Higher order coherence

Figure 1. Schematic diagram for the setup of the Young's Double Slit Experiment.

In Young's double slit experiment, light from a light source is allowed to pass through two pinholes separated by some distance, and a screen is placed some distance away from the pinholes where the interference between the light waves is observed (Figure. 1). Young's double slit experiment demonstrates the dependence of interference on coherence, specifically on the first-order correlation. This experiment is equivalent to the Mach–Zehnder interferometer with the caveat that Young's double slit experiment is concerned with spatial coherence, while the Mach–Zehnder interferometer relies on temporal coherence.[2]

The intensity measured at the position ${\displaystyle \mathbf {r} }$ at time ${\displaystyle t}$ is

${\displaystyle \langle I\rangle =\langle |E^{+}(\mathbf {r} ,t)|^{2}\rangle =\langle I\rangle =I_{1}+I_{2}+2{\sqrt {I_{1}I_{2}}}|\gamma ^{(1)}(x_{1},x_{2})|cos\phi (x_{1},x_{2})}$.

Light field has highest degree of coherence when the corresponding interference pattern has the maximum contrast on the screen. The fringe contrast is defined as ${\displaystyle V={\frac {I_{max}-I_{min}}{I_{max}+I_{min}}}}$.

Classically, ${\displaystyle I_{min}^{max}=I_{1}+I_{2}\pm 2{\sqrt {I_{1}I_{2}}}|\gamma ^{(1)}(x_{1},x_{2})|}$ and hence ${\displaystyle V={\frac {2{\sqrt {I_{1}I_{2}}}|\gamma ^{(1)}(x_{1},x_{2})|}{I_{1}+I_{2}}}}$. As coherence is the ability to interfere visibility and coherence are linked:

${\displaystyle |\gamma ^{(1)}(x_{1},x_{2})|=1}$ means highest contrast, complete coherence

${\displaystyle 0<|\gamma ^{(1)}(x_{1},x_{2})|<1}$ means partial fringe visibility, partial coherence

${\displaystyle |\gamma ^{(1)}(x_{1},x_{2})|=0}$ means no contrast, complete incoherence.[2][3]

Classically, the electric field at a position ${\displaystyle \mathbf {r} }$, is the sum of electric field components from at the two pinholes ${\displaystyle \mathbf {r} _{1}}$ and ${\displaystyle \mathbf {r} _{2}}$ earlier times ${\displaystyle t_{1},t_{2}}$ respectably i.e. ${\displaystyle E^{+}(\mathbf {r} ,t)=E^{+}(\mathbf {r_{1}} ,t_{1})+E^{+}(\mathbf {r} _{2},t_{2})}$. Correspondingly, in the quantum description the electric field operators are similarly related, ${\displaystyle {\hat {E}}^{+}(\mathbf {r} ,t)={\hat {E}}^{+}(\mathbf {r_{1}} ,t_{1})+{\hat {E}}^{+}(\mathbf {r} _{2},t_{2})}$. This implies

${\displaystyle I=Tr[\rho {\hat {E}}^{-}(\mathbf {r} ,t){\hat {E}}^{+}(\mathbf {r} ,t)]=I_{1}+I_{2}+2{\sqrt {I_{1}I_{2}}}|g^{(1)}(x_{1},x_{2})|cos\phi (x_{1},x_{2})}$.

The intensity fluctuates as a function of position i.e. the quantum mechanical treatment also predicts interference fringes. Moreover, in accordance to the intuitive understanding of coherence i.e. ability to interfere, the interference patterns depend on the first-order correlation function ${\displaystyle g^{(1)}}$.[1] Comparing this to the classical intensity, we note that the only difference is that the classical normalized correlation ${\displaystyle \gamma ^{(1)}}$ is now replaced by the quantum correlation ${\displaystyle g^{(1)}}$. Even the computations here look strikingly similar to the ones that might be done classically.[2] However, the quantum interference that occurs in this process is fundamentally different from the classical interference of electromagnetic waves. Quantum interference occurs when two possible histories, given a particular initial and final state, interfere. In this experiment, given an initial state of the photon before the pinhole and it final state at the screen, the two possible histories correspond to the two pinholes through which the photon could have passed. Hence, quantum mechanically, here the photon is interfering with itself. Such interference of different histories, however, occurs only when the observer has no specific way of determining which of the different histories actually occurred. If the system is observed to determine the path of the photon, then on average the interference of amplitudes will vanish.[1]

Figure 2. A schematic diagram for the setup for Hanbury Brown and Twiss's original experiment.

