Interference of Light in a Michelson-Morley Interferometer: A Quantum Optical Approach

Abstract

The temporal coherence interference properties of light as revealed by single detector intensity measurements in a Michelson-Morley interferometer (MMI) is often described in terms of classical optics. We show, in a pedagogical manner, how such features of light also can be understood in terms of a more general quantum-optics framework. If a thermal reference source is used in the MMI local oscillator port in combination with a thermal source in the signal port, the interference pattern revealed by single detector intensity measurements shows a distinctive dependence on the differences in the temperature of the two sources. A related method has actually been used to perform high-precision measurements of the cosmic microwave background radiation. The general quantum-optics framework allows us to consider any initial quantum state. As an example, we consider the interference of single photons as a tool to determine the peak angular-frequency of a single-photon pulse interfering with a single-photon reference pulse. A similar consideration for laser pulses, in terms of coherent states, leads to a different response in the detector. The MMI experimental setup is therefore an example of an optical device where one, in terms of intensity measurements, can exhibit the difference between classical and quantum-mechanical light.

1. Introduction

In 2006, Smoot and Mather shared the Nobel Prize in physics “for their discovery of the black-body form and anisotropy of the cosmic microwave background radiation (CMB)” [1]. These exciting discoveries were a breakthrough in modern cosmology by the CMB anisotropy and the strong validation of the black-body spectrum as predicted by the Big Bang theory. The discovery of the black-body form of the CMB spectrum and the high-precision measurement of the CMB temperature (see e.g., [2]) relied heavily on the so-called Far-Infrared Absolute Spectrophotometer (FIRAS) [3] on board the Cosmic Background Explorer (COBE) [4, 5]. In short, the FIRAS is a Michelson-Morley interferometer enabling a comparison of the interference patterns between an observed source and a reference black-body source on board the COBE satellite.

In this paper, we will make use of Glauber’s theory for photon detection [6, 7] (for a guide to the early literature see e.g., [8–10] and for textbook accounts see e.g., [11–14]) together with elementary quantum mechanics to show how the principles of the FIRAS can be understood in a straight-forward manner using a quantum-optics frame-work. Interference phenomena in classical optics are described in terms of classical electro-magnetic fields. Such classical fields can be obtained in terms of expectation values of the observable electromagnetic field operators using very special quantum states, that is, coherent states (see e.g., [8–10]). Quantum states with a fixed number of photons, like single-photon states, have, however, no average electromagnetic field, and the interference effects of such states therefore requires an extension of classical considerations. The use of quantum optics enables us to actually consider arbitrary quantum states of the source and reference source, thereby extending the realm of classical optics.

In Section 2, we recapitulate the principles of Glaubers photon detection theory [6, 7] and the transformation properties of a quantum field in a beamsplitter (see e.g., [15–17]). The Glauber theory of optical coherence is by now well established and plays a central role in fundamental studies of quantum interference effects of photon quantum states (see e.g., [18–20]). In Section 3, we consider temporal interference effects in the Michelson-Morley interferometer for pure quantum states like single-photon states as well as classical states corresponding to coherent states (see e.g., [8–10]). In Section 4, we explain the principles of interference of thermal light in the Michelson-Morley interferometer by using only vacuum as the reference source and reproduce known expressions. Here, we also show that the detection intensity obtained with a vacuum reference source does not depend on the details of signal quantum state considered, thermal or not, but only on the average number of photons present. In Section 5, we consider the full system with an observed thermal source combined with a thermal reference source. With the results obtained, we then recover the basic principle of FIRAS and how it may be used as a high-precision thermometer. In Section 6, we, finally, give some concluding remarks.

Our presentation extends a recent presentation by Donges [21] and illustrates, for example, that a quantum-mechanical treatment directly leads to the concept of a thermal coherence length without explicitly making use of results from classical physics like the Wiener-Khintchine theorem as in [21] and that any quantum state can be considered.

2. The Michelson-Morley Interferometer

We consider the classical Michelson-Morley interferometer (MMI) as illustrated in Figure 1, where the so-called temporal coherence properties of the radiation field are probed (for an early account see e.g., [22]). Figure 1 is a simplified and schematic version of the FIRAS experimental setup. In order to understand the appearance of interference effects in the MMI, we first discuss the separate parts of the MMI before we consider the full setup with the presence of a reference beam.

Figure 1 

Schematic drawing of the MMI setup considered in this paper. The thick and diagonal line is representing a beam splitter with a transmission coefficient 𝑇. The movable mirrors 1 and 2 are identical. The numbers in the parenthesis are referring to the indices used to label the mode operators of the various light beams as discussed in the main text. The signal port (1) and the local oscillator port LO (0) are prepared with various quantum states, like Fock states, coherent states, or states with random phases like thermal states. The output mode (4) will not be considered in the present paper.

