Classical light vs. nonclassical light: characterizations and interesting applications

Abstract

We briefly review the ideas that have shaped modern optics and have led to various applications of light ranging from spectroscopy to astrophysics, and street lights to quantum communication. The review is primarily focused on the modern applications of classical light and nonclassical light. Specific attention has been given to the applications of squeezed, antibunched, and entangled states of radiation field. Applications of Fock states (especially single photon states) in the field of quantum communication are also discussed.

Acknowledgements

AP thanks Department of Science and Technology (DST), India for the support provided through the project number EMR/2015/000393. He also thanks Kishore Thapliyal, S. Aravinda and J. Banerji for their interest in this work. AG thanks the National Academy of Sciences India (NASI), for supporting the present work through the M. N. Saha Distinguished Fellowship.

Notes

No potential conflict of interest was reported by the authors.

1 Interested readers may freely read Maxwell’s original paper at http://www.jstor.org/stable/pdf/108892.pdf.

2 Everyday, we see that relative velocity of two cars that approach each other with the same speed is double of the individual speed. This is in accordance with the Galilean transformation, but according to Maxwell’s equation, light would always move with a constant velocity c in free space. Thus, if we send light from two torches in the opposite direction, their relative velocity would still remain c. This was in sharp contrast with the Galilean transformation.

3 Planck’s paper cited here as [3] was published in 1901, but the paper contains following note- In other form reported in the German Physical Society (Deutsche Physikalische Gesellschaft) in the meetings of October 19 and December 14, 1900, published in Verh. Dtsch. Phys. Ges. Berlin, (1900) 2, 202 and 237. An English translation of Verh. Dtsch. Phys. Ges. Berlin, (1900) 2, 237 is available at http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20(1900),%20Distribution%20Law.pdf.

4 It may be noted that Bohr model also originated in an effort to explain the origin of lights of certain wavelengths (as was observed in Lyman, Balmer, Paschen, Bracket and Pfund series.

5 The main problem in defining a wave function of photon in position space arises because of the fact that it cannot be localized in position space as it has a definite momentum.

6 Interested readers may read about Abraham–Minkowski dilemma in detail to know the origin of this interesting problem. About a century ago, Abraham and Minkowski gave two different expressions for momentum of light in a medium. To understand the dilemma, at the single photon level, we may note that for free space momentum of a photon is and it’s unambiguous, but for a medium having refractive index n, there are two competing expressions for photon momentum: and . Both are used, and thus the open question is: Which one of these two expressions is correct? Apparently, the problem arises because even in the classical optics, there is no universally accepted definition for the electromagnetic momentum in a dispersive medium.

7 It is interesting to note that Nobel laureate Raman, provided a semiclassical explanation of the Compton effect in Ref. [23].

8 The word “quantum” means discrete. In quantum mechanics, we have Hermitian operators for all the physical observables. These operators satisfy eigenvalue equations, where the eigenfunctions are the wave functions. Obtained eigenvalues corresponding to any operator is discrete and on a particular measurement, we can obtain only one of those eigenvalues as the value of the corresponding physical observable. Thus, in quantum mechanics, we obtain discrete values for an observable, in other words, in quantum mechanics allowed values of physical observables get quantized. Historically, at the beginning of quantum mechanics (say, between 1925 and 1926), it was restricted to the quantization of the motion of particles, only, and in all the early works of the founder fathers of quantum mechanics (e.g. Schrodinger, Heisenberg, Dirac), electromagnetic field was treated classically. Later, in 1927, Dirac quantized electromagnetic field [29], subsequently, Jordan and Wigner developed a formalism in which particles are also represented by quantized fields. This led to quantum field theory, which has been formulated in the language of second quantization.

9 It may be noted that for the finite dimensional Hilbert space, these definitions are not equivalent, and any finite superposition of Fock states is always nonclassical.

10 In our notation, a Poissonian distribution is one which follows and . Here, can be easily recognized as a coherent state by noting that

11 Note that this description is valid for any quantum state and it’s not restricted to the quantum states of radiation field.

