# Why can two atoms can be entangled?

As a qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization.

since entanglement is a quantum phenomena so I can understand that electron/photons can be entangled but I am not able to understand why 2 atoms can be entangled, do atoms show two-level quantum-mechanical system?

• Entanglement is a property of multiparticle systems. Number of levels does not matter. Jul 19, 2022 at 7:29
• what is or isn't entangled are degrees of freedom, more than the "particles" themselves. You can have entanglement between the inner degrees of freedom of a single particle, for example. See e.g. physics.stackexchange.com/a/530389/58382, physics.stackexchange.com/a/531084/58382, quantumcomputing.stackexchange.com/a/21915/55. So, say, there is no problem in having the spin degrees of freedom of a pair of atoms being entangled... or the positions of the atoms can be entangled, etc.
– glS
Jul 19, 2022 at 12:23

If you have already accepted that qubits can be entangled, then you can see that atoms can be entangled too by realizing that we can encode qubits into atoms. We can do this by selecting a pair of orthonormal states of an atom and declaring them to be our computational basis. The subspace spanned by the two states constitutes (an encoding of) a qubit.

For example, we can take a hydrogen atom and define

$$|0\rangle := |s^1p^0\rangle\\ |1\rangle := |s^0p^1\rangle\tag1$$

where $$|s^up^v\rangle$$ denotes the state of the atom with $$u\in\{0,1\}$$ spin-up electrons on the first $$s$$ orbital and $$v\in\{0,1\}$$ spin-up electrons on for example the first $$p_x$$ orbital. Since these states correspond to different angular momentum the spectral theorem implies that they are orthogonal.

Now, take the entangled state $$\frac{|01\rangle+|10\rangle}{\sqrt2}$$ of two qubits. We can write down a corresponding state of two atoms as

$$\frac{|s^1p^0\rangle|s^0p^1\rangle+|s^0p^1\rangle|s^1p^0\rangle}{\sqrt2}\tag2$$

which is an entangled state of two hydrogen atoms.

There are many other (better) ways of encoding qubits in atoms. For example, we could use electron or nuclear spin, orbital angular momentum, fine structure etc.

What is relevant is whether the two systems that we want to see entangled can be protected against decoherence. As such, the whole world is quantum mechanical, but, we can see characteristically quantum mechanical behavior in objects that are not decohered by the environment. One way to prevent decoherence from happening is isolating the system from the environment well enough (in the sense of not letting it interact sufficiently with the environment). So, as long as atoms can be isolated well enough from the environment, they can be entangled.