Direct Measurement of the Quantum Wavefunction

The wavefunction $\Psi$, also known as the ‘quantum state’, is considerably more difficult to measure than the state of a classical particle, which is determined simply by measuring its position $X$ and momentum $P$. According to the Heisenberg Uncertainty Principle, in quantum theory a precise measurement of $X$ disturbs the particle’s wavefunction and forces a subsequent measurement of $P$ to become random. Thus we learn nothing of the particle’s momentum. Indeed, it is impossible to determine a completely unknown wavefunction of single system Wootters1982 .

Consider instead performing a measurement of $X$ on an ensemble of particles, all with the same $\Psi$. The probability of getting result $X=x$ is $\left|\Psi(x)\right|^{2}$. Similarly, the probability of $P=p$ would be $\left|\Phi(p)\right|^{2}$, where $\Phi(p)$ is the Fourier transform of $\Psi(x).$ Even these two probability distributions are not enough to determine $\Psi(x)$ unambiguously (see the 1d phase retrieval problem Trebino2002 ). Instead, one must reconstruct $\Psi$ by performing a large set of distinct measurements (e.g. $Q(\theta)=X\cos(\theta)+P\sin(\theta)$), and then estimating a $\Psi$ that is most compatible with the measurement results. This method is known as quantum state tomography Vogel1989 ; Smithey1993 ; Breitenbach1997 ; White1999 ; Hofheinz2009 . In contrast, we introduce a method to measure $\Psi$ of an ensemble directly. Here, by ‘direct’ we mean that the method is free from complicated sets of measurements and computations; the average raw signal originating from where the wavefunction is being probed is simply proportional to its real and imaginary components at that point. The method rests upon the sequential measurement of two complementary variables of the system.

At the center of the direct measurement method is a reduction to the disturbance induced by the first measurement. Consider the measurement of an arbitrary variable $A$. In general, measurement can be seen as the coupling between an apparatus and a physical system that results in the translation of a pointer. The pointer position indicates the result of a measurement. In a technique known as ‘weak measurement’, one reduces the coupling strength and this correspondingly reduces the disturbance created by the measurement Aharonov1988 ; Ritchie1991 ; Resch2004 ; Smith2004 ; Pryde2005 ; Mir2007 ; Hosten2008 ; Dixon2009 ; Lundeen2009 ; Aharonov2010 . This strategy also compromises measurement precision but this can be regained by averaging. The average of the weak measurement is simply the expectation value $\left\langle\Psi\right|A\left|\Psi\right\rangle$, indicated by an average position shift of the pointer proportional to this amount.

A distinguishing feature of weak measurement is that it does not disturb a subsequent normal (or ‘strong’) measurement of another observable $C$ in the limit where the coupling vanishes. For the particular ensemble subset that gave outcome $C=c,$ one can derive the average of the weak measurement of $A$. In the limit of zero interaction strength, this is called the Weak Value and is given Aharonov1988 by,

Selecting a particular subset of an ensemble based on a subsequent measurement outcome is known as ‘post-selection’, and is a common tool in quantum information processing Knill2001 ; Duan2001 .

Unlike the standard expectation value $\left\langle A\right\rangle$, the Weak Value $\left\langle A\right\rangle_{W}$ can be a complex number. This seemingly strange result can be shown to have a simple physical manifestation: the pointer’s position is shifted by Re$\left\langle A\right\rangle_{W}$ and receives a momentum kick of Im$\left\langle A\right\rangle_{W}$ Aharonov1990 ; Lundeen2005 ; Jozsa2007 . The complex nature of the Weak Value suggests that it could be used to indicate both the real and imaginary parts of the wavefunction.

Returning to our example of a single particle, consider the weak measurement of position ($A=\pi_{x}\equiv\left|x\right\rangle\left\langle x\right|$) followed by a strong measurement of momentum giving $P=p$. In this case the Weak Value is

In the case $p=0$, this simplifies to

where $k=1/\Phi(0)$ is a constant (which can be eliminated later by normalizing the wavefunction). The average result of the weak measurement of $\pi_{x}$ is proportional to the wavefunction of the particle at $x$. Scanning the weak measurement through $x$ gives the complete wavefunction. At each $x$, the observed position and momentum shifts of the measurement pointer are proportional to Re$\Psi(x)$ and Im$\Psi(x)$ , respectively. In short, by reducing the disturbance induced by measuring $X$ and then measuring $P$ normally, we measure the wavefunction of the single particle.

As an experimental example, we performed a direct measurement of the transverse spatial wavefunction of a photon. Considering a photon travelling along the $Z$ direction, we directly measure the $X$ wavefunction of the photon, sometimes called the ‘spatial mode’ (see the Supplementary Discussion). The Wigner function of the spatial mode of a classical beam has been measured directly but not for a single photon state Mukamel2003 ; Smith2005 .

We produce a stream of photons in one of two ways, either by attenuating a laser beam or by generating single photons through spontaneous parametric downconversion (SPDC) (see the Supplementary Methods for details). The photons have a center wavelength of $\lambda=783$nm or $800$nm, respectively. The experiment (details and schematic in Fig. 1) can be divided into four sequential steps: preparation of the transverse wavefunction, weak measurement of the transverse position of the photon, post-selection of those photons with zero transverse momenta, and readout of the weak measurement.