2.1. Glauber’s Theory of Photon Detection

Let us outline a simple, but not unrealistic, model of a photon detector situated at the space-time point (^→𝑥,𝑡). In this simplified model of a photon detection process [6, 7], the detection of a photon is described by an annihilation of a photon at the detector which modifies the initial state |in⟩ as follows: ||in⟩→𝐸^→(+)^→𝑥,𝑡||in⟩.(1) Here the time-evolution is implemented in the Heisenberg picture instead of a Schrödinger picture with space-and time-dependent “wave-functions” for photons. The Heisenberg picture seems actually to be mandatory since position-dependent wave functions appear to be ill defined for massless particles with a nonzero spin (see e.g., [23]). The observable electric field operator in Heisenberg picture, 𝐸^→(^→𝑥,𝑡)=∑𝐸_𝑚→𝑚(^→𝑥,𝑡), is then described in terms of a suitable normal mode expansion, indexed by mode the number 𝑚, as a superposition of positive and negative angular-frequency contributions as follows: 𝐸^→_𝑚^→𝑥,𝑡=𝐸^→𝑚(+)^→𝑥,𝑡+𝐸^→𝑚(−)^→𝑥,𝑡,(2) where 𝐸^→𝑚(+)(^→𝑥,𝑡) (𝐸^→𝑚(−)(^→𝑥,𝑡)) contains an annihilation (creation) operator for a photon with mode number 𝑚. According to the basic Born rule in quantum mechanics, the probability to detect the system in a final state |out⟩, after the single-photon absorption process, is then proportional to |⟨out|𝐸^→(+)(^→𝑥,𝑡)|in⟩|². Since the exact details of the final states are, in general, unknown we sum over all possible final states |out⟩, that is, we consider ∑_out|⟨out|𝐸^→(+)(^→𝑥,𝑡)|in⟩|²=⟨in|𝐸^→(−)(^→𝑥,𝑡)𝐸^→(+)(^→𝑥,𝑡)|in⟩, where use has been made of the completeness of all possible final states, that is, ∑_out|out⟩⟨out|=1. In general, we also have to consider not only a pure initial quantum state but also a quantum state as described by a density matrix. This leads to a description of the observed intensity 𝐼 which we write in the form 𝐼=Tr𝜌𝐸_in^→(−)^→𝑥,𝑡𝐸^→(+)^→𝑥,𝑡,(3) where 𝜌_in is the density matrix describing the initial state. Since we will not be interested in the absolute normalization of the observed intensity 𝐼, we can neglect normalization constants that may enter into 𝐼. It is a remarkable fact that an analysis of single-photon interference in a Young interferometer using such a quantum-mechanical description of the photon detection process [24], as well as a proper experimental investigation of single-photon interference [25, 26], is fairly recent in the history physics.

For reasons of clarity, we will now consider a typical plane wave normal mode expansion of the electromagnetic field observable 𝐸^→(^→𝑥,𝑡) in terms of a mode sum over wave-vectors ^→𝑘 and polarization-vector labels 𝜆, that is, we have for its positive angular-frequency part that𝐸^→(+)^→𝑥,𝑡=𝑖_→𝑘𝜆ℏ𝜔_𝑘2𝑉𝜀_0→𝜀_→𝑘𝜆𝑎_→𝑘𝜆𝑒^{𝑖→𝑘⋅→𝑥−𝑖𝜔𝑘(𝑡−𝑡0)},(4) with 𝜔_𝑘=𝑐|^→𝑘|. The annihilation operator 𝑎_→𝑘𝜆, and the corresponding unit polarization vector ^→𝜀_→𝑘𝜆, denotes the normal mode considered. 𝑡₀ is a suitable initial-time parameter which will be used, in a manner to be clarified below, to express the dependence on various propagation times for the optical paths as given in the MMI setup, a procedure as is well known in classical optics (see e.g., [12, 22]). If the direction of the light beams considered is well defined, the dependence of the detector position ^→𝑥 can be neglected in the expression for 𝐼. In general, spatial modulations of the measured intensity are expected [27, 28]. A theoretical analysis of such effects along lines as discussed in the literature (see e.g., [29, 30]) will, however, not enlighten the issues we are addressing in the present paper. 𝑉 is a quantization volume that will be allowed to be arbitrarily large at an appropriate late stage of our calculations.

Since the dependence of normalization constants will be irrelevant, and since we will only consider polarization-independent optical devices, we make use of a scalar notation. We therefore suppress the wave vector and polarization labels and with 𝜔≡𝜔_𝑘 we write 𝑎(𝜔)≡𝑎_→𝑘𝜆 such that [𝑎(𝜔),𝑎^†(𝜔)]=𝛿_𝜔𝜔 in terms of a discrete Kronecker delta 𝛿_𝜔𝜔. We also make use of the following convenient notation [12] for the positive angular-frequency part of the electric field at the position of the detector at time 𝑡: 𝐸⁽⁺⁾(𝑡)=𝑖_𝜔√𝛿𝜔𝜔^1/2𝑎(𝜔)𝑒^{𝑖𝜙(𝑡)},(5) with the prescription ∑_𝜔𝛿𝜔→∫𝑑𝜔𝜔^𝑑−1 in the infinite volume limit 𝑉→∞ and where 𝑑 is the number of space-dimensions. Here 𝜙(𝑡)≡−𝜔(𝑡−𝜏) now denotes a mode-dependent optical phase which explicitly takes the source-detection propagation time into account in terms of the time-delay 𝜏, which will be evaluated in detail for the MMI-setup below. In (5), the mode operator 𝑎(𝜔) (≡𝑎₅(𝜔) in Figure 1) will later be expressed in terms of a superposition of the input mode operators (≡𝑎₀(𝜔) and ≡𝑎₁(𝜔) in Figure 1). If the detector time 𝑡 enters explicitly into the detection intensity equation (3), we will perform a time average which corresponds to a finite detector time-resolution window. The corresponding time average of the observed intensity 𝐼≡𝐼(𝑡) will be denoted by ⟨𝐼⟩, that is, ⟨𝐼⟩=1𝑇_int^{𝑇int/2−𝑇}_int/2𝑑𝑡𝐼(𝑡),(6) where the time 𝑇_int of integration, as for example, the time during which an actual measurement proceeds, is chosen to be sufficiently large in comparison with typical time scales of intensity fluctuations. The time-averaged observed intensity ⟨𝐼⟩ will in general, as we will see explicitly below, be a function of a time-delay 𝜏 depending on the actual experimental set up.

