Common Symplectic Transformations

There are a number of symplectic transformations on Gaussian states that have particular importance. For a single-mode Gaussian state the most important transformations are displacement, rotation, and squeezing.

The displacement operation is described by the displacement operator defined in Eq. (1.19), which is the complex version of the Weyl operator. It has no effect on the covariance matrix of a Gaussian state and only changes the mean values of the quadratures by a displacement ¯x → x¯ +d, where α = (x0 +ip0) and d = (x0, p0)T. Application of an

arbitrary displacement operation onto the vacuum results in the class of coherent states. The squeezing operation is described by the squeezing operator defined in Eq. (1.28). It transforms the covariance matrix asV→S(r)VS(r)T, where

S(r)≡ e−r ₀ 0 er (1.63)

is the symplectic map describing the squeezing operation. Application of the squeezing operation to the vacuum results in a state with zero mean values of the quadratures and covariance matrix V=S(2r), where the variance in one quadrature is reduced below the vacuum level and the variance is increased in the other. In other words, applying the squeezing operation to the vacuum creates the class of squeezed states.

Phase rotation of a Gaussian state results in the mixing of quadratures, and therefore introduces correlations between the quadratures of a single mode. It is described by the symplectic map R(θ)≡ cosθ sinθ −sinθ cosθ . (1.64)

It has no effect on the mean value of the quadratures, and can accurately be thought of as a rotation of the Wigner function. Combining the rotation operation with the squeezing operation allows for squeezing in any direction.

Any general one-mode Gaussian state can be created by applying a combination of squeezing, displacement and rotation to a thermal state [216]. This demonstrates the importance of the three described operations. Thermal states have a covariance matrix

1.3. Gaussian States

general one-mode Gaussian state is therefore a state with mean dand covariance matrix

V= (2¯n+ 1)R(θ)S(2r)R(θ)T, (1.65) where ¯n= 0 gives a general pure one-mode Gaussian state.

1.3.4 Two-mode Gaussian States

Now we want to study two-mode Gaussian states and study correlations between different light modes. A general two-mode Gaussian state is created by interactions between general one-mode Gaussian states. For this thesis, the most important transformation describing an interaction between two modes is the beamsplitter interaction. This transformation is described by the operator

B(θ) = exp[θ(ˆa†ˆb−ˆaˆb†)], (1.66) where ˆa and ˆb are the annihilation operators of the two modes and θ is related to the transmissivity of the beamsplitter by the relation τ = cos2θ. In terms of covariance matrices, the symplectic matrix describing the interaction is

B(τ) = √ τI √1−τI −√1−τI √τI . (1.67)

The beamsplitter operation hybridises the two input modes and each of the output modes can be thought of as a superposition of the two input modes, with the weighting of the superposition dependent on the transmissivity of the beamsplitter. The beamsplitter operation is a passive operation, as can be seen by the linearity of the beamsplitter operator in Eq. (1.66), which means that it preserves the photon number of the input beams. As well as describing the mixing of two modes, the beamsplitter is also useful to describe loss in a Gaussian channel, where the reflectivity ρ = (1−τ) quantifies the loss and the reflected mode represents the lost part of the beam.

The partial trace is another operation on multi-mode Gaussian states that is of par- ticular interest. Given a two mode state ˆρAB the partial trace over modeB has the action

trB( ˆρAB) = ˆρA, where ˆρA is the state of mode A. The partial trace removes any infor-

mation about modeB and just leaves the marginal state of modeA. The partial trace is often used to describe loss, as it models the elimination of part of a state. The definition of the partial trace can be trivially extended to include more modes and tracing over dif- ferent modes. It is useful to note that applying the partial trace to a pure state, in general results in a mixed state, except for the case that the traced out mode is uncorrelated with the remaining mode or modes. The covariance matrix of a state after a partial trace is simply the covariance matrix of the state before the partial trace, but with the entries related to the traced-out mode deleted, which results in a covariance matrix with reduced dimension.

An important two-mode Gaussian state is the two-mode squeezed state, with applica- tions in many quantum optics experiments [28]. It can be created by mixing two orthog- onally squeezed states on a balanced (τ = 1/2) beamsplitter. This results in a state with zero mean and a covariance matrix of the form

S2(r) = cosh(2r)I sinh(2r)σz sinh(2r)σz cosh(2r)I . (1.68)

This state is also known as an Einstein-Podoslki-Rosen (EPR) state due to the nature of the correlations between the quadratures of the two modes. Note that this is the standard form of the EPR state, and the original squeezed modes could be at any angle, as long as they are orthogonal, resulting in a state with a different covariance matrix but the same entanglement properties. In the limit of r → ∞ the result is an ideal EPR state with perfect correlations between the modes, i.e., ˆxA = ˆxB and ˆpA=−pˆB. Finally, it is

interesting to note that taking the partial trace of an EPR state results in a thermal state for the remaining mode as can be seen from Eq. (1.68) and the definition of a thermal state. Due to this, it can be said that the EPR state is the purification of a thermal state.