12 Although it is usually referred to as Glauber–Sudarshan P-function, and the related formulation as the Glauber–Sudarshan P-representation and Glauber won 2005 Nobel prize in Physics for developing this formalism, it is a bit controversial. Many scientists and Sudarshan himself often argue that this representation that provide correct quantum mechanical theory of optical coherence was actually developed by Sudarshan, and was later adopted by Glauber, who coined the term P-representation. As P-representation or diagonal representation played crucial role in the development of the non-classical optics, this debate about the origin of P-representation is in existence since long. However, it resurfaced in 2005–2006, when Glauber won the Nobel prize in Physics for this formulation, but Sudarshan missed it and wrote a strong letter of objection to the Nobel committee (for a short description of the controversy, interested readers may see [32–34]). To us it appears that Nobel committee gave more credit to Glauber’s 1963 paper [35] published in February 1963, over Sudarshan’s more powerful work [36] published in April 1963. However, P-representation or diagonal representation (or, equivalently optical equivalence theorem) was actually developed by Sudarshan and it would have been more appropriate it call it Sudarshan diagonal representation or sudarshan’s -representation as he had used in place of in his pioneering work. In fact, in Equation (4) of Ref. [36], Sudarshan expressed density function as where he considered as quantum state. Almost five months later, in Section VII of Ref. [37], Glauber reintroduced diagonal representation of Sudarshan as P-representation. Note that Equation (7.6) of [37] is the same as Equation (7(7) ) given above. For a clear and chronological description of the events that happened in 1963, see [38].

13 There exists an interesting paper by Kiesel et al. [40] in which experimental determination of a well-behaved P-function is reported for a single-photon added thermal state. However, the method cannot be generalized as P-functions of nonclassical states are not always well-behaved. Further, to the best of our knowledge this is the only work that reports experimental determination of P-function.

14 Although, coherent state and squeezed state were discovered in the early years of quantum mechanics, their importance was realized much later. Consequently, Schrodinger and Kennard did not receive much credit for these discoveries. In this context, Nieto made following very interesting remark in [52] – “To be popular in physics you have to either be good or lucky. Sometimes it is better to be lucky. But if you are going to be good, perhaps you shouldn’t be too good.”

15 An entangled state is a quantum state of a composite system which cannot be expressed as a tensor product of the component systems (sub-systems) that constitute the composite system. Specifically, if the composite state , where and represent arbitrary states of subsystem A and B, then is considered to be entangled, otherwise it is called separable. Thus, a two photon state is entangled, but the state is separable. Here and denote horizontal and vertical states of polarization, respectively.

16 In [173], the author had described the scheme as a scheme for QKD, but a careful look into the scheme easily reveals that the scheme proposed in [173] can be easily modified to obtain a scheme for quantum secure direct communication.

17 This point can be understood clearly with an example. Consider coin tossing mechanism, where we expect random output for a fair coin. However, if we know the force applied on the coin, air drag, height of the coin when thrown up, etc., we can solve the equation of motion and obtain the outcome. Thus, the apparent randomness is due to our lack of knowledge about some parameters involved, and the random numbers created through coin tossing or any other classical mechanism can only be pseudo random number. In contrast, quantum random numbers are truly random as they appear due to the inherent probabilistic nature of the quantum mechanics.

18 In DIQKD, security of the schemes does not depend on the perfection of the devices used for implementation of the schemes. Specifically, even if a corrupted device is used by the sender and the receiver, the generated key would remain unconditionally secure. To implement any scheme of completely DIQKD one would require photon detectors with 100% efficiency, which is not available now. Thus implementation of DIQKD is not possible now. However, a weaker version of this which makes the security of keys independent from the measurement devices used by the receiver and sender is called MDIQKD, and the same can be implemented with the existing technology.

19 Quantum teleportation is a very interesting process that nicely illustrates the power of quantum mechanics. In this scheme, an unknown quantum state is transferred using prior shared entanglement and classical communication, but the state cannot be found in the channels that connect the sender and the receiver.

20 Due to the similarity of this two-way communication task with a telephone, this type of scheme is also referred to as quantum telephone [197] and quantum conversation [198].

Additional information

Funding

This work was supported by Department of Science and Technology (DST), India, Science and Engineering Research Board [project number EMR/2015/000393]; National Academy of Sciences India (NASI).