For a finite quantization volume 𝑉, 𝜔 can be regarded to be discrete and it may be assumes that 𝑑=1. When appropriate we will, however, also consider 𝑑=3 in order to compare with related results in the literature [21, 22]. Our main results will, however, not be very sensitive to the choice of 𝑑. Using the same notation as above, a single-photon quantum state |𝑓⟩, with an angular-frequency distribution given by 𝑓≡𝑓(𝜔), is then given by ||𝑓⟩=_𝜔√𝛿𝜔𝑓(𝜔)||1_𝜔⟩≡𝑓,𝑎^†||0⟩,(7) where |1_𝜔⟩=𝑎^†(𝜔)|0⟩ denotes a single-photon state with angular-frequency 𝜔, normalized according to ⟨1_𝜔|1_𝜔⟩=𝛿_𝜔𝜔, and where |0⟩ denotes the vacuum state. We also make use of the notation (𝑓,𝑎^†)=(𝑓^∗,𝑎)^†≡∑_𝜔√𝛿𝜔𝑓(𝜔)𝑎^†(𝜔). The state |𝑓⟩ above is an eigenstate of the number operator 𝑁=∑_𝜔𝑎^†(𝜔)𝑎(𝜔), that is, a Fock state, with, of course, an eigenvalue corresponding to one particle present. Normalization of the state |𝑓⟩ for 𝑑=1 therefore corresponds to ⟨𝑓∣𝑓⟩=1=_𝜔𝛿𝜔𝑓^∗(𝜔)𝑓(𝜔)=^∞0𝑑𝜔||𝑓(𝜔)||²,(8) in the large-volume 𝑉 limit. In order to make our presentation quantitative, we will, for reasons of simplicity, consider real-valued, single-photon, angular-frequency distributions 𝑓(𝜔) such that 𝑓(𝜔)=1𝑁exp−𝜔−𝜔²2𝜎²,(9) with a mean angular-frequency 𝜔 and width 𝜎 and where the normalization constant 𝑁 is given by ||𝑁||²=𝜎√𝜋21+√2𝜋^0𝜔/𝜎𝑑𝑥𝑒^−𝑥2.(10)

This choice of frequency distribution makes it possible to actually carry out all relevant expressions analytically. In obtaining the properly normalized expression Equation (9), we keep 𝜔≥0. It may, however, sometimes be possible to extend the range of angular frequencies to arbitrarily negative values in (8), so that |𝑁|²=𝜎√𝜋, with an exponential small error, which makes some of the expressions more tractable and transparent. With our choice of beam parameters below, it turns out that such an approximation plays only a minor role with regard to our numerical evaluations where we make use of 𝜔≥0. It is now clear that ⟨𝑓|𝐸^→(^→𝑥,𝑡)|𝑓⟩≡0, that is, a single-photon state has no average electromagnetic field.

There are, however, quantum states with a nonzero expectation value of the electromagnetic field operator 𝐸^→(^→𝑥,𝑡) are like coherent states. Conventional coherent states |𝑓⟩_𝑐, as expressed in terms of the single-photon distribution 𝑓, can for example, be obtained using a multimode displacement operator (see e.g., [8–10]), that is, ||𝑓⟩_𝑐=𝑒^{(𝑓,𝑎†)−(𝑓∗,𝑎)}||0⟩=𝑒^{−⟨𝑓∣𝑓⟩/2}𝑒^{(𝑓,𝑎†)}||0⟩(11) such that 𝑎(𝜔)||𝑓⟩_𝑐=(𝛿𝜔)^1/2𝑓(𝜔)||𝑓⟩_𝑐,(12) and hence ⟨𝑓|_𝑐|𝑁||𝑓⟩_𝑐=_𝜔𝛿𝜔||𝑓(𝜔)||².(13) We also observe that the expectation value of the single-component electromagnetic field operator 𝐸⁽⁺⁾(𝑡) as defined in (5) is given by ⟨𝑓|_𝑐|𝐸⁽⁺⁾(𝑡)^→𝑥,𝑡||𝑓⟩_𝑐=𝑖_𝜔𝛿𝜔𝜔^1/2𝑓(𝜔)𝑒^{𝑖𝜙(𝑡)},(14) which therefore corresponds to a classical electromagnetic field. Classical interference effects as expressed in terms of such expectation values, which are nonlinearly dependent of the quantum states considered, are, of course, not directly linked to quantum-mechanical interference effects which are linearly dependent of quantum states. The use of quantum optics clarifies the connection between these views on interference effects.

The input quantum states to be considered in Figure 1, will, of course, correspond to the replacements 𝑎(𝜔)→𝑎₀(𝜔) and 𝑎(𝜔)→𝑎₁(𝜔), respectively in the definition of the states considered above.

