Nonclassicality of thermal radiation

Lars M Johansen

Department of Technology, Buskerud University College, N-3601 Kongsberg, Norway

Email: lars.m.johansen@hibu.no

Received 21 June 2004, accepted for publication 23 September 2004
Published 15 October 2004

Glauber and Sudarshan have demonstrated that any density matrix can be expanded diagonally in terms of coherent states [1, 2],

The weight function P(α) is known as the Glauber-Sudarshan P-distribution. It has been proposed that all quantum states for which the P-distribution is a probability distribution may be considered essentially classical [1, 3]. Put differently, this means that any state which can be expressed as a classical mixture of coherent states should be regarded as essentially classical. Several measures of `nonclassicality' have been introduced on the basis of this criterion [4-7].

The term `nonclassical' may be applied in a variety of settings. Our intent here is to investigate under what conditions a classical stochastic theory may serve as a model describing physical phenomena. As such, we use the word `nonclassical' as a label for experiments where classical physics fails as a model.

There is no doubt that coherent states are the most `classical-like' of all pure states of the harmonic oscillator [1, 8-11]. Therefore, it is tempting to regard any classical mixture of coherent states as `classical'. Essentially, positive P-distributions serve as a stochastic c-number model for normally ordered expectations. According to the `optical equivalence theorem' [1, 2], the expectation of any normally ordered operator can be expressed as an integral over the corresponding c-number expression,

Therefore, whenever P(α) has the properties of a probability distribution, one may say that a classical model exists for the expectation of any normal ordered operator product.

In the literature there exist some comments that quantum states with well behaved P-functions may also be nonclassical. In [12] it is said that states with a small photon number are always nonclassical, with reference to the correspondence principle. However, in [13] it is stated (by the same author) that `once the spectral distribution is given, blackbody radiation also has a good classical description, no matter how small the photon occupation numbers may be '.

Due to properties of the P-distribution, a non-negative, well behaved P-distribution ensures that the Wigner distribution is non-negative. However, in this letter we shall be concerned with the lesser known Rivier ordering and the associated Terletsky-Margenau-Hill (TMH) distribution [14, 15]. We shall see that in a specific type of measurement known as `weak measurements' [16] no classical underpinning can be given for thermal states of low occupation number.

The P-distribution for a thermal state is

where is the expected photon number. This distribution has the properties of a probability distribution, hence this is essentially a classical state according to the Glauber criterion. In this letter, we shall nevertheless demonstrate that weak measurements [16] on thermal radiation cannot be modelled by a classical c-number theory when the occupation number is sufficiently low. This is the case, e.g., for blackbody radiation at the dominant wavelength determined by Wien's displacement law. We demonstrate that this nonclassical effect can only be found in quantum systems for which the TMH quasi-probability distribution [14, 15] is negative. On the other hand, for quantum states for which the TMH distribution is positive, a classical stochastic model exists for weak measurements. We shall see that this is approximately the case for thermal radiation of high occupation number.

In a standard, projective measurement, the uncertainty of the pointer is initially very small so that the pointer can distinguish different eigenvalues of the observable [17]. In the weak measurement scheme, the pointer is assumed to be in a pure, Gaussian state of large uncertainty. In this way, the pointer cannot discern the different eigenvalues of the observable. The theory of weak measurements has been generalized to arbitrary states of object and pointer in [18]. Generally, a weak measurement of an observable conditioned on the outcome of projective measurement of an observable yields the real part of the weak value

Here is the density operator of the object.

We now consider a weak measurement of postselected on , where and are quadrature operators. By inserting the completeness relation

in equation (4) it is easily shown [19] that the weak value of postselected on is a conditional moment of the standard ordered distribution [20]

where

is a `conditional' standard ordered quasi-probability phase space distribution, and where

is the standard ordered phase space distribution [20]. The standard ordered distribution is the complex conjugate of the Kirkwood distribution [21]. These are of course not proper probability distributions since they may be complex. However, they yield correct marginal distributions when integrated over either phase space variable, just as for the Wigner distribution.

In a weak measurement, the pointer readout is proportional to the real part of the weak value [16, 18]. We may express the real part of (pⁿ)_w as

where

is a conditional TMH distribution and where

is the TMH distribution. This distribution is real, but may be negative. It also gives the correct marginal distributions. Both the Wigner, standard ordered, anti-standard (or Kirkwood) and the TMH distribution belong to a parametrized class of probability distributions introduced by Cohen [22].

A classical analysis of weak measurements [18] shows that the pointer displacement is proportional to a weak value which can be expressed in exactly the same form as equation (6) or (9), but over a positive phase space distribution. Therefore, formally we may say that a classical model for the pointer displacement exists whenever the TMH distribution has the properties of a probability distribution. It follows straightforwardly that for a classical model, the weak value cannot exceed the range of the classical observable pⁿ. In particular, the weak value of a positive observable pⁿ cannot be negative in a classical model.

