Abstract
In this manuscript, we explore the effects of continuous measurements upon the quantized electromagnetic field through a series of simple examples. For this purpose, we consider the SrinivasDavies model to describe the optical field dynamics probed continuously by a photodetector. Through the application of this continuous photodetection model to some specific situations, it is possible to cover some basic concepts of quantum mechanics such as the principle of superposition, the collapse of the wave function, the probabilistic character of the possible outcomes associated with projective measurements, as well as some advanced topics such as the description of open quantum systems, irreversible processes in quantum mechanics, decoherence and dissipation associated with nonunitary evolution of quantum systems. Besides, we also consider the important concept of entanglement between two electromagnetic fields and how it is affected by the photodetection process. This work aims to provide complementary material for undergraduate and graduate students interested in the effects of measuring devices acting on quantum systems.
Keywords:
Continuous measurements; Photodetection; Quantized optical fields
1. Introduction
The postulates of Quantum Mechanics (QM), following the Copenhagen interpretation, establish the foundations to describe nature at the microscopic scale. Inside the QM formalism, it is possible to describe atoms, molecules as well as light and their mutual interactions [^{1}[1] C. CohenTannoudji, B. Diu, Bernard and F. Laloë, Quantum Mechanics (Willey, New Jersey, 1977).,^{2}[2] J.J. Sakurai, Modern Quantum mechanics (AddisonWesley Reading, Boston, 1994)]. Since the birth of QM, the investigations about coherence, quantum dynamics, projective measurements, and the intrinsic probabilistic nature of the outcomes associated with these kinds of measurement, are extremely important to check the veracity of the postulates and, therefore, of great interest to physicists concerned about the foundations of QM. Until now there is no experimental evidence that demonstrates any inconsistency with the postulates or the Copenhagen interpretation of QM [^{3}[3] P. Shadbolt, J.C.F. Mathews, A. Laing and J.L. O'brien, Nature Physics 10, 278 (2014) ^{}[4] A. Aspect, Phys. Rev. D. 14, 1944 (1976).^{5}[5] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47, 460 (1981).].
In this paper, we are interested in exploring the concepts raised above, considering continuous measurements applied to the quantized optical field via photodetection. Despite an extensive literature [^{6}[6] M.D. Srinivas and E.B. Davies, Optica Acta 28, 981 (1981). ^{}[7] M.D. Srinivas, Pramana 47, 1 (1996) ^{}[8] M.D. Srinivas, in: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes (Springer, New York, 1984), p. 356. ^{}[9] M. Ueda, N. Imoto and T. Ogawa, Phys. Rev. A. 41, 3891 (1990). ^{}[10] N. Imoto, M. Ueda and T. Ogawa, Phys. Rev. A. 41, 4127 (1990). ^{}[11] M. Ban, Phys. Rev. A. 51, 1604 (1995). ^{}[12] V. Peřinová, A. Lukš and J. Křepelka, Phys. Rev. A. 54, 821 (1996). ^{}[13] B. Masashi, Phys. Lett. A 235, 209 (1997). ^{}[14] V. Peřinová and A. Lukš, Progress in Optics 40, 115 (2000).^{15}[15] D.F. Walls and G.J. Milburn, Quantum Optics (Springer, Berlin, 2008).] about this subject, we care about being quite pedagogical to help advanced undergraduate and graduate students interested in the effects of measuring devices acting on quantum systems. This kind of problem is becoming increasingly important given the significant quantumbased technologies advancements [^{16}[16] M.A. Nielsen and I. Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2010). ^{}[17] J.P. Dowling and G.J. Milburn, Philos. Trans. R. Soc. A. 361, 1655 (2003). ^{}[18] A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S.T. Glaser, F. Jelezko et al., New J. Phys. 20, 80201 (2018). ^{}[19] H.M. Wiseman and G.J. Milburn, Quantum measurement and control (Cambridge University Press, New York, 2009). ^{}[20] T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe and J.L. O'Brien, Nature 464, 45 (2010). ^{}[21] I. Georgescu and F. Nori, Phys. World 25, 16 (2012). ^{}[22] I.A. Walmsley, Science 348, 525 (2015). ^{}[23] D. Browne, S. Bose, F. Mintert and M.S. Kim, Prog. Quantum. Electron. 54, 2 (2017). ^{}[24] S. Ritter and J. Stuhler, PhotonicsViews 16, 75 (2019). ^{}[25] J. Wang, F. Sciarrino, A. Laing and M.G. Thompson, Nat. Photonics, 1 (2019). ^{}[26] M. Leduc and S. Tanzilli, Photoniques 3, 46 (2019). ^{}[27] C. Monroe, M.G. Raymer and J. Taylor, Science 364, 440 (2019). ^{}[28] Y. Yamamoto, M. Sasaki and H. Takesue, Quantum Science and Technology 4, 020502 (2019). ^{}[29] P.A. Moreau, E. Toninelli, T. Gregory and M.J. Padgett, Nat. Rev. Phys. 1, 367 (2019). ^{}[30] O.S. MagañaLoaiza and R.W. Boyd, Rep. Prog. Phys. 82, 124401 (2019).^{31}[31] L. Gyongyosi and S. Imre, Comput. Sci. Rev. 31, 51 (2019).].