2.2. Transformation in the Beam Splitter

Next, we consider a dispersionless beam splitter with frequency-independent transmittance 𝑇 and reflectance 𝑅. If we assume a prefect beam splitter, where all light is either reflected or transmitted, we have 𝑅+𝑇=1. The input annihilation operators 𝑎₀(𝜔) and 𝑎₁(𝜔) of the beam splitter will then transform according to (see e.g., [15–17]) 𝑎𝑎₂(𝜔)=√𝑇𝑎₀(𝜔)+𝑖√𝑅𝑎₁(𝜔),₃(𝜔)=√𝑇𝑎₁(𝜔)+𝑖√𝑅𝑎₀(𝜔),(15) where 𝑎₀(𝜔) and 𝑎₁(𝜔) are the LO and signal port mode annihilation operators, and 𝑎₂(𝜔),𝑎₃(𝜔) the output annihilation mode operators corresponding to the transmitted and reflected modes, respectively. The transformation rules for the corresponding creation operators follows from (15) by Hermitian conjugation. The two independent sets of output annihilation and creation operators will then obey the required canonical commutation relations [𝑎₂(𝜔),𝑎^†2(𝜔)]=[𝑎₃(𝜔),𝑎^†3(𝜔)]=1. The phase shift between the transmitted and reflected part, described by the complex numbers in (15) will play an important role below as is also the case in the famous Hong-Ou-Mandel two-photon correlation experiment [31] and related investigations (see e.g., [32–35]).

A light beam arriving at the beam splitter after being reflected in the mirrors 1 and 2 will, in general, be phase shifted, that is, expressed in terms of mode operators this process corresponds to the propagation time replacements in (5): 𝑎𝑎₂₃(𝜔)𝑒^{𝑖𝜙(𝑡)}→𝑎₂(𝜔)𝑒^{𝑖𝜙2(𝑡)},(𝜔)𝑒^{𝑖𝜙(𝑡)}→𝑎₃(𝜔)𝑒^{𝑖𝜙3(𝑡)},(16) due to difference in optical path lengths with 𝜙₂=−𝜔(𝑡−𝜏₂),𝜙₃=−𝜔(𝑡−𝜏₃)in terms of time-delays 𝜏₂ and 𝜏₃. The reflections at the identical mirrors in the MMI setup (Figure 1) will also introduce phase-shifts, but they are equal for the two light beams and can therefore be neglected all together. A light beam passing through the beam splitter after reflection at the mirrors will, again, transform according to (15) and we therefore, finally, obtain an expression for the mode operator describing incident light on the photon detector in (5), that is, 𝑎₅(𝜔)𝑒^{𝑖𝜙(𝑡)}→√𝑇𝑎₂(𝜔)𝑒^{𝑖𝜙2(𝑡)}+𝑖√𝑅𝑎₃(𝜔)𝑒^{𝑖𝜙3(𝑡)}=𝑎+₀(𝑎𝜔₁)𝑇𝑒^{𝑖𝜙2(𝑡)}−𝑅𝑒^{𝑖𝜙3(𝑡)}(𝜔)𝑖√𝑇𝑅𝑒^{𝑖𝜙2(𝑡)}+𝑖√𝑇𝑅𝑒^{𝑖𝜙3(𝑡)}.(17) For reasons of clarity, we will consider a 50/50 beam splitter, which also makes interference effects dependent only on the phases present, that is, we make the choice 𝑇=𝑅=1/2. The corresponding electromagnetic field operator to be used in Glauber’s theory of photon detection, is therefore given by 𝐸⁽⁺⁾(𝑡)=𝑖_𝜔12√𝛿𝜔𝜔^1/2×𝑎+₀𝑖(𝑎𝜔₁)𝑒^{𝑖𝜙2(𝑡)}−𝑒^{𝑖𝜙3(𝑡)}(𝜔)𝑒^{𝑖𝜙2(𝑡)}+𝑒^{𝑖𝜙3(𝑡)}.(18)

