Classicalization of Quantum State of Detector by Amplification Process
It has been shown that a macroscopic system being in a high-temperature thermal coherent state can be, in principle, driven into a non-classical state by coupling to a microscopic system. Therefore, thermal coherent states do not truly represent the classical limit of quantum description. Here, we study the classical limit of quantum state of a more relevant macroscopic system, namely the pointer of a detector, after the phase-preserving linear amplification process. In particular, we examine to what extent it is possible to find the corresponding amplified state in a superposition state, by coupling the pointer to a qubit system. We demonstrate quantitatively that the amplification process is able to produce the classical limit of quantum state of the pointer, offering a route to a classical state in a sense of not to be projected into a superposition state.
Decoherence theory attempts to give an account of how classical behaviour emerges from the quantum description by invoking the notion of environment Zurek (2003); Schlosshauer (2005). But, prior to that, the question is what is the classical limit of quantum state of a macroscopic system111Note that we distinguish between ‘macroscopic’ (‘microscopic’) and ‘classical” (‘quantum’). The former refer to the system whereas the later refers to the description. in the first place? As Bell said: “What exactly qualifies some physical systems to play the role of measurer?” Bell (2004). More relevant to the present work, we refine this question: what exactly qualifies a quantum state to represent the physical state of an isolated classical object, e.g. a measurement apparatus or a cat for that matter?
Since the advent of quantum mechanics, there have been several proposals as to how the classical behaviour emerges from the underling quantum dynamics (for a review see Landsman (2006)). The first one, proposed by Bohr as a correspondence principle, claimed that the classical behaviour emerges in the limit of large quantum numbers Bohr (1923). However, it turned out that this is not true in general. As a counterexample, a microscopic oscillator in a energy state with large quantum number doesn’t faithfully represent the classical behaviour. This motivated Schrödinger to propose an alternative quantum description of a “classical-like” state of the oscillator, the so-called “coherent state”, whose dynamics closely resembles that of the classical one Schrödinger (1926).
Despite the promising classical features of coherent states, because of the linearity of Schrödinger equation, a system prepared in a coherent state can be driven into a superposition state by coupling to a microscopic system Schleich et al. (1991); Bužek et al. (1992); Chai (1992); Janszky et al. (1993, 1995); Szabo et al. (1996); Wiseman and Vaccaro (2002a, b); Johansen (2004a). In fact, the non-classical features of coherent states are even recognized to be useful in quantum information science Cirel’son (1980); Grosshans et al. (2003). The problem remains even if we consider a more classical state; a high-temperature thermal coherent state Johansen (2004b); Jeong and Ralph (2006, 2007); Zheng (2007); Jeong et al. (2008). It has been also demonstrated that such seemingly classical states violate the Leggett-Garg inequality Asadian et al. (2014). Therefore, it seems that the thermalization process by itself is not sufficient to classicalize a macroscopic system.
In the early 1980s, it was suggested that the amplification process in detectors might have a crucial role in the quantum-to-classical transition Machida and Namiki (1980a, b); Araki (1980); Namiki et al. (1998). The development of MASERs as possible amplifiers triggered a flurry of interest in the quantum description of amplification process in 1960s Louisell et al. (1961); Heffner (1962); Haus and Mullen (1962); Gordon et al. (1963a, b), leading to the realization that a linear amplifier unavoidably adds noise to the input signal. This fundamental limit is expressed formally as a bound on the second moment of the added noise Caves (1982); Clerk et al. (2010). In this context, quantum non-demolition measurements are designed to circumvent the limitations imposed by such limit when performing repeated measurements of quantum states Braginskii (1988); Caves et al. (1980); Bocko and Onofrio (1996). The quantum limit on the entire distribution of the added noise was provided just recently Caves et al. (2012). Notably, a realistic amplifier transforms the quantum state of detector into a form which is not necessarily equivalent to the Gaussian distribution of the thermal coherent state. The amplified state is supposed to represent the pointer state from which the measurement result is read-out, and thus we expect to have a definite value at the macroscopic scale. Therefore, we believe that it is timely to revisit this problem in the light of the recent advances in the amplifiers and see if the quantum description of a realistic amplifier yields the most “classical-like” quantum state.