To take into account the action of the detector on the optical field, we use the theoretical photodetection model developed by Srinivas and Davies [^{6}[6] M.D. Srinivas and E.B. Davies, Optica Acta 28, 981 (1981).,^{8}[8] M.D. Srinivas, in: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes (Springer, New York, 1984), p. 356.,^{15}[15] D.F. Walls and G.J. Milburn, Quantum Optics (Springer, Berlin, 2008).]. The SrinivasDavies model is based on the mathematical framework called quantum operations [16][16] M.A. Nielsen and I. Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2010). which permits not only to calculate the evolution of closed systems but also the evolution of open quantum systems. It is important to learn and dominate techniques that describe the dynamics of quantum systems interacting with the environment since in real life there is no such thing as closed system, and to describe real processes we have to take into account the influence of the rest of the Universe upon the system of interest.
The SrinivasDavies model is presented in Sec. 2. In Sec. 3, to illustrate basic and general results, we review the simplest case where the SrinivasDavies model can be applied: just onemode of the optical field. For an arbitrary initial state, we not only present the dynamics of the field probed by the detector (the conditioned and unconditioned states) and the photocounting probability distribution (the probability distribution associated with the number of photons counted by the detector over a time period t), but we also present detailed calculations. For concreteness, we review two specific cases, the number and the coherent states, pointing out the differences and the similarities between them when they are probed by a photodetector. In Sec. 4, we consider another example given by a superposition of two coherent states and analyze the dependence on the number of photons counted upon the dynamics of two special cases: the odd and even superpositions of coherent states. In Sec. 5, we consider two noninteracting optical fields initially prepared as an entangled state of twomode coherent states. There we assume two scenarios: one in which just one mode is continuously probed, showing how the combined state evolves under the influence of a local detection, and another where both modes are probed by two independent photodetectors with different detection rates. In order to guide the readers, the calculations needed to obtain the main results concerning the dynamics of the probability distributions and the conditioned and unconditioned states are detailed in the appendices. Finally, in Sec. 6 a summary is presented.
2. SrinivasDavies continuous photocounting model
The SrinivasDavies continuous photocounting model (SDmodel) is based on the following statement: whenever a system in an arbitrary state ρ is subject to a process (instantaneous or not) and a certain outcome is observed as a result, the state that emerges from the process will be of the form [16][16] M.A. Nielsen and I. Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2010).
The operator ρ is the density operator, also known as the statistical operator, and it represents the physical state of the system [^{1}[1] C. CohenTannoudji, B. Diu, Bernard and F. Laloë, Quantum Mechanics (Willey, New Jersey, 1977).,^{2}[2] J.J. Sakurai, Modern Quantum mechanics (AddisonWesley Reading, Boston, 1994),^{32}[32] K. Blum, Density matrix theory and applications (Springer, New York, 2012).]. The map ε in the equation (1) is what is called a quantum operation; it can represent an unitary evolution of a closed system, the collapse of a state due to a measurement associated with some observable, a nonunitary evolution of a system interacting with the environment, or a more general process that encompasses nonunitary evolution between multiple instantaneous projective measurements. In mathematical language, just for the sake of formality, an operation ε is a linear positive transformation on the space of all trace class operators on the Hilbert space
Another mathematical particularity that we consider relevant to mention is that the SDmodel satisfies a semigroup structure which means that the dynamics of the system is irreversible in time. A semigroup is an algebraic structure that requires only the associative property between the elements of a set provided with a binary operation rule, and it does not need to have the identity nor the inverse operations. Furthermore, if the system in the state ρ is subject to a sequence of two experiments with outcomes corresponding to the operations
with
Here, we are interested in describing the action of the photodetection on a quantized electromagnetic field. In this context, the process of photodetection represented by the SDmodel is characterized by a set of operations
The state above is the evolved state conditioned to the number k of photons counted by the detector over a time interval
depending only on the time interval
which is the probability of the event “k photons counted by the detector during the time lapse t”. These axioms are the following:
ensures that
i.e., the probability of the detector records a photocounting at the very beginning of time is extremely unlikely and it goes to zero in the limit
The operation
where Y is the generator of the semigroup dynamics given by divisible maps
with
In terms of the operations
Since the operation
where
Considering a canonical photoncounting process, where the operation
where the Hilbert space operator R is defined by
These choices given by equations (8), (14) and (15) reflect the complementary property of the two mutually exclusive probabilities
that is, the complementarity between the probability associated to the event “to count 1 photon during an infinitesimal interval of time
The probability
3. Singlemode free field photocounting statistics
In this section we consider the formalism developed by Srinivas and Davies applied to the simplest possible case: onemode of the quantized electromagnetic field. Although this example is found in the original Srinivas and Davies paper [6][6] M.D. Srinivas and E.B. Davies, Optica Acta 28, 981 (1981)., we include it here for the sake of completeness, containing further discussions illustrating the results and the basic techniques that will be useful in the following sections. The evolution of such system is generated by the Hamiltonian [36][36] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York,1973).
