--- abstract: 'The spin of a photon in the interaction with a magnetic field is determined by a combination of two polarizations. The quantum-mechanical nature of the spins of individual photons can be measured using weak values. We present an experimentally accessible weak value of the total spin of a photon interacting with an arbitrary magnetic field. We demonstrate that the weak value gives access to the polarization density matrix and to the Wigner distribution function of the spin degree of freedom of a photon interacting with a magnetic field.' address: - 'Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit[ä]{}t Jena, Max-Wien-Platz 1, D-07743 Jena, Germany' - 'Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa City, Chiba 277-8581, Japan' - 'Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit[ä]{}t Jena, Max-Wien-Platz 1, D-07743 Jena, Germany' author: - Michael Schreiber - Harald Weinfurter - Masahiro Takeoka - Yong Zheng title: 'The weak value of the total spin of a photon interacting with a magnetic field' --- Introduction {#sec1} ============ Weak values have recently attracted much attention as a new tool in quantum physics [@Aharonov1988; @Ritchie1991; @Hosten2008; @Kocsis2011]. The weak values can be seen as the result of measurements performed after the measurement devices have returned to their original state. They appear if the coupling between the preselected state and the postselected state is weak. The theoretical framework of weak values is a generalization of the standard quantum mechanics [@Aharonov1990]. It allows for results that are apparently contradicting with standard quantum theory but are in perfect agreement with experiments and may be useful for describing phenomena where the standard quantum theory fails. Examples for this are the Aharonov-Bohm effect [@Tonomura1989; @Tonomura1990], weak measurements in which the measuring interaction becomes very strong at certain times [@Hosten2008; @Kocsis2011; @Dressel2013], and the Wigner distribution function of a photon [@Aharonov1986; @Duck1989]. A direct experimental implementation of the weak values is given by so-called weak measurement tomography [@Lundeen2009]. In such an experiment, the postselected state is, similar to the experiment performed by Aharonov [*et al.*]{}, reconstructed by performing a sequence of strong measurements. Weak values are obtained from the recorded outcomes of the weak measurement [@Aharonov1991]. The weak value of a Hermitian operator $\hat{A}$ is defined by [@Aharonov1988; @Ritchie1991; @Hosten2008; @Kocsis2011] $$\langle \hat{A} \rangle_w = \frac{\langle\psi_f|\hat{A}|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle} = \frac{\sum_m \langle\psi_f|\hat{P}_m|\psi_i\rangle A_m}{\sum_m \langle\psi_f|\hat{P}_m|\psi_i\rangle}, \label{wv}$$ where $|\psi_i\rangle$ and $|\psi_f\rangle$ are the initial and final states, and $\hat{P}_m$ denotes the projection operators onto the final states $|\psi_m\rangle = |\psi_f\rangle \hat{P}_m|\psi_i\rangle/\langle\psi_f|\hat{P}_m|\psi_i\rangle$. In weak measurement tomography, weak values of Hermitian operators are used to reconstruct the postselected state. The weak values do not give direct access to the preselected state [@Lundeen2009; @Lundeen2009b], which needs to be known in order to calculate the weak values. Here, we present an experimentally accessible preselected state for which the weak values of any Hermitian operator can be evaluated directly. In general, such a preselected state is not an eigenstate of $\hat{A}$, but a (possibly mixed) statistical mixture of states that can be prepared from the preselected state. We use this statistical mixture as the preselected state to obtain the weak values of the components of the total spin operator of a photon interacting with an arbitrary magnetic field. The total spin of a photon is a classical property and its quantum nature is usually not considered [@Mandel1995]. However, it has been observed that the total spin of a photon can be used to increase the violation of the Bell inequalities [@Kwiat1995]. Recently, a direct link between classical and quantum properties has been established for classical (i.e., optical) fields [@Kagalwala2013; @Ghose2013], demonstrating how the quantum nature of optical properties manifests itself classically. It was shown that the quantum mechanical measurement of the total spin gives different results than the standard optical measurement scheme, giving rise to a non-classical correlation [@Kagalwala2013; @Ghose2013]. Moreover, the total spin of a single photon in a quantum superposition of spin states can exhibit its quantum nature by violating a Bell inequality. In the present article, we first give an explicit expression of the total spin operator and of its components. We show that the weak values of the components can be directly measured, using weak values. Moreover, we give an expression for the weak value of the total spin of a photon interacting with a magnetic field, and use this expression to calculate the weak values of the components of the spin. We find that in the weak limit, the measured quantity becomes proportional to the magnetic field. Furthermore, we show that the distribution of the measured quantity in space is proportional to the magnetic field squared. This is shown by direct calculation, and also intuitively explained in analogy to the distribution of an electric dipole moment of a single atom induced by a magnetic field [@Aharonov1986; @Duck1989]. We finally show how the experimentally accessible weak value can be used to reconstruct the total spin density matrix of a photon interacting with a magnetic field. The reconstructed density matrix corresponds to the spin component of a photon propagating through the magnetic field. The article is organized as follows: In Sec. \[sec2\], we define the total spin of a photon. In Sec. \[sec3\], we discuss our results. Sec. \[sec4\] gives a detailed derivation of the weak value for the total spin of a photon, and Sec. \[sec5\] a discussion of the total spin density matrix of a photon. The total spin operator of a photon {#sec2} =================================== The spin operator of a photon is $$\hat{S}_z = \frac{1}{2}(\hat{a}^\dag\hat{a} - \hat{b}^\dag\hat{b}),$$ where $\hat{a}$ ($\hat{b}$) is the annihilation operator of a photon with an intrinsic angular momentum quantum number $m_a$ ($m_b$). The index $z$ refers to the $z$ component of the spin operator $\hat{S}_z$. The eigenvalues of the spin operator $\hat{S}_z$ are $\pm 1/2$. The spin of a photon can be measured using a magnetic field oriented along the $z$ axis. The magnetic field interacts with the orbital angular momentum component of the photon by the magnetic moment of the photon and couples to the spin component of the photon via the Zeeman effect. The spin interaction with a magnetic field can be described by the Hamiltonian [@Bialynicki-Birula1994; @Bialynicki-Birula1996] $$H = \frac{\hat{m}_a}{2}\gamma B_z - \frac{\hat{m}_b}{2}\gamma B_z + \gamma B_x(\hat{a}^\dag\hat{a} + \hat{b}^\dag\hat{b}) +\frac{1}{2}\kappa(\hat{a}^\dag\hat{a} - \hat{b}^\dag\hat{b}), \label{H}$$ where $B_x$, $B_z$ are the components of the magnetic field along the $x$ and $z$ axis, respectively, $\kappa$ is the magnitude of the photon-photon interaction, and $\gamma$ is the gyromagnetic factor. A photon with momentum ${\bf k}$, an intrinsic angular momentum of $m$, and a spin $s$ can be written as [@Bialynicki-Birula1994; @Bialynicki-Birula1996] $$a_s^\dag({\bf k}) = \sum_{\bf n} \varepsilon({\bf n},{\bf k}) e^{i{\bf n}{\bf r}} a_{s,{\bf n}}^\dag, \label{photon_decomp}$$ where $\varepsilon({\bf n},{\bf k})$ is the spin-dependent component of the electric field associated with the mode ${\bf n}$, and $a_{s,{\bf n}}^\dag$ is a photon creation operator in the mode ${\bf n}$ with polarization $s$. The mode ${\bf n}$ is either a circular or linear polarized mode of a standing wave for which $${\bf k} = k_n (\cos\varphi_n