Solving Problems with Timedependent Hamiltonians¶
Methods for Writing TimeDependent Operators¶
In the previous examples of quantum evolution,
we assumed that the systems under consideration were described by timeindependent Hamiltonians.
However, many systems have explicit time dependence in either the Hamiltonian,
or the collapse operators describing coupling to the environment, and sometimes both components might depend on time.
The timeevolutions solvers
qutip.mesolve
, qutip.mcsolve
, qutip.sesolve
, qutip.bloch_redfield.brmesolve
qutip.stochastic.ssesolve
, qutip.stochastic.photocurrent_sesolve
, qutip.stochastic.smesolve
, and qutip.stochastic.photocurrent_mesolve
are all capable of handling timedependent Hamiltonians and collapse terms.
There are, in general, three different ways to implement timedependent problems in QuTiP:
Function based: Hamiltonian / collapse operators expressed using [qobj, func] pairs, where the timedependent coefficients of the Hamiltonian (or collapse operators) are expressed using Python functions.
String (Cython) based: The Hamiltonian and/or collapse operators are expressed as a list of [qobj, string] pairs, where the timedependent coefficients are represented as strings. The resulting Hamiltonian is then compiled into C code using Cython and executed.
Array Based: The Hamiltonian and/or collapse operators are expressed as a list of [qobj, np.array] pairs. The arrays are 1 dimensional and dtype are complex or float. They must contain one value for each time in the tlist given to the solver. Cubic spline interpolation will be used between the given times.
Hamiltonian function (outdated): The Hamiltonian is itself a Python function with timedependence. Collapse operators must be time independent using this input format.
Given the multiple choices of input style, the first question that arrises is which option to choose?
In short, the function based method (option #1) is the most general,
allowing for essentially arbitrary coefficients expressed via userdefined functions.
However, by automatically compiling your system into C++ code,
the second option (string based) tends to be more efficient and will run faster
[This is also the only format that is supported in the qutip.bloch_redfield.brmesolve
solver].
Of course, for small system sizes and evolution times, the difference will be minor.
Although this method does not support all timedependent coefficients that one can think of,
it does support essentially all problems that one would typically encounter.
Timedependent coefficients using any of the following functions,
or combinations thereof (including constants) can be compiled directly into C++code:
'abs', 'acos', 'acosh', 'arg', 'asin', 'asinh', 'atan', 'atanh', 'conj',
'cos', 'cosh','exp', 'erf', 'zerf', 'imag', 'log', 'log10', 'norm', 'pi',
'proj', 'real', 'sin', 'sinh', 'sqrt', 'tan', 'tanh'
In addition, QuTiP supports cubic spline based interpolation functions [Modeling NonAnalytic and/or Experimental TimeDependent Parameters using Interpolating Functions].
If you require mathematical functions other than those listed above,
it is possible to call any of the functions in the NumPy library using the prefix np.
before the function name in the string, i.e. 'np.sin(t)'
and scipy.special
imported as spe
.
This includes a wide range of functionality, but comes with a small overhead created by going from C++>Python>C++.
Finally option #4, expressing the Hamiltonian as a Python function, is the original method for time dependence in QuTiP 1.x. This method is somewhat less efficient then the previously mentioned ones. However, in contrast to the other options this method can be used in implementing timedependent Hamiltonians that cannot be expressed as a function of constant operators with timedependent coefficients.
A collection of examples demonstrating the simulation of timedependent problems can be found on the tutorials web page.
Function Based Time Dependence¶
A very general way to write a timedependent Hamiltonian or collapse operator is by using Python functions as the timedependent coefficients. To accomplish this, we need to write a Python function that returns the timedependent coefficient. Additionally, we need to tell QuTiP that a given Hamiltonian or collapse operator should be associated with a given Python function. To do this, one needs to specify operatorfunction pairs in list format: [Op, py_coeff]
, where Op
is a given Hamiltonian or collapse operator and py_coeff
is the name of the Python function representing the coefficient. With this format, the form of the Hamiltonian for both mesolve
and mcsolve
is:
>>> H = [H0, [H1, py_coeff1], [H2, py_coeff2], ...]
where H0
is a timeindependent Hamiltonian, while H1
and H2
are timedependent. The same format can be used for collapse operators:
>>> c_ops = [[C0, py_coeff0], C1, [C2, py_coeff2], ...]