where
which naturally implies that
Clearly, from equation (19), the role of the annihilation operator is to remove a quantum of light from the field changing its state from one with n photons to another with
The number operator
Naturally, we can express the most general onemode optical field state as follows
where
Now, if we consider a detector probing the optical field, every time that one photon is detected (by absorption) the field loses that photon to the detector. To represent this absorption process mathematically it is reasonable to write the super operator J through the following expression
where γ is a parameter related with the detector's efficiency and represents the coupling between the detector and the field. A microscopic model for a detector operating by photon absorption justifying the choice in the equation (25) can be found in [10][10] N. Imoto, M. Ueda and T. Ogawa, Phys. Rev. A. 41, 4127 (1990).. The equation (25) implies via equation (15) that
Then, the generator of the nonunitary dynamics in which the singlemode optical field is subject between two consecutive photocountings is given by
Besides, from the equations (25) and (13) we have
where
which implies that
With the help of the following theorem (see [36][36] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York,1973).): If A and B are two noncommuting operators and ξ an arbitrary parameter, then
it is possible to calculate the equations (29) and (30), which gives:
With the equations (33) and (34), the operation (31) becomes
The calculus of the multiple integral above is presented in the Appendix B which gives the following result
Therefore, the state (3) and the probability distribution (5) in terms of the fundamental operations J (which absorves one photon from the field by the detector) and
The state (37) is obtained after k photons have been detected by the measuring device during the probing time t. It represents our knowledge about the optical field state after the continuous photodetection process has ended. It is called conditioned state since it is conditioned to (our knowledge about) the number of photons that were counted. Now, if we want to know how likely is this particular outcome, that is, if we want to know how probable is to count k photons during the probing time t, the answer is given by the equation (38). Considering an initial state of the form (24), the probability (38) can be written as follows
where
As we can see in the equation (39), for
The conditioned state (37) is the state that emerges from the continuous photodetection process when it is known how many photons were counted during the probing time t; however, if it is not known how many photons were counted during the probing time we usually describe our lack of information about the state through the unconditioned state, which is the state obtained averaging the conditioned state (37) over all possible k counts weighted by the photocouting probability distribution (39)
The equations (37), (39) and (41) can be applied for any initial onemode optical field state. Let us then illustrate the applicability of these equations for some concrete examples. Let us begin with two well known particular cases, the number and the coherent states [^{8}[8] M.D. Srinivas, in: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes (Springer, New York, 1984), p. 356.,^{36}[36] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York,1973).,^{39}[39] R.J. Glauber, Phys. Rev. 131, 2766 (1963).].
In the first example, if we prepare the initial optical field in a state where we know for sure how many photons are contained initially in the field, that is, a number state
and
What can we conclude from the conditioned and unconditioned states (42) and (44)? And what can we say about the probability distribution (43)? First of all, we can see that the conditioned state (42) does not depend on time at all, depending solely on the number of photons counted by the detector. It represents the collapse of the initial state
The time dependence of the photocounting probability distribution
As we can see in the Fig. 1, each possible outcome has its own time dependent probability distribution
Now, let us suppose that it is not known how many photons were counted during the process. Let us say that, for some reason, we only know that the detector has counted some number of photons but we do not know that number. In this case, we are unable to say what the optical field state is for sure but, instead, we could say that the state of the field can be
The second example of a singlemode initial state considered in this section is the coherent state [39][39] R.J. Glauber, Phys. Rev. 131, 2766 (1963).. This important representation of the optical field can be written as an infinity superposition of Fock states [36][36] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York,1973).,
The photon number probability distribution for the coherent state follows a Poisson distribution
with an average number of photons given by
From the equations (45) and (24) we have
where the sum in the equation (48) converges to the following expression (see Appendix D)
with
It is interesting to notice that the coherent state is not affected by the operation J during the photocounting process since it is an eigenvector of the annihilation operator,
with complex eigenvalue
Hence, it does not matter if one knows or not how many photons are counted during the photocounting process, the coherent state will remain pure and coherent all the time, differently from the number state where the lack of information about how many photons are counted is manifested as a mixed state with statistical weights, as expressed by equation (44).