3. Interference of Fock States and Coherent States

Let us now specifically consider single photons in the signal and local ports, that is, the following initial Fock state as follows: ||in⟩=||𝑓_𝑠⟩⊗||𝑓_𝑙𝑜⟩,(19) with single-photon angular-frequency distributions according to (9). In the numerical results to be presented, we will use the same spectral widths 𝜎 but with 𝜔→𝜔_𝑠=3𝜎 for the signal port and 𝜔→𝜔_𝑙𝑜=3.15𝜎 or 𝜔_𝑙𝑜=2.85𝜎 for the local oscillator port. We now observe that the action of the electromagnetic field operator 𝐸⁽⁺⁾(𝑡), as given by (18), on this initial state is given by 𝐸⁽⁺⁾(𝑡)||in⟩=𝑖_𝜔12𝛿𝜔𝜔^1/2×𝑖+𝑓𝑓_𝑠(_𝑙𝑜𝜔)𝑒^{𝑖𝜙2(𝑡)}1+𝑒^𝑖𝜔𝜏||0⟩⊗||𝑓_𝑙𝑜⟩(𝜔)𝑒^{𝑖𝜙2(𝑡)}1−𝑒^𝑖𝜔𝜏||𝑓_𝑠⟩⊗||0⟩,(20) where we have made use of the fact that 𝜙₃(𝑡)−𝜙₂(𝑡)=𝜔(𝜏₃−𝜏₂)≡𝜔𝜏, which now defines the optical time-delay 𝜏. The nonnormalized state 𝐸⁽⁺⁾(𝑡)|in⟩ describes an entangled combination of the two orthogonal state vectors |0⟩⊗|𝑓_𝑙𝑜⟩ and |𝑓_𝑠⟩⊗|0⟩ corresponding to the absorption of one photon from the signal source or from the local oscillator, respectively. Apart from an overall normalization constant, each expansion coefficient, that is, the corresponding probability amplitude, contains information about the single-photon interference for the signal or the local oscillator photon, respectively. As we have discussed above, the probability for the detection of one photon will, according to Glauber’s theory of photon detection, be proportional to the matrix element ⟨in|𝐸⁽⁻⁾(𝑡)𝐸⁽⁺⁾(𝑡)|in⟩, that is, the absolute modulus square |𝐸⁽⁺⁾(𝑡)|in⟩|² of the nonnormalized state 𝐸⁽⁺⁾(𝑡)|in⟩. A time average over the time 𝑡 according to (6) with 𝑇_int≫1/𝜎 leads to a Dirac 𝛿(𝜔−𝜔′)-function and therefore makes any double-sum over frequencies into a single-sum. In the large-volume limit and for 𝑑=1, we then obtain ⟨𝐼⟩(𝜏)=12𝑇_int^∞0×𝜔||𝑓𝑑𝜔2𝜋_𝑠+𝜔||𝑓(𝜔_𝑙𝑜)||²(1+cos𝜔𝜏)(𝜔)||²(1−cos𝜔𝜏).(21) The single-photon detection intensity ⟨𝐼⟩(𝜏) is then a sum of two single-photon interference patterns. We notice the appearance of a relative minus sign in the local oscillator photon contribution, which has its origin in the relative phase-shift between the transmission-reflection and transmission-transmission coefficients of a beamsplitter alluded to in connection with (15) and (17). If 𝜔_𝑠=𝜔_𝑙𝑜 then the detector cannot distinguish between a photon from the signal or the local source with the same spectral widths 𝜎 and the two interference effects will exactly cancel against each other with the result that ⟨𝐼⟩(𝜏) will be independent of 𝜏.

If 𝜔_𝑠,𝜔_𝑙𝑜≫𝜎, then, within a good numerical approximation, we can first replace the linear 𝜔 dependence in (21) with 𝜔_𝑠 and 𝜔_𝑙𝑜 in front of the corresponding angular-frequency distributions, and then extend the integration to include arbitrarily negative angular-frequencies. One then finds that ⟨𝐼⟩(0)≃2𝜋𝜔_𝑠/𝑇_int and ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)≃211+𝜔_𝑙𝑜𝜔_𝑠+𝑒^{−𝜎2𝜏2/4}cos𝜏𝜔_𝑠−𝜔_𝑙𝑜𝜔_𝑠cos𝜏𝜔_𝑙𝑜,(22) where the interference effects are exponentially sensitive to the spectral width 𝜎 of the single-photon angular-distributions in a fashion similar to the spectral width dependence in the famous Hong-Ou-Mandel two-photon experiment [31]. The asymptotic value of ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) is given by (1+𝜔_𝑙𝑜/𝜔_𝑠)/2. In Figure 2, we exhibit ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) according to (21) with the choice as in (19). With a given reference distribution |𝑓_𝑙𝑜(𝜔)|² of the local oscillator one can, for example, now in principle infer the common spectral width 𝜎 of the single-photon sources as well as the corresponding angular-frequency 𝜔_𝑠. A related experimental situation is actually discussed in [36] for a general single-photon state, that is, not necessarily a pure quantum state.

Figure 2 

The normalized single-photon intensity ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) as a function of the dimensionless time-delay 𝜏𝜎 for the case of single-photon states in the signal and the LO ports with the same spectral width 𝜎 but with different mean frequencies. Two different examples are plotted with 𝜔𝑠=3𝜎: 𝜔𝑙𝑜=3.15𝜎 with (solid line) or 𝜔𝑙𝑜=2.85𝜎 (dashed line). The asymptotic values of ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) can be obtained from the expression (22) in the main text, that is, (1+𝜔𝑙𝑜/𝜔𝑠)/2.

In the case of coherent states in the signal and local ports, with single-particle state angular-frequency distributions 𝑓_𝑠(𝜔) and 𝑓_𝑙𝑜(𝜔) as above, (20) is now modified according to 𝐸⁽⁺⁾(𝑡)||in⟩=𝑖_𝜔12𝛿𝜔𝜔^1/2×𝑖+𝑓𝑓_𝑠(_𝑙𝑜𝜔)𝑒^{𝑖𝜙2(𝑡)}1+𝑒^𝑖𝜔𝜏(𝜔)𝑒^{𝑖𝜙2(𝑡)}1−𝑒^𝑖𝜔𝜏||𝑓_𝑠⟩_𝑐⊗||𝑓_𝑙𝑜⟩_𝑐,(23) which is not an entangled state. This is so since the removal of a photon from a coherent state does not change the state since it actually contains infinitely many photons, that is, 𝑎(𝜔)|𝑓⟩_𝑐=(𝛿𝜔)^1/2𝑓(𝜔)|𝑓⟩_𝑐according to (12). The intensity ⟨𝐼⟩(𝜏) according to (21) is, for real-valued single-photon distributions 𝑓_𝑠(𝜔) and 𝑓_𝑙𝑜(𝜔), therefore replaced by ⟨𝐼⟩(𝜏)=12𝑇_int^∞0𝑑𝜔𝜔2+𝜋||𝑓||𝑓_𝑠(𝜔)||²(1+cos𝜔𝜏)_𝑙𝑜−2𝑓_𝑠(𝜔)||²(1−cos𝜔𝜏)(𝜔)𝑓_𝑙𝑜(𝜔)sin𝜔𝜏,(24) which now contains an additional interference contribution as compared with (21). In Figure 3, we exhibit ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) with coherent states generated by the same choice of single-photon states as in Figure 2. The interference pattern is, as in Figure 2, sensitive to the actual angular-frequency distributions. With a given reference distribution |𝑓_𝑙𝑜(𝜔)|² of the local oscillator, one can now infer the common spectral width 𝜎 of the coherent state sources as well as the corresponding angular-frequency 𝜔_𝑠. By a comparison with Figure 2, we conclude that the MMI setup is sensitive to the actual form of the initial quantum states despite the fact that we are only considering single-photon detection processes. With the same approximations leading to (22), we find an additional contribution to ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0), that is, ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)≃121+𝜔_𝑙𝑜𝜔_𝑠1−𝑒^{−𝛿𝜔2/4𝜎2−𝜎2𝜏2/4}+𝑒_{−𝜎2𝜏2/4}sin𝜔𝜏cos𝜏𝜔_𝑠−𝜔_𝑙𝑜𝜔_𝑠cos𝜏𝜔_𝑙𝑜,(25) with 𝛿𝜔≡(𝜔_𝑠−𝜔_𝑙𝑜) and 𝜔≡(𝜔_𝑙𝑜+𝜔_𝑠)/2. In particular, and more important, we notice that when 𝜔≡𝜔_𝑙𝑜=𝜔_𝑠 we can write ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)≃1−𝑒^{−𝜎2𝜏2/4}sin𝜏𝜔,(26) which now explicitly depends on 𝜏 in contrast to the initial Fock state considerations above in which case ⟨𝐼⟩(𝜏)≃⟨𝐼⟩(0).

Figure 3 

The normalized intensity ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) as a function of the time-delay 𝜏𝜎 for the case of coherent states as generated by the same single-photon states as in Figure 2. Two different examples are plotted with 𝜔𝑠=3𝜎: 𝜔𝑙𝑜=3.15𝜎 with (solid line) or 𝜔𝑙𝑜=2.85𝜎 (dashed line). The asymptotic values of ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) are as in Figure 2.

4. Interference of Thermal Light in the Michelson-Morley Interferometer

4.1. Thermal Light

For the readers convenience, let us first outline a simple and quantum mechanical description of thermal black body radiation at an absolute temperature 𝑇. Black body radiation is, basically, light with random phases. For a single mode with angular-frequency 𝜔 and if |𝑛⟩=𝑎^†(𝜔)^𝑛|0⟩/√𝑛! denotes a 𝑛-photon state, the density matrix describing the thermal light is given by 𝜌(𝜔)=^∞_𝑛=0𝑝_𝑛(𝜔)||𝑛_𝜔⟩⟨𝑛_𝜔||,(27) in terms of the Bose-Einstein distribution: 𝑝_𝑛(𝜔)=𝑛(𝜔,𝑇)1+𝑛(𝜔,𝑇)^𝑛11+𝑛(𝜔,𝑇),(28) with 𝑛(𝜔,𝑇)=1expℏ𝜔/𝑘_𝐵𝑇−1.(29)

The state 𝜌(𝜔) corresponds to an extreme value of the von Neumann entropy 𝑆, that is, 𝑆=−𝑘_𝐵𝑇Tr[𝜌(𝜔)ln𝜌(𝜔)],(30) subjected to the constraints (see e.g., [13, 22]): Tr𝑎_†Tr[𝜌(𝜔)]=1,(𝜔)𝑎(𝜔)𝜌(𝜔)=𝑛(𝜔,𝑇).(31) The random, or chaotic, nature of the quantum state 𝜌(𝜔) corresponds to a phase-independent Glauber-Sudarshan 𝒫(𝛼)-representation (see e.g., [37–42]) in terms of a coherent state, that is, 𝜌(𝜔)=𝑑²𝛼𝒫(𝛼)|𝛼⟩⟨𝛼|,(32) using a single-mode coherent state |𝛼⟩=exp(𝛼𝑎^†(𝜔)−𝛼^∗𝑎(𝜔))|0⟩. For thermal light, one finds that 𝒫(𝛼)=1𝜋𝑛(𝜔,𝑇)exp−𝑛(|𝜔𝛼|,²𝑇),(33) which obeys the normalization condition: Tr[𝜌(𝜔)]=𝑑²𝛼𝒫(𝛼)=1,(34) as well as Tr𝑎^†(𝜔)𝑎(𝜔)𝜌(𝜔)=𝑑²𝛼𝑃(𝛼)|𝛼|²=𝑛(𝜔,𝑇).(35)

A multimode system at thermal equilibrium is then described in terms of a tensor product 𝜌(𝑇)=⊗_𝜔𝜌(𝜔), where we have performed the replacement 𝛼→𝛼(𝜔) in (32) and (33). The Glauber-Sudarshan 𝒫(𝛼)-representation for the state 𝜌(𝑇) is now, in particular, useful in our considerations since the response in a single-photon detector can be obtained immediately from the previous results for coherent light using an average procedure.

4.2. Thermal Light in the Signal Port and Vacuum in the Local Oscillator Port

We are now in the position to consider a density matrix describing a system where we have thermal light in the signal port and vacuum in the LO port, that is, the initial density matrix 𝜌_in of the total system is given by 𝜌_in=𝜌(𝑇)⊗(|0⟩⟨0|)_𝑙𝑜.(36) By making use of [25], with 𝑓_𝑠(𝜔)→𝛼(𝜔) and 𝑓_𝑙𝑜(𝜔)→0, and then performing an average over 𝛼(𝜔) according to (35), we immediately obtain the following result: ⟨𝐼⟩(𝜏)=12𝑇_int^∞0𝑑𝜔2𝜋𝜔^𝑑𝑛(𝜔,𝑇)(1+cos𝜔𝜏),(37) in 𝑑 space dimensions.

In passing, we notice the important fact that (37) is, in fact, valid for any physical quantum state of the form (36), where 𝜌(𝑇) is replaced with any density matrix 𝜌(𝜔) not necessarily thermal, due to the generality of the Glauber-Sudarshan 𝑃(𝛼)-representative equation (32). The interference effects as exhibited by the MMI setup, therefore, actually only depends on the, in general, angular-frequency dependent mean number Tr[𝑎^†(𝜔)𝑎(𝜔)𝜌(𝜔)]=𝑛(𝜔) and not on other features of the actual quantum state 𝜌(𝜔).

By a straightforward change of the variable of integration in (37), and with 𝑎≡𝜏𝑘_𝐵𝑇/ℏ, we then find for a thermal quantum state 𝜌(𝑇) that ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)=121+𝐽1(𝑑)^∞0𝑑𝑥𝑥^𝑑cos(𝑎𝑥)exp(𝑥)−1,(38) where 𝐽(𝑑) can be expressed in terms of gamma and Riemann’s 𝜁 functions, that is, 𝐽(𝑑)=Γ(1+𝑑)𝜁(𝑑+1)=Γ(1+𝑑)^∞_𝑛=11𝑛^𝑑+1.(39) Particular values are 𝐽(1)=𝜋²/6 and 𝐽(3)=𝜋⁴/15. In the case of 𝑑=3, we, therefore, recover the well-known expression for ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) [22] as also discussed in [21]. In, for example, 𝑑=3 it is actually possible to carry out the relevant integral in the expression equation (38) for ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) analytically with the result ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)=211+152+sicnohsh(𝑎𝜋(2)𝑎⁴𝜋)−3(𝑎𝜋)⁴.(40)

Equation (40) shows that interference effects have a power-law sensitivity for larger 𝑎. In Figure 4, we show ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) for varying values of 𝑎 in the case with 𝑑=3 and one infers a characteristic coherence length 𝑙_𝑐 of thermal light in the MMI of the form: 𝑙_𝑐≃1.5ℏ𝑐𝑘_𝐵𝑇,(41) as also discussed in [21].

Figure 4 

The normalized intensity ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) as a function of 𝑎≡𝜏𝑘𝐵𝑇0/ℏ for the case of thermal light in the signal port and vacuum in the LO port and for 𝑑=3. We infer a characteristic thermal coherence time 𝜏𝑐 in terms of 𝑎≃1.5 (dashed vertical line), that is, 𝜏𝑐≃1.5ℏ/𝑘𝐵𝑇.

5. Thermal Light in the Signal and Local Ports

As we have seen above, with a vacuum in the LO port and with a signal thermal source the single-photon detection process exhibits an interference pattern. We now investigate what happens if we have thermal light with a temperature 𝑇₀ in the LO port and thermal light with temperature 𝑇₁ in the signal port. For this setup, the corresponding initial density matrix becomes 𝜌_in=𝜌𝑇₁⊗𝜌𝑇₀.(42) By making use of this density matrix as well as the same methods as described in Section 4.2 by performing independent averages over 𝑓_𝑠(𝜔) and 𝑓_𝑙𝑜(𝜔) in (24) according to (35), we immediately obtain, in the large-volume limit, the following result: ⟨𝐼⟩(𝜏)=12𝑇_int^∞0𝑑𝜔𝜔^𝑑2𝜋+𝑛𝜔,𝑇₁(1+cos𝜔𝜏)𝑛𝜔,𝑇₀(1−cos𝜔𝜏).(43) In obtaining this result, we notice that the last term in (24), suitably extended to complex-valued 𝑓_𝑠(𝜔) and 𝑓_𝑙𝑜(𝜔), averages to zero due to the chaotic nature, that is, phase-independence of thermal light according to (32) and (33). The relative intensity then takes the form (𝑑=3) as follows: ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)=12+115+𝑇𝑇⁰₁⁴2𝜋⁴^∞0𝑑𝑥𝑥³×cos𝑎₁exp(𝑥)−1𝑥−𝑇⁰𝑇₁⁴cos𝑎₀𝑥,(44) where 𝑎₀=𝜏𝑘_𝐵𝑇₀/ℏ and 𝑎₁=𝜏𝑘_𝐵𝑇₁/ℏ. The integrals in (44) can, again, be solved analytically in a fashion similar to the integral in (38), that is, ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)=211+𝑇𝑇⁰₁⁴1−152+cosh2𝑎⁰𝜋+15sinh𝑎₀𝜋⁴22+cosh2𝑎¹𝜋sinh𝑎₁𝜋⁴.(45) Due to the presence of hyperbolic functions in (45), we observe that ⟨𝐼⟩(𝜏)/⟨𝐼⟩(0) approaches its asymptotic value (1+(𝑇₀/𝑇₁)⁴)/2 exponentially fast as a function of the time-delay 𝜏 in contrast to the power-law dependence in (40). It is now of interest to study the behavior of (45) when the temperatures 𝑇₀ and 𝑇₁ of the local oscillator and signal, respectively are varied. In Figure 5, we exhibit the interference when 𝑇₀ is slightly smaller or larger than 𝑇₁ as a function of the parameter 𝜏𝑘_𝐵𝑇₁/ℏ for 𝑑=3. The corresponding interference, of course, disappears when the two temperatures are equal. The sensitivity of the interference pattern with regard to the difference in temperatures of the source and the reference temperature, that is, of the local source, constitutes the basic ingredient of the FIRAS setup. We also observe that the coherence length 𝑐 for this MMI setup is roughly the same as in Section 4.2 since the two temperatures of the source considered are close to each other.

Figure 5 

The same as in Figure 4 but with thermal light in both ports with 𝑇1=1.01𝑇0 (upper curve), 𝑇1=0.99𝑇0 (lower curve), and 𝑇1=𝑇0 (solid line). The vertical dashed line corresponds to the thermal coherence length 𝑙𝑐=𝑐𝜏𝑐 with 𝜏𝑐𝑘𝐵𝑇1/ℏ≃1.5.

6. Conclusions and Final Remarks

We have seen that the interference of thermal light in the Michelson-Morley interferometer can be described, in a straightforward manner, by making use of Glauber’s theory of photon detection and elementary quantum theory of the electromagnetic field. Furthermore, we have seen the emergence of a natural coherence length _𝑐≃1.5ℏ𝑐/𝑘_𝐵𝑇 of thermal light in the MMI. The result for thermal light in both the signal and the local oscillator ports shows that the interference pattern is sensitive to the difference in temperature of the two sources. This is the basic principle used by the FIRAS on board the COBE satellite in order to perform high-precision measurements of the temperature and the spectrum of the cosmic micro-wave background radiation.

It may first appear surprising that nonmonochromatic and chaotic/thermal light, with random phases, exhibits interference effects since it can be argued that a quantum state with a fixed number of photons has an undetermined “quantum phase” (see e.g., [43–48]). A mixed quantum state of such photon-number eigenstates therefore have no intrinsic and well-defined phase. As we, however, have seen the interference effects are naturally obtained in terms of the phase-dependent normal-mode expansion of the quantized electromagnetic field and its corresponding timeevolution rather than in terms of a Schrödinger picture and possible phases of quantum states (see e.g., the comments in [49]).

Since we have been considering initial quantum states in terms of a fixed number of photons as well as “classical” states, corresponding to coherent states with an infinite number of photons present, a quantum-mechanical language is mandatory. The signal and local ports of the MMI setup corresponds to independent input sources. It is, of course, a well-known experimental fact that independent photon sources can give rise to interference effects (see e.g., [50–55]). Despite the fact that such interference effects are well established, the interpretation of them can, nevertheless, gives rise to interesting issues regarding the very fundamental aspects of the quantum-mechanical world (see e.g., [56]) when considering, in particular, interference effects using single-photon sources.

We have seen that for multimode systems the quantum nature of these independent sources actually affects the nature of the single-photon intensity measurements. We have already mentioned that the angular-frequency distribution of a single photon can be measured using a similar experimental setup as the MMI considered in the present paper [36]. With a vacuum state in the local oscillator port and a single-photon signal angular-frequency distribution 𝑓(𝜔) of the form considered in (9), one finds, using (21), that ⟨𝐼⟩(𝜏)⟨𝐼⟩(0)≃211+𝑒^{−𝜎2𝜏2/4}cos𝜏𝜔_𝑠.(46) We conclude by noticing that a characteristic exponential behavior as in (46) for single-photon interference has recently been observed by measuring the photoluminescence signal of a single quantum dot [57] using a Michelson-Morley interferometer (for a related study of interference effects of dissimilar photon sources, see [58]).

Acknowledgments

This research has made use of NASA’s Astrophysics Data System. The research was supported by the Research Council of Norway through Grants 170935/V30 and FRINAT-191564, and by NTNU. One of the authors (B.-S. Skagerstain) wishes to thank Professor F. G. Scholtz for a generous and stimulating hospitality during a joint NITheP and Stias, Stellenbosch (SA), workshop in 2009, Professors M. Reid and D. V. Ahluwalia, University of Christchurch, NZ, J. R. Klauder, University of Florida, Gainesville, USA, P. S. Riseborough, Temple University, USA., the TH Division at CERN, and the Centre for Advanced Study (CAS), Oslo, for hospitality when the present work was in progress.