In this work, we analyse the classicality of the pointer state after the amplification process, by looking at the possibility of projecting them into superposition states (see FIG. 3). In particular, we focus on a mathematical model put forward by Caves and co-workers to examine phase-preserving linear amplifiers Caves et al. (2012). In essence, the amplification of the input mode of the pointer requires it to be coupled to an external mode, called ancillary mode, which adds noise to the output mode. The state of the ancillary mode determines the effect of the added noise on the input state. Our results show that a non-ideal (realistic) amplifier is able to produce classical states.
Ii Classical Limit of Quantum State
What is exactly the classical limit of quantum state of the pointer? Suppose that the pointer at the microscopic scale is prepared in a superposition of two energy states. According to the decoherence theory, given the appropriate interaction between the microscopic system and the the surrounding environment, the reduced state of the pointer is an incoherent mixture of coherent states, e.g.,
which is widely considered as a classical mixture of distinct states. From quantum-mechanical point of view, however, it can be equally well represented by
where we defined . A priorily quantum mechanics is unbiased toward any of these two inequivalent representations; as there is no unique ensemble decomposition of the mixed states. This so-called basis ambiguity Schlosshauer (2005); Bassi and Ghirardi (2003) might look a mathematical problem, but it has physical consequences. Obviously, for , a measurement in basis results non-classical states222The same argument applies even to a thermal coherent state Jeong and Ralph (2006). However, the representation of the pointer’s state in all bases should be equivalent, because all dynamical operators of the pointer as a macroscopic system commutes with each other.
Form the above argument, we conclude that the classical state of the pointer is realized if two conditions are fulfilled. First, if we expand the state in a given basis, there should be no interference between constituting states. This is a necessary but insufficient condition, because constituting states in another basis might have non-zero interference. This means that although there is no interference between constituting states, each can still be projected to a superposition state. Therefore, we add a second condition, asserting that there should be no interference for each constituting state.
To fulfil the second condition, the state should contains a sufficient amount of noise, hindering a generation of entanglement with a genuine microscopic quantum state, and yet maintaining a well-defined pointer position in the coarse-grained macroscopic scale, expected from the macroscopic pointer. In fact, the classical state does not reveal the detailed quantum state of the pointer. This means that there might be many quantum states, in the fine-grained scale, basically producing the same classical state of the pointer. Therefore, the classical state of the pointer is represented by a probabilistic mixture of all these consistent quantum states.Here, since the classical signal does not distinguish between the micro-states, we assign equal probability to them. This produces a top-flatted probability distribution (see FIG. 1). We can define the corresponding classical state as
This is in accordance with the idea that coarse-grained measurements give rise to classical behaviour Kofler and Brukner (2007).
We proved that this noisy, top-flat distribution destroys the interference effect in all bases in the appendix.
But how the classical state (3) is realized in the first place? The recent advances on quantum description of realistic amplifiers, motivated us to look at the classicality of amplified states. Amplification process not only amplifies the input signal but also adds noise to the input state, producing a macroscopic signal which is noisy at the microscopic scale, and yet having well-defined pointer position state at the coarse-grained macroscopic scale. It is worth mentioning that the resulting amplified state is not in an equivalent form to a thermal coherent state. Therefore, it is intriguing to see if, unlike a thermal coherent state, its probability of being found in a superposition state, vanishes. Before investigating this problem let us briefly review the amplification process.
Iii Amplification Process
The setting for our analysis is a bosonic mode , called primary mode, which is to undergo amplification process. The type of amplification one typically thinks of in physics is linear amplification, which means that the output mode is linearly related to the input mode (e.g. being multiplied by some fixed amplitude gain ). Here, we shall deal with linear amplification, since it is straightforward to treat mathematically. Also we wish to amplify both quadratures of the input mode with the same gain. This type of amplification is often referred to as phase-preserving amplification (FIG. 2).