Here we have demonstrated that the ordering of timedependent and timeindependent terms does not matter. In addition, any or all of the collapse operators may be timedependent.
Note
While, in general, you can arrange timedependent and timeindependent terms in any order you like, it is best to place all timeindependent terms first.
As an example, we will look at an example that has a timedependent Hamiltonian of the form \(H=H_{0}f(t)H_{1}\) where \(f(t)\) is the timedependent driving strength given as \(f(t)=A\exp\left[\left( t/\sigma \right)^{2}\right]\). The following code sets up the problem
ustate = basis(3, 0)
excited = basis(3, 1)
ground = basis(3, 2)
N = 2 # Set where to truncate Fock state for cavity
sigma_ge = tensor(qeye(N), ground * excited.dag()) # g><e
sigma_ue = tensor(qeye(N), ustate * excited.dag()) # u><e
a = tensor(destroy(N), qeye(3))
ada = tensor(num(N), qeye(3))
c_ops = [] # Build collapse operators
kappa = 1.5 # Cavity decay rate
c_ops.append(np.sqrt(kappa) * a)
gamma = 6 # Atomic decay rate
c_ops.append(np.sqrt(5*gamma/9) * sigma_ue) # Use Rb branching ratio of 5/9 e>u
c_ops.append(np.sqrt(4*gamma/9) * sigma_ge) # 4/9 e>g
t = np.linspace(15, 15, 100) # Define time vector
psi0 = tensor(basis(N, 0), ustate) # Define initial state
state_GG = tensor(basis(N, 1), ground) # Define states onto which to project
sigma_GG = state_GG * state_GG.dag()
state_UU = tensor(basis(N, 0), ustate)
sigma_UU = state_UU * state_UU.dag()
g = 5 # coupling strength
H0 = g * (sigma_ge.dag() * a + a.dag() * sigma_ge) # timeindependent term
H1 = (sigma_ue.dag() + sigma_ue) # timedependent term
Given that we have a single timedependent Hamiltonian term, and constant collapse terms, we need to specify a single Python function for the coefficient \(f(t)\). In this case, one can simply do
def H1_coeff(t, args):
return 9 * np.exp((t / 5.) ** 2)
In this case, the return value depends only on time. However, when specifying Python functions for coefficients, the function must have (t,args) as the input variables, in that order. Having specified our coefficient function, we can now specify the Hamiltonian in list format and call the solver (in this case qutip.mesolve
)
H = [H0,[H1, H1_coeff]]
output = mesolve(H, psi0, t, c_ops, [ada, sigma_UU, sigma_GG])
We can call the Monte Carlo solver in the exact same way (if using the default ntraj=500
):
output = mcsolve(H, psi0, t, c_ops, [ada, sigma_UU, sigma_GG])
The output from the master equation solver is identical to that shown in the examples, the Monte Carlo however will be noticeably off, suggesting we should increase the number of trajectories for this example. In addition, we can also consider the decay of a simple Harmonic oscillator with timevarying decay rate
kappa = 0.5
def col_coeff(t, args): # coefficient function
return np.sqrt(kappa * np.exp(t))
N = 10 # number of basis states
a = destroy(N)
H = a.dag() * a # simple HO
psi0 = basis(N, 9) # initial state
c_ops = [[a, col_coeff]] # timedependent collapse term
times = np.linspace(0, 10, 100)
output = mesolve(H, psi0, times, c_ops, [a.dag() * a])
Using the args variable¶
In the previous example we hardcoded all of the variables, driving amplitude \(A\) and width \(\sigma\), with their numerical values. This is fine for problems that are specialized, or that we only want to run once. However, in many cases, we would like to change the parameters of the problem in only one location (usually at the top of the script), and not have to worry about manually changing the values on each run. QuTiP allows you to accomplish this using the keyword args
as an input to the solvers. For instance, instead of explicitly writing 9 for the amplitude and 5 for the width of the gaussian driving term, we can make use of the args
variable
def H1_coeff(t, args):
return args['A'] * np.exp((t/args['sigma'])**2)
or equivalently,
def H1_coeff(t, args):
A = args['A']
sig = args['sigma']
return A * np.exp((t / sig) ** 2)
where args
is a Python dictionary of key: value
pairs args = {'A': a, 'sigma': b}
where a
and b
are the two parameters for the amplitude and width, respectively. Of course, we can always hardcode the values in the dictionary as well args = {'A': 9, 'sigma': 5}
, but there is much more flexibility by using variables in args
. To let the solvers know that we have a set of args to pass we append the args
to the end of the solver input:
output = mesolve(H, psi0, times, c_ops, [a.dag() * a], args={'A': 9, 'sigma': 5})
or to keep things looking pretty
args = {'A': 9, 'sigma': 5}
output = mesolve(H, psi0, times, c_ops, [a.dag() * a], args=args)
Once again, the Monte Carlo solver qutip.mcsolve
works in an identical manner.