The photon number probability distribution associated with the evolved state of the field,
Figure (2) shows the photon number probability distribution dynamics given by the equation (52) for different times and for
The photon number probability distribution
The photodetection probability distribution given by equation (49) is depicted in the Fig. (3) for the same average number of photons
The photocounting probability distribution
From an experimental point of view, the optical field intensities (or even individual photons) are not measured directly but rather indirectly, measuring the photocurrent produced and amplified by the detector. The photocurrent statistics reflects the statistical properties of light. For a better understanding of the photodetection process in real experiments let us consider the simplest optical setup needed to measure photon statistics. This setup needs three basic devices: a source of light, a photodetector and a discriminator. The source of light can be a hightquantumefficiency light emitting diode (LED), which is suited for the producing of a quantized light beam. A good photodetector candidate which is wellsuited for laboratory applications, enabling individual photons to be detected when the incident flux of light is low and the optical field needs to be treated quantum mechanically, are the photomultiplier tubes (PMTs), which are photodetectors that absorves and transforms incident photons, via photoelectric effect, into measurable electric pulses (the photocurrents). Basically, there are three steps inside a PMT: first, photons coming from the light source strike a photocathode inside the vacuum tube ejecting electrons from its surface via photoelectric effect; second, the photoelectrons ejected from the photocathode surface (the first emission) are directed towards the electron multiplier for amplification of the initial signal. The electron multiplier is built by several electrodes called dynodes, each held at a more positive electric potential than the preceding one. The multiplication of electrons and consequently amplification of the photocurrent signal is the result of multiple electron secondary emissions that take place on the dynodes surfaces due to successive collisions of the preceding electrons ejected from the preceding dynodes onto the subsequent ones. Finally, in the third and last step inside the PMT, the large number of electrons that reach the anode results in a sharp current pulse which is straightforward to detect with the help of the discriminator, a device that selects (count) the pulses that have energy above some threshold energy value and discards the pulses that have energy bellow this threshold. The discriminator is important because it separates photocounts from noise. The important thing to notice here is that the statistics of emitted photoelectrons preserves the statistical properties of the probability distribution of absorbed photons of a given state. The number of photoelectron pulses computed within a fixed interval length
4. Superposition of coherent states
In this section we analyse the effect of the photodetection over a singlemode superposition of two coherent states
where the normalization factor is
and
For large amplitudes the state given by equation (54) are known in the literature as cat states [47][47] C.C. Gerry and P.L. Knight, Am. J. Phys. 65, 964 (1997). and this is so because for large amplitudes the two states that compose the superposition can be considered practically distinguishable states with small overlapping wave functions (like the superposition of two macroscopic states representing the cat alive and dead in the famous Schrödinger's cat gedanken experiment). Although these states are hard to be produced in the laboratory, they have been generated for small amplitudes [48][48] A. Ourjoumtsev, R. TualleBrouri, J. Laurat and P. Grangier, Science 312, 83 (2006).. Two special cases are obtained when we consider
where
are different from zero, respectively, only for n even (in the case of even state) or only for n odd (in the case of odd state). Figure (4) shows the even and odd photon number probability distributions for
The photon number probability distributions
Now, let us analyse what happens when we perform continuous photodetection over the even or odd singlemode coherent superposition. The conditioned state (37) for an initial even superposition,
where the normalization factor is given by
If the initial state is the odd superposition it is straightforward to notice that
where
Through the conditioned states (58) and (60) it is possible to calculate the photon number probability distributions of the optical field
The plus (minus) sign inside the brackets in the equation (61) is related with the even (odd) probability distribution. In addition, the conditioned state can change from an even state to an odd state depending on whether
The photocounting probability
where
The probability distribution
5. Entangled coherent state
Let us consider now the photocounting effects over a twomode entangled state of the optical field. Entanglement is a genuine quantum property without classical counterpart where two or more components of a combined system share a joint state that is not separable as a tensor product of the individual subsystems' states [^{49}[49] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).,^{50}[50] J.M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73, 565 (2001).]. Moreover, entanglement is a very important resource for a variety of quantum computation and quantum information tasks [16][16] M.A. Nielsen and I. Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2010).. In general, entangled states between two (or more) particles are generated by letting the particles interact with each other [32][32] K. Blum, Density matrix theory and applications (Springer, New York, 2012).. Several processes have been proposed to generate entanglement between two or more optical fields. The process of spontaneous parametric downconversion [15][15] D.F. Walls and G.J. Milburn, Quantum Optics (Springer, Berlin, 2008)., for example, is a way for the generation of entangled photon pairs. Here we are not interested in how the two modes were entangled and we simply suppose that they have interacted sometime in the past, becoming entangled after the interaction. The Hamiltonian describing the free evolution after the interaction is given by
where
We are going to consider the following initial twomode coherent entangled state [51][51] B.C. Sanders, Phys. Rev. A. 45, 6811 (1992).