An ensemble of photons with wavefunction $\Psi(x)$ is emitted from a single mode (SM) fiber and collimated. We will begin by directly measuring this wavefunction (described in detail in Fig. 1). We then further test our method by inducing known magnitude and phase changes to the photons here to prepare a series of modified wavefunctions.

We weakly measure the transverse position of the photon by coupling it to an internal degree of freedom of the photon, its polarization. This allows us to use the linear polarization angle of the photon as the pointer. At a position $x$ where we wish to measure $\pi_{x}=\left|x\right\rangle\left\langle x\right|$ we rotate the linear polarization of the light by $\varphi.$ Consider if $\varphi$ is set to $90^{\circ}$. In this case, one can perfectly discriminate whether a photon had position $x$ because it is possible to perfectly discriminate between orthogonal polarizations, $0^{\circ}$ and $90^{\circ}$. This is a strong measurement. Reducing the strength of the measurement corresponds to reducing $\varphi$, which makes it impossible to discriminate with certainty whether any particular photon had $X=x.$ The benefit of this reduction in precision is a commensurate reduction in the disturbance to the wavefunction of the single photon.

We then use a Fourier Transform lens and a slit to post-select only those photons with $p=0$. This constitutes the strong measurement of $P$.

In this subset of the photon ensemble, we find the average value of our weak measurement of $\pi_{x}$. The average rotation of the pointer, the linear polarization, is proportional to the real part of the Weak Value. Its complementary pointer variable, the rotation in the circular polarization basis, is proportional to the imaginary part of the Weak Value Lundeen2005 . Formally, if we treat the initial polarization as a spin-1/2 spin down vector, then the weak value is given by

where $\sigma_{x}$ and $\sigma_{y}$ are the Pauli x and y matrices, respectively, and $\left|s\right\rangle$ is the final polarization state of the pointer Lundeen2005 . We measure the $\sigma_{x}$ and $\sigma_{y}$ expectation values by sending the photons through either through a half-wave plate or quarter-wave plate, respectively, and then through a polarizing beamsplitter (PBS). Thus, we read out Re$\Psi(x)$ (half-wave plate) and Im$\Psi(x)$ (quarter-wave plate) from the signal imbalance between detectors 1 and 2 at the outputs of the PBS.

With $\varphi=20^{\circ}$, we scan our measurement of $\pi_{x}$ in 1 mm steps and find the Weak Value $\left\langle\pi_{x}\right\rangle_{W}$ at each step. In this way, we directly measure the photon transverse wavefunction, $\Psi(x)=\left|\Psi(x)\right|\exp\left(i\phi(x)\right)$. We normalize the $\sigma_{x}$ and $\sigma_{y}$ measurements by the same factor, so that $\int\left|\Psi(x)\right|^{2}dx=1,$which eliminates the proportionality constant, $\sin\varphi/\Phi(0).$

To confirm our direct measurement method, we test it on a series of different wavefunctions. Using our SPDC single photon source, we start by measuring the initial truncated Gaussian wavefunction (Fig. 2) described in Fig. 1. Switching to the laser source of photons, we then modify the magnitude, and then the phase, of the initial wavefunction with an apodized filter and glass plate, respectively, to create two new test $\Psi$ (Fig. 3). We conduct more quantitative modification of the wavefunction phase by introducing a series of phase gradients and then phase curvatures (Fig. 4). For all the test wavefunctions, we have found good agreement between the expected and measured wavefunction, including its phase and magnitude (see the Figure Captions for details).

We now describe how the technique of weak measurement can be used to directly measure the quantum state of an arbitrary quantum system. We have the freedom to measure the quantum state in any chosen basis $\left\{\left|a\right\rangle\right\}$ (associated with observable $A$) of the system. The method entails weakly measuring a projector in this basis, e.g. $\pi_{a}\equiv\left|a\right\rangle\left\langle a\right|$, and post-selecting on a particular value $b_{0}$ of the complementary observable $B$ (See the Supplementary Discussion for a precise definition of complementarity). In this case, the Weak Value is

where $v$ is a constant, independent of $a$. Thus the Weak Value is proportional to the amplitude of state $\left|a\right\rangle$ in the quantum state. Stepping $a$ through all the states in the $A$ basis directly gives the quantum state represented in that basis

This is the general theoretical result of this paper. It shows that in any physical system one can directly measure the quantum state of that system by scanning a weak measurement through a basis and appropriately post-selecting in the complementary basis.

Weak measurement necessarily trades efficiency for accuracy or precision. A comparison of our method to current tomographic reconstruction techniques will require careful consideration of the signal to noise ratio in a given system. In order to increase this ratio in the direct measurement of the photon spatial wavefunction, future experiments will investigate the simultaneous post-selection of many transverse momenta.

In our direct measurement method, the wavefunction manifests itself as shifts of the pointer of the measurement apparatus. In this sense, the method provides a simple and unambiguous Operational definition Bridgman1927 of the quantum state: it is the average result of a weak measurement of a variable followed by a strong measurement of the complementary variable. We anticipate that the simplicity of the method will make feasible the full characterization of quantum systems (e.g. atomic orbitals, molecular wavefunctions Itatani2004 , ultrafast quantum wavepackets Dudovich2006 ) previously unamenable to it. The method can also be viewed as a transcription of quantum state of the system to that of the pointer, a potentially useful protocol for quantum information.