A perfect phase-preserving linear amplifier transforms the input mode directly to the output one: . However, this transformation violates unitarity. Physically, this means that amplification of the primary mode requires it to be coupled to an external mode , called ancillary mode, which adds noise to the output mode. When referred to the input, the output noise is constrained as Caves (1982). The minimum added noise, corresponding to the lower bound, is the half-quantum of vacuum noise. The amplifier working with the minimum added noise is called an ideal amplifier. The simplest model of such an amplifier is provided by a parametric amplifier Mollow and Glauber (1967a, b); Collett and Walls (1988). The ideal amplified state of the input state is given by Caves et al. (2012)
where is the two-mode squeezing operator, with the amplitude gain being , and is the positive density operator of the ancillary mode. The density operator is diagonal in the number basis
where ‘’s are the corresponding eigenvalues. Note that for an ideal amplifier, we have .
The amplified state for the complete distribution of the added noise is given by the amplifier map Caves et al. (2012)
The superoperator amplifies the input state with the amplitude gain . For a coherent input state, the output of is just a displaced coherent state: . The superoperator adds a noise to the output state by smearing out a phase-space distribution into a broader distribution
where marks the slot where the input to the superoperator goes and is the displacement operator for the mode ; . The real-valued function is called the smearing function. This function is independent of the input state, but it depends on the gain as
Note that the smearing function of an ideal linear amplifier is isomorphic to the Glauber-Sudarshan P function of the thermal coherent state , where is the variance and is the displacement in the phase-space. We safely assume that before amplification due to the internal dynamics of the macroscopic system, the system is evolved to the most classical pure state, i.e. a coherent state .
Our analysis here is mathematically based on the optical states. Nonetheless, the generalization of our approach to the corresponding mechanical states is straightforward. For example, a superconducting qubit can manifest macroscopic distinguishable states by injecting currents of opposite verses. The corresponding wave functions are effectively Gaussian states. They are isomorphic to the coherent states in our analysis.
Iv A case study: A pointer interacting with a two-level system
Now we check whether an amplified state is classical in the sense we defined in the second section. The state of the pointer after amplification is given as
We consider as the initial state of the pointer, interacting with a qubit system, as illustrated in Fig. 3.
The qubit is prepared in state , where and are the spin-up and spin-down states of the qubit in the direction. Qubit being in interacts with the pointer prepared in the amplified coherent state , with the interaction Hamiltonian being , where is the coupling strength. After , the resulting state is given by
where and . Upon the qubit measurement on the basis , the detector’s state is projected into a superposition state
is the probability of finding the amplifier in the corresponding superposition state, .
The probability distribution of diagonal and off-diagonal elements of can be obtained as and , respectively. The two peaks along axis are well-separated and represent the pointer positions, if the measurement has been performed in basis. Interference fringes along axis are a typical signature of quantum superposition between macroscopically distinct states. It is worth mentioning that the interference pattern indicates the generated quantum entanglement between the qubit and the amplified mode proceeding the projective measurement on the qubit. Therefore, any amplified state truly representing the classical limit should suppress the generated entanglement with the qubit. The amplitude and the pattern of peaks and interference fringes depend on the choice of ancillary eigenvalues . The only constraint imposed by quantum mechanics is to guarantee that is a valid density operator, s should be non-negative333Carlton M. Caves, Private Communication.. It is not yet clear in detail how s are parameterized in actual “non-ideal” amplifiers. Nonetheless, we found that the most appropriate choice to ensure the emergence of classicality is (see Fig. 4).
The peaks and interference fringes for a high-temperature thermal coherent state and the corresponding amplified coherent states for ancillary mode with one, two and three available states are plotted in FIG. 5, for an optimized choice of s.
Notably, the non-ideal amplification process for certain range of parameters produces a probability distribution (see Fig. 5(e,g)) which has similar top-flatted shape with that of our heuristic model illustrated in FIG. 1.