String Format Method¶
Note
You must have Cython installed on your computer to use this format. See Installation for instructions on installing Cython.
The stringbased timedependent format works in a similar manner as the previously discussed Python function method. That being said, the underlying code does something completely different. When using this format, the strings used to represent the timedependent coefficients, as well as Hamiltonian and collapse operators, are rewritten as Cython code using a code generator class and then compiled into C code. The details of this metaprogramming will be published in due course. However, in short, this can lead to a substantial reduction in time for complex timedependent problems, or when simulating over long intervals.
Like the previous method, the stringbased format uses a list pair format [Op, str]
where str
is now a string representing the timedependent coefficient. For our first example, this string would be '9 * exp((t / 5.) ** 2)'
. The Hamiltonian in this format would take the form:
ustate = basis(3, 0)
excited = basis(3, 1)
ground = basis(3, 2)
N = 2 # Set where to truncate Fock state for cavity
sigma_ge = tensor(qeye(N), ground * excited.dag()) # g><e
sigma_ue = tensor(qeye(N), ustate * excited.dag()) # u><e
a = tensor(destroy(N), qeye(3))
ada = tensor(num(N), qeye(3))
c_ops = [] # Build collapse operators
kappa = 1.5 # Cavity decay rate
c_ops.append(np.sqrt(kappa) * a)
gamma = 6 # Atomic decay rate
c_ops.append(np.sqrt(5*gamma/9) * sigma_ue) # Use Rb branching ratio of 5/9 e>u
c_ops.append(np.sqrt(4*gamma/9) * sigma_ge) # 4/9 e>g
t = np.linspace(15, 15, 100) # Define time vector
psi0 = tensor(basis(N, 0), ustate) # Define initial state
state_GG = tensor(basis(N, 1), ground) # Define states onto which to project
sigma_GG = state_GG * state_GG.dag()
state_UU = tensor(basis(N, 0), ustate)
sigma_UU = state_UU * state_UU.dag()
g = 5 # coupling strength
H0 = g * (sigma_ge.dag() * a + a.dag() * sigma_ge) # timeindependent term
H1 = (sigma_ue.dag() + sigma_ue) # timedependent term
H = [H0, [H1, '9 * exp((t / 5) ** 2)']]
Notice that this is a valid Hamiltonian for the stringbased format as exp
is included in the above list of suitable functions. Calling the solvers is the same as before:
output = mesolve(H, psi0, t, c_ops, [a.dag() * a])
We can also use the args
variable in the same manner as before, however we must rewrite our string term to read: 'A * exp((t / sig) ** 2)'
H = [H0, [H1, 'A * exp((t / sig) ** 2)']]
args = {'A': 9, 'sig': 5}
output = mesolve(H, psi0, times, c_ops, [a.dag()*a], args=args)
Important
Naming your args
variables exp
, sin
, pi
etc. will cause errors when using the stringbased format.
Collapse operators are handled in the exact same way.
