Eq. (1) for the period $\tau$ and then the appropriate projection is made. Various orders of correlators indicate the number of performed measurements. In all of these maps, the violation of the LGIs both above (areas with red color) and below (areas with blue color) the allowed classical values is shown, for details see Eq. (4).
Fig. 3. Measurement (A). $K_3-K_6$ correlators as a function of time delay $\tau = t_j-t_i$ between projective measurements and strength of excitation $\beta$, where $\Pi _{+}= \vert B_3\rangle \langle B_3 \vert$.
Fig. 4. Measurement (A). $K_3-K_6$ correlators as a function of time delay $\tau = t_j-t_i$ between projective measurements and strength of excitation $\beta$, where $\Pi _{+}= \vert B_1\rangle \langle B_1\vert$.
Fig. 5. Measurement (A). $K_3-K_6$ correlators as a function of time delay $\tau = t_j-t_i$ between projective measurements and strength of excitation $\beta$, where $\Pi _{+}= \vert B_2\rangle \langle B_2 \vert$.
Fig. 6. Measurement (B). $K_3-K_6$ correlators as a function of time delay $\tau = t_j-t_i$ between projective measurements and strength of excitation $\beta$, where $\Pi _{+}= \vert B_1\rangle \langle B_1\vert +\vert B_3\rangle \langle B_3 \vert$ and $\Pi _{-}=\mathbb {I}-\Pi _{+}$.
We can see that different types of measurements result in different abilities to violate the LGIs.
In the A-type measurement we can distinguish between a specific Bell state and any other state of the system. We can see that, for this type of measurement, the appropriate choice of projection operator $\Pi _{+}$ leads to the violation of LGIs of different orders. The proper choice of the Bell state in our system is crucial here. This can be seen when we analyze the differences in the correlators values corresponding to different $\Pi _+$ projectors in Figs. 3–5 devoted in turn to the effects of projections on the Bell states $|B_3\rangle$, $|B_1\rangle$, and $|B_2\rangle$.
The violation of the LGIs by projecting onto the state $\vert B_3\rangle$ and using the correlators up to the sixth order is demonstrated in Fig. 3. For smaller values of the external excitation $\beta$, the correlators exceed their border values at any time delay between the subsequent measurements only for values less than $5.3\pi$. When the excitation strength $\beta$ increases, the LGIs are violated only for specific values of time delay $\tau$. It holds that the larger the value of time delay $\tau$ is, the less probable the LGI violation is. This is clearly demonstrated in the graphs for higher-order correlators ($K_4$ - $K_6$) drawn in Fig. 3, but also the lowest-order correlator $K_2$ indicates that it is more difficult to violate the LGIs for stronger excitations.
Projections on the other Bell states whose performance is quantified in Figs. 4 and 5 indicate considerable smaller violations of the LGIs. Also the correlators of higher-than-the-third order exhibit only weak tendency to violate the LGIs.
The B-type measurements in our system are much less sensitive in indicating the LGI violation because they say whether the system is found in a superposition of a given pair of the Bell states (11) or not. For the considered coupled system, it is relatively easy to show that performing this type of measurement leads to the LGI violation. We show in Fig. 6 that for any order of correlators as well as arbitrarily strong external excitation $\beta$ the LGI violation can be obtained easily. In the maps of Fig. 6 one can observe certain kind of periodicity with respect to the time delay $\tau$. Also, the LGIs violation can be demonstrated for both sides of the inequalities (4): Apart from the violation of the relation $K_n\leq n-2$ for weaker excitations $\beta$ one can also observe the violation of the relations $-n\leq K_n$ and $-(n-1)\leq K_n$ for odd and even $n>2$ orders respectively. We note that in the A-type measurements, all of the correlators mainly violate the right-hand side of the appropriate inequalities.
5. Conclusions
We have considered temporal correlations in the nonlinear system of linearly coupled Kerr-type oscillators. These correlations were described via the appropriate Leggett-Garg inequalities written in Eq. (6). The violation of the Leggett-Garg inequalities means that either the concept of macrorealism is not valid or noninvasive measurements cannot be performed on the considered system. For each Leggett-Garg inequality, a projection on a specific quantum state has to be given to define the measurement. We have considered two different types of measurements found in the definition of Leggett-Garg inequalities: (A) Projections on a specific Bell state and (B) projection on a superposition of the Bell states. Different projections are endowed with different sensitivities for violating the Leggett-Garg inequalities.
Using both types of projections, the violation of the Leggett-Garg inequalities has been demonstrated using different time delays and even higher excitation strengths applying the correlators with increasing orders.
As it is easy to find the coupled nonlinear oscillators in the states that are the superpositions of the Bell states the simplest observation of the violation of the Leggett-Garg inequalities is expected with the projections onto these states.
Funding
Polish Minister of Education and Science (003/RID/2018/19); Narodowe Centrum Nauki (DEC- 2019/34/A/ST2/00081).
Acknowledgment
J. K. Kalaga and W. Leoński acknowledge the support of project no. 003/RID/2018/19, funding amount 11 936 596.10 PLN, from the program of the Polish Minister of Education and Science under the name "Regional Initiative of Excellence" in 2019–023. A. Kowalewska-Kudłaszyk acknowledges the support of the Maestro Grant No. DEC-2019/34/A/ST2/00081 of the Polish National Science Centre (NCN).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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