According to FIG. 5, as we include the non-ideal terms in the ideal phase-preserving linear amplifier, the interference effects are weakened gradually. This also can be verified, using a quantitative measure of macroscopicity. Lee and Jeong introduced a general and inclusive measure of macroscopicity in the phase space as Lee and Jeong (2011)
where is the Wigner function of state and is the number of modes. The measure is plotted for amplified coherent states with the first term, with first two terms and with first three terms in FIG. 6. As we expected, this shows that with a large gain , as we include the non-ideal terms in the ideal amplifier, the probability of macroscopic quantum superpositions decreases.
To ensure that the non-ideal linear amplifier has a non-thermal effect, we compare the interference fringes appeared in the high-temperature thermal coherent state with those of a non-ideal amplified coherent state with equal purity, . For a thermal state with , the amplifier gain of the corresponding amplified coherent state with the first term, the first two terms and the first three terms are , and , respectively. For a fixed purity, the interference fringes for an amplifier with the corresponding terms are plotted in FIG. 7. The suppression of interference shows that including non-ideal terms to an ideal amplifier has a non-thermal effect. One can imagine that the inclusion of many of such terms in the actual detector completely vanishes interference.
V Discussion & concluding remarks
It is of significance to come up with a form of quantum state representing the state of a macroscopic system, and resolving the problem of basis ambiguity, without invoking the entanglement with the environment or modifying the laws of quantum mechanics. That is why, we look at the amplifier, as it is supposed to describe the pointer’s state, and examine the basis ambiguity of the amplified state via a non-classical evolution sketched in FIG. 1. Our results show that an amplifier state can demonstrate a behavior which is not similar to that of a thermal coherent state when it is coupled to a microscopic state, yet having well-defined pointer position state at the coarse-grained macroscopic scale. Nonetheless, we stress that to achieve a more conclusive result we need to examine a non-ideal amplifier state with realistic parameters. To our best knowledge this is yet to be fully identified, and this issue requires further research.
We would like to thank Dr. Saleh Rahimi-Keshari for his instructive comments. FTG acknowledges the financial support of Iranian National Science Foundation (INSF) for this work. *
Appendix A Resolution of Basis Ambiguity Problem for Classical State
Here we prove that the classical state (3) is the same in all bases. In order to make more connection to the classical phase space description, we shift to the Wigner representation
where is the displacement operator. As the summation is over all coherent states which lie in a single slot in the coarse-grained scale we have
where is a coarse-grained phase amplitude which has classical dynamics. The mixed state corresponding to (1) would then be
We may expand a new basis in terms of the old one as
The mixed state (3) can be equivalently decomposed in the new basis
where the corresponding Wigner representation is
The interference term is negligible, thus the decomposition in the new basis (6) is approximately the decomposition in the old basis (3). This means that the classicalized state of the pointer leads to a unique ensemble decomposition at coarse-grained scale. Therefore, and emerge dynamically as the unique coarse-grained or classical preferred basis.
- Zurek (2003) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
- Schlosshauer (2005) M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2005).
- Bell (2004) J. S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge University Press, 2004).
- Landsman (2006) N. P. Landsman, Handbook of the Philosophy of Science 2, 417 (2006).
- Bohr (1923) N. Bohr, Zeitschrift für Physik 13, 117 (1923).
- Schrödinger (1926) E. Schrödinger, Naturwissenschaften 14, 664 (1926).
- Schleich et al. (1991) W. Schleich, M. Pernigo, and F. Le Kien, Phys. Rev. A 44, 2172 (1991).
- Bužek et al. (1992) V. Bužek, A. Vidiella-Barranco, and P. L. Knight, Phys. Rev. A 45, 6570 (1992).
- Chai (1992) C.-l. Chai, Phys. Rev. A 46, 7187 (1992).
- Janszky et al. (1993) J. Janszky, P. Domokos, and P. Adam, Phys. Rev. A 48, 2213 (1993).
- Janszky et al. (1995) J. Janszky, P. Domokos, S. Szabó, and P. Adam, Phys. Rev. A 51, 4191 (1995).