In the Hanbury Brown and Twiss experiment (Figure 2.), a light beam is split using a beam splitter and then detected by detectors, which are equidistant from the beam splitter. Subsequently, signal measured by the second detector is delayed by time ${\displaystyle \tau }$ and the coincidence rate between the original and delayed signal is counted. This experiment correlates intensities, ${\displaystyle |E^{+}(\mathbf {r} ,t+\tau )E^{+}(\mathbf {r} ,t)|^{2}}$, rather than electric fields and hence measures the second order correlation function

${\displaystyle G_{c}^{(2)}(t,t+\tau ,t+\tau ,t)=\langle E^{-}(t)E^{-}(t+\tau )E^{+}(t+\tau )E^{+}(t)\rangle }$.

Under the assumption of stationary statistics, at a given position, the normalized correlation function is

${\displaystyle g^{(2)}={\frac {\langle {\hat {E}}^{-}(0){\hat {E}}^{-}(\tau ){\hat {E}}^{+}(\tau ){\hat {E}}^{+}(0)\rangle }{\langle {\hat {E}}^{-}(0){\hat {E}}^{+}(0)\rangle \langle {\hat {E}}^{-}(\tau ){\hat {E}}^{+}(\tau )\rangle }}}$

${\displaystyle g^{(2)}}$ here measures the probability of coincidence of two photons being detected with a time difference ${\displaystyle \tau }$.[2]

For all varieties of chaotic light, the following relationship between the first order and second-order coherences holds:

${\displaystyle g^{(2)}(\tau )=1+|g^{(1)}(\tau )|^{2}}$.

This relationship is true for both the classical and quantum correlation functions. Moreover, as ${\displaystyle |g^{(1)}(\tau )|}$ always takes a value between 0 and 1, for a chaotic light beam, ${\displaystyle 1\leq g^{(2)}\leq 2}$. The light source used by Hanbury Brown and Twiss was stellar light which is chaotic. Hanbury Brown and Twiss used this result to compute the first order coherence from their measurement of the second order coherence. The observed second order coherence the curve was as shown in figure 2.[5]

For Gaussian light source ${\displaystyle g^{(1)}=e^{-i\omega _{0}\tau -{\frac {\pi }{2}}({\frac {\tau }{\tau _{0}}})^{2}}}$. Often a Gaussian light source is chaotic and consequently,

Figure 3. The second order coherence for stellar light as measured in the Hanbury Brown and Twiss experiment as a function of the time delay introduced between the signals ${\displaystyle \tau /\tau _{0}}$, where ${\displaystyle \tau _{0}}$ is the coherence length.

${\displaystyle g^{(2)}(\tau )=1+e^{-{\frac {\pi }{2}}({\frac {\tau }{\tau _{0}}})^{2}}}$.

This model fits the observation that was done by Hanbury Brown and Twiss using stellar light as demonstrated in figure 3. If thermal light was used instead of stellar light in the same setup, then we would see a different function for the second order coherence.[5] Thermal light can be modeled to be a Lorentzian power spectrum centered around frequency ${\displaystyle \omega _{0}}$, which means ${\displaystyle \langle E^{*}(0)E(\tau )\rangle =E_{0}^{2}e^{-|\tau |/\tau _{0}}}$, where ${\displaystyle \tau _{0}}$ is the coherence length of the beam. Correspondingly, ${\displaystyle g^{(1)}=e^{-i\omega _{0}\tau -|\tau |/\tau _{0}}}$ and ${\displaystyle g^{(2)}(\tau )=1+e^{-2|\tau |/\tau _{0}}}$. The second-order coherence for stellar (Gaussian), thermal (Lorentzian) and coherent light is shown in Figure 4. Note that when stellar/thermal light beam is first order coherent i.e. ${\displaystyle g^{(1)}(0)=1}$, the second order coherence is 2, meaning at zero time delay chaotic light right is first order coherent but not second order coherent.[3][5]