We now demonstrate that this condition is violated by thermal states of low occupation number, and hence that weak measurements on such states cannot be modelled by a stochastic theory where quadratures are c-numbers.

By using equation (1), we may express the standard ordered distribution in terms of the P-distribution,

By using the quadrature representation of a coherent state  [23]

and by inserting the P-distribution for a thermal state in equation (12), it can be shown that the standard ordered distribution for a thermal state is

where

is the variance of each quadrature.

Thermal radiation is often well described by the spectral distribution of blackbody radiation. For blackbody radiation the expected occupation number is

At maximal irradiance, as given by Wien's displacement law, the occupation number is of the order 10 ^- 2. The TMH distribution [15], which is the real part of the standard ordered distribution, has been plotted in figure 1 for an occupation number of 10 ^- 2, and in figure 2 at an occupation number of unity. It is clearly negative in the former case, and it is practically a classical distribution in the latter case. This behaviour affects the nonclassicality which can be demonstrated in weak measurements.

Figure 1. The TMH distribution for thermal radiation at maximum irradiance given by Wien's displacement law, i.e., for an occupation number of . The distribution is virtually identical to that of vacuum [24, 19]. There are distinctly negative regions, which lead to nonclassical behaviour.

Figure 2. The TMH distribution for thermal radiation for an occupation number of . The negative regions have practically disappeared.

By inserting equation (14) in (6) we find that the weak value of is

This is an inverted parabolic curve which is negative for . Thus, for sufficiently large q, the weak value of is negative for any thermal state. A classical radiation model cannot reproduce this result.

The standard ordered distribution yields correct marginals when integrated over. Thus, the marginal distribution for the quadrature is

This is a standard Gaussian distribution with variance σ². The probability of observing a negative weak value is

This function has been plotted as a function of in figure 3. We see that the result is nonclassical essentially for . For occupation numbers typical of the Planckian spectrum there is a significant probability for observing a nonclassical negative weak value of .

Figure 3. The probability of observing a negative weak value of for thermal radiation as a function of the occupation number . At maximum irradiance of blackbody radiation, given by Wien's displacement law, the occupation number is of the order 10 - 2, which gives a significant probability of nonclassicality.

In principle, weak measurements employ the same interaction Hamiltonians as strong (or projective) measurements [25, 18]. If a strong measurement can be performed with a specific interaction Hamiltonian, then a weak measurement can be performed with the same interaction type provided that the interaction strength is sufficiently weak.

At present, no feasible quantum optical experimental arrangement is known to the author to implement the experiment presented here^Note1 . It could be possible to observe the nonclassical effect found in this letter for a massive particle in thermal equilibrium in a harmonic oscillator potential. If a detector is placed at a distance exceeding from its equilibrium position, a weak measurement of kinetic energy should yield a negative value on average. However, this requires observation in the low temperature regime. For a temperature of 1 µK, the required oscillator frequency is of the order of 100 kHz.

A comment can be made on the usage of pointer states. It has recently been shown that weak measurements can be performed with arbitrary pointer states, with the only restriction that the current density of the pointer state must vanish [18]. This means that a weak interaction between a thermal pointer state and another thermal object state may yield a pointer reading which cannot be explained in terms of a classical c-number model of radiation!

One might object that the nonclassicality found in these experiments is produced by the experimental arrangement itself and is not a property of the state. After all, we are considering a joint measurement of two noncommuting observables, and such observables are known to affect each other. For example, if two strong measurements are performed in series, the first measurement will collapse the state and the second measurement will simply reflect properties of the collapsed state. However, the situation is quite different in weak measurements. The interaction between the object and the pointer system can be made arbitrarily weak, so that the postselection measurement is not influenced by the weak measurement. Therefore, the probability distribution for the postselection observable is preserved during the measurement interaction [16, 18].

It is also worth noticing that, in the c-number representation that we use, it is the representation of the state, the TMH distribution, which has nonclassical properties. The TMH distribution gives a complete description of the state, and is equivalent to any other state representation, such as the Wigner distribution or the density operator. In the TMH representation, the representation of the weakly measured observable, , is represented by the c-number p², and hence is represented just as in classical theory. Therefore, failure of the classical model is due to a nonclassical representation of the state and not of the observable. Furthermore, it is of interest to note that the nonclassical effect demonstrated here disappears for high occupation numbers. In such cases, the representation of the state becomes essentially classical. High occupation numbers appear for high temperatures. For blackbody radiation at visible wavelengths, for example, an occupation number of unity is achieved first at temperatures of several tens of thousands of kelvin. The occupation number of sunlight therefore falls well within the nonclassical regime.

In conclusion, we have demonstrated that weak measurements on thermal states of low occupation number cannot be modelled by a stochastic c-number theory. A source such as sunlight is in fact sufficient to produce this effect. Low occupation numbers are typically found in most forms of naturally occurring radiation. The nonclassical effect is related to negativity of the TMH distribution. The classical limit is regained for high occupation numbers, where the TMH distribution is approximately non-negative.

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Note1