- Szabo et al. (1996) S. Szabo, P. Adam, J. Janszky, and P. Domokos, Phys. Rev. A 53, 2698 (1996).
- Wiseman and Vaccaro (2002a) H. M. Wiseman and J. A. Vaccaro, Phys. Rev. A 65, 043605 (2002a).
- Wiseman and Vaccaro (2002b) H. M. Wiseman and J. A. Vaccaro, Phy. Rev. A 65, 043606 (2002b).
- Johansen (2004a) L. M. Johansen, Phys. Let. A 329, 184 (2004a).
- Cirel’son (1980) B. S. Cirel’son, Lett. Math. Phys. 4, 93 (1980).
- Grosshans et al. (2003) F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, Nature 421, 238 (2003).
- Johansen (2004b) L. M. Johansen, J. Opt. B: Quantum Semiclassical Opt. 6, L21 (2004b).
- Jeong and Ralph (2006) H. Jeong and T. C. Ralph, Phys. Rev. Lett. 97, 100401 (2006).
- Jeong and Ralph (2007) H. Jeong and T. C. Ralph, Phys. Rev. A 76, 042103 (2007).
- Zheng (2007) S.-B. Zheng, Phys. Rev. A 75, 032114 (2007).
- Jeong et al. (2008) H. Jeong, J. Lee, and H. Nha, J. Opt. Soc. Am. B 25, 1025 (2008).
- Asadian et al. (2014) A. Asadian, C. Brukner, and P. Rabl, Phys. Rev. Lett. 112, 190402 (2014).
- Machida and Namiki (1980a) S. Machida and M. Namiki, Prog. Theor. Phys. 63, 1457 (1980a).
- Machida and Namiki (1980b) S. Machida and M. Namiki, Prog. Theor. Phys. 63, 1833 (1980b).
- Araki (1980) H. Araki, Progr. Theor. Phys. 64, 719 (1980).
- Namiki et al. (1998) M. Namiki, H. Nakazato, and S. Pascazio, Decoherence and quantum measurements (World Scientific, 1998).
- Louisell et al. (1961) W. Louisell, A. Yariv, and A. Siegman, Phys. Rev. 124, 1646 (1961).
- Heffner (1962) H. Heffner, Proceedings of the IRE 50, 1604 (1962).
- Haus and Mullen (1962) H. A. Haus and J. Mullen, Phys. Rev. 128, 2407 (1962).
- Gordon et al. (1963a) J. Gordon, L. Walker, and W. Louisell, Phys. Rev. 130, 806 (1963a).
- Gordon et al. (1963b) J. Gordon, W. H. Louisell, and L. Walker, Phys. Rev. 129, 481 (1963b).
- Caves (1982) C. M. Caves, Phys. Rev. D 26, 1817 (1982).
- Clerk et al. (2010) A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
- Braginskii (1988) V. B. Braginskii, Physics-Uspekhi 31, 836 (1988).
- Caves et al. (1980) C. M. Caves, K. S. Thorne, R. W. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980).
- Bocko and Onofrio (1996) M. F. Bocko and R. Onofrio, Rev. Mod. Phys. 68, 755 (1996).
- Caves et al. (2012) C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, Phys. Rev. A 86, 063802 (2012).
- Bassi and Ghirardi (2003) A. Bassi and G. Ghirardi, Phys. Rep. 379, 257 (2003).
- Kofler and Brukner (2007) J. Kofler and C. Brukner, Phys. Rev. Lett. 99, 180403 (2007).
- Mollow and Glauber (1967a) B. Mollow and R. Glauber, Phys. Rev. 160, 1076 (1967a).
- Mollow and Glauber (1967b) B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1967b).
- Collett and Walls (1988) M. Collett and D. Walls, Phys. Rev. Lett. 61, 2442 (1988).
- Lee and Jeong (2011) C.-W. Lee and H. Jeong, Phys. Rev. Lett. 106, 220401 (2011).