Modeling NonAnalytic and/or Experimental TimeDependent Parameters using Interpolating Functions¶
Sometimes it is necessary to model a system where the timedependent parameters are nonanalytic functions, or are derived from experimental data (i.e. a collection of data points). In these situations, one can use interpolating functions as an approximate functional form for input into a timedependent solver. QuTiP includes its own custom cubic spline interpolation class qutip.interpolate.Cubic_Spline
to provide this functionality. To see how this works, lets first generate some noisy data:
t = np.linspace(15, 15, 100)
func = lambda t: 9*np.exp((t / 5)** 2)
noisy_func = lambda t: func(t)+(0.05*func(t))*np.random.randn(t.shape[0])
noisy_data = noisy_func(t)
plt.figure()
plt.plot(t, func(t))
plt.plot(t, noisy_data, 'o')
plt.show()
To turn these data points into a function we call the QuTiP qutip.interpolate.Cubic_Spline
class using the first and last domain time points, t[0]
and t[1]
, respectively, as well as the entire array of data points:
S = Cubic_Spline(t[0], t[1], noisy_data)
plt.figure()
plt.plot(t, func(t))
plt.plot(t, noisy_data, 'o')
plt.plot(t, S(t), lw=2)
plt.show()
Note that, at present, only equally spaced real or complex data sets can be accommodated. This cubic spline class S
can now be pasted to any of the mesolve
, mcsolve
, or sesolve
functions where one would normally input a timedependent function or stringrepresentation. Taking the problem from the previous section as an example. We would make the replacement:
H = [H0, [H1, '9 * exp((t / 5) ** 2)']]
to
H = [H0, [H1, S]]
When combining interpolating functions with other Python functions or strings, the interpolating class will automatically pick the appropriate method for calling the class. That is to say that, if for example, you have other timedependent terms that are given in the stringformat, then the cubic spline representation will also be passed in a stringcompatible format. In the stringformat, the interpolation function is compiled into ccode, and thus is quite fast. This is the default method if no other timedependent terms are present.
Accesing the state from solver¶
New in QuTiP 4.4
The state of the system, the ket vector or the density matrix,
is available to timedependent Hamiltonian and collapse operators in args
.
Some keys of the argument dictionary are understood by the solver to be values
to be updated with the evolution of the system.
The state can be obtained in 3 forms: Qobj
, vector (1d np.array
), matrix (2d np.array
),
expectation values and collapse can also be obtained.
Preparation 
usage 
Notes 

state as Qobj 


The ket or density matrix as a Qobj with 
state as matrix 


The state as a matrix, equivalent to 
state as vector 


The state as a vector, equivalent to 
expectation value 


Expectation value of the operator 
collpases 


List of collapse,
each collapse is a tuple of the pair 
Here psi0
is the initial value used for tests before the evolution begins.
qutip.bloch_redfield.brmesolve
does not support these arguments.
Reusing TimeDependent Hamiltonian Data¶
Note
This section covers a specialized topic and may be skipped if you are new to QuTiP.
When repeatedly simulating a system where only the timedependent variables, or initial state change, it is possible to reuse the Hamiltonian data stored in QuTiP and there by avoid spending time needlessly preparing the Hamiltonian and collapse terms for simulation. To turn on the the reuse features, we must pass a qutip.solver.Options
object with the rhs_reuse
flag turned on. Instructions on setting flags are found in Setting Options for the Dynamics Solvers. For example, we can do
H = [H0, [H1, 'A * exp((t / sig) ** 2)']]
args = {'A': 9, 'sig': 5}
output = mcsolve(H, psi0, times, c_ops, [a.dag()*a], args=args)
opts = Options(rhs_reuse=True)
args = {'A': 10, 'sig': 3}
output = mcsolve(H, psi0, times, c_ops, [a.dag()*a], args=args, options=opts)
The second call to qutip.mcsolve
does not reorganize the data, and in the case of the string format, does not recompile the Cython code. For the small system here, the savings in computation time is quite small, however, if you need to call the solvers many times for different parameters, this savings will obviously start to add up.