What Is the Relationship Between Photons and Electromagnetic Field Wave Packets?
HaifeiA Wave Packet Can Correspond to a Photon
Example: A photon can be in a state of coherent superposition of different frequencies: \(|\psi\rangle=\sum_{k}c_k|k\rangle,\quad \sum_k|c_k|^2=1\). In this case, the photon can manifest as a wave packet.
You can imagine an atom de-exciting and producing a photon; this photon will of course manifest as a wave packet.
Some may argue: if all nonideal factors are excluded, then the linewidth of this photon depends only on natural broadening (lifetime), and it can be regarded as having a single frequency. It therefore has poor localization and cannot be called a wave packet. This is indeed true.
However, if one considers single photons generated by pulsed pumping and parametric processes in the low-gain regime, their natural linewidth is itself very large. In this case, they are indeed in a coherent superposition of different frequencies and manifest as well-localized wave packets in the time domain.
Definition: Single-Photon State
A state of the form \(\sum_{k}c_k|0,\cdots,\underbrace{1}_{k-\text{th}},\cdots,0\rangle\) is called a single-photon state, where \(|0,\cdots,\underbrace{1}_{k-\text{th}},\cdots,0\rangle\) denotes that there is one photon in the k-th mode, and \(\sum_{k}|c_k|^2=1\).
Intuitively, a coherent superposition of states each having one photon in a different mode is still a single-photon state.
However, a Photon Is Not Necessarily a Wave Packet
Counterexample 1: After passing through an NPBS (non-polarizing beam splitter), a photon simultaneously takes two paths: a transmitted path and a reflected path. At this point, the photon is nonlocal.
Counterexample 2: After passing through a PBS (polarizing beam splitter), a circularly polarized photon simultaneously takes two paths: a transmitted path and a reflected path. At this point, the photon is nonlocal. If path is regarded as a degree of freedom of the Hilbert space, then the photon is in an entangled state of path and polarization.
Counterexample 3: A photon can also consist of two wave packets along the same path: split a photon into two paths using a PBS, then recombine the two paths using another PBS. If the optical path lengths of the two paths differ, then after recombination, the photon becomes two wave packets, one following the other.
Counterexample 4: A photon can even interfere with itself: first split the photon into two wave packets, then adjust the optical path lengths of the two paths to be approximately equal, and finally overlap the two wave packets again. This is a Michelson/Mach–Zehnder interferometer at the single-photon level.
More Interesting Photon States
The above three counterexamples are all related to spatiotemporal modes and are well-known facts. Below are several other interesting states that may refresh your understanding of photons.
Example 1: Frequency superposition state
As stated above, a photon can be in a superposition of different frequencies. More importantly, a photon can also be two wave packets rather than one wave packet in the frequency domain. For example, a photon can be in a superposition of wavelengths 1557nm and 1563nm; such states can be prepared experimentally[1].
Example 2: Electric-field superposition state (Schrödinger cat state)
When the photon number is very small, the electric and magnetic fields also have significant uncertainty (just as momentum and position have uncertainty). The electric and magnetic fields form a pair of canonical variables.
Schrödinger cat states can be prepared experimentally, in which the electric field is in a superposition of \(|\alpha\rangle\) and \(|-\alpha\rangle\). In this case, if you measure the electric field at a certain point (along the polarization direction), you may obtain either a positive or a negative value.
When the photon number is relatively large, such states are very fragile and rapidly decohere into a statistical mixture of \(|\alpha\rangle\) and \(|-\alpha\rangle\).
At lower photon-number levels, such states can be prepared experimentally[2].
Example 3: Frequency-entangled state
Two photons can also be in a frequency-entangled state: \(|\psi\rangle=\frac{1}{2}(|\mu\rangle \otimes|\nu\rangle+|\nu\rangle \otimes|\mu\rangle)\). In brief, if the wavelength of one photon is observed to be 1557nm, then the wavelength of the other photon immediately collapses to 1663nm, and vice versa[1].
Example 4: Number-entangled state
NOON states can be prepared experimentally: \(|\psi\rangle=\frac{1}{2}(|N,0\rangle + |0,N\rangle)\), meaning that if one mode has N photons, then the other mode has no photons, and vice versa. Moreover, the number can only be 0 or N, not any other value.
These two modes can be spatial modes. In this case, you can imagine a group of photons in a superposition of all moving left and all moving right. If you detect N photons on one side, then you immediately know that there are no photons at the distant other end, and vice versa.
Of course, when the photon number is relatively large, such states are rather fragile, and it is difficult to ensure that the photons maintain stable phase relations with one another.
In short, photons have many degrees of freedom. Any degree of freedom you can think of can be superposed and entangled. However, when the photon number is large, there are three difficulties: preparation is difficult, decoherence occurs easily, and characterization is difficult (state tomography).
A photon can be a wave packet, or it may not be a wave packet; a wave packet can be a photon, or it may not be a photon. The key is to understand coherent superposition in quantum mechanics: \(\sum_{k}c_k|0,\cdots,\underbrace{1}_{k-\text{th}},\cdots,0\rangle\).
References
- ^abShukhin, A., Hurvitz, I., Trajtenberg-Mills, S., Arie, A. & Eisenberg, H. Two-dimensional Control of a Biphoton Joint Spectrum. Preprint at http://arxiv.org/abs/2311.09660 (2023).
- ^Lvovsky, A. I. et al. Production and applications of non-Gaussian quantum states of light. Preprint at http://arxiv.org/abs/2006.16985 (2020).