with the corresponding density operator
where
Here, we consider two different cases: firstly only the mode 2 is probed and secondly both modes are probed. Let us see what happens when only the mode 2 is probed by the detector while the mode 1 evolves freely. Since the two modes do not interact, the field operators of one of the modes commute with the operators of the other mode and the dynamics of the twomode coherent optical field is given by
where
where
For
with the mode 2 reaching the vacuum state and the mode 1 becoming the following conditioned state
which can be the even or the odd state superposition depending on the parity of k.
Since we are considering the detection just upon the mode 2, the photodetection probability can be obtained directly by inserting the initial probability distribution of mode 2
into equation (39), which gives the following result
where
The probability distribution
The conditioned state when both modes are simultaneously probed by two independent detectors are also derived. We assume that each detector has its own detection rate,
where
where
6. Summary
In this paper we consider the effects of continuous measurements over the quantum state of physical systems. In particular, we use the theory of continuous photodetection proposed by Srinivas and Davies to calculate the state of the electromagnetic field conditioned to continuous photocounting. We have explored two situations: (1) a single mode electromagnetic field where three different initial states are considered: a number state, a coherent state, and a superposition of coherent states; (2) a twomode entangled state of the electromagnetic field where two scenarios were explored: one in which just one of the modes is continuously probed by a photo detector, and another one where both modes are probed by two independent detectors with different detection rates. In addition, in order to guide the readers, the calculations needed to obtain the main results concerning the dynamics of the probability distributions and the conditioned states are detailed in the appendices.
The article has a pedagogical purpose and we believe that it can be useful for students and teachers. Readers interested in further applications and other approaches may find a extensive material in the references [^{9}[9] M. Ueda, N. Imoto and T. Ogawa, Phys. Rev. A. 41, 3891 (1990).,^{11}[11] M. Ban, Phys. Rev. A. 51, 1604 (1995). ^{}[12] V. Peřinová, A. Lukš and J. Křepelka, Phys. Rev. A. 54, 821 (1996). ^{}[13] B. Masashi, Phys. Lett. A 235, 209 (1997).^{14}[14] V. Peřinová and A. Lukš, Progress in Optics 40, 115 (2000).,^{37}[37] B.R. Mollow, Phys. Rev. 168, 1896 (1968).,^{38}[38] M.O. Scully and W.E. Lamb Jr., Phys. Rev. 179, 368 (1969).,^{52}[52] M. Ueda, Phys. Rev. A 41, 3875 (1990). ^{}[53] M. Ueda, N. Imoto and T. Ogawa, Phys. Rev. A. 41, 6331 (1990). ^{}[54] T. Ogawa, M. Ueda and N. Imoto, Phys. Rev. A. 43, 6458 (1991). ^{}[55] C.T. Lee, Quant. Optic J. Eur. Opt. Soc. B. 6, 397 (1994). ^{}[56] S.J. van Enk and O. Hirota, Phys. Rev. A. 64, 022313 (2001). ^{}[57] P. Warszawski and H.M. Wiseman, J. Opt. B Quantum Semiclassical Opt. 5, 1 (2001). ^{}[58] M.C. Oliveira, S.S. Mizrahi and V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 5, S271 (2003). ^{}[59] G.A. Prataviera and M.C. Oliveira, Phys. Rev. A. 70, 011602 (2004). ^{}[60] K. Jacobs and D.A. Steck, Contemp. Phys. 47, 279 (2006).^{61}[61] L.G.E. Arruda, G.A. Prataviera and M.C. Oliveira, Ann. Phys. 389, 30 (2018).].
Supplementary material
The following online material is available for this article:
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Publication Dates

Publication in this collection
17 July 2020 
Date of issue
2020
History

Received
20 Jan 2020 
revreceived
10 Apr 2020 
Accepted
14 Apr 2020