yvec : array_like y-coordinates at which to calculate the Wigner function. Here we suppose the pump mode is initially in a coherent state, while the signal and idler modes are in the ground state. As soon as one W_mn Wigner function is, calculated, the corresponding contribution is added to the total Wigner, function, weighted by the corresponding element in the density matrix, Using Laguerre polynomials from scipy to evaluate the Wigner function for, the density matrices :math:`|m>

In general, this evolution is governed by the Schrödinger equation (5) with a non-Hermitian effective Hamiltonian, where again, the Cn are collapse operators, each corresponding to a separate irreversible process with rate γn. demonstrated below. Calculates tensor product from list of input operators or states. functions they can for example be negative. QuTiP provides an object-oriented framework for representing generic quantum systems, and for performing calculations and simulations on such systems. these distributions are also great for demonstrating if a quantum state is J. L. O’Brien, A. Furusawa, ", Using an iterative method to evaluate the wigner functions for the Fock, :math:`W = \sum_{mn} \\rho_{mn} W_{mn}` where :math:`W_{mn}` is the Wigner. Phys. arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. The derivation of Eq. The state of an open quantum system is therefore described in terms of ensemble averaged states using the density matrix formalism. D. Seljebotn, K. Smith, If you are

IN NO EVENT SHALL THE COPYRIGHT. It is a straightforward exercise to implement an arbitrary time-dependence by changing the definition of the Hamiltonian callback function, or by modifying the time-independent part of the Hamiltonian to correspond to a more complicated quantum system. J. Casanova, G. Romero, Licensing provisions: GPLv3 We can solve it by writing \(\chi\) and the superoperator propagator as \([N^4]\) vectors, and likewise write the superoperator product \(\tilde{B}_m\tilde{B}_n^\dagger\) as a \([N^4\times N^4]\) matrix \(M\): Note that to obtain \(\chi\) with this method we have to construct a matrix \(M\) with a size that is the square of the size of the superoperator for the system. Sometimes, it may also be useful to directly visualizing the underlying matrix For computational purposes it is useful to expand the PDE in a set of basis functions that span the Hilbert space of the Hamiltonian, and to write the equation in matrix and vector form.

# you want to decrease that number to ~50 first. After discussing with the authors, I now understand what they were trying to show with their calculations, and was actually quite impressed with the knowledge of the main author with respect to numerical simulations. xvec : array_like x-coordinates at which to calculate the Wigner function. Since QuTiP is written in python IV: Using the renormalized state from step III as the new initial condition at time τ, draw a new random number, and repeat the above procedure until the final simulation time is reached. 3 we show the scaling of the elapsed time for a typical simulation, here chosen to be the Heisenberg spin-chain.

For the largest dimensionality considered in Fig. [GCC 4.6.3]. # this list of conditions and the following disclaimer. )` where, :math:`c_L = \sum_n \\rho_{n,L+n} LL_n^L` where, :math:`LL_n^L = (-1)^n sqrt(L!n!/(L+n)!)

In the common case of iterating over a list of parameters phi-coordinates at which to calculate the Q function. where g is the coupling strength. To do so go to. Here, the environment is continuously monitored, resulting in a series of quantum jumps in the system wave function, conditioned on the increase in information gained about the state of the system via the environmental measurements Haroche and Raimond (2006). To fix this dilemma, QuTiP includes a nonlinear colormap function qutip.visualization.wigner_cmap 8 we plot the expectation values for the three modes of the trilinear Hamiltonian (18), with corresponding damping rates, γa=0.2, γb=0.8, γc=0.2, for the initial state given by Eq. state the problem in such a form, that we have a function iterating purposes. Sign up to our mailing list for occasional updates.

Program Title: QuTiP: The Quantum Toolbox in Python In this Jaynes-Cumming problem we are interested in the excitation number of the cavity and the atom, and as such can then define a list of expectation value operators. Note that the left hand side of Eq.

photon number for a single atom couple to a cavity as a function of To find the steady-state of the system in QuTiP all we need to do is define the Hamiltonian and collapse operators and call: is the steady state density matrix. x-coordinates at which to calculate the Wigner function.

The 'fft' method evaluates the Fourier, transform of the density matrix.

We can then write, where \(\tilde{\rho}\) is the vector representation of the density matrix \(\rho\). consistency. to find it if you do a search for qutip. ", Using an iterative method to evaluate the wigner functions for the Fock, :math:`W = \sum_{mn} \\rho_{mn} W_{mn}` where :math:`W_{mn}` is the Wigner. (2011), and data visualization capabilities Hunter (2007), that largely are implemented in efficient compiled code. The Qobj class is the QuTiP representation of quantum operators and state vectors. L. Dale, J. W. Harlow, 3rd edition, 2004. as possible. 10. Values representing the Q-function calculated over the specified range, # Q-function for a pure state: Q = ||^2 / pi, # |alpha> = the coherent state with amplitude alpha. Here we will try to reproduce Figure #1 from Qian, Quantum Signatures of the Optomechanical Instability. It takes a ket or a density matrix as input, together with arrays that define the ranges of the phase-space coordinates (in the x-y plane). The Qobj class internally maintains a record of the principal attributes of the quantum object it represents.

curious how I did that, or want to create your own environment, take a As your job gets longer and longer and as you submit New J. Phys. y-coordinates at which to calculate the Wigner function. (11). To generate subplots (a)-(h), all we need to do is calculate this steady-state for each value of detuning and coupling strength given in the paper. Opt.

70 The 'fft' method accepts only an xvec input for the x-coordinate. Rev. Photonics (2010). (2). making these negative values stand out in a figure is helpful for both analysis and publication From a Qobj instance one may also calculate items such as the expectation value (expect) for an arbitrary operator with the QuTiP function, find the fidelity (fidelity) between two density matrices Nielsen and Chuang (2000), or calculate the Wigner function (wigner) of a quantum state. (2006). Starting from a pure state |ψ(0)⟩: I: Choose a random number r between zero and one, representing the probability that a quantum jump occurs. 1) may be inspected by displaying the string representation of the object returned by its str method. Mod. In absence of a quantum computer Buluta et al. which, in this case, shows that the Qobj instance rho0_qubit is a 2×2, Hermitian quantum operator representing a balanced coherent superposition of its two basis states. In this section we display the QuTiP codes underlying the calculations performed in generating Figures 2, 4, 6, 7, 8, 9, and 10. 13 (2011) A complete overview of QuTiP is given on its website Nation and Johansson (2011).

467 (2010) 297. At this point we are ready to let QuTiP calculate the time-evolution of the system. look at the very last section of this post. 2), in the exploration of open quantum dynamics will be the focus of Sec. with cavity frequency ωc, cavity decay rate κ, and where ⟨n⟩th is the steady state thermal occupation number. Instead, we will focus on the key data structures, and numerical routines underlying the majority of calculations.

M. Lenander, E. Lucero, method : string {'clenshaw', 'iterative', 'laguerre', 'fft'}, Select method 'clenshaw' 'iterative', 'laguerre', or 'fft', where 'clenshaw', and 'iterative' use an iterative method to evaluate the Wigner functions for density, matrices :math:`|m>

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