Classically, we can think of a light beam as having a probability distribution as a function of mode amplitudes, ${\displaystyle P(\{\alpha _{k}\})}$ and in that case, the second order correlation function ${\displaystyle G^{(2)}(\tau ,0)=\langle E^{-}(\tau )E^{+}(\tau )E^{-}(0)E^{+}(0)\rangle \int P(\{\alpha _{k}\})E^{*}(\tau )E(\tau )E^{*}(0)E(0)d\{\alpha _{k}\}}$. If we assume that the quantum state of the setup is ${\displaystyle \rho =\int d\{\alpha _{k}\}P(\{\alpha _{k}\})|\{\alpha _{k}\}\rangle \langle \{\alpha _{k}\}|}$, then the quantum mechanical correlation function,${\displaystyle G^{(2)}(\tau ,0)=Tr[\rho {\hat {E}}^{-}(\tau ){\hat {E}}^{+}(\tau ){\hat {E}}^{-}(0){\hat {E}}^{+}(0)]=\int P(\{\alpha _{k}\})E^{*}(\tau )E(\tau )E^{*}(0)E(0)d\{\alpha _{k}\}}$, which is same as the classical result.[6]

Figure 4. The second order coherence for thermal, stellar and coherent light as a function of time delay. ${\displaystyle \tau _{0}}$ is the coherence length of the light beam.

Similar to the case of Young's double slit experiment, the classical and the quantum description lead to the same result, but that does not mean that two descriptions are equivalent. Classically, the light beams arrives as an electromagnetic wave and interferes owing to the superposition principle. The quantum description is not as straightforward. To understand the subtleties in the quantum description, assume that photons from the source are emitted independent of each other at the source and that the photons are not split by the beam splitter. When the intensity of the source is set to be very low, such that only one photon might be detected at any time, accounting for the fact that there might be accidental coincidences, which are statistically independent of time, the coincidence counter should not change with respect to the time difference. However, as shown in Figure 3., for stellar light ${\displaystyle g^{(2)}(\tau )=1+e^{-2|\tau |/\tau _{0}}}$, so without any time delay ${\displaystyle g^{(2)}(0)=2}$ and with a large time delay ${\displaystyle \lim _{\tau \rightarrow \infty }g^{(2)}(\tau )=1}$. Hence, even when there was no time-delay the photons from the source were arriving in pairs! This effect is termed photon bunching. Moreover, if a laser light was used at the source instead of chaotic light, then second order coherence would be independent of the time delay. HBT's experiment allows for a fundamentally distinction in the way in which photons are emitted from a laser compared to a natural light source. Such a distinction is not captured by the classical description on wave interference.[1]

A field is called mth order coherent if there exists a function ${\displaystyle E(x)}$ such that all correlation functions for ${\displaystyle n<m}$ factorize. Notationally, this means

${\displaystyle G^{(n)}(x_{1},...,x_{2n})=\prod \limits _{j=1}^{n}E^{*}(j)E^{*}(j+1)}$

This factorizability of all ${\displaystyle n<m}$ correlation functions implies that ${\displaystyle |G^{(n)}(x_{1},...,x_{2n})|^{2}=\prod \limits _{j=1}^{2n}G^{(1)}(x_{j},x_{j})}$. As ${\displaystyle g^{(n)}(x_{1},...,x_{2n})}$ was defined to be ${\displaystyle {\frac {G^{(n)}(x_{1},...,x_{n};x_{n},...,x_{1})}{\prod \limits _{j=1}^{n}G^{(1)}(x_{j},x_{j})}}}$, it follows that ${\displaystyle |g^{(n)}(x_{1},...,x_{2n})|=1}$ for ${\displaystyle n<m}$, if the field is m-coherent.[7] For a m-coherent field, the ${\displaystyle m}$ photons being detected will be detected statistically independent of each other.[1]

Given an upper bound on how many photons can be present in the field, there is an upper bound on the Mth coherence the field can have. This is because ${\displaystyle {\hat {E}}^{+}}$ is proportional the annihilation operator. To see this, begin with a mixed state for the field ${\displaystyle \sum \limits _{n,m}c_{n,m}|n\rangle \langle m|}$. If this sum has an upper limit on n, m i.e. ${\displaystyle M>n,m}$, ${\displaystyle Tr[\rho {\hat {E}}^{+}(x_{1})...{\hat {E}}^{+}(x_{p})]}$ is proportional to ${\displaystyle Tr[\sum \limits _{n,m}c_{n,m}{\hat {a}}...({\text{p times}})...{\hat {a}}|n\rangle \langle m|]=\sum \limits _{n,m}c_{n,m}\langle m|{\hat {a}}...({\text{p times}})...{\hat {a}}|n\rangle =0}$ for ${\displaystyle p>m}$. This result would be unintuitive in a classical description, but fortunately such a case has no classical counterpart because we cannot put an upper bound on the number of photons in the classical case.[1]

When dealing with classical optics, physicists often employ the assumption that the statistics of the system are stationary. This means that while the observations might fluctuate, the underlying statistics of the system remains constant as time progresses. The quantum analogue of stationary statistics is to require that the density operator, which contains the information about the wavefunction, commutes with the Hamiltonian. Owing to Schrödinger equation, ${\displaystyle {\frac {d\rho }{dt}}={\frac {-i}{h}}[H,\rho ]}$, stationary statistics implies that the density operator is independent of time. Consequently, in${\displaystyle G^{(n)}(x_{1},...,x_{2n})}$, owing to the cyclicity of the trace, we can transform the time independence of the density operator in the Schrödinger picture to the time independence of ${\displaystyle {\hat {E}}^{+}}$ and ${\displaystyle {\hat {E}}^{-}}$, in the Heisenberg picture, giving us

${\displaystyle G^{(n)}(x_{1},...,x_{2n})=Tr[{\hat {\rho }}{\hat {E}}^{-}(x_{1},t)....{\hat {E}}^{-}(x_{n},t){\hat {E}}^{+}(x_{n+1},t)...{\hat {E}}^{+}(x_{2n,t})]}$ ${\displaystyle =Tr[{\hat {\rho }}{\hat {E}}^{-}(x_{1},t+\tau )....{\hat {E}}^{-}(x_{n},t+\tau ){\hat {E}}^{+}(x_{n+1},t+\tau )...{\hat {E}}^{+}(x_{2n},t+\tau )]}$.

This means that under the assumption that the underlying statistics of the system are stationary, the nth order correlation functions do not change when every time argument is translated by the same amount. In other words, rather than looking at actual times, the correlation function is only concerned with the ${\displaystyle 2n-1}$ time differences.[1]

Coherent state are quantum mechanical states that have the maximal coherence and have the most "classical"-like behavior. A coherent state is defined as the quantum mechanical state that is the eigenstate of the electric field operator ${\displaystyle {\hat {E}}^{+}}$. As ${\displaystyle {\hat {E}}^{+}}$ is directly proportional to the annihilation operator the coherent state is an eigenstate of the annihilation operator. Given a coherent state ${\displaystyle |\alpha \rangle }$, ${\displaystyle G^{(n)}(x_{1},...,x_{2n})=Tr[\sum \limits _{n,m}c_{n,m}{\hat {a}}...{\hat {a}}|\alpha \rangle \langle \alpha |]=\langle \alpha |{\hat {a}}...({\text{p times}})...{\hat {a}}|\alpha \rangle =1}$. Consequently, coherent states have all orders of coherences as